Nuprl Lemma : eqmod_functionality_wrt_eqmod
∀m,m',a,a',b,b':ℤ. (a ≡ a' mod m) ⇒ (b ≡ b' mod m) ⇒ (a ≡ b mod m ⇐⇒ a' ≡ b' mod m') supposing m = m' ∈ ℤ
Proof
Definitions occuring in Statement :
eqmod: a ≡ b mod m,
uimplies: b supposing a,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
uimplies: b supposing a,
member: t ∈ T,
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
prop: ℙ,
uall: ∀[x:A]. B[x],
rev_implies: P ⇐ Q,
subtype_rel: A ⊆r B
Lemmas referenced :
eqmod_wf,
equal-wf-base,
int_subtype_base,
eqmod_inversion,
eqmod_transitivity
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
Error :isect_memberFormation_alt,
cut,
introduction,
axiomEquality,
hypothesis,
thin,
rename,
independent_pairFormation,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
Error :universeIsType,
intEquality,
applyEquality,
sqequalRule,
equalitySymmetry,
hyp_replacement,
applyLambdaEquality,
dependent_functionElimination,
independent_functionElimination,
because_Cache
Latex:
\mforall{}m,m',a,a',b,b':\mBbbZ{}.
(a \mequiv{} a' mod m) {}\mRightarrow{} (b \mequiv{} b' mod m) {}\mRightarrow{} (a \mequiv{} b mod m \mLeftarrow{}{}\mRightarrow{} a' \mequiv{} b' mod m') supposing m = m'
Date html generated:
2019_06_20-PM-02_24_18
Last ObjectModification:
2018_09_26-PM-05_58_20
Theory : num_thy_1
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