Theorem isnv 30147

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⟩ ∈ CVecOLD ∧ 𝑁:𝑋⟶ℝ ∧ ∀𝑥 ∈ 𝑋 (((𝑁‘𝑥) = 0 → 𝑥 = 𝑍) ∧ ∀𝑦 ∈ ℂ (𝑁‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑁‘𝑥)) ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥𝐺𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦)))))

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

Step	Hyp	Ref	Expression
1		nvex 30146	. 2 ⊢ (⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ NrmCVec → (𝐺 ∈ V ∧ 𝑆 ∈ V ∧ 𝑁 ∈ V))
2		vcex 30113	. . . . 5 ⊢ (⟨𝐺, 𝑆⟩ ∈ CVecOLD → (𝐺 ∈ V ∧ 𝑆 ∈ V))
3	2	adantr 480	. . . 4 ⊢ ((⟨𝐺, 𝑆⟩ ∈ CVecOLD ∧ 𝑁:𝑋⟶ℝ) → (𝐺 ∈ V ∧ 𝑆 ∈ V))
4		isnv.1	. . . . . . 7 ⊢ 𝑋 = ran 𝐺
5	2	simpld 494	. . . . . . . 8 ⊢ (⟨𝐺, 𝑆⟩ ∈ CVecOLD → 𝐺 ∈ V)
6		rnexg 7899	. . . . . . . 8 ⊢ (𝐺 ∈ V → ran 𝐺 ∈ V)
7	5, 6	syl 17	. . . . . . 7 ⊢ (⟨𝐺, 𝑆⟩ ∈ CVecOLD → ran 𝐺 ∈ V)
8	4, 7	eqeltrid 2836	. . . . . 6 ⊢ (⟨𝐺, 𝑆⟩ ∈ CVecOLD → 𝑋 ∈ V)
9		fex 7230	. . . . . 6 ⊢ ((𝑁:𝑋⟶ℝ ∧ 𝑋 ∈ V) → 𝑁 ∈ V)
10	8, 9	sylan2 592	. . . . 5 ⊢ ((𝑁:𝑋⟶ℝ ∧ ⟨𝐺, 𝑆⟩ ∈ CVecOLD) → 𝑁 ∈ V)
11	10	ancoms 458	. . . 4 ⊢ ((⟨𝐺, 𝑆⟩ ∈ CVecOLD ∧ 𝑁:𝑋⟶ℝ) → 𝑁 ∈ V)
12		df-3an 1088	. . . 4 ⊢ ((𝐺 ∈ V ∧ 𝑆 ∈ V ∧ 𝑁 ∈ V) ↔ ((𝐺 ∈ V ∧ 𝑆 ∈ V) ∧ 𝑁 ∈ V))
13	3, 11, 12	sylanbrc 582	. . 3 ⊢ ((⟨𝐺, 𝑆⟩ ∈ CVecOLD ∧ 𝑁:𝑋⟶ℝ) → (𝐺 ∈ V ∧ 𝑆 ∈ V ∧ 𝑁 ∈ V))
14	13	3adant3 1131	. 2 ⊢ ((⟨𝐺, 𝑆⟩ ∈ CVecOLD ∧ 𝑁:𝑋⟶ℝ ∧ ∀𝑥 ∈ 𝑋 (((𝑁‘𝑥) = 0 → 𝑥 = 𝑍) ∧ ∀𝑦 ∈ ℂ (𝑁‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑁‘𝑥)) ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥𝐺𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦)))) → (𝐺 ∈ V ∧ 𝑆 ∈ V ∧ 𝑁 ∈ V))
15		isnv.2	. . 3 ⊢ 𝑍 = (GId‘𝐺)
16	4, 15	isnvlem 30145	. 2 ⊢ ((𝐺 ∈ V ∧ 𝑆 ∈ V ∧ 𝑁 ∈ V) → (⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ NrmCVec ↔ (⟨𝐺, 𝑆⟩ ∈ CVecOLD ∧ 𝑁:𝑋⟶ℝ ∧ ∀𝑥 ∈ 𝑋 (((𝑁‘𝑥) = 0 → 𝑥 = 𝑍) ∧ ∀𝑦 ∈ ℂ (𝑁‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑁‘𝑥)) ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥𝐺𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦))))))
17	1, 14, 16	pm5.21nii 378	1 ⊢ (⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ NrmCVec ↔ (⟨𝐺, 𝑆⟩ ∈ CVecOLD ∧ 𝑁:𝑋⟶ℝ ∧ ∀𝑥 ∈ 𝑋 (((𝑁‘𝑥) = 0 → 𝑥 = 𝑍) ∧ ∀𝑦 ∈ ℂ (𝑁‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑁‘𝑥)) ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥𝐺𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦)))))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2105  ∀wral 3060  Vcvv 3473  ⟨cop 4634   class class class wbr 5148  ran crn 5677  ⟶wf 6539  ‘cfv 6543  (class class class)co 7412  ℂcc 11114  ℝcr 11115  0cc0 11116   + caddc 11119   · cmul 11121   ≤ cle 11256  abscabs 15188  GIdcgi 30025  CVec_OLDcvc 30093  NrmCVeccnv 30119

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7729

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7415  df-oprab 7416  df-vc 30094  df-nv 30127

This theorem is referenced by:  isnvi  30148  nvi  30149