Nuprl Lemma : uhg-identity1
∀[t1,t2,t3,t4:ℝ].
((((((r1 + (t1 * t2)) * (t3 + t4)) - (r1 + (t3 * t4)) * (t1 + t2))
* (((r1 + (t1 * t3)) * (t2 + t4)) - (r1 + (t2 * t4)) * (t1 + t3)))
+ ((r(2) * ((t3 * t4) - t1 * t2)) * r(2) * ((t2 * t4) - t1 * t3)))
= ((((r1 - t1 * t2) * (t3 + t4)) - (r1 - t3 * t4) * (t1 + t2))
* (((r1 - t1 * t3) * (t2 + t4)) - (r1 - t2 * t4) * (t1 + t3))))
Proof
Definitions occuring in Statement :
rsub: x - y,
req: x = y,
rmul: a * b,
radd: a + b,
int-to-real: r(n),
real: ℝ,
uall: ∀[x:A]. B[x],
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
all: ∀x:A. B[x],
req_int_terms: t1 ≡ t2,
false: False,
not: ¬A,
top: Top
Lemmas referenced :
req_witness,
rmul_wf,
rsub_wf,
int-to-real_wf,
radd_wf,
real_wf,
itermSubtract_wf,
itermAdd_wf,
itermMultiply_wf,
itermConstant_wf,
itermVar_wf,
req-iff-rsub-is-0,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_add_lemma,
real_term_value_mul_lemma,
real_term_value_const_lemma,
real_term_value_var_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
because_Cache,
natural_numberEquality,
hypothesis,
hypothesisEquality,
independent_functionElimination,
sqequalRule,
isect_memberEquality,
productElimination,
independent_isectElimination,
dependent_functionElimination,
approximateComputation,
lambdaEquality,
int_eqEquality,
intEquality,
voidElimination,
voidEquality
Latex:
\mforall{}[t1,t2,t3,t4:\mBbbR{}].
((((((r1 + (t1 * t2)) * (t3 + t4)) - (r1 + (t3 * t4)) * (t1 + t2))
* (((r1 + (t1 * t3)) * (t2 + t4)) - (r1 + (t2 * t4)) * (t1 + t3)))
+ ((r(2) * ((t3 * t4) - t1 * t2)) * r(2) * ((t2 * t4) - t1 * t3)))
= ((((r1 - t1 * t2) * (t3 + t4)) - (r1 - t3 * t4) * (t1 + t2))
* (((r1 - t1 * t3) * (t2 + t4)) - (r1 - t2 * t4) * (t1 + t3))))
Date html generated:
2017_10_05-AM-00_16_50
Last ObjectModification:
2017_06_17-AM-10_06_19
Theory : inner!product!spaces
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