Nuprl Lemma : implies-equal-div
∀[a,c:ℤ]. ∀[b,d:ℤ-o]. (a ÷ b) = (c ÷ d) ∈ ℤ supposing (d * 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,
nequal: a ≠ b ∈ T ,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
all: ∀x:A. B[x],
top: Top,
and: P ∧ Q,
prop: ℙ,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
true: True,
squash: ↓T,
decidable: Dec(P),
or: P ∨ Q,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q)
Lemmas referenced :
div_rem_sum,
mul_preserves_eq,
mul_cancel_in_eq,
divide_wfa,
int_entire_a,
int_nzero_properties,
full-omega-unsat,
intformand_wf,
intformeq_wf,
itermVar_wf,
itermConstant_wf,
intformnot_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_not_lemma,
int_formula_prop_wf,
int_subtype_base,
nequal_wf,
set_subtype_base,
remainder_wfa,
squash_wf,
true_wf,
int_nzero_wf,
decidable__equal_int,
itermMultiply_wf,
int_term_value_mul_lemma,
equal_wf,
istype-universe,
rem-mul,
subtype_rel_self,
iff_weakening_equal,
add-is-int-iff,
multiply-is-int-iff,
itermAdd_wf,
int_term_value_add_lemma,
false_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
because_Cache,
equalityTransitivity,
equalitySymmetry,
hypothesis,
setElimination,
rename,
independent_isectElimination,
dependent_set_memberEquality_alt,
multiplyEquality,
lambdaFormation_alt,
natural_numberEquality,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
dependent_functionElimination,
isect_memberEquality_alt,
voidElimination,
sqequalRule,
independent_pairFormation,
universeIsType,
equalityIstype,
applyEquality,
baseClosed,
sqequalBase,
intEquality,
inhabitedIsType,
baseApply,
closedConclusion,
axiomEquality,
isectIsTypeImplies,
imageElimination,
unionElimination,
imageMemberEquality,
instantiate,
universeEquality,
productElimination,
pointwiseFunctionality,
promote_hyp
Latex:
\mforall{}[a,c:\mBbbZ{}]. \mforall{}[b,d:\mBbbZ{}\msupminus{}\msupzero{}]. (a \mdiv{} b) = (c \mdiv{} d) supposing (d * a) = (b * c)
Date html generated:
2020_05_19-PM-09_41_16
Last ObjectModification:
2019_10_16-PM-03_47_23
Theory : int_2
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