hdmap1vallem - Mathbox for Norm Megill

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Theorem hdmap1vallem 41798

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 41797	. . 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 6863	. 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 6875	. . . 4 ⊢ 𝑄 ∈ V
19		riotaex 7351	. . . 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 4542	. . 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 6870	. . . . 5 ⊢ (𝑥 = 𝑇 → ((2^nd ‘𝑥) = 0 ↔ (2^nd ‘𝑇) = 0 ))
22		fveq2 6861	. . . . . . . . . 10 ⊢ (𝑥 = 𝑇 → (2^nd ‘𝑥) = (2^nd ‘𝑇))
23	22	sneqd 4604	. . . . . . . . 9 ⊢ (𝑥 = 𝑇 → {(2^nd ‘𝑥)} = {(2^nd ‘𝑇)})
24	23	fveq2d 6865	. . . . . . . 8 ⊢ (𝑥 = 𝑇 → (𝑁‘{(2^nd ‘𝑥)}) = (𝑁‘{(2^nd ‘𝑇)}))
25	24	fveqeq2d 6869	. . . . . . 7 ⊢ (𝑥 = 𝑇 → ((𝑀‘(𝑁‘{(2^nd ‘𝑥)})) = (𝐽‘{ℎ}) ↔ (𝑀‘(𝑁‘{(2^nd ‘𝑇)})) = (𝐽‘{ℎ})))
26		2fveq3 6866	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑇 → (1^st ‘(1^st ‘𝑥)) = (1^st ‘(1^st ‘𝑇)))
27	26, 22	oveq12d 7408	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑇 → ((1^st ‘(1^st ‘𝑥)) − (2^nd ‘𝑥)) = ((1^st ‘(1^st ‘𝑇)) − (2^nd ‘𝑇)))
28	27	sneqd 4604	. . . . . . . . . 10 ⊢ (𝑥 = 𝑇 → {((1^st ‘(1^st ‘𝑥)) − (2^nd ‘𝑥))} = {((1^st ‘(1^st ‘𝑇)) − (2^nd ‘𝑇))})
29	28	fveq2d 6865	. . . . . . . . 9 ⊢ (𝑥 = 𝑇 → (𝑁‘{((1^st ‘(1^st ‘𝑥)) − (2^nd ‘𝑥))}) = (𝑁‘{((1^st ‘(1^st ‘𝑇)) − (2^nd ‘𝑇))}))
30	29	fveq2d 6865	. . . . . . . 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 6866	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑇 → (2^nd ‘(1^st ‘𝑥)) = (2^nd ‘(1^st ‘𝑇)))
32	31	oveq1d 7405	. . . . . . . . . 10 ⊢ (𝑥 = 𝑇 → ((2^nd ‘(1^st ‘𝑥))𝑅ℎ) = ((2^nd ‘(1^st ‘𝑇))𝑅ℎ))
33	32	sneqd 4604	. . . . . . . . 9 ⊢ (𝑥 = 𝑇 → {((2^nd ‘(1^st ‘𝑥))𝑅ℎ)} = {((2^nd ‘(1^st ‘𝑇))𝑅ℎ)})
34	33	fveq2d 6865	. . . . . . . 8 ⊢ (𝑥 = 𝑇 → (𝐽‘{((2^nd ‘(1^st ‘𝑥))𝑅ℎ)}) = (𝐽‘{((2^nd ‘(1^st ‘𝑇))𝑅ℎ)}))
35	30, 34	eqeq12d 2746	. . . . . . 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 7349	. . . . 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 4518	. . . 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 2730	. . . 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 6969	. . 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 2765	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 1540 ∈ wcel 2109 Vcvv 3450 ifcif 4491 {csn 4592 ↦ cmpt 5191 × cxp 5639 ‘cfv 6514 ℩crio 7346 (class class class)co 7390 1^st c1st 7969 2^nd c2nd 7970 Basecbs 17186 0_gc0g 17409 -_gcsg 18874 LSpanclspn 20884 LHypclh 39985 DVecHcdvh 41079 LCDualclcd 41587 mapdcmpd 41625 HDMap1chdma1 41792

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-rep 5237 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-ral 3046 df-rex 3055 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4875 df-iun 4960 df-br 5111 df-opab 5173 df-mpt 5192 df-id 5536 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-riota 7347 df-ov 7393 df-hdmap1 41794

This theorem is referenced by: hdmap1val 41799

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