Theorem vcass 30591

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 30585	. . . . 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 3071	. . . . . . . . . 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 3071	. . . . . . . 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 3071	. . . . . 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 1135	. . . . 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 7364	. . . . . 6 ⊢ (𝑥 = 𝐶 → ((𝑦 · 𝑧)𝑆𝑥) = ((𝑦 · 𝑧)𝑆𝐶))
14		oveq2 7364	. . . . . . 7 ⊢ (𝑥 = 𝐶 → (𝑧𝑆𝑥) = (𝑧𝑆𝐶))
15	14	oveq2d 7372	. . . . . 6 ⊢ (𝑥 = 𝐶 → (𝑦𝑆(𝑧𝑆𝑥)) = (𝑦𝑆(𝑧𝑆𝐶)))
16	13, 15	eqeq12d 2750	. . . . 5 ⊢ (𝑥 = 𝐶 → (((𝑦 · 𝑧)𝑆𝑥) = (𝑦𝑆(𝑧𝑆𝑥)) ↔ ((𝑦 · 𝑧)𝑆𝐶) = (𝑦𝑆(𝑧𝑆𝐶))))
17		oveq1 7363	. . . . . . 7 ⊢ (𝑦 = 𝐴 → (𝑦 · 𝑧) = (𝐴 · 𝑧))
18	17	oveq1d 7371	. . . . . 6 ⊢ (𝑦 = 𝐴 → ((𝑦 · 𝑧)𝑆𝐶) = ((𝐴 · 𝑧)𝑆𝐶))
19		oveq1 7363	. . . . . 6 ⊢ (𝑦 = 𝐴 → (𝑦𝑆(𝑧𝑆𝐶)) = (𝐴𝑆(𝑧𝑆𝐶)))
20	18, 19	eqeq12d 2750	. . . . 5 ⊢ (𝑦 = 𝐴 → (((𝑦 · 𝑧)𝑆𝐶) = (𝑦𝑆(𝑧𝑆𝐶)) ↔ ((𝐴 · 𝑧)𝑆𝐶) = (𝐴𝑆(𝑧𝑆𝐶))))
21		oveq2 7364	. . . . . . 7 ⊢ (𝑧 = 𝐵 → (𝐴 · 𝑧) = (𝐴 · 𝐵))
22	21	oveq1d 7371	. . . . . 6 ⊢ (𝑧 = 𝐵 → ((𝐴 · 𝑧)𝑆𝐶) = ((𝐴 · 𝐵)𝑆𝐶))
23		oveq1 7363	. . . . . . 7 ⊢ (𝑧 = 𝐵 → (𝑧𝑆𝐶) = (𝐵𝑆𝐶))
24	23	oveq2d 7372	. . . . . 6 ⊢ (𝑧 = 𝐵 → (𝐴𝑆(𝑧𝑆𝐶)) = (𝐴𝑆(𝐵𝑆𝐶)))
25	22, 24	eqeq12d 2750	. . . . 5 ⊢ (𝑧 = 𝐵 → (((𝐴 · 𝑧)𝑆𝐶) = (𝐴𝑆(𝑧𝑆𝐶)) ↔ ((𝐴 · 𝐵)𝑆𝐶) = (𝐴𝑆(𝐵𝑆𝐶))))
26	16, 20, 25	rspc3v 3590	. . . 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 1127	. 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 1086   = wceq 1541   ∈ wcel 2113  ∀wral 3049   × cxp 5620  ran crn 5623  ⟶wf 6486  ‘cfv 6490  (class class class)co 7356  1^st c1st 7929  2^nd c2nd 7930  ℂcc 11022  1c1 11025   + caddc 11027   · cmul 11029  AbelOpcablo 30568  CVec_OLDcvc 30582

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pr 5375  ax-un 7678

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-rab 3398  df-v 3440  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-if 4478  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-br 5097  df-opab 5159  df-mpt 5178  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-fv 6498  df-ov 7359  df-1st 7931  df-2nd 7932  df-vc 30583

This theorem is referenced by:  vcz  30599  nvsass  30652