Theorem vcass 30586

Description: Associative law for the scalar product of a complex vector space. (Contributed by NM, 3-Nov-2006.) (New usage is discouraged.)

Hypotheses

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

Assertion

Ref	Expression
vcass	⊢ ((𝑊 ∈ CVecOLD ∧ (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ 𝑋)) → ((𝐴 · 𝐵)𝑆𝐶) = (𝐴𝑆(𝐵𝑆𝐶)))

Proof of Theorem vcass

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		vciOLD.1	. . . . . 6 ⊢ 𝐺 = (1st ‘𝑊)
2		vciOLD.2	. . . . . 6 ⊢ 𝑆 = (2nd ‘𝑊)
3		vciOLD.3	. . . . . 6 ⊢ 𝑋 = ran 𝐺
4	1, 2, 3	vciOLD 30580	. . . . 5 ⊢ (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		simpr 484	. . . . . . . . . . 11 ⊢ ((((𝑦 + 𝑧)𝑆𝑥) = ((𝑦𝑆𝑥)𝐺(𝑧𝑆𝑥)) ∧ ((𝑦 · 𝑧)𝑆𝑥) = (𝑦𝑆(𝑧𝑆𝑥))) → ((𝑦 · 𝑧)𝑆𝑥) = (𝑦𝑆(𝑧𝑆𝑥)))
6	5	ralimi 3083	. . . . . . . . . 10 ⊢ (∀𝑧 ∈ ℂ (((𝑦 + 𝑧)𝑆𝑥) = ((𝑦𝑆𝑥)𝐺(𝑧𝑆𝑥)) ∧ ((𝑦 · 𝑧)𝑆𝑥) = (𝑦𝑆(𝑧𝑆𝑥))) → ∀𝑧 ∈ ℂ ((𝑦 · 𝑧)𝑆𝑥) = (𝑦𝑆(𝑧𝑆𝑥)))
7	6	adantl 481	. . . . . . . . 9 ⊢ ((∀𝑧 ∈ 𝑋 (𝑦𝑆(𝑥𝐺𝑧)) = ((𝑦𝑆𝑥)𝐺(𝑦𝑆𝑧)) ∧ ∀𝑧 ∈ ℂ (((𝑦 + 𝑧)𝑆𝑥) = ((𝑦𝑆𝑥)𝐺(𝑧𝑆𝑥)) ∧ ((𝑦 · 𝑧)𝑆𝑥) = (𝑦𝑆(𝑧𝑆𝑥)))) → ∀𝑧 ∈ ℂ ((𝑦 · 𝑧)𝑆𝑥) = (𝑦𝑆(𝑧𝑆𝑥)))
8	7	ralimi 3083	. . . . . . . 8 ⊢ (∀𝑦 ∈ ℂ (∀𝑧 ∈ 𝑋 (𝑦𝑆(𝑥𝐺𝑧)) = ((𝑦𝑆𝑥)𝐺(𝑦𝑆𝑧)) ∧ ∀𝑧 ∈ ℂ (((𝑦 + 𝑧)𝑆𝑥) = ((𝑦𝑆𝑥)𝐺(𝑧𝑆𝑥)) ∧ ((𝑦 · 𝑧)𝑆𝑥) = (𝑦𝑆(𝑧𝑆𝑥)))) → ∀𝑦 ∈ ℂ ∀𝑧 ∈ ℂ ((𝑦 · 𝑧)𝑆𝑥) = (𝑦𝑆(𝑧𝑆𝑥)))
9	8	adantl 481	. . . . . . 7 ⊢ (((1𝑆𝑥) = 𝑥 ∧ ∀𝑦 ∈ ℂ (∀𝑧 ∈ 𝑋 (𝑦𝑆(𝑥𝐺𝑧)) = ((𝑦𝑆𝑥)𝐺(𝑦𝑆𝑧)) ∧ ∀𝑧 ∈ ℂ (((𝑦 + 𝑧)𝑆𝑥) = ((𝑦𝑆𝑥)𝐺(𝑧𝑆𝑥)) ∧ ((𝑦 · 𝑧)𝑆𝑥) = (𝑦𝑆(𝑧𝑆𝑥))))) → ∀𝑦 ∈ ℂ ∀𝑧 ∈ ℂ ((𝑦 · 𝑧)𝑆𝑥) = (𝑦𝑆(𝑧𝑆𝑥)))
10	9	ralimi 3083	. . . . . 6 ⊢ (∀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 ∀𝑦 ∈ ℂ ∀𝑧 ∈ ℂ ((𝑦 · 𝑧)𝑆𝑥) = (𝑦𝑆(𝑧𝑆𝑥)))
11	10	3ad2ant3 1136	. . . . 5 ⊢ ((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 ∀𝑦 ∈ ℂ ∀𝑧 ∈ ℂ ((𝑦 · 𝑧)𝑆𝑥) = (𝑦𝑆(𝑧𝑆𝑥)))
12	4, 11	syl 17	. . . 4 ⊢ (𝑊 ∈ CVecOLD → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ ℂ ∀𝑧 ∈ ℂ ((𝑦 · 𝑧)𝑆𝑥) = (𝑦𝑆(𝑧𝑆𝑥)))
13		oveq2 7439	. . . . . 6 ⊢ (𝑥 = 𝐶 → ((𝑦 · 𝑧)𝑆𝑥) = ((𝑦 · 𝑧)𝑆𝐶))
14		oveq2 7439	. . . . . . 7 ⊢ (𝑥 = 𝐶 → (𝑧𝑆𝑥) = (𝑧𝑆𝐶))
15	14	oveq2d 7447	. . . . . 6 ⊢ (𝑥 = 𝐶 → (𝑦𝑆(𝑧𝑆𝑥)) = (𝑦𝑆(𝑧𝑆𝐶)))
16	13, 15	eqeq12d 2753	. . . . 5 ⊢ (𝑥 = 𝐶 → (((𝑦 · 𝑧)𝑆𝑥) = (𝑦𝑆(𝑧𝑆𝑥)) ↔ ((𝑦 · 𝑧)𝑆𝐶) = (𝑦𝑆(𝑧𝑆𝐶))))
17		oveq1 7438	. . . . . . 7 ⊢ (𝑦 = 𝐴 → (𝑦 · 𝑧) = (𝐴 · 𝑧))
18	17	oveq1d 7446	. . . . . 6 ⊢ (𝑦 = 𝐴 → ((𝑦 · 𝑧)𝑆𝐶) = ((𝐴 · 𝑧)𝑆𝐶))
19		oveq1 7438	. . . . . 6 ⊢ (𝑦 = 𝐴 → (𝑦𝑆(𝑧𝑆𝐶)) = (𝐴𝑆(𝑧𝑆𝐶)))
20	18, 19	eqeq12d 2753	. . . . 5 ⊢ (𝑦 = 𝐴 → (((𝑦 · 𝑧)𝑆𝐶) = (𝑦𝑆(𝑧𝑆𝐶)) ↔ ((𝐴 · 𝑧)𝑆𝐶) = (𝐴𝑆(𝑧𝑆𝐶))))
21		oveq2 7439	. . . . . . 7 ⊢ (𝑧 = 𝐵 → (𝐴 · 𝑧) = (𝐴 · 𝐵))
22	21	oveq1d 7446	. . . . . 6 ⊢ (𝑧 = 𝐵 → ((𝐴 · 𝑧)𝑆𝐶) = ((𝐴 · 𝐵)𝑆𝐶))
23		oveq1 7438	. . . . . . 7 ⊢ (𝑧 = 𝐵 → (𝑧𝑆𝐶) = (𝐵𝑆𝐶))
24	23	oveq2d 7447	. . . . . 6 ⊢ (𝑧 = 𝐵 → (𝐴𝑆(𝑧𝑆𝐶)) = (𝐴𝑆(𝐵𝑆𝐶)))
25	22, 24	eqeq12d 2753	. . . . 5 ⊢ (𝑧 = 𝐵 → (((𝐴 · 𝑧)𝑆𝐶) = (𝐴𝑆(𝑧𝑆𝐶)) ↔ ((𝐴 · 𝐵)𝑆𝐶) = (𝐴𝑆(𝐵𝑆𝐶))))
26	16, 20, 25	rspc3v 3638	. . . 4 ⊢ ((𝐶 ∈ 𝑋 ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ ℂ ∀𝑧 ∈ ℂ ((𝑦 · 𝑧)𝑆𝑥) = (𝑦𝑆(𝑧𝑆𝑥)) → ((𝐴 · 𝐵)𝑆𝐶) = (𝐴𝑆(𝐵𝑆𝐶))))
27	12, 26	syl5 34	. . 3 ⊢ ((𝐶 ∈ 𝑋 ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝑊 ∈ CVecOLD → ((𝐴 · 𝐵)𝑆𝐶) = (𝐴𝑆(𝐵𝑆𝐶))))
28	27	3coml 1128	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ 𝑋) → (𝑊 ∈ CVecOLD → ((𝐴 · 𝐵)𝑆𝐶) = (𝐴𝑆(𝐵𝑆𝐶))))
29	28	impcom 407	1 ⊢ ((𝑊 ∈ CVecOLD ∧ (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ 𝑋)) → ((𝐴 · 𝐵)𝑆𝐶) = (𝐴𝑆(𝐵𝑆𝐶)))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1540   ∈ wcel 2108  ∀wral 3061   × cxp 5683  ran crn 5686  ⟶wf 6557  ‘cfv 6561  (class class class)co 7431  1^st c1st 8012  2^nd c2nd 8013  ℂcc 11153  1c1 11156   + caddc 11158   · cmul 11160  AbelOpcablo 30563  CVec_OLDcvc 30577

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-ov 7434  df-1st 8014  df-2nd 8015  df-vc 30578

This theorem is referenced by:  vcz  30594  nvsass  30647