Nuprl Lemma : rcos-rabs
∀x:ℝ. (rcos(|x|) = rcos(x))
Proof
Definitions occuring in Statement :
rcos: rcos(x),
rabs: |x|,
req: x = y,
real: ℝ,
all: ∀x:A. B[x]
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
prop: ℙ,
or: P ∨ Q,
implies: P ⇒ Q,
not: ¬A,
false: False,
stable: Stable{P},
uimplies: b supposing a,
guard: {T},
uiff: uiff(P;Q),
and: P ∧ Q,
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
real_wf,
stable_req,
rcos_wf,
rabs_wf,
false_wf,
rless_wf,
int-to-real_wf,
not_wf,
req_wf,
istype-void,
minimal-double-negation-hyp-elim,
minimal-not-not-excluded-middle,
rminus_wf,
rleq_weakening_rless,
rcos-rminus,
req_functionality,
rcos_functionality,
rabs-of-nonpos,
req_weakening,
not-rless,
rabs-of-nonneg
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
universeIsType,
cut,
introduction,
extract_by_obid,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
sqequalRule,
unionEquality,
natural_numberEquality,
functionEquality,
functionIsType,
unionIsType,
because_Cache,
independent_isectElimination,
independent_functionElimination,
unionElimination,
voidElimination,
dependent_functionElimination,
productElimination
Latex:
\mforall{}x:\mBbbR{}. (rcos(|x|) = rcos(x))
Date html generated:
2019_10_30-AM-11_41_33
Last ObjectModification:
2019_05_17-PM-03_10_22
Theory : reals_2
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