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Theorem isnv 27798
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 27797 . 2 (⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ NrmCVec → (𝐺 ∈ V ∧ 𝑆 ∈ V ∧ 𝑁 ∈ V))
2 vcex 27764 . . . . 5 (⟨𝐺, 𝑆⟩ ∈ CVecOLD → (𝐺 ∈ V ∧ 𝑆 ∈ V))
32adantr 472 . . . 4 ((⟨𝐺, 𝑆⟩ ∈ CVecOLD𝑁:𝑋⟶ℝ) → (𝐺 ∈ V ∧ 𝑆 ∈ V))
4 isnv.1 . . . . . . 7 𝑋 = ran 𝐺
52simpld 477 . . . . . . . 8 (⟨𝐺, 𝑆⟩ ∈ CVecOLD𝐺 ∈ V)
6 rnexg 7265 . . . . . . . 8 (𝐺 ∈ V → ran 𝐺 ∈ V)
75, 6syl 17 . . . . . . 7 (⟨𝐺, 𝑆⟩ ∈ CVecOLD → ran 𝐺 ∈ V)
84, 7syl5eqel 2844 . . . . . 6 (⟨𝐺, 𝑆⟩ ∈ CVecOLD𝑋 ∈ V)
9 fex 6655 . . . . . 6 ((𝑁:𝑋⟶ℝ ∧ 𝑋 ∈ V) → 𝑁 ∈ V)
108, 9sylan2 492 . . . . 5 ((𝑁:𝑋⟶ℝ ∧ ⟨𝐺, 𝑆⟩ ∈ CVecOLD) → 𝑁 ∈ V)
1110ancoms 468 . . . 4 ((⟨𝐺, 𝑆⟩ ∈ CVecOLD𝑁:𝑋⟶ℝ) → 𝑁 ∈ V)
12 df-3an 1074 . . . 4 ((𝐺 ∈ V ∧ 𝑆 ∈ V ∧ 𝑁 ∈ V) ↔ ((𝐺 ∈ V ∧ 𝑆 ∈ V) ∧ 𝑁 ∈ V))
133, 11, 12sylanbrc 701 . . 3 ((⟨𝐺, 𝑆⟩ ∈ CVecOLD𝑁:𝑋⟶ℝ) → (𝐺 ∈ V ∧ 𝑆 ∈ V ∧ 𝑁 ∈ V))
14133adant3 1127 . 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 27796 . 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 367 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 196  wa 383  w3a 1072   = wceq 1632  wcel 2140  wral 3051  Vcvv 3341  cop 4328   class class class wbr 4805  ran crn 5268  wf 6046  cfv 6050  (class class class)co 6815  cc 10147  cr 10148  0cc0 10149   + caddc 10152   · cmul 10154  cle 10288  abscabs 14194  GIdcgi 27675  CVecOLDcvc 27744  NrmCVeccnv 27770
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-8 2142  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-rep 4924  ax-sep 4934  ax-nul 4942  ax-pr 5056  ax-un 7116
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ne 2934  df-ral 3056  df-rex 3057  df-reu 3058  df-rab 3060  df-v 3343  df-sbc 3578  df-csb 3676  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-nul 4060  df-if 4232  df-sn 4323  df-pr 4325  df-op 4329  df-uni 4590  df-iun 4675  df-br 4806  df-opab 4866  df-mpt 4883  df-id 5175  df-xp 5273  df-rel 5274  df-cnv 5275  df-co 5276  df-dm 5277  df-rn 5278  df-res 5279  df-ima 5280  df-iota 6013  df-fun 6052  df-fn 6053  df-f 6054  df-f1 6055  df-fo 6056  df-f1o 6057  df-fv 6058  df-ov 6818  df-oprab 6819  df-vc 27745  df-nv 27778
This theorem is referenced by:  isnvi  27799  nvi  27800
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