hdmap1vallem - Mathbox for Norm Megill

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Theorem hdmap1vallem 41779

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 41778	. . 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 6908	. 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 6920	. . . 4 ⊢ 𝑄 ∈ V
19		riotaex 7391	. . . 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 4580	. . 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 6915	. . . . 5 ⊢ (𝑥 = 𝑇 → ((2^nd ‘𝑥) = 0 ↔ (2^nd ‘𝑇) = 0 ))
22		fveq2 6906	. . . . . . . . . 10 ⊢ (𝑥 = 𝑇 → (2^nd ‘𝑥) = (2^nd ‘𝑇))
23	22	sneqd 4642	. . . . . . . . 9 ⊢ (𝑥 = 𝑇 → {(2^nd ‘𝑥)} = {(2^nd ‘𝑇)})
24	23	fveq2d 6910	. . . . . . . 8 ⊢ (𝑥 = 𝑇 → (𝑁‘{(2^nd ‘𝑥)}) = (𝑁‘{(2^nd ‘𝑇)}))
25	24	fveqeq2d 6914	. . . . . . 7 ⊢ (𝑥 = 𝑇 → ((𝑀‘(𝑁‘{(2^nd ‘𝑥)})) = (𝐽‘{ℎ}) ↔ (𝑀‘(𝑁‘{(2^nd ‘𝑇)})) = (𝐽‘{ℎ})))
26		2fveq3 6911	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑇 → (1^st ‘(1^st ‘𝑥)) = (1^st ‘(1^st ‘𝑇)))
27	26, 22	oveq12d 7448	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑇 → ((1^st ‘(1^st ‘𝑥)) − (2^nd ‘𝑥)) = ((1^st ‘(1^st ‘𝑇)) − (2^nd ‘𝑇)))
28	27	sneqd 4642	. . . . . . . . . 10 ⊢ (𝑥 = 𝑇 → {((1^st ‘(1^st ‘𝑥)) − (2^nd ‘𝑥))} = {((1^st ‘(1^st ‘𝑇)) − (2^nd ‘𝑇))})
29	28	fveq2d 6910	. . . . . . . . 9 ⊢ (𝑥 = 𝑇 → (𝑁‘{((1^st ‘(1^st ‘𝑥)) − (2^nd ‘𝑥))}) = (𝑁‘{((1^st ‘(1^st ‘𝑇)) − (2^nd ‘𝑇))}))
30	29	fveq2d 6910	. . . . . . . 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 6911	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑇 → (2^nd ‘(1^st ‘𝑥)) = (2^nd ‘(1^st ‘𝑇)))
32	31	oveq1d 7445	. . . . . . . . . 10 ⊢ (𝑥 = 𝑇 → ((2^nd ‘(1^st ‘𝑥))𝑅ℎ) = ((2^nd ‘(1^st ‘𝑇))𝑅ℎ))
33	32	sneqd 4642	. . . . . . . . 9 ⊢ (𝑥 = 𝑇 → {((2^nd ‘(1^st ‘𝑥))𝑅ℎ)} = {((2^nd ‘(1^st ‘𝑇))𝑅ℎ)})
34	33	fveq2d 6910	. . . . . . . 8 ⊢ (𝑥 = 𝑇 → (𝐽‘{((2^nd ‘(1^st ‘𝑥))𝑅ℎ)}) = (𝐽‘{((2^nd ‘(1^st ‘𝑇))𝑅ℎ)}))
35	30, 34	eqeq12d 2750	. . . . . . 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 632	. . . . . 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 7389	. . . . 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 4556	. . . 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 2734	. . . 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 7013	. . 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 586	. 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 2774	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 1536 ∈ wcel 2105 Vcvv 3477 ifcif 4530 {csn 4630 ↦ cmpt 5230 × cxp 5686 ‘cfv 6562 ℩crio 7386 (class class class)co 7430 1^st c1st 8010 2^nd c2nd 8011 Basecbs 17244 0_gc0g 17485 -_gcsg 18965 LSpanclspn 20986 LHypclh 39966 DVecHcdvh 41060 LCDualclcd 41568 mapdcmpd 41606 HDMap1chdma1 41773

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-hdmap1 41775

This theorem is referenced by: hdmap1val 41780

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