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Theorem vcidOLD 28347

Description: Identity element for the scalar product of a complex vector space. (Contributed by NM, 3-Nov-2006.) Obsolete theorem, use clmvs1 23698 together with cvsclm 23731 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 28344	. . 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 486	. . . . 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 3128	. . . 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 1132	. . 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 7143	. . . 4 ⊢ (𝑥 = 𝐴 → (1𝑆𝑥) = (1𝑆𝐴))
10		id 22	. . . 4 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴)
11	9, 10	eqeq12d 2814	. . 3 ⊢ (𝑥 = 𝐴 → ((1𝑆𝑥) = 𝑥 ↔ (1𝑆𝐴) = 𝐴))
12	11	rspccva 3570	. 2 ⊢ ((∀𝑥 ∈ 𝑋 (1𝑆𝑥) = 𝑥 ∧ 𝐴 ∈ 𝑋) → (1𝑆𝐴) = 𝐴)
13	8, 12	sylan 583	1 ⊢ ((𝑊 ∈ CVec_OLD ∧ 𝐴 ∈ 𝑋) → (1𝑆𝐴) = 𝐴)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ∀wral 3106 × cxp 5517 ran crn 5520 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 1^st c1st 7669 2^nd c2nd 7670 ℂcc 10524 1c1 10527 + caddc 10529 · cmul 10531 AbelOpcablo 28327 CVec_OLDcvc 28341

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-fv 6332 df-ov 7138 df-1st 7671 df-2nd 7672 df-vc 28342

This theorem is referenced by: vc2OLD 28351 vc0 28357 vcm 28359 nvsid 28410

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