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Theorem isnv 28374
Description: The predicate "is a normed complex vector space." (Contributed by NM, 5-Jun-2008.) (New usage is discouraged.)
Hypotheses
Ref Expression
isnv.1 𝑋 = ran 𝐺
isnv.2 𝑍 = (GId‘𝐺)
Assertion
Ref Expression
isnv (⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ NrmCVec ↔ (⟨𝐺, 𝑆⟩ ∈ CVecOLD𝑁:𝑋⟶ℝ ∧ ∀𝑥𝑋 (((𝑁𝑥) = 0 → 𝑥 = 𝑍) ∧ ∀𝑦 ∈ ℂ (𝑁‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑁𝑥)) ∧ ∀𝑦𝑋 (𝑁‘(𝑥𝐺𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦)))))
Distinct variable groups:   𝑥,𝑦,𝐺   𝑥,𝑁,𝑦   𝑥,𝑆,𝑦   𝑥,𝑋,𝑦
Allowed substitution hints:   𝑍(𝑥,𝑦)

Proof of Theorem isnv
StepHypRef Expression
1 nvex 28373 . 2 (⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ NrmCVec → (𝐺 ∈ V ∧ 𝑆 ∈ V ∧ 𝑁 ∈ V))
2 vcex 28340 . . . . 5 (⟨𝐺, 𝑆⟩ ∈ CVecOLD → (𝐺 ∈ V ∧ 𝑆 ∈ V))
32adantr 483 . . . 4 ((⟨𝐺, 𝑆⟩ ∈ CVecOLD𝑁:𝑋⟶ℝ) → (𝐺 ∈ V ∧ 𝑆 ∈ V))
4 isnv.1 . . . . . . 7 𝑋 = ran 𝐺
52simpld 497 . . . . . . . 8 (⟨𝐺, 𝑆⟩ ∈ CVecOLD𝐺 ∈ V)
6 rnexg 7592 . . . . . . . 8 (𝐺 ∈ V → ran 𝐺 ∈ V)
75, 6syl 17 . . . . . . 7 (⟨𝐺, 𝑆⟩ ∈ CVecOLD → ran 𝐺 ∈ V)
84, 7eqeltrid 2915 . . . . . 6 (⟨𝐺, 𝑆⟩ ∈ CVecOLD𝑋 ∈ V)
9 fex 6965 . . . . . 6 ((𝑁:𝑋⟶ℝ ∧ 𝑋 ∈ V) → 𝑁 ∈ V)
108, 9sylan2 594 . . . . 5 ((𝑁:𝑋⟶ℝ ∧ ⟨𝐺, 𝑆⟩ ∈ CVecOLD) → 𝑁 ∈ V)
1110ancoms 461 . . . 4 ((⟨𝐺, 𝑆⟩ ∈ CVecOLD𝑁:𝑋⟶ℝ) → 𝑁 ∈ V)
12 df-3an 1085 . . . 4 ((𝐺 ∈ V ∧ 𝑆 ∈ V ∧ 𝑁 ∈ V) ↔ ((𝐺 ∈ V ∧ 𝑆 ∈ V) ∧ 𝑁 ∈ V))
133, 11, 12sylanbrc 585 . . 3 ((⟨𝐺, 𝑆⟩ ∈ CVecOLD𝑁:𝑋⟶ℝ) → (𝐺 ∈ V ∧ 𝑆 ∈ V ∧ 𝑁 ∈ V))
14133adant3 1128 . 2 ((⟨𝐺, 𝑆⟩ ∈ CVecOLD𝑁:𝑋⟶ℝ ∧ ∀𝑥𝑋 (((𝑁𝑥) = 0 → 𝑥 = 𝑍) ∧ ∀𝑦 ∈ ℂ (𝑁‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑁𝑥)) ∧ ∀𝑦𝑋 (𝑁‘(𝑥𝐺𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦)))) → (𝐺 ∈ V ∧ 𝑆 ∈ V ∧ 𝑁 ∈ V))
15 isnv.2 . . 3 𝑍 = (GId‘𝐺)
164, 15isnvlem 28372 . 2 ((𝐺 ∈ V ∧ 𝑆 ∈ V ∧ 𝑁 ∈ V) → (⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ NrmCVec ↔ (⟨𝐺, 𝑆⟩ ∈ CVecOLD𝑁:𝑋⟶ℝ ∧ ∀𝑥𝑋 (((𝑁𝑥) = 0 → 𝑥 = 𝑍) ∧ ∀𝑦 ∈ ℂ (𝑁‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑁𝑥)) ∧ ∀𝑦𝑋 (𝑁‘(𝑥𝐺𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦))))))
171, 14, 16pm5.21nii 382 1 (⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ NrmCVec ↔ (⟨𝐺, 𝑆⟩ ∈ CVecOLD𝑁:𝑋⟶ℝ ∧ ∀𝑥𝑋 (((𝑁𝑥) = 0 → 𝑥 = 𝑍) ∧ ∀𝑦 ∈ ℂ (𝑁‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑁𝑥)) ∧ ∀𝑦𝑋 (𝑁‘(𝑥𝐺𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398  w3a 1083   = wceq 1537  wcel 2114  wral 3125  Vcvv 3473  cop 4549   class class class wbr 5042  ran crn 5532  wf 6327  cfv 6331  (class class class)co 7133  cc 10513  cr 10514  0cc0 10515   + caddc 10518   · cmul 10520  cle 10654  abscabs 14573  GIdcgi 28252  CVecOLDcvc 28320  NrmCVeccnv 28346
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pr 5306  ax-un 7439
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-ral 3130  df-rex 3131  df-reu 3132  df-rab 3134  df-v 3475  df-sbc 3753  df-csb 3861  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4270  df-if 4444  df-sn 4544  df-pr 4546  df-op 4550  df-uni 4815  df-iun 4897  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5436  df-xp 5537  df-rel 5538  df-cnv 5539  df-co 5540  df-dm 5541  df-rn 5542  df-res 5543  df-ima 5544  df-iota 6290  df-fun 6333  df-fn 6334  df-f 6335  df-f1 6336  df-fo 6337  df-f1o 6338  df-fv 6339  df-ov 7136  df-oprab 7137  df-vc 28321  df-nv 28354
This theorem is referenced by:  isnvi  28375  nvi  28376
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