Nuprl Lemma : r2-det-is-dot-product
∀[a,b,c:ℝ^2]. (|abc| = a - b⋅λi.if (i =z 0) then -(c - b 1) else c - b 0 fi )
Proof
Definitions occuring in Statement :
r2-det: |pqr|,
dot-product: x⋅y,
real-vec-sub: X - Y,
real-vec: ℝ^n,
req: x = y,
rminus: -(x),
ifthenelse: if b then t else f fi ,
eq_int: (i =z j),
uall: ∀[x:A]. B[x],
apply: f a,
lambda: λx.A[x],
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
real-vec-sub: X - Y,
dot-product: x⋅y,
r2-det: |pqr|,
subtract: n - m,
nat: ℕ,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
not: ¬A,
implies: P ⇒ Q,
prop: ℙ,
real-vec: ℝ^n,
int_seg: {i..j-},
all: ∀x:A. B[x],
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
uimplies: b supposing a,
subtype_rel: A ⊆r B,
lelt: i ≤ j < k,
less_than: a < b,
squash: ↓T,
true: True,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
so_lambda: λ2x.t[x],
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
so_apply: x[s],
nequal: a ≠ b ∈ T ,
eq_int: (i =z j),
itermConstant: "const",
req_int_terms: t1 ≡ t2,
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
req_witness,
r2-det_wf,
dot-product_wf,
false_wf,
le_wf,
real-vec-sub_wf,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
rminus_wf,
real-vec_wf,
lelt_wf,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
int_seg_wf,
rsub_wf,
radd_wf,
rmul_wf,
rsum_wf,
decidable__lt,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermVar_wf,
itermConstant_wf,
itermAdd_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_term_value_add_lemma,
int_formula_prop_wf,
intformeq_wf,
int_formula_prop_eq_lemma,
decidable__le,
intformle_wf,
int_formula_prop_le_lemma,
equal-wf-base,
int_subtype_base,
real_term_polynomial,
itermSubtract_wf,
itermMultiply_wf,
itermMinus_wf,
int-to-real_wf,
real_term_value_const_lemma,
real_term_value_sub_lemma,
real_term_value_add_lemma,
real_term_value_mul_lemma,
real_term_value_var_lemma,
real_term_value_minus_lemma,
req-iff-rsub-is-0,
req_functionality,
req_weakening,
rsum-split-first,
radd_functionality,
rsum-single
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
dependent_set_memberEquality,
natural_numberEquality,
independent_pairFormation,
lambdaFormation,
because_Cache,
lambdaEquality,
setElimination,
rename,
unionElimination,
equalityElimination,
productElimination,
independent_isectElimination,
applyEquality,
imageMemberEquality,
baseClosed,
equalityTransitivity,
equalitySymmetry,
dependent_pairFormation,
promote_hyp,
dependent_functionElimination,
instantiate,
cumulativity,
independent_functionElimination,
voidElimination,
isect_memberEquality,
int_eqEquality,
intEquality,
voidEquality,
computeAll,
addEquality,
setEquality
Latex:
\mforall{}[a,b,c:\mBbbR{}\^{}2]. (|abc| = a - b\mcdot{}\mlambda{}i.if (i =\msubz{} 0) then -(c - b 1) else c - b 0 fi )
Date html generated:
2017_10_03-AM-11_40_36
Last ObjectModification:
2017_04_11-PM-05_29_09
Theory : reals
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