hdmap1vallem - Mathbox for Norm Megill

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Theorem hdmap1vallem 39738

Description: Value of preliminary map from vectors to functionals in the closed kernel dual space. (Contributed by NM, 15-May-2015.)

**Hypotheses**
Ref	Expression
hdmap1val.h	⊢ 𝐻 = (LHyp‘𝐾)
hdmap1fval.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
hdmap1fval.v	⊢ 𝑉 = (Base‘𝑈)
hdmap1fval.s	⊢ − = (-_g‘𝑈)
hdmap1fval.o	⊢ 0 = (0_g‘𝑈)
hdmap1fval.n	⊢ 𝑁 = (LSpan‘𝑈)
hdmap1fval.c	⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
hdmap1fval.d	⊢ 𝐷 = (Base‘𝐶)
hdmap1fval.r	⊢ 𝑅 = (-_g‘𝐶)
hdmap1fval.q	⊢ 𝑄 = (0_g‘𝐶)
hdmap1fval.j	⊢ 𝐽 = (LSpan‘𝐶)
hdmap1fval.m	⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)
hdmap1fval.i	⊢ 𝐼 = ((HDMap1‘𝐾)‘𝑊)
hdmap1fval.k	⊢ (𝜑 → (𝐾 ∈ 𝐴 ∧ 𝑊 ∈ 𝐻))
hdmap1val.t	⊢ (𝜑 → 𝑇 ∈ ((𝑉 × 𝐷) × 𝑉))

**Assertion**
Ref	Expression
hdmap1vallem	⊢ (𝜑 → (𝐼‘𝑇) = if((2^nd ‘𝑇) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2^nd ‘𝑇)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1^st ‘(1^st ‘𝑇)) − (2^nd ‘𝑇))})) = (𝐽‘{((2^nd ‘(1^st ‘𝑇))𝑅ℎ)})))))

Distinct variable groups: 𝐶,ℎ 𝐷,ℎ ℎ,𝐽 ℎ,𝑀 ℎ,𝑁 𝑈,ℎ ℎ,𝑉 𝑇,ℎ Allowed substitution hints: 𝜑(ℎ) 𝐴(ℎ) 𝑄(ℎ) 𝑅(ℎ) 𝐻(ℎ) 𝐼(ℎ) 𝐾(ℎ) − (ℎ) 𝑊(ℎ) 0 (ℎ)

Proof of Theorem hdmap1vallem Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		hdmap1val.h	. . . 4 ⊢ 𝐻 = (LHyp‘𝐾)
2		hdmap1fval.u	. . . 4 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
3		hdmap1fval.v	. . . 4 ⊢ 𝑉 = (Base‘𝑈)
4		hdmap1fval.s	. . . 4 ⊢ − = (-_g‘𝑈)
5		hdmap1fval.o	. . . 4 ⊢ 0 = (0_g‘𝑈)
6		hdmap1fval.n	. . . 4 ⊢ 𝑁 = (LSpan‘𝑈)
7		hdmap1fval.c	. . . 4 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
8		hdmap1fval.d	. . . 4 ⊢ 𝐷 = (Base‘𝐶)
9		hdmap1fval.r	. . . 4 ⊢ 𝑅 = (-_g‘𝐶)
10		hdmap1fval.q	. . . 4 ⊢ 𝑄 = (0_g‘𝐶)
11		hdmap1fval.j	. . . 4 ⊢ 𝐽 = (LSpan‘𝐶)
12		hdmap1fval.m	. . . 4 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)
13		hdmap1fval.i	. . . 4 ⊢ 𝐼 = ((HDMap1‘𝐾)‘𝑊)
14		hdmap1fval.k	. . . 4 ⊢ (𝜑 → (𝐾 ∈ 𝐴 ∧ 𝑊 ∈ 𝐻))
15	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14	hdmap1fval 39737	. . 3 ⊢ (𝜑 → 𝐼 = (𝑥 ∈ ((𝑉 × 𝐷) × 𝑉) ↦ if((2^nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2^nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1^st ‘(1^st ‘𝑥)) − (2^nd ‘𝑥))})) = (𝐽‘{((2^nd ‘(1^st ‘𝑥))𝑅ℎ)}))))))
16	15	fveq1d 6758	. 2 ⊢ (𝜑 → (𝐼‘𝑇) = ((𝑥 ∈ ((𝑉 × 𝐷) × 𝑉) ↦ if((2^nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2^nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1^st ‘(1^st ‘𝑥)) − (2^nd ‘𝑥))})) = (𝐽‘{((2^nd ‘(1^st ‘𝑥))𝑅ℎ)})))))‘𝑇))
17		hdmap1val.t	. . 3 ⊢ (𝜑 → 𝑇 ∈ ((𝑉 × 𝐷) × 𝑉))
18	10	fvexi 6770	. . . 4 ⊢ 𝑄 ∈ V
19		riotaex 7216	. . . 4 ⊢ (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2^nd ‘𝑇)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1^st ‘(1^st ‘𝑇)) − (2^nd ‘𝑇))})) = (𝐽‘{((2^nd ‘(1^st ‘𝑇))𝑅ℎ)}))) ∈ V
20	18, 19	ifex 4506	. . 3 ⊢ if((2^nd ‘𝑇) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2^nd ‘𝑇)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1^st ‘(1^st ‘𝑇)) − (2^nd ‘𝑇))})) = (𝐽‘{((2^nd ‘(1^st ‘𝑇))𝑅ℎ)})))) ∈ V
21		fveqeq2 6765	. . . . 5 ⊢ (𝑥 = 𝑇 → ((2^nd ‘𝑥) = 0 ↔ (2^nd ‘𝑇) = 0 ))
22		fveq2 6756	. . . . . . . . . 10 ⊢ (𝑥 = 𝑇 → (2^nd ‘𝑥) = (2^nd ‘𝑇))
23	22	sneqd 4570	. . . . . . . . 9 ⊢ (𝑥 = 𝑇 → {(2^nd ‘𝑥)} = {(2^nd ‘𝑇)})
24	23	fveq2d 6760	. . . . . . . 8 ⊢ (𝑥 = 𝑇 → (𝑁‘{(2^nd ‘𝑥)}) = (𝑁‘{(2^nd ‘𝑇)}))
25	24	fveqeq2d 6764	. . . . . . 7 ⊢ (𝑥 = 𝑇 → ((𝑀‘(𝑁‘{(2^nd ‘𝑥)})) = (𝐽‘{ℎ}) ↔ (𝑀‘(𝑁‘{(2^nd ‘𝑇)})) = (𝐽‘{ℎ})))
26		2fveq3 6761	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑇 → (1^st ‘(1^st ‘𝑥)) = (1^st ‘(1^st ‘𝑇)))
27	26, 22	oveq12d 7273	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑇 → ((1^st ‘(1^st ‘𝑥)) − (2^nd ‘𝑥)) = ((1^st ‘(1^st ‘𝑇)) − (2^nd ‘𝑇)))
28	27	sneqd 4570	. . . . . . . . . 10 ⊢ (𝑥 = 𝑇 → {((1^st ‘(1^st ‘𝑥)) − (2^nd ‘𝑥))} = {((1^st ‘(1^st ‘𝑇)) − (2^nd ‘𝑇))})
29	28	fveq2d 6760	. . . . . . . . 9 ⊢ (𝑥 = 𝑇 → (𝑁‘{((1^st ‘(1^st ‘𝑥)) − (2^nd ‘𝑥))}) = (𝑁‘{((1^st ‘(1^st ‘𝑇)) − (2^nd ‘𝑇))}))
30	29	fveq2d 6760	. . . . . . . 8 ⊢ (𝑥 = 𝑇 → (𝑀‘(𝑁‘{((1^st ‘(1^st ‘𝑥)) − (2^nd ‘𝑥))})) = (𝑀‘(𝑁‘{((1^st ‘(1^st ‘𝑇)) − (2^nd ‘𝑇))})))
31		2fveq3 6761	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑇 → (2^nd ‘(1^st ‘𝑥)) = (2^nd ‘(1^st ‘𝑇)))
32	31	oveq1d 7270	. . . . . . . . . 10 ⊢ (𝑥 = 𝑇 → ((2^nd ‘(1^st ‘𝑥))𝑅ℎ) = ((2^nd ‘(1^st ‘𝑇))𝑅ℎ))
33	32	sneqd 4570	. . . . . . . . 9 ⊢ (𝑥 = 𝑇 → {((2^nd ‘(1^st ‘𝑥))𝑅ℎ)} = {((2^nd ‘(1^st ‘𝑇))𝑅ℎ)})
34	33	fveq2d 6760	. . . . . . . 8 ⊢ (𝑥 = 𝑇 → (𝐽‘{((2^nd ‘(1^st ‘𝑥))𝑅ℎ)}) = (𝐽‘{((2^nd ‘(1^st ‘𝑇))𝑅ℎ)}))
35	30, 34	eqeq12d 2754	. . . . . . 7 ⊢ (𝑥 = 𝑇 → ((𝑀‘(𝑁‘{((1^st ‘(1^st ‘𝑥)) − (2^nd ‘𝑥))})) = (𝐽‘{((2^nd ‘(1^st ‘𝑥))𝑅ℎ)}) ↔ (𝑀‘(𝑁‘{((1^st ‘(1^st ‘𝑇)) − (2^nd ‘𝑇))})) = (𝐽‘{((2^nd ‘(1^st ‘𝑇))𝑅ℎ)})))
36	25, 35	anbi12d 630	. . . . . 6 ⊢ (𝑥 = 𝑇 → (((𝑀‘(𝑁‘{(2^nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1^st ‘(1^st ‘𝑥)) − (2^nd ‘𝑥))})) = (𝐽‘{((2^nd ‘(1^st ‘𝑥))𝑅ℎ)})) ↔ ((𝑀‘(𝑁‘{(2^nd ‘𝑇)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1^st ‘(1^st ‘𝑇)) − (2^nd ‘𝑇))})) = (𝐽‘{((2^nd ‘(1^st ‘𝑇))𝑅ℎ)}))))
37	36	riotabidv 7214	. . . . 5 ⊢ (𝑥 = 𝑇 → (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2^nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1^st ‘(1^st ‘𝑥)) − (2^nd ‘𝑥))})) = (𝐽‘{((2^nd ‘(1^st ‘𝑥))𝑅ℎ)}))) = (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2^nd ‘𝑇)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1^st ‘(1^st ‘𝑇)) − (2^nd ‘𝑇))})) = (𝐽‘{((2^nd ‘(1^st ‘𝑇))𝑅ℎ)}))))
38	21, 37	ifbieq2d 4482	. . . 4 ⊢ (𝑥 = 𝑇 → if((2^nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2^nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1^st ‘(1^st ‘𝑥)) − (2^nd ‘𝑥))})) = (𝐽‘{((2^nd ‘(1^st ‘𝑥))𝑅ℎ)})))) = if((2^nd ‘𝑇) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2^nd ‘𝑇)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1^st ‘(1^st ‘𝑇)) − (2^nd ‘𝑇))})) = (𝐽‘{((2^nd ‘(1^st ‘𝑇))𝑅ℎ)})))))
39		eqid 2738	. . . 4 ⊢ (𝑥 ∈ ((𝑉 × 𝐷) × 𝑉) ↦ if((2^nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2^nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1^st ‘(1^st ‘𝑥)) − (2^nd ‘𝑥))})) = (𝐽‘{((2^nd ‘(1^st ‘𝑥))𝑅ℎ)}))))) = (𝑥 ∈ ((𝑉 × 𝐷) × 𝑉) ↦ if((2^nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2^nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1^st ‘(1^st ‘𝑥)) − (2^nd ‘𝑥))})) = (𝐽‘{((2^nd ‘(1^st ‘𝑥))𝑅ℎ)})))))
40	38, 39	fvmptg 6855	. . 3 ⊢ ((𝑇 ∈ ((𝑉 × 𝐷) × 𝑉) ∧ if((2^nd ‘𝑇) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2^nd ‘𝑇)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1^st ‘(1^st ‘𝑇)) − (2^nd ‘𝑇))})) = (𝐽‘{((2^nd ‘(1^st ‘𝑇))𝑅ℎ)})))) ∈ V) → ((𝑥 ∈ ((𝑉 × 𝐷) × 𝑉) ↦ if((2^nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2^nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1^st ‘(1^st ‘𝑥)) − (2^nd ‘𝑥))})) = (𝐽‘{((2^nd ‘(1^st ‘𝑥))𝑅ℎ)})))))‘𝑇) = if((2^nd ‘𝑇) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2^nd ‘𝑇)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1^st ‘(1^st ‘𝑇)) − (2^nd ‘𝑇))})) = (𝐽‘{((2^nd ‘(1^st ‘𝑇))𝑅ℎ)})))))
41	17, 20, 40	sylancl 585	. 2 ⊢ (𝜑 → ((𝑥 ∈ ((𝑉 × 𝐷) × 𝑉) ↦ if((2^nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2^nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1^st ‘(1^st ‘𝑥)) − (2^nd ‘𝑥))})) = (𝐽‘{((2^nd ‘(1^st ‘𝑥))𝑅ℎ)})))))‘𝑇) = if((2^nd ‘𝑇) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2^nd ‘𝑇)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1^st ‘(1^st ‘𝑇)) − (2^nd ‘𝑇))})) = (𝐽‘{((2^nd ‘(1^st ‘𝑇))𝑅ℎ)})))))
42	16, 41	eqtrd 2778	1 ⊢ (𝜑 → (𝐼‘𝑇) = if((2^nd ‘𝑇) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2^nd ‘𝑇)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1^st ‘(1^st ‘𝑇)) − (2^nd ‘𝑇))})) = (𝐽‘{((2^nd ‘(1^st ‘𝑇))𝑅ℎ)})))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2108 Vcvv 3422 ifcif 4456 {csn 4558 ↦ cmpt 5153 × cxp 5578 ‘cfv 6418 ℩crio 7211 (class class class)co 7255 1^st c1st 7802 2^nd c2nd 7803 Basecbs 16840 0_gc0g 17067 -_gcsg 18494 LSpanclspn 20148 LHypclh 37925 DVecHcdvh 39019 LCDualclcd 39527 mapdcmpd 39565 HDMap1chdma1 39732

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-ral 3068 df-rex 3069 df-reu 3070 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-op 4565 df-uni 4837 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-id 5480 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-riota 7212 df-ov 7258 df-hdmap1 39734

This theorem is referenced by: hdmap1val 39739

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