Nuprl Lemma : qmul_inv
∀[r:ℚ]. (r * 1/r) = 1 ∈ ℚ supposing ¬(r = 0 ∈ ℚ)
Proof
Definitions occuring in Statement :
qinv: 1/r,
qmul: r * s,
rationals: ℚ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
not: ¬A,
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
not: ¬A,
subtype_rel: A ⊆r B,
uiff: uiff(P;Q),
and: P ∧ Q,
prop: ℙ,
rev_uimplies: rev_uimplies(P;Q),
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
nat_plus: ℕ+,
cand: A c∧ B,
implies: P ⇒ Q,
false: False,
qdiv: (r/s),
ifthenelse: if b then t else f fi ,
btrue: tt,
bfalse: ff,
mk-rational: mk-rational(a;b),
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
decidable: Dec(P),
or: P ∨ Q
Lemmas referenced :
assert-qeq,
qmul_wf,
qinv_wf,
int-subtype-rationals,
assert_wf,
qeq_wf2,
not_wf,
equal-wf-T-base,
rationals_wf,
q-elim,
nat_plus_properties,
equal-wf-base,
int_subtype_base,
qdiv_wf,
qeq-elim,
qmul-elim,
qinv-elim,
neg_assert_of_eq_int,
mk-rational_wf,
satisfiable-full-omega-tt,
intformand_wf,
intformeq_wf,
itermMultiply_wf,
itermVar_wf,
itermConstant_wf,
intformnot_wf,
int_formula_prop_and_lemma,
int_formula_prop_eq_lemma,
int_term_value_mul_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_not_lemma,
int_formula_prop_wf,
nequal_wf,
mul_nzero,
intformless_wf,
int_formula_prop_less_lemma,
assert_of_eq_int,
decidable__equal_int
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
because_Cache,
independent_isectElimination,
hypothesis,
addLevel,
impliesFunctionality,
natural_numberEquality,
applyEquality,
sqequalRule,
productElimination,
baseClosed,
independent_pairFormation,
isect_memberEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
setElimination,
rename,
hyp_replacement,
Error :applyLambdaEquality,
functionEquality,
independent_functionElimination,
lambdaFormation,
levelHypothesis,
promote_hyp,
impliesLevelFunctionality,
voidElimination,
baseApply,
closedConclusion,
isintReduceTrue,
multiplyEquality,
dependent_set_memberEquality,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
voidEquality,
computeAll,
unionElimination
Latex:
\mforall{}[r:\mBbbQ{}]. (r * 1/r) = 1 supposing \mneg{}(r = 0)
Date html generated:
2016_10_25-AM-11_50_58
Last ObjectModification:
2016_07_12-AM-07_47_41
Theory : rationals
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