Nuprl Lemma : equal-iff-diff-zero
∀x,y:ℤ. uiff(x = y ∈ ℤ;(x - y) = 0 ∈ ℤ)
Proof
Definitions occuring in Statement :
uiff: uiff(P;Q),
all: ∀x:A. B[x],
subtract: n - m,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
member: t ∈ T,
decidable: Dec(P),
or: P ∨ Q,
uall: ∀[x:A]. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
implies: P ⇒ Q,
not: ¬A,
top: Top,
prop: ℙ
Lemmas referenced :
subtract_wf,
equal_wf,
int_formula_prop_wf,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_term_value_subtract_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermConstant_wf,
itermVar_wf,
itermSubtract_wf,
intformeq_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__equal_int
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
independent_pairFormation,
isect_memberFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
because_Cache,
hypothesis,
unionElimination,
isectElimination,
natural_numberEquality,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
hypothesisEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
computeAll
Latex:
\mforall{}x,y:\mBbbZ{}. uiff(x = y;(x - y) = 0)
Date html generated:
2016_05_14-AM-07_20_29
Last ObjectModification:
2016_01_07-PM-03_59_53
Theory : int_2
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