### Nuprl Lemma : qmul_assoc

`∀[r,s,t:ℚ].  (((r * s) * t) = (r * s * t) ∈ ℚ)`

Proof

Definitions occuring in Statement :  qmul: `r * s` rationals: `ℚ` uall: `∀[x:A]. B[x]` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uiff: `uiff(P;Q)` and: `P ∧ Q` rev_uimplies: `rev_uimplies(P;Q)` uimplies: `b supposing a` all: `∀x:A. B[x]` exists: `∃x:A. B[x]` nat_plus: `ℕ+` cand: `A c∧ B` not: `¬A` implies: `P `` Q` subtype_rel: `A ⊆r B` iff: `P `⇐⇒` Q` int_nzero: `ℤ-o` nequal: `a ≠ b ∈ T ` satisfiable_int_formula: `satisfiable_int_formula(fmla)` false: `False` prop: `ℙ` qmul: `r * s` qeq: `qeq(r;s)` so_lambda: `λ2x.t[x]` so_apply: `x[s]` callbyvalueall: callbyvalueall has-value: `(a)↓` has-valueall: `has-valueall(a)` ifthenelse: `if b then t else f fi ` bfalse: `ff` decidable: `Dec(P)` or: `P ∨ Q`
Lemmas referenced :  assert-qeq qmul_wf q-elim nat_plus_properties iff_weakening_uiff assert_wf qeq_wf2 int-subtype-rationals equal-wf-base rationals_wf int_subtype_base istype-assert qdiv-int-elim full-omega-unsat intformand_wf intformeq_wf itermVar_wf itermConstant_wf intformless_wf istype-int int_formula_prop_and_lemma int_formula_prop_eq_lemma int_term_value_var_lemma int_term_value_constant_lemma int_formula_prop_less_lemma int_formula_prop_wf nequal_wf valueall-type-has-valueall product-valueall-type int-valueall-type evalall-reduce assert_of_eq_int decidable__equal_int intformnot_wf itermMultiply_wf int_formula_prop_not_lemma int_term_value_mul_lemma qdiv_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation_alt cut introduction extract_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality hypothesis productElimination independent_pairFormation independent_isectElimination dependent_functionElimination setElimination rename lambdaFormation_alt independent_functionElimination applyEquality sqequalRule closedConclusion natural_numberEquality baseClosed because_Cache dependent_set_memberEquality_alt approximateComputation dependent_pairFormation_alt lambdaEquality_alt int_eqEquality Error :memTop,  universeIsType voidElimination equalityIstype inhabitedIsType sqequalBase equalitySymmetry intEquality productEquality independent_pairEquality callbyvalueReduce multiplyEquality unionElimination hyp_replacement applyLambdaEquality

Latex:
\mforall{}[r,s,t:\mBbbQ{}].    (((r  *  s)  *  t)  =  (r  *  s  *  t))

Date html generated: 2020_05_20-AM-09_13_22
Last ObjectModification: 2020_01_24-PM-03_56_21

Theory : rationals

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