Nuprl Lemma : rmul*_functionality
∀[x,y,u,v:ℝ*]. (x = y ⇒ u = v ⇒ x * u = y * v)
Proof
Definitions occuring in Statement :
rmul*: x * y,
req*: x = y,
real*: ℝ*,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
rmul*: x * y,
member: t ∈ T,
prop: ℙ,
uimplies: b supposing a,
and: P ∧ Q,
req*: x = y,
exists: ∃x:A. B[x],
nat: ℕ,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
all: ∀x:A. B[x],
rfun*2: f*(x;y),
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
so_apply: x[s],
real*: ℝ*,
int_upper: {i...},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
req*_wf,
real*_wf,
rfun*2_wf,
rmul_wf,
real_wf,
req_witness,
req_wf,
req_weakening,
false_wf,
le_wf,
int_upper_wf,
all_wf,
int_upper_subtype_nat,
subtype_rel_self,
nat_wf,
rmul_comm,
req*_functionality,
rfun*2_functionality,
req*_weakening,
req_functionality,
rmul_functionality
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
because_Cache,
lambdaEquality,
productElimination,
sqequalRule,
independent_functionElimination,
productEquality,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
dependent_pairFormation,
dependent_set_memberEquality,
natural_numberEquality,
independent_pairFormation,
setElimination,
rename,
applyEquality
Latex:
\mforall{}[x,y,u,v:\mBbbR{}*]. (x = y {}\mRightarrow{} u = v {}\mRightarrow{} x * u = y * v)
Date html generated:
2018_05_22-PM-03_16_35
Last ObjectModification:
2017_10_06-PM-03_47_15
Theory : reals_2
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