### Nuprl Lemma : implies-equal-div2

`∀[a,c:ℤ]. ∀[b:ℤ-o].  (a ÷ b) = c ∈ ℤ supposing a = (b * c) ∈ ℤ`

Proof

Definitions occuring in Statement :  int_nzero: `ℤ-o` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` divide: `n ÷ m` multiply: `n * m` int: `ℤ` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uimplies: `b supposing a` int_nzero: `ℤ-o` true: `True` nequal: `a ≠ b ∈ T ` not: `¬A` implies: `P `` Q` sq_type: `SQType(T)` all: `∀x:A. B[x]` guard: `{T}` false: `False` prop: `ℙ` decidable: `Dec(P)` or: `P ∨ Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` top: `Top` and: `P ∧ Q` subtype_rel: `A ⊆r B` so_lambda: `λ2x.t[x]` so_apply: `x[s]`
Lemmas referenced :  implies-equal-div subtype_base_sq int_subtype_base nequal_wf int_nzero_properties decidable__equal_int full-omega-unsat intformand_wf intformnot_wf intformeq_wf itermMultiply_wf itermConstant_wf itermVar_wf istype-int int_formula_prop_and_lemma istype-void int_formula_prop_not_lemma int_formula_prop_eq_lemma int_term_value_mul_lemma int_term_value_constant_lemma int_term_value_var_lemma int_formula_prop_wf div_one set_subtype_base int_nzero_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation_alt introduction cut extract_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality dependent_set_memberEquality_alt natural_numberEquality lambdaFormation_alt instantiate cumulativity intEquality independent_isectElimination hypothesis dependent_functionElimination equalityTransitivity equalitySymmetry independent_functionElimination voidElimination equalityIstype inhabitedIsType baseClosed sqequalBase universeIsType setElimination rename because_Cache unionElimination approximateComputation dependent_pairFormation_alt lambdaEquality_alt int_eqEquality isect_memberEquality_alt sqequalRule independent_pairFormation applyEquality baseApply closedConclusion axiomEquality isectIsTypeImplies

Latex:
\mforall{}[a,c:\mBbbZ{}].  \mforall{}[b:\mBbbZ{}\msupminus{}\msupzero{}].    (a  \mdiv{}  b)  =  c  supposing  a  =  (b  *  c)

Date html generated: 2020_05_19-PM-09_41_23
Last ObjectModification: 2019_10_16-PM-04_24_14

Theory : int_2

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