Theorem vcdir 31195

Description: Distributive 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
vcdir	⊢ ((𝑊 ∈ CVecOLD ∧ (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ 𝑋)) → ((𝐴 + 𝐵)𝑆𝐶) = ((𝐴𝑆𝐶)𝐺(𝐵𝑆𝐶)))

Proof of Theorem vcdir

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 31190	. . . . 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		simpl 488	. . . . . . . . . . 11 ⊢ ((((𝑦 + 𝑧)𝑆𝑥) = ((𝑦𝑆𝑥)𝐺(𝑧𝑆𝑥)) ∧ ((𝑦 · 𝑧)𝑆𝑥) = (𝑦𝑆(𝑧𝑆𝑥))) → ((𝑦 + 𝑧)𝑆𝑥) = ((𝑦𝑆𝑥)𝐺(𝑧𝑆𝑥)))
6	5	ralimi 3103	. . . . . . . . . 10 ⊢ (∀𝑧 ∈ ℂ (((𝑦 + 𝑧)𝑆𝑥) = ((𝑦𝑆𝑥)𝐺(𝑧𝑆𝑥)) ∧ ((𝑦 · 𝑧)𝑆𝑥) = (𝑦𝑆(𝑧𝑆𝑥))) → ∀𝑧 ∈ ℂ ((𝑦 + 𝑧)𝑆𝑥) = ((𝑦𝑆𝑥)𝐺(𝑧𝑆𝑥)))
7	6	adantl 487	. . . . . . . . 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) = ((𝑦𝑆𝑥)𝐺(𝑧𝑆𝑥)))
8	7	ralimi 3103	. . . . . . . 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) = ((𝑦𝑆𝑥)𝐺(𝑧𝑆𝑥)))
9	8	adantl 487	. . . . . . 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) = ((𝑦𝑆𝑥)𝐺(𝑧𝑆𝑥)))
10	9	ralimi 3103	. . . . . 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) = ((𝑦𝑆𝑥)𝐺(𝑧𝑆𝑥)))
11	10	3ad2ant3 1153	. . . . 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) = ((𝑦𝑆𝑥)𝐺(𝑧𝑆𝑥)))
12	4, 11	syl 18	. . . 4 ⊢ (𝑊 ∈ CVecOLD → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ ℂ ∀𝑧 ∈ ℂ ((𝑦 + 𝑧)𝑆𝑥) = ((𝑦𝑆𝑥)𝐺(𝑧𝑆𝑥)))
13		oveq2 7448	. . . . . 6 ⊢ (𝑥 = 𝐶 → ((𝑦 + 𝑧)𝑆𝑥) = ((𝑦 + 𝑧)𝑆𝐶))
14		oveq2 7448	. . . . . . 7 ⊢ (𝑥 = 𝐶 → (𝑦𝑆𝑥) = (𝑦𝑆𝐶))
15		oveq2 7448	. . . . . . 7 ⊢ (𝑥 = 𝐶 → (𝑧𝑆𝑥) = (𝑧𝑆𝐶))
16	14, 15	oveq12d 7458	. . . . . 6 ⊢ (𝑥 = 𝐶 → ((𝑦𝑆𝑥)𝐺(𝑧𝑆𝑥)) = ((𝑦𝑆𝐶)𝐺(𝑧𝑆𝐶)))
17	13, 16	eqeq12d 2779	. . . . 5 ⊢ (𝑥 = 𝐶 → (((𝑦 + 𝑧)𝑆𝑥) = ((𝑦𝑆𝑥)𝐺(𝑧𝑆𝑥)) ↔ ((𝑦 + 𝑧)𝑆𝐶) = ((𝑦𝑆𝐶)𝐺(𝑧𝑆𝐶))))
18		oveq1 7447	. . . . . . 7 ⊢ (𝑦 = 𝐴 → (𝑦 + 𝑧) = (𝐴 + 𝑧))
19	18	oveq1d 7455	. . . . . 6 ⊢ (𝑦 = 𝐴 → ((𝑦 + 𝑧)𝑆𝐶) = ((𝐴 + 𝑧)𝑆𝐶))
20		oveq1 7447	. . . . . . 7 ⊢ (𝑦 = 𝐴 → (𝑦𝑆𝐶) = (𝐴𝑆𝐶))
21	20	oveq1d 7455	. . . . . 6 ⊢ (𝑦 = 𝐴 → ((𝑦𝑆𝐶)𝐺(𝑧𝑆𝐶)) = ((𝐴𝑆𝐶)𝐺(𝑧𝑆𝐶)))
22	19, 21	eqeq12d 2779	. . . . 5 ⊢ (𝑦 = 𝐴 → (((𝑦 + 𝑧)𝑆𝐶) = ((𝑦𝑆𝐶)𝐺(𝑧𝑆𝐶)) ↔ ((𝐴 + 𝑧)𝑆𝐶) = ((𝐴𝑆𝐶)𝐺(𝑧𝑆𝐶))))
23		oveq2 7448	. . . . . . 7 ⊢ (𝑧 = 𝐵 → (𝐴 + 𝑧) = (𝐴 + 𝐵))
24	23	oveq1d 7455	. . . . . 6 ⊢ (𝑧 = 𝐵 → ((𝐴 + 𝑧)𝑆𝐶) = ((𝐴 + 𝐵)𝑆𝐶))
25		oveq1 7447	. . . . . . 7 ⊢ (𝑧 = 𝐵 → (𝑧𝑆𝐶) = (𝐵𝑆𝐶))
26	25	oveq2d 7456	. . . . . 6 ⊢ (𝑧 = 𝐵 → ((𝐴𝑆𝐶)𝐺(𝑧𝑆𝐶)) = ((𝐴𝑆𝐶)𝐺(𝐵𝑆𝐶)))
27	24, 26	eqeq12d 2779	. . . . 5 ⊢ (𝑧 = 𝐵 → (((𝐴 + 𝑧)𝑆𝐶) = ((𝐴𝑆𝐶)𝐺(𝑧𝑆𝐶)) ↔ ((𝐴 + 𝐵)𝑆𝐶) = ((𝐴𝑆𝐶)𝐺(𝐵𝑆𝐶))))
28	17, 22, 27	rspc3v 3595	. . . 4 ⊢ ((𝐶 ∈ 𝑋 ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ ℂ ∀𝑧 ∈ ℂ ((𝑦 + 𝑧)𝑆𝑥) = ((𝑦𝑆𝑥)𝐺(𝑧𝑆𝑥)) → ((𝐴 + 𝐵)𝑆𝐶) = ((𝐴𝑆𝐶)𝐺(𝐵𝑆𝐶))))
29	12, 28	syl5 35	. . 3 ⊢ ((𝐶 ∈ 𝑋 ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝑊 ∈ CVecOLD → ((𝐴 + 𝐵)𝑆𝐶) = ((𝐴𝑆𝐶)𝐺(𝐵𝑆𝐶))))
30	29	3coml 1145	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ 𝑋) → (𝑊 ∈ CVecOLD → ((𝐴 + 𝐵)𝑆𝐶) = ((𝐴𝑆𝐶)𝐺(𝐵𝑆𝐶))))
31	30	impcom 413	1 ⊢ ((𝑊 ∈ CVecOLD ∧ (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ 𝑋)) → ((𝐴 + 𝐵)𝑆𝐶) = ((𝐴𝑆𝐶)𝐺(𝐵𝑆𝐶)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147  ∀wral 3080   × cxp 5653  ran crn 5656  ⟶wf 6551  ‘cfv 6555  (class class class)co 7440  1^st c1st 8020  2^nd c2nd 8021  ℂcc 11222  1c1 11225   + caddc 11227   · cmul 11229  AbelOpcablo 31173  CVec_OLDcvc 31187

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pr 5395  ax-un 7772

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-iota 6509  df-fun 6557  df-fn 6558  df-f 6559  df-fv 6563  df-ov 7443  df-1st 8022  df-2nd 8023  df-vc 31188

This theorem is used by:  vc2OLD  31197  vc0  31203  vcm  31205  nvdir  31260