Theorem isvciOLD 30666

Description: Properties that determine a complex vector space. (Contributed by NM, 5-Nov-2006.) Obsolete version of iscvsi 25106. (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 1124	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ 𝑋) → (𝑦𝑆(𝑥𝐺𝑧)) = ((𝑦𝑆𝑥)𝐺(𝑦𝑆𝑧)))
7	6	3expa 1119	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ ℂ) ∧ 𝑧 ∈ 𝑋) → (𝑦𝑆(𝑥𝐺𝑧)) = ((𝑦𝑆𝑥)𝐺(𝑦𝑆𝑧)))
8	7	ralrimiva 3130	. . . . . . 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 1126	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (((𝑦 + 𝑧)𝑆𝑥) = ((𝑦𝑆𝑥)𝐺(𝑧𝑆𝑥)) ∧ ((𝑦 · 𝑧)𝑆𝑥) = (𝑦𝑆(𝑧𝑆𝑥))))
13	12	3expa 1119	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ ℂ) ∧ 𝑧 ∈ ℂ) → (((𝑦 + 𝑧)𝑆𝑥) = ((𝑦𝑆𝑥)𝐺(𝑧𝑆𝑥)) ∧ ((𝑦 · 𝑧)𝑆𝑥) = (𝑦𝑆(𝑧𝑆𝑥))))
14	13	ralrimiva 3130	. . . . . . 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 3130	. . . . 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 3054	. . 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 30633	. . . . . 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 30607	. . . 4 ⊢ 𝑋 = ran 𝐺
23	22	isvcOLD 30665	. . 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 1343	. 2 ⊢ ⟨𝐺, 𝑆⟩ ∈ CVecOLD
25	1, 24	eqeltri 2833	1 ⊢ 𝑊 ∈ CVecOLD

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3052  ⟨cop 4574   × cxp 5622  dom cdm 5624  ⟶wf 6488  (class class class)co 7360  ℂcc 11027  1c1 11030   + caddc 11032   · cmul 11034  GrpOpcgr 30575  AbelOpcablo 30630  CVec_OLDcvc 30644

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682  ax-cnex 11085

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7363  df-grpo 30579  df-ablo 30631  df-vc 30645

This theorem is referenced by:  cncvcOLD  30669  hilvc  31248  hhssnv  31350