`∀[r,s:ℚ].  (r + s ∈ ℚ)`

Proof

Definitions occuring in Statement :  qadd: `r + s` rationals: `ℚ` uall: `∀[x:A]. B[x]` member: `t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` rationals: `ℚ` quotient: `x,y:A//B[x; y]` and: `P ∧ Q` all: `∀x:A. B[x]` implies: `P `` Q` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]` uimplies: `b supposing a` b-union: `A ⋃ B` tunion: `⋃x:A.B[x]` bool: `𝔹` unit: `Unit` ifthenelse: `if b then t else f fi ` pi2: `snd(t)` qadd: `r + s` so_lambda: `λ2x.t[x]` so_apply: `x[s]` int_nzero: `ℤ-o` callbyvalueall: callbyvalueall has-value: `(a)↓` has-valueall: `has-valueall(a)` btrue: `tt` qeq: `qeq(r;s)` bfalse: `ff` uiff: `uiff(P;Q)` rev_uimplies: `rev_uimplies(P;Q)` sq_type: `SQType(T)` guard: `{T}` subtype_rel: `A ⊆r B` nequal: `a ≠ b ∈ T ` not: `¬A` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` false: `False` prop: `ℙ` decidable: `Dec(P)` or: `P ∨ Q`
Lemmas referenced :  rationals_wf quotient-member-eq b-union_wf int_nzero_wf equal-wf-T-base bool_wf qeq_wf qeq-equiv valueall-type-has-valueall bunion-valueall-type int-valueall-type product-valueall-type istype-int set-valueall-type nequal_wf evalall-reduce isint-int eqtt_to_assert eq_int_wf assert_of_eq_int subtype_base_sq int_subtype_base mul-distributes-right mul-commutes add-commutes btrue_wf subtype_rel_b-union-left subtype_rel_b-union-right trivial-equal bfalse_wf int_entire_a int_nzero_properties full-omega-unsat intformand_wf intformeq_wf itermVar_wf itermConstant_wf intformnot_wf int_formula_prop_and_lemma int_formula_prop_eq_lemma int_term_value_var_lemma int_term_value_constant_lemma int_formula_prop_not_lemma int_formula_prop_wf ifthenelse_wf mul-swap mul-associates mul_preserves_eq decidable__equal_int itermAdd_wf itermMultiply_wf int_term_value_add_lemma int_term_value_mul_lemma
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation_alt introduction cut sqequalHypSubstitution pointwiseFunctionalityForEquality extract_by_obid hypothesis sqequalRule pertypeElimination promote_hyp thin productElimination equalityTransitivity equalitySymmetry inhabitedIsType lambdaFormation_alt rename isectElimination intEquality productEquality lambdaEquality_alt hypothesisEquality baseClosed independent_isectElimination dependent_functionElimination imageElimination unionElimination equalityElimination independent_functionElimination because_Cache natural_numberEquality callbyvalueReduce isintReduceTrue addEquality independent_pairEquality multiplyEquality setElimination Error :memTop,  instantiate cumulativity equalityIstype productIsType sqequalBase universeIsType axiomEquality isect_memberEquality_alt isectIsTypeImplies applyEquality imageMemberEquality dependent_pairEquality_alt dependent_set_memberEquality_alt approximateComputation dependent_pairFormation_alt int_eqEquality independent_pairFormation voidElimination universeEquality

Latex:
\mforall{}[r,s:\mBbbQ{}].    (r  +  s  \mmember{}  \mBbbQ{})

Date html generated: 2020_05_20-AM-09_12_51
Last ObjectModification: 2019_12_31-PM-08_19_57

Theory : rationals

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