Nuprl Lemma : eqmod_weakening
∀m,a,b:ℤ. a ≡ b mod m supposing a = b ∈ ℤ
Proof
Definitions occuring in Statement :
eqmod: a ≡ b mod m,
uimplies: b supposing a,
all: ∀x:A. B[x],
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
eqmod: a ≡ b mod m,
all: ∀x:A. B[x],
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
prop: ℙ,
top: Top,
subtract: n - m,
divides: b | a,
exists: ∃x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False
Lemmas referenced :
equal-wf-base,
int_subtype_base,
divides_wf,
subtract_wf,
minus-one-mul,
add-mul-special,
zero-mul,
decidable__equal_int,
full-omega-unsat,
intformnot_wf,
intformeq_wf,
itermConstant_wf,
itermMultiply_wf,
itermVar_wf,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_constant_lemma,
int_term_value_mul_lemma,
int_term_value_var_lemma,
int_formula_prop_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
lambdaFormation,
Error :isect_memberFormation_alt,
cut,
introduction,
axiomEquality,
hypothesis,
thin,
rename,
Error :universeIsType,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
intEquality,
hypothesisEquality,
applyEquality,
isect_memberEquality,
voidElimination,
voidEquality,
hyp_replacement,
equalitySymmetry,
applyLambdaEquality,
dependent_pairFormation,
natural_numberEquality,
dependent_functionElimination,
because_Cache,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
lambdaEquality,
int_eqEquality,
baseClosed,
baseApply,
closedConclusion
Latex:
\mforall{}m,a,b:\mBbbZ{}. a \mequiv{} b mod m supposing a = b
Date html generated:
2019_06_20-PM-02_24_09
Last ObjectModification:
2018_09_26-PM-05_58_22
Theory : num_thy_1
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