Theorem isvciOLD 30507

Description: Properties that determine a complex vector space. (Contributed by NM, 5-Nov-2006.) Obsolete version of iscvsi 25078. (New usage is discouraged.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
isvciOLD.1	⊢ 𝐺 ∈ AbelOp
isvciOLD.2	⊢ dom 𝐺 = (𝑋 × 𝑋)
isvciOLD.3	⊢ 𝑆:(ℂ × 𝑋)⟶𝑋
isvciOLD.4	⊢ (𝑥 ∈ 𝑋 → (1𝑆𝑥) = 𝑥)
isvciOLD.5	⊢ ((𝑦 ∈ ℂ ∧ 𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) → (𝑦𝑆(𝑥𝐺𝑧)) = ((𝑦𝑆𝑥)𝐺(𝑦𝑆𝑧)))
isvciOLD.6	⊢ ((𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ ∧ 𝑥 ∈ 𝑋) → ((𝑦 + 𝑧)𝑆𝑥) = ((𝑦𝑆𝑥)𝐺(𝑧𝑆𝑥)))
isvciOLD.7	⊢ ((𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ ∧ 𝑥 ∈ 𝑋) → ((𝑦 · 𝑧)𝑆𝑥) = (𝑦𝑆(𝑧𝑆𝑥)))
isvciOLD.8	⊢ 𝑊 = ⟨𝐺, 𝑆⟩

Assertion

Ref	Expression
isvciOLD	⊢ 𝑊 ∈ CVecOLD

Distinct variable groups:   𝑥,𝑦,𝑧,𝐺   𝑥,𝑆,𝑦,𝑧   𝑥,𝑋,𝑦,𝑧

Allowed substitution hints:   𝑊(𝑥,𝑦,𝑧)

Proof of Theorem isvciOLD

Step	Hyp	Ref	Expression
1		isvciOLD.8	. 2 ⊢ 𝑊 = ⟨𝐺, 𝑆⟩
2		isvciOLD.1	. . 3 ⊢ 𝐺 ∈ AbelOp
3		isvciOLD.3	. . 3 ⊢ 𝑆:(ℂ × 𝑋)⟶𝑋
4		isvciOLD.4	. . . . 5 ⊢ (𝑥 ∈ 𝑋 → (1𝑆𝑥) = 𝑥)
5		isvciOLD.5	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ℂ ∧ 𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) → (𝑦𝑆(𝑥𝐺𝑧)) = ((𝑦𝑆𝑥)𝐺(𝑦𝑆𝑧)))
6	5	3com12 1123	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ 𝑋) → (𝑦𝑆(𝑥𝐺𝑧)) = ((𝑦𝑆𝑥)𝐺(𝑦𝑆𝑧)))
7	6	3expa 1118	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ ℂ) ∧ 𝑧 ∈ 𝑋) → (𝑦𝑆(𝑥𝐺𝑧)) = ((𝑦𝑆𝑥)𝐺(𝑦𝑆𝑧)))
8	7	ralrimiva 3132	. . . . . . 7 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ ℂ) → ∀𝑧 ∈ 𝑋 (𝑦𝑆(𝑥𝐺𝑧)) = ((𝑦𝑆𝑥)𝐺(𝑦𝑆𝑧)))
9		isvciOLD.6	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ ∧ 𝑥 ∈ 𝑋) → ((𝑦 + 𝑧)𝑆𝑥) = ((𝑦𝑆𝑥)𝐺(𝑧𝑆𝑥)))
10		isvciOLD.7	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ ∧ 𝑥 ∈ 𝑋) → ((𝑦 · 𝑧)𝑆𝑥) = (𝑦𝑆(𝑧𝑆𝑥)))
11	9, 10	jca 511	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ ∧ 𝑥 ∈ 𝑋) → (((𝑦 + 𝑧)𝑆𝑥) = ((𝑦𝑆𝑥)𝐺(𝑧𝑆𝑥)) ∧ ((𝑦 · 𝑧)𝑆𝑥) = (𝑦𝑆(𝑧𝑆𝑥))))
12	11	3comr 1125	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (((𝑦 + 𝑧)𝑆𝑥) = ((𝑦𝑆𝑥)𝐺(𝑧𝑆𝑥)) ∧ ((𝑦 · 𝑧)𝑆𝑥) = (𝑦𝑆(𝑧𝑆𝑥))))
13	12	3expa 1118	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ ℂ) ∧ 𝑧 ∈ ℂ) → (((𝑦 + 𝑧)𝑆𝑥) = ((𝑦𝑆𝑥)𝐺(𝑧𝑆𝑥)) ∧ ((𝑦 · 𝑧)𝑆𝑥) = (𝑦𝑆(𝑧𝑆𝑥))))
14	13	ralrimiva 3132	. . . . . . 7 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ ℂ) → ∀𝑧 ∈ ℂ (((𝑦 + 𝑧)𝑆𝑥) = ((𝑦𝑆𝑥)𝐺(𝑧𝑆𝑥)) ∧ ((𝑦 · 𝑧)𝑆𝑥) = (𝑦𝑆(𝑧𝑆𝑥))))
15	8, 14	jca 511	. . . . . 6 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ ℂ) → (∀𝑧 ∈ 𝑋 (𝑦𝑆(𝑥𝐺𝑧)) = ((𝑦𝑆𝑥)𝐺(𝑦𝑆𝑧)) ∧ ∀𝑧 ∈ ℂ (((𝑦 + 𝑧)𝑆𝑥) = ((𝑦𝑆𝑥)𝐺(𝑧𝑆𝑥)) ∧ ((𝑦 · 𝑧)𝑆𝑥) = (𝑦𝑆(𝑧𝑆𝑥)))))
16	15	ralrimiva 3132	. . . . 5 ⊢ (𝑥 ∈ 𝑋 → ∀𝑦 ∈ ℂ (∀𝑧 ∈ 𝑋 (𝑦𝑆(𝑥𝐺𝑧)) = ((𝑦𝑆𝑥)𝐺(𝑦𝑆𝑧)) ∧ ∀𝑧 ∈ ℂ (((𝑦 + 𝑧)𝑆𝑥) = ((𝑦𝑆𝑥)𝐺(𝑧𝑆𝑥)) ∧ ((𝑦 · 𝑧)𝑆𝑥) = (𝑦𝑆(𝑧𝑆𝑥)))))
17	4, 16	jca 511	. . . 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))))))
18	17	rgen 3053	. . 3 ⊢ ∀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)))))
19		ablogrpo 30474	. . . . . 6 ⊢ (𝐺 ∈ AbelOp → 𝐺 ∈ GrpOp)
20	2, 19	ax-mp 5	. . . . 5 ⊢ 𝐺 ∈ GrpOp
21		isvciOLD.2	. . . . 5 ⊢ dom 𝐺 = (𝑋 × 𝑋)
22	20, 21	grporn 30448	. . . 4 ⊢ 𝑋 = ran 𝐺
23	22	isvcOLD 30506	. . 3 ⊢ (⟨𝐺, 𝑆⟩ ∈ 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)))))))
24	2, 3, 18, 23	mpbir3an 1342	. 2 ⊢ ⟨𝐺, 𝑆⟩ ∈ CVecOLD
25	1, 24	eqeltri 2830	1 ⊢ 𝑊 ∈ CVecOLD

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2108  ∀wral 3051  ⟨cop 4607   × cxp 5652  dom cdm 5654  ⟶wf 6526  (class class class)co 7403  ℂcc 11125  1c1 11128   + caddc 11130   · cmul 11132  GrpOpcgr 30416  AbelOpcablo 30471  CVec_OLDcvc 30485

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 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7727  ax-cnex 11183

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-iota 6483  df-fun 6532  df-fn 6533  df-f 6534  df-f1 6535  df-fo 6536  df-f1o 6537  df-fv 6538  df-ov 7406  df-grpo 30420  df-ablo 30472  df-vc 30486

This theorem is referenced by:  cncvcOLD  30510  hilvc  31089  hhssnv  31191