Theorem isvcOLD 28614

Description: The predicate "is a complex vector space." (Contributed by NM, 31-May-2008.) Obsolete version of iscvsp 23979. (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 28613	. 2 ⊢ (⟨𝐺, 𝑆⟩ ∈ CVecOLD → (𝐺 ∈ V ∧ 𝑆 ∈ V))
2		elex 3416	. . . . 5 ⊢ (𝐺 ∈ AbelOp → 𝐺 ∈ V)
3	2	adantr 484	. . . 4 ⊢ ((𝐺 ∈ AbelOp ∧ 𝑆:(ℂ × 𝑋)⟶𝑋) → 𝐺 ∈ V)
4		cnex 10775	. . . . . . 7 ⊢ ℂ ∈ V
5		ablogrpo 28582	. . . . . . . 8 ⊢ (𝐺 ∈ AbelOp → 𝐺 ∈ GrpOp)
6		isvcOLD.1	. . . . . . . . 9 ⊢ 𝑋 = ran 𝐺
7		rnexg 7660	. . . . . . . . 9 ⊢ (𝐺 ∈ GrpOp → ran 𝐺 ∈ V)
8	6, 7	eqeltrid 2835	. . . . . . . 8 ⊢ (𝐺 ∈ GrpOp → 𝑋 ∈ V)
9	5, 8	syl 17	. . . . . . 7 ⊢ (𝐺 ∈ AbelOp → 𝑋 ∈ V)
10		xpexg 7513	. . . . . . 7 ⊢ ((ℂ ∈ V ∧ 𝑋 ∈ V) → (ℂ × 𝑋) ∈ V)
11	4, 9, 10	sylancr 590	. . . . . 6 ⊢ (𝐺 ∈ AbelOp → (ℂ × 𝑋) ∈ V)
12		fex 7020	. . . . . 6 ⊢ ((𝑆:(ℂ × 𝑋)⟶𝑋 ∧ (ℂ × 𝑋) ∈ V) → 𝑆 ∈ V)
13	11, 12	sylan2 596	. . . . 5 ⊢ ((𝑆:(ℂ × 𝑋)⟶𝑋 ∧ 𝐺 ∈ AbelOp) → 𝑆 ∈ V)
14	13	ancoms 462	. . . 4 ⊢ ((𝐺 ∈ AbelOp ∧ 𝑆:(ℂ × 𝑋)⟶𝑋) → 𝑆 ∈ V)
15	3, 14	jca 515	. . 3 ⊢ ((𝐺 ∈ AbelOp ∧ 𝑆:(ℂ × 𝑋)⟶𝑋) → (𝐺 ∈ V ∧ 𝑆 ∈ V))
16	15	3adant3 1134	. 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 28612	. 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 383	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 209   ∧ wa 399   ∧ w3a 1089   = wceq 1543   ∈ wcel 2112  ∀wral 3051  Vcvv 3398  ⟨cop 4533   × cxp 5534  ran crn 5537  ⟶wf 6354  (class class class)co 7191  ℂcc 10692  1c1 10695   + caddc 10697   · cmul 10699  GrpOpcgr 28524  AbelOpcablo 28579  CVec_OLDcvc 28593

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2708  ax-rep 5164  ax-sep 5177  ax-nul 5184  ax-pow 5243  ax-pr 5307  ax-un 7501  ax-cnex 10750

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2728  df-clel 2809  df-nfc 2879  df-ne 2933  df-ral 3056  df-rex 3057  df-reu 3058  df-rab 3060  df-v 3400  df-sbc 3684  df-csb 3799  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4224  df-if 4426  df-pw 4501  df-sn 4528  df-pr 4530  df-op 4534  df-uni 4806  df-iun 4892  df-br 5040  df-opab 5102  df-mpt 5121  df-id 5440  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-iota 6316  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-ov 7194  df-ablo 28580  df-vc 28594

This theorem is referenced by:  isvciOLD  28615