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

Theorem isvcOLD 30608
Description: The predicate "is a complex vector space." (Contributed by NM, 31-May-2008.) Obsolete version of iscvsp 25175. (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 ∧ ∀𝑥𝑋 ((1𝑆𝑥) = 𝑥 ∧ ∀𝑦 ∈ ℂ (∀𝑧𝑋 (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
StepHypRef Expression
1 vcex 30607 . 2 (⟨𝐺, 𝑆⟩ ∈ CVecOLD → (𝐺 ∈ V ∧ 𝑆 ∈ V))
2 elex 3499 . . . . 5 (𝐺 ∈ AbelOp → 𝐺 ∈ V)
32adantr 480 . . . 4 ((𝐺 ∈ AbelOp ∧ 𝑆:(ℂ × 𝑋)⟶𝑋) → 𝐺 ∈ V)
4 cnex 11234 . . . . . . 7 ℂ ∈ V
5 ablogrpo 30576 . . . . . . . 8 (𝐺 ∈ AbelOp → 𝐺 ∈ GrpOp)
6 isvcOLD.1 . . . . . . . . 9 𝑋 = ran 𝐺
7 rnexg 7925 . . . . . . . . 9 (𝐺 ∈ GrpOp → ran 𝐺 ∈ V)
86, 7eqeltrid 2843 . . . . . . . 8 (𝐺 ∈ GrpOp → 𝑋 ∈ V)
95, 8syl 17 . . . . . . 7 (𝐺 ∈ AbelOp → 𝑋 ∈ V)
10 xpexg 7769 . . . . . . 7 ((ℂ ∈ V ∧ 𝑋 ∈ V) → (ℂ × 𝑋) ∈ V)
114, 9, 10sylancr 587 . . . . . 6 (𝐺 ∈ AbelOp → (ℂ × 𝑋) ∈ V)
12 fex 7246 . . . . . 6 ((𝑆:(ℂ × 𝑋)⟶𝑋 ∧ (ℂ × 𝑋) ∈ V) → 𝑆 ∈ V)
1311, 12sylan2 593 . . . . 5 ((𝑆:(ℂ × 𝑋)⟶𝑋𝐺 ∈ AbelOp) → 𝑆 ∈ V)
1413ancoms 458 . . . 4 ((𝐺 ∈ AbelOp ∧ 𝑆:(ℂ × 𝑋)⟶𝑋) → 𝑆 ∈ V)
153, 14jca 511 . . 3 ((𝐺 ∈ AbelOp ∧ 𝑆:(ℂ × 𝑋)⟶𝑋) → (𝐺 ∈ V ∧ 𝑆 ∈ V))
16153adant3 1131 . 2 ((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)))))) → (𝐺 ∈ V ∧ 𝑆 ∈ V))
176isvclem 30606 . 2 ((𝐺 ∈ V ∧ 𝑆 ∈ V) → (⟨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))))))))
181, 16, 17pm5.21nii 378 1 (⟨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)))))))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  wral 3059  Vcvv 3478  cop 4637   × cxp 5687  ran crn 5690  wf 6559  (class class class)co 7431  cc 11151  1c1 11154   + caddc 11156   · cmul 11158  GrpOpcgr 30518  AbelOpcablo 30573  CVecOLDcvc 30587
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-ablo 30574  df-vc 30588
This theorem is referenced by:  isvciOLD  30609
  Copyright terms: Public domain W3C validator