Nuprl Lemma : implies-ip-triangle
∀rv:InnerProductSpace. ∀a,b,c,a':Point. (a_b_a' ⇒ ab=a'b ⇒ ab=cb ⇒ c # a ⇒ c # a' ⇒ Δ(a;b;c))
Proof
Definitions occuring in Statement :
ip-triangle: Δ(a;b;c),
ip-between: a_b_c,
ip-congruent: ab=cd,
inner-product-space: InnerProductSpace,
ss-sep: x # y,
ss-point: Point,
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
ip-triangle: Δ(a;b;c),
member: t ∈ T,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
req: x = y,
ip-congruent: ab=cd,
prop: ℙ,
guard: {T},
uimplies: b supposing a,
and: P ∧ Q,
rv-sub: x - y,
rv-minus: -x,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
ss-eq: x ≡ y,
not: ¬A,
or: P ∨ Q,
exists: ∃x:A. B[x],
rsub: x - y,
nat: ℕ,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
rneq: x ≠ y,
less_than: a < b,
squash: ↓T,
true: True
Lemmas referenced :
ip-triangle-lemma,
rv-sub_wf,
inner-product-space_subtype,
ss-sep_wf,
real-vector-space_subtype1,
subtype_rel_transitivity,
inner-product-space_wf,
real-vector-space_wf,
separation-space_wf,
ip-congruent_wf,
ip-between_wf,
ss-point_wf,
ss-eq_wf,
rv-add_wf,
rv-mul_wf,
int-to-real_wf,
radd_wf,
rmul_wf,
rv-minus_wf,
rv-0_wf,
rv-norm_wf,
real_wf,
rleq_wf,
req_wf,
rv-ip_wf,
uiff_transitivity,
ss-eq_functionality,
rv-add_functionality,
ss-eq_weakening,
rv-mul-linear,
rv-add-assoc,
rv-mul-mul,
rv-add-swap,
rv-mul-add-alt,
rv-mul_functionality,
req_transitivity,
radd_functionality,
rmul-int,
req_weakening,
radd-int,
rv-mul0,
rv-0-add,
rless_functionality,
rv-norm_functionality,
ss-sep-symmetry,
rv-norm-positive,
rv-sep-iff,
false_wf,
or_wf,
minimal-double-negation-hyp-elim,
minimal-not-not-excluded-middle,
ip-between-iff,
rsub_wf,
ip-congruent_functionality,
rv-sub_functionality,
rv-norm-equal-iff,
rminus_wf,
rv-mul-1-add,
req_inversion,
rminus-as-rmul,
rmul_functionality,
rminus-radd,
radd_comm,
rminus-rminus,
rmul-minus,
rmul_over_rminus,
rminus_functionality,
rmul-distrib,
rmul-one-both,
req_functionality,
rv-ip_functionality,
rv-add-comm,
rv-mul-1-add-alt,
radd-ac,
radd-assoc,
radd-zero-both,
rv-ip-mul,
rv-ip-mul2,
rnexp_wf,
le_wf,
rv-norm-squared,
rnexp-positive,
equal_wf,
rless_wf,
rdiv_wf,
rless-int,
rmul-assoc,
rmul_preserves_req,
rmul-distrib2,
rmul_comm,
rmul-ac,
radd-preserves-req,
radd-rminus-both,
rmul-zero-both,
rmul-rdiv-cancel2,
rsub_functionality,
rmul-rdiv-cancel,
uiff_transitivity3,
squash_wf,
true_wf,
rminus-int,
rv-mul-add,
rmul-int-rdiv,
uiff_transitivity2,
ip-between_functionality,
ss-eq_inversion,
ip-between-same,
rv-norm0,
rv-norm-is-zero,
rv-mul-0
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
isectElimination,
applyEquality,
hypothesis,
sqequalRule,
because_Cache,
independent_functionElimination,
instantiate,
independent_isectElimination,
natural_numberEquality,
minusEquality,
lambdaEquality,
setElimination,
rename,
setEquality,
productEquality,
productElimination,
functionEquality,
unionElimination,
promote_hyp,
multiplyEquality,
addEquality,
dependent_set_memberEquality,
independent_pairFormation,
equalityTransitivity,
equalitySymmetry,
inrFormation,
imageMemberEquality,
baseClosed,
addLevel,
impliesFunctionality,
imageElimination,
voidElimination
Latex:
\mforall{}rv:InnerProductSpace. \mforall{}a,b,c,a':Point. (a\_b\_a' {}\mRightarrow{} ab=a'b {}\mRightarrow{} ab=cb {}\mRightarrow{} c \# a {}\mRightarrow{} c \# a' {}\mRightarrow{} \mDelta{}(a;b;c))
Date html generated:
2017_10_04-PM-11_59_36
Last ObjectModification:
2017_03_10-PM-07_05_55
Theory : inner!product!spaces
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