hdmap1vallem - Mathbox for Norm Megill

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Theorem hdmap1vallem 37087

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 37086	. . 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 6193	. 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		fvex 6201	. . . . 5 ⊢ (0_g‘𝐶) ∈ V
19	10, 18	eqeltri 2697	. . . 4 ⊢ 𝑄 ∈ V
20		riotaex 6615	. . . 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
21	19, 20	ifex 4156	. . 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
22		fveq2 6191	. . . . . 6 ⊢ (𝑥 = 𝑇 → (2^nd ‘𝑥) = (2^nd ‘𝑇))
23	22	eqeq1d 2624	. . . . 5 ⊢ (𝑥 = 𝑇 → ((2^nd ‘𝑥) = 0 ↔ (2^nd ‘𝑇) = 0 ))
24	22	sneqd 4189	. . . . . . . . . 10 ⊢ (𝑥 = 𝑇 → {(2^nd ‘𝑥)} = {(2^nd ‘𝑇)})
25	24	fveq2d 6195	. . . . . . . . 9 ⊢ (𝑥 = 𝑇 → (𝑁‘{(2^nd ‘𝑥)}) = (𝑁‘{(2^nd ‘𝑇)}))
26	25	fveq2d 6195	. . . . . . . 8 ⊢ (𝑥 = 𝑇 → (𝑀‘(𝑁‘{(2^nd ‘𝑥)})) = (𝑀‘(𝑁‘{(2^nd ‘𝑇)})))
27	26	eqeq1d 2624	. . . . . . 7 ⊢ (𝑥 = 𝑇 → ((𝑀‘(𝑁‘{(2^nd ‘𝑥)})) = (𝐽‘{ℎ}) ↔ (𝑀‘(𝑁‘{(2^nd ‘𝑇)})) = (𝐽‘{ℎ})))
28		fveq2 6191	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑇 → (1^st ‘𝑥) = (1^st ‘𝑇))
29	28	fveq2d 6195	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑇 → (1^st ‘(1^st ‘𝑥)) = (1^st ‘(1^st ‘𝑇)))
30	29, 22	oveq12d 6668	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑇 → ((1^st ‘(1^st ‘𝑥)) − (2^nd ‘𝑥)) = ((1^st ‘(1^st ‘𝑇)) − (2^nd ‘𝑇)))
31	30	sneqd 4189	. . . . . . . . . 10 ⊢ (𝑥 = 𝑇 → {((1^st ‘(1^st ‘𝑥)) − (2^nd ‘𝑥))} = {((1^st ‘(1^st ‘𝑇)) − (2^nd ‘𝑇))})
32	31	fveq2d 6195	. . . . . . . . 9 ⊢ (𝑥 = 𝑇 → (𝑁‘{((1^st ‘(1^st ‘𝑥)) − (2^nd ‘𝑥))}) = (𝑁‘{((1^st ‘(1^st ‘𝑇)) − (2^nd ‘𝑇))}))
33	32	fveq2d 6195	. . . . . . . 8 ⊢ (𝑥 = 𝑇 → (𝑀‘(𝑁‘{((1^st ‘(1^st ‘𝑥)) − (2^nd ‘𝑥))})) = (𝑀‘(𝑁‘{((1^st ‘(1^st ‘𝑇)) − (2^nd ‘𝑇))})))
34	28	fveq2d 6195	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑇 → (2^nd ‘(1^st ‘𝑥)) = (2^nd ‘(1^st ‘𝑇)))
35	34	oveq1d 6665	. . . . . . . . . 10 ⊢ (𝑥 = 𝑇 → ((2^nd ‘(1^st ‘𝑥))𝑅ℎ) = ((2^nd ‘(1^st ‘𝑇))𝑅ℎ))
36	35	sneqd 4189	. . . . . . . . 9 ⊢ (𝑥 = 𝑇 → {((2^nd ‘(1^st ‘𝑥))𝑅ℎ)} = {((2^nd ‘(1^st ‘𝑇))𝑅ℎ)})
37	36	fveq2d 6195	. . . . . . . 8 ⊢ (𝑥 = 𝑇 → (𝐽‘{((2^nd ‘(1^st ‘𝑥))𝑅ℎ)}) = (𝐽‘{((2^nd ‘(1^st ‘𝑇))𝑅ℎ)}))
38	33, 37	eqeq12d 2637	. . . . . . 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)})))
39	27, 38	anbi12d 747	. . . . . 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)}))))
40	39	riotabidv 6613	. . . . 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)}))))
41	23, 40	ifbieq2d 4111	. . . 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)})))))
42		eqid 2622	. . . 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)})))))
43	41, 42	fvmptg 6280	. . 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)})))))
44	17, 21, 43	sylancl 694	. 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)})))))
45	16, 44	eqtrd 2656	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 384 = wceq 1483 ∈ wcel 1990 Vcvv 3200 ifcif 4086 {csn 4177 ↦ cmpt 4729 × cxp 5112 ‘cfv 5888 ℩crio 6610 (class class class)co 6650 1^st c1st 7166 2^nd c2nd 7167 Basecbs 15857 0_gc0g 16100 -_gcsg 17424 LSpanclspn 18971 LHypclh 35270 DVecHcdvh 36367 LCDualclcd 36875 mapdcmpd 36913 HDMap1chdma1 37081

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-hdmap1 37083

This theorem is referenced by: hdmap1val 37088

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