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Theorem vcidOLD 27991

Description: Identity element for the scalar product of a complex vector space. (Contributed by NM, 3-Nov-2006.) Obsolete theorem, use clmvs1 23300 together with cvsclm 23333 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 27988	. . 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 476	. . . . 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 3134	. . . 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 1126	. . 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 6930	. . . 4 ⊢ (𝑥 = 𝐴 → (1𝑆𝑥) = (1𝑆𝐴))
10		id 22	. . . 4 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴)
11	9, 10	eqeq12d 2793	. . 3 ⊢ (𝑥 = 𝐴 → ((1𝑆𝑥) = 𝑥 ↔ (1𝑆𝐴) = 𝐴))
12	11	rspccva 3510	. 2 ⊢ ((∀𝑥 ∈ 𝑋 (1𝑆𝑥) = 𝑥 ∧ 𝐴 ∈ 𝑋) → (1𝑆𝐴) = 𝐴)
13	8, 12	sylan 575	1 ⊢ ((𝑊 ∈ CVec_OLD ∧ 𝐴 ∈ 𝑋) → (1𝑆𝐴) = 𝐴)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 386 ∧ w3a 1071 = wceq 1601 ∈ wcel 2107 ∀wral 3090 × cxp 5353 ran crn 5356 ⟶wf 6131 ‘cfv 6135 (class class class)co 6922 1^st c1st 7443 2^nd c2nd 7444 ℂcc 10270 1c1 10273 + caddc 10275 · cmul 10277 AbelOpcablo 27971 CVec_OLDcvc 27985

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226

This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3400 df-sbc 3653 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4672 df-br 4887 df-opab 4949 df-mpt 4966 df-id 5261 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-fv 6143 df-ov 6925 df-1st 7445 df-2nd 7446 df-vc 27986

This theorem is referenced by: vc2OLD 27995 vc0 28001 vcm 28003 nvsid 28054

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