Theorem isvcOLD 29827

Description: The predicate "is a complex vector space." (Contributed by NM, 31-May-2008.) Obsolete version of iscvsp 24643. (New usage is discouraged.) (Proof modification is discouraged.)

Hypothesis

Ref	Expression
isvcOLD.1	⊢ 𝑋 = ran 𝐺

Assertion

Ref	Expression
isvcOLD	⊢ (⟨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)))))))

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

Allowed substitution hint:   𝑋(𝑦)

Proof of Theorem isvcOLD

Step	Hyp	Ref	Expression
1		vcex 29826	. 2 ⊢ (⟨𝐺, 𝑆⟩ ∈ CVecOLD → (𝐺 ∈ V ∧ 𝑆 ∈ V))
2		elex 3492	. . . . 5 ⊢ (𝐺 ∈ AbelOp → 𝐺 ∈ V)
3	2	adantr 481	. . . 4 ⊢ ((𝐺 ∈ AbelOp ∧ 𝑆:(ℂ × 𝑋)⟶𝑋) → 𝐺 ∈ V)
4		cnex 11190	. . . . . . 7 ⊢ ℂ ∈ V
5		ablogrpo 29795	. . . . . . . 8 ⊢ (𝐺 ∈ AbelOp → 𝐺 ∈ GrpOp)
6		isvcOLD.1	. . . . . . . . 9 ⊢ 𝑋 = ran 𝐺
7		rnexg 7894	. . . . . . . . 9 ⊢ (𝐺 ∈ GrpOp → ran 𝐺 ∈ V)
8	6, 7	eqeltrid 2837	. . . . . . . 8 ⊢ (𝐺 ∈ GrpOp → 𝑋 ∈ V)
9	5, 8	syl 17	. . . . . . 7 ⊢ (𝐺 ∈ AbelOp → 𝑋 ∈ V)
10		xpexg 7736	. . . . . . 7 ⊢ ((ℂ ∈ V ∧ 𝑋 ∈ V) → (ℂ × 𝑋) ∈ V)
11	4, 9, 10	sylancr 587	. . . . . 6 ⊢ (𝐺 ∈ AbelOp → (ℂ × 𝑋) ∈ V)
12		fex 7227	. . . . . 6 ⊢ ((𝑆:(ℂ × 𝑋)⟶𝑋 ∧ (ℂ × 𝑋) ∈ V) → 𝑆 ∈ V)
13	11, 12	sylan2 593	. . . . 5 ⊢ ((𝑆:(ℂ × 𝑋)⟶𝑋 ∧ 𝐺 ∈ AbelOp) → 𝑆 ∈ V)
14	13	ancoms 459	. . . 4 ⊢ ((𝐺 ∈ AbelOp ∧ 𝑆:(ℂ × 𝑋)⟶𝑋) → 𝑆 ∈ V)
15	3, 14	jca 512	. . 3 ⊢ ((𝐺 ∈ AbelOp ∧ 𝑆:(ℂ × 𝑋)⟶𝑋) → (𝐺 ∈ V ∧ 𝑆 ∈ V))
16	15	3adant3 1132	. 2 ⊢ ((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)))))) → (𝐺 ∈ V ∧ 𝑆 ∈ V))
17	6	isvclem 29825	. 2 ⊢ ((𝐺 ∈ V ∧ 𝑆 ∈ V) → (⟨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))))))))
18	1, 16, 17	pm5.21nii 379	1 ⊢ (⟨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)))))))

Syntax hints:   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  ∀wral 3061  Vcvv 3474  ⟨cop 4634   × cxp 5674  ran crn 5677  ⟶wf 6539  (class class class)co 7408  ℂcc 11107  1c1 11110   + caddc 11112   · cmul 11114  GrpOpcgr 29737  AbelOpcablo 29792  CVec_OLDcvc 29806

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724  ax-cnex 11165

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7411  df-ablo 29793  df-vc 29807

This theorem is referenced by:  isvciOLD  29828