Theorem vcidOLD 31172

Description: Identity element for the scalar product of a complex vector space. (Contributed by NM, 3-Nov-2006.) Obsolete theorem, use clmvs1 25420 together with cvsclm 25453 instead. (New usage is discouraged.) (Proof modification is discouraged.)

Hypotheses

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

Assertion

Ref	Expression
vcidOLD	⊢ ((𝑊 ∈ CVecOLD ∧ 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 = (1st ‘𝑊)
2		vciOLD.2	. . . 4 ⊢ 𝑆 = (2nd ‘𝑊)
3		vciOLD.3	. . . 4 ⊢ 𝑋 = ran 𝐺
4	1, 2, 3	vciOLD 31169	. . 3 ⊢ (𝑊 ∈ CVecOLD → (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 488	. . . . 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 3099	. . . 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 1153	. . 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 18	. 2 ⊢ (𝑊 ∈ CVecOLD → ∀𝑥 ∈ 𝑋 (1𝑆𝑥) = 𝑥)
9		oveq2 7429	. . . 4 ⊢ (𝑥 = 𝐴 → (1𝑆𝑥) = (1𝑆𝐴))
10		id 23	. . . 4 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴)
11	9, 10	eqeq12d 2776	. . 3 ⊢ (𝑥 = 𝐴 → ((1𝑆𝑥) = 𝑥 ↔ (1𝑆𝐴) = 𝐴))
12	11	rspccva 3575	. 2 ⊢ ((∀𝑥 ∈ 𝑋 (1𝑆𝑥) = 𝑥 ∧ 𝐴 ∈ 𝑋) → (1𝑆𝐴) = 𝐴)
13	8, 12	sylan 592	1 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → (1𝑆𝐴) = 𝐴)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144  ∀wral 3076   × cxp 5648  ran crn 5651  ⟶wf 6535  ‘cfv 6539  (class class class)co 7421  1^st c1st 8000  2^nd c2nd 8001  ℂcc 11202  1c1 11205   + caddc 11207   · cmul 11209  AbelOpcablo 31152  CVec_OLDcvc 31166

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390  ax-un 7752

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-iota 6494  df-fun 6541  df-fn 6542  df-f 6543  df-fv 6547  df-ov 7424  df-1st 8002  df-2nd 8003  df-vc 31167

This theorem is used by:  vc2OLD  31176  vc0  31182  vcm  31184  nvsid  31235