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Theorem isvciOLD 30612
Description: Properties that determine a complex vector space. (Contributed by NM, 5-Nov-2006.) Obsolete version of iscvsi 25181. (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 (𝑥𝑋 → (1𝑆𝑥) = 𝑥)
isvciOLD.5 ((𝑦 ∈ ℂ ∧ 𝑥𝑋𝑧𝑋) → (𝑦𝑆(𝑥𝐺𝑧)) = ((𝑦𝑆𝑥)𝐺(𝑦𝑆𝑧)))
isvciOLD.6 ((𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ ∧ 𝑥𝑋) → ((𝑦 + 𝑧)𝑆𝑥) = ((𝑦𝑆𝑥)𝐺(𝑧𝑆𝑥)))
isvciOLD.7 ((𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ ∧ 𝑥𝑋) → ((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
StepHypRef 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 (𝑥𝑋 → (1𝑆𝑥) = 𝑥)
5 isvciOLD.5 . . . . . . . . . 10 ((𝑦 ∈ ℂ ∧ 𝑥𝑋𝑧𝑋) → (𝑦𝑆(𝑥𝐺𝑧)) = ((𝑦𝑆𝑥)𝐺(𝑦𝑆𝑧)))
653com12 1123 . . . . . . . . 9 ((𝑥𝑋𝑦 ∈ ℂ ∧ 𝑧𝑋) → (𝑦𝑆(𝑥𝐺𝑧)) = ((𝑦𝑆𝑥)𝐺(𝑦𝑆𝑧)))
763expa 1118 . . . . . . . 8 (((𝑥𝑋𝑦 ∈ ℂ) ∧ 𝑧𝑋) → (𝑦𝑆(𝑥𝐺𝑧)) = ((𝑦𝑆𝑥)𝐺(𝑦𝑆𝑧)))
87ralrimiva 3152 . . . . . . 7 ((𝑥𝑋𝑦 ∈ ℂ) → ∀𝑧𝑋 (𝑦𝑆(𝑥𝐺𝑧)) = ((𝑦𝑆𝑥)𝐺(𝑦𝑆𝑧)))
9 isvciOLD.6 . . . . . . . . . . 11 ((𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ ∧ 𝑥𝑋) → ((𝑦 + 𝑧)𝑆𝑥) = ((𝑦𝑆𝑥)𝐺(𝑧𝑆𝑥)))
10 isvciOLD.7 . . . . . . . . . . 11 ((𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ ∧ 𝑥𝑋) → ((𝑦 · 𝑧)𝑆𝑥) = (𝑦𝑆(𝑧𝑆𝑥)))
119, 10jca 511 . . . . . . . . . 10 ((𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ ∧ 𝑥𝑋) → (((𝑦 + 𝑧)𝑆𝑥) = ((𝑦𝑆𝑥)𝐺(𝑧𝑆𝑥)) ∧ ((𝑦 · 𝑧)𝑆𝑥) = (𝑦𝑆(𝑧𝑆𝑥))))
12113comr 1125 . . . . . . . . 9 ((𝑥𝑋𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (((𝑦 + 𝑧)𝑆𝑥) = ((𝑦𝑆𝑥)𝐺(𝑧𝑆𝑥)) ∧ ((𝑦 · 𝑧)𝑆𝑥) = (𝑦𝑆(𝑧𝑆𝑥))))
13123expa 1118 . . . . . . . 8 (((𝑥𝑋𝑦 ∈ ℂ) ∧ 𝑧 ∈ ℂ) → (((𝑦 + 𝑧)𝑆𝑥) = ((𝑦𝑆𝑥)𝐺(𝑧𝑆𝑥)) ∧ ((𝑦 · 𝑧)𝑆𝑥) = (𝑦𝑆(𝑧𝑆𝑥))))
1413ralrimiva 3152 . . . . . . 7 ((𝑥𝑋𝑦 ∈ ℂ) → ∀𝑧 ∈ ℂ (((𝑦 + 𝑧)𝑆𝑥) = ((𝑦𝑆𝑥)𝐺(𝑧𝑆𝑥)) ∧ ((𝑦 · 𝑧)𝑆𝑥) = (𝑦𝑆(𝑧𝑆𝑥))))
158, 14jca 511 . . . . . 6 ((𝑥𝑋𝑦 ∈ ℂ) → (∀𝑧𝑋 (𝑦𝑆(𝑥𝐺𝑧)) = ((𝑦𝑆𝑥)𝐺(𝑦𝑆𝑧)) ∧ ∀𝑧 ∈ ℂ (((𝑦 + 𝑧)𝑆𝑥) = ((𝑦𝑆𝑥)𝐺(𝑧𝑆𝑥)) ∧ ((𝑦 · 𝑧)𝑆𝑥) = (𝑦𝑆(𝑧𝑆𝑥)))))
1615ralrimiva 3152 . . . . 5 (𝑥𝑋 → ∀𝑦 ∈ ℂ (∀𝑧𝑋 (𝑦𝑆(𝑥𝐺𝑧)) = ((𝑦𝑆𝑥)𝐺(𝑦𝑆𝑧)) ∧ ∀𝑧 ∈ ℂ (((𝑦 + 𝑧)𝑆𝑥) = ((𝑦𝑆𝑥)𝐺(𝑧𝑆𝑥)) ∧ ((𝑦 · 𝑧)𝑆𝑥) = (𝑦𝑆(𝑧𝑆𝑥)))))
174, 16jca 511 . . . 4 (𝑥𝑋 → ((1𝑆𝑥) = 𝑥 ∧ ∀𝑦 ∈ ℂ (∀𝑧𝑋 (𝑦𝑆(𝑥𝐺𝑧)) = ((𝑦𝑆𝑥)𝐺(𝑦𝑆𝑧)) ∧ ∀𝑧 ∈ ℂ (((𝑦 + 𝑧)𝑆𝑥) = ((𝑦𝑆𝑥)𝐺(𝑧𝑆𝑥)) ∧ ((𝑦 · 𝑧)𝑆𝑥) = (𝑦𝑆(𝑧𝑆𝑥))))))
1817rgen 3069 . . 3 𝑥𝑋 ((1𝑆𝑥) = 𝑥 ∧ ∀𝑦 ∈ ℂ (∀𝑧𝑋 (𝑦𝑆(𝑥𝐺𝑧)) = ((𝑦𝑆𝑥)𝐺(𝑦𝑆𝑧)) ∧ ∀𝑧 ∈ ℂ (((𝑦 + 𝑧)𝑆𝑥) = ((𝑦𝑆𝑥)𝐺(𝑧𝑆𝑥)) ∧ ((𝑦 · 𝑧)𝑆𝑥) = (𝑦𝑆(𝑧𝑆𝑥)))))
19 ablogrpo 30579 . . . . . 6 (𝐺 ∈ AbelOp → 𝐺 ∈ GrpOp)
202, 19ax-mp 5 . . . . 5 𝐺 ∈ GrpOp
21 isvciOLD.2 . . . . 5 dom 𝐺 = (𝑋 × 𝑋)
2220, 21grporn 30553 . . . 4 𝑋 = ran 𝐺
2322isvcOLD 30611 . . 3 (⟨𝐺, 𝑆⟩ ∈ CVecOLD ↔ (𝐺 ∈ AbelOp ∧ 𝑆:(ℂ × 𝑋)⟶𝑋 ∧ ∀𝑥𝑋 ((1𝑆𝑥) = 𝑥 ∧ ∀𝑦 ∈ ℂ (∀𝑧𝑋 (𝑦𝑆(𝑥𝐺𝑧)) = ((𝑦𝑆𝑥)𝐺(𝑦𝑆𝑧)) ∧ ∀𝑧 ∈ ℂ (((𝑦 + 𝑧)𝑆𝑥) = ((𝑦𝑆𝑥)𝐺(𝑧𝑆𝑥)) ∧ ((𝑦 · 𝑧)𝑆𝑥) = (𝑦𝑆(𝑧𝑆𝑥)))))))
242, 3, 18, 23mpbir3an 1341 . 2 𝐺, 𝑆⟩ ∈ CVecOLD
251, 24eqeltri 2840 1 𝑊 ∈ CVecOLD
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1087   = wceq 1537  wcel 2108  wral 3067  cop 4654   × cxp 5698  dom cdm 5700  wf 6569  (class class class)co 7448  cc 11182  1c1 11185   + caddc 11187   · cmul 11189  GrpOpcgr 30521  AbelOpcablo 30576  CVecOLDcvc 30590
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-cnex 11240
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-ov 7451  df-grpo 30525  df-ablo 30577  df-vc 30591
This theorem is referenced by:  cncvcOLD  30615  hilvc  31194  hhssnv  31296
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