Nuprl Lemma : hyptrans_add
∀[rv:InnerProductSpace]. ∀[e,x:Point]. ∀[t,s:ℝ].
hyptrans(rv;e;t + s;x) ≡ hyptrans(rv;e;t;hyptrans(rv;e;s;x)) supposing e^2 = r1
Proof
Definitions occuring in Statement :
hyptrans: hyptrans(rv;e;t;x),
rv-ip: x ⋅ y,
inner-product-space: InnerProductSpace,
req: x = y,
radd: a + b,
int-to-real: r(n),
real: ℝ,
ss-eq: x ≡ y,
ss-point: Point,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
and: P ∧ Q,
ss-eq: x ≡ y,
not: ¬A,
implies: P ⇒ Q,
false: False,
subtype_rel: A ⊆r B,
guard: {T},
prop: ℙ,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
le: A ≤ B,
less_than': less_than'(a;b),
cand: A c∧ B,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
req_int_terms: t1 ≡ t2,
top: Top
Lemmas referenced :
hyptrans_decomp,
ss-sep_wf,
real-vector-space_subtype1,
inner-product-space_subtype,
subtype_rel_transitivity,
inner-product-space_wf,
real-vector-space_wf,
separation-space_wf,
hyptrans_wf,
radd_wf,
req_wf,
rv-ip_wf,
int-to-real_wf,
real_wf,
ss-point_wf,
ss-eq_wf,
rv-add_wf,
rv-mul_wf,
rmul_wf,
sinh_wf,
rsqrt_wf,
radd-non-neg,
rleq-int,
false_wf,
rv-ip-nonneg,
rleq_wf,
itermSubtract_wf,
itermMultiply_wf,
itermVar_wf,
req-iff-rsub-is-0,
itermAdd_wf,
ss-eq_weakening,
uiff_transitivity,
ss-eq_functionality,
hyptrans_functionality,
req_weakening,
hyptrans_lemma,
ss-eq_transitivity,
rv-add_functionality,
rv-mul_functionality,
req_transitivity,
rmul_functionality,
sinh_functionality,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma,
real_term_value_add_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_isectElimination,
hypothesis,
productElimination,
sqequalRule,
lambdaEquality,
because_Cache,
isectElimination,
applyEquality,
instantiate,
natural_numberEquality,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
voidElimination,
independent_functionElimination,
independent_pairFormation,
lambdaFormation,
dependent_set_memberEquality,
setElimination,
rename,
setEquality,
productEquality,
approximateComputation,
int_eqEquality,
intEquality,
voidEquality
Latex:
\mforall{}[rv:InnerProductSpace]. \mforall{}[e,x:Point]. \mforall{}[t,s:\mBbbR{}].
hyptrans(rv;e;t + s;x) \mequiv{} hyptrans(rv;e;t;hyptrans(rv;e;s;x)) supposing e\^{}2 = r1
Date html generated:
2017_10_05-AM-00_27_59
Last ObjectModification:
2017_06_21-PM-02_30_26
Theory : inner!product!spaces
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