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Theorem vcidOLD 30593

Description: Identity element for the scalar product of a complex vector space. (Contributed by NM, 3-Nov-2006.) Obsolete theorem, use clmvs1 25140 together with cvsclm 25173 instead. (New usage is discouraged.) (Proof modification is discouraged.)

**Hypotheses**
Ref	Expression
vciOLD.1	⊢ 𝐺 = (1^st ‘𝑊)
vciOLD.2	⊢ 𝑆 = (2^nd ‘𝑊)
vciOLD.3	⊢ 𝑋 = ran 𝐺

**Assertion**
Ref	Expression
vcidOLD	⊢ ((𝑊 ∈ CVec_OLD ∧ 𝐴 ∈ 𝑋) → (1𝑆𝐴) = 𝐴)

Proof of Theorem vcidOLD Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		vciOLD.1	. . . 4 ⊢ 𝐺 = (1^st ‘𝑊)
2		vciOLD.2	. . . 4 ⊢ 𝑆 = (2^nd ‘𝑊)
3		vciOLD.3	. . . 4 ⊢ 𝑋 = ran 𝐺
4	1, 2, 3	vciOLD 30590	. . 3 ⊢ (𝑊 ∈ CVec_OLD → (𝐺 ∈ AbelOp ∧ 𝑆:(ℂ × 𝑋)⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ((1𝑆𝑥) = 𝑥 ∧ ∀𝑦 ∈ ℂ (∀𝑧 ∈ 𝑋 (𝑦𝑆(𝑥𝐺𝑧)) = ((𝑦𝑆𝑥)𝐺(𝑦𝑆𝑧)) ∧ ∀𝑧 ∈ ℂ (((𝑦 + 𝑧)𝑆𝑥) = ((𝑦𝑆𝑥)𝐺(𝑧𝑆𝑥)) ∧ ((𝑦 · 𝑧)𝑆𝑥) = (𝑦𝑆(𝑧𝑆𝑥)))))))
5		simpl 482	. . . . 5 ⊢ (((1𝑆𝑥) = 𝑥 ∧ ∀𝑦 ∈ ℂ (∀𝑧 ∈ 𝑋 (𝑦𝑆(𝑥𝐺𝑧)) = ((𝑦𝑆𝑥)𝐺(𝑦𝑆𝑧)) ∧ ∀𝑧 ∈ ℂ (((𝑦 + 𝑧)𝑆𝑥) = ((𝑦𝑆𝑥)𝐺(𝑧𝑆𝑥)) ∧ ((𝑦 · 𝑧)𝑆𝑥) = (𝑦𝑆(𝑧𝑆𝑥))))) → (1𝑆𝑥) = 𝑥)
6	5	ralimi 3081	. . . 4 ⊢ (∀𝑥 ∈ 𝑋 ((1𝑆𝑥) = 𝑥 ∧ ∀𝑦 ∈ ℂ (∀𝑧 ∈ 𝑋 (𝑦𝑆(𝑥𝐺𝑧)) = ((𝑦𝑆𝑥)𝐺(𝑦𝑆𝑧)) ∧ ∀𝑧 ∈ ℂ (((𝑦 + 𝑧)𝑆𝑥) = ((𝑦𝑆𝑥)𝐺(𝑧𝑆𝑥)) ∧ ((𝑦 · 𝑧)𝑆𝑥) = (𝑦𝑆(𝑧𝑆𝑥))))) → ∀𝑥 ∈ 𝑋 (1𝑆𝑥) = 𝑥)
7	6	3ad2ant3 1134	. . 3 ⊢ ((𝐺 ∈ AbelOp ∧ 𝑆:(ℂ × 𝑋)⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ((1𝑆𝑥) = 𝑥 ∧ ∀𝑦 ∈ ℂ (∀𝑧 ∈ 𝑋 (𝑦𝑆(𝑥𝐺𝑧)) = ((𝑦𝑆𝑥)𝐺(𝑦𝑆𝑧)) ∧ ∀𝑧 ∈ ℂ (((𝑦 + 𝑧)𝑆𝑥) = ((𝑦𝑆𝑥)𝐺(𝑧𝑆𝑥)) ∧ ((𝑦 · 𝑧)𝑆𝑥) = (𝑦𝑆(𝑧𝑆𝑥)))))) → ∀𝑥 ∈ 𝑋 (1𝑆𝑥) = 𝑥)
8	4, 7	syl 17	. 2 ⊢ (𝑊 ∈ CVec_OLD → ∀𝑥 ∈ 𝑋 (1𝑆𝑥) = 𝑥)
9		oveq2 7439	. . . 4 ⊢ (𝑥 = 𝐴 → (1𝑆𝑥) = (1𝑆𝐴))
10		id 22	. . . 4 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴)
11	9, 10	eqeq12d 2751	. . 3 ⊢ (𝑥 = 𝐴 → ((1𝑆𝑥) = 𝑥 ↔ (1𝑆𝐴) = 𝐴))
12	11	rspccva 3621	. 2 ⊢ ((∀𝑥 ∈ 𝑋 (1𝑆𝑥) = 𝑥 ∧ 𝐴 ∈ 𝑋) → (1𝑆𝐴) = 𝐴)
13	8, 12	sylan 580	1 ⊢ ((𝑊 ∈ CVec_OLD ∧ 𝐴 ∈ 𝑋) → (1𝑆𝐴) = 𝐴)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ∀wral 3059 × cxp 5687 ran crn 5690 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 1^st c1st 8011 2^nd c2nd 8012 ℂcc 11151 1c1 11154 + caddc 11156 · cmul 11158 AbelOpcablo 30573 CVec_OLDcvc 30587

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-fv 6571 df-ov 7434 df-1st 8013 df-2nd 8014 df-vc 30588

This theorem is referenced by: vc2OLD 30597 vc0 30603 vcm 30605 nvsid 30656

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