MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  isvciOLD Structured version   Visualization version   GIF version

Theorem isvciOLD 30608
Description: Properties that determine a complex vector space. (Contributed by NM, 5-Nov-2006.) Obsolete version of iscvsi 25175. (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 1122 . . . . . . . . 9 ((𝑥𝑋𝑦 ∈ ℂ ∧ 𝑧𝑋) → (𝑦𝑆(𝑥𝐺𝑧)) = ((𝑦𝑆𝑥)𝐺(𝑦𝑆𝑧)))
763expa 1117 . . . . . . . 8 (((𝑥𝑋𝑦 ∈ ℂ) ∧ 𝑧𝑋) → (𝑦𝑆(𝑥𝐺𝑧)) = ((𝑦𝑆𝑥)𝐺(𝑦𝑆𝑧)))
87ralrimiva 3143 . . . . . . 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 1124 . . . . . . . . 9 ((𝑥𝑋𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (((𝑦 + 𝑧)𝑆𝑥) = ((𝑦𝑆𝑥)𝐺(𝑧𝑆𝑥)) ∧ ((𝑦 · 𝑧)𝑆𝑥) = (𝑦𝑆(𝑧𝑆𝑥))))
13123expa 1117 . . . . . . . 8 (((𝑥𝑋𝑦 ∈ ℂ) ∧ 𝑧 ∈ ℂ) → (((𝑦 + 𝑧)𝑆𝑥) = ((𝑦𝑆𝑥)𝐺(𝑧𝑆𝑥)) ∧ ((𝑦 · 𝑧)𝑆𝑥) = (𝑦𝑆(𝑧𝑆𝑥))))
1413ralrimiva 3143 . . . . . . 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 3143 . . . . 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 3060 . . 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 30575 . . . . . 6 (𝐺 ∈ AbelOp → 𝐺 ∈ GrpOp)
202, 19ax-mp 5 . . . . 5 𝐺 ∈ GrpOp
21 isvciOLD.2 . . . . 5 dom 𝐺 = (𝑋 × 𝑋)
2220, 21grporn 30549 . . . 4 𝑋 = ran 𝐺
2322isvcOLD 30607 . . 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 1340 . 2 𝐺, 𝑆⟩ ∈ CVecOLD
251, 24eqeltri 2834 1 𝑊 ∈ CVecOLD
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1536  wcel 2105  wral 3058  cop 4636   × cxp 5686  dom cdm 5688  wf 6558  (class class class)co 7430  cc 11150  1c1 11153   + caddc 11155   · cmul 11157  GrpOpcgr 30517  AbelOpcablo 30572  CVecOLDcvc 30586
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-cnex 11208
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-ov 7433  df-grpo 30521  df-ablo 30573  df-vc 30587
This theorem is referenced by:  cncvcOLD  30611  hilvc  31190  hhssnv  31292
  Copyright terms: Public domain W3C validator