Theorem isnv 31093

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 31092	. 2 ⊢ (⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ NrmCVec → (𝐺 ∈ V ∧ 𝑆 ∈ V ∧ 𝑁 ∈ V))
2		vcex 31059	. . . . 5 ⊢ (⟨𝐺, 𝑆⟩ ∈ CVecOLD → (𝐺 ∈ V ∧ 𝑆 ∈ V))
3	2	adantr 486	. . . 4 ⊢ ((⟨𝐺, 𝑆⟩ ∈ CVecOLD ∧ 𝑁:𝑋⟶ℝ) → (𝐺 ∈ V ∧ 𝑆 ∈ V))
4		isnv.1	. . . . . . 7 ⊢ 𝑋 = ran 𝐺
5	2	simpld 500	. . . . . . . 8 ⊢ (⟨𝐺, 𝑆⟩ ∈ CVecOLD → 𝐺 ∈ V)
6		rnexg 7899	. . . . . . . 8 ⊢ (𝐺 ∈ V → ran 𝐺 ∈ V)
7	5, 6	syl 18	. . . . . . 7 ⊢ (⟨𝐺, 𝑆⟩ ∈ CVecOLD → ran 𝐺 ∈ V)
8	4, 7	eqeltrid 2864	. . . . . 6 ⊢ (⟨𝐺, 𝑆⟩ ∈ CVecOLD → 𝑋 ∈ V)
9		fex 7225	. . . . . 6 ⊢ ((𝑁:𝑋⟶ℝ ∧ 𝑋 ∈ V) → 𝑁 ∈ V)
10	8, 9	sylan2 605	. . . . 5 ⊢ ((𝑁:𝑋⟶ℝ ∧ ⟨𝐺, 𝑆⟩ ∈ CVecOLD) → 𝑁 ∈ V)
11	10	ancoms 464	. . . 4 ⊢ ((⟨𝐺, 𝑆⟩ ∈ CVecOLD ∧ 𝑁:𝑋⟶ℝ) → 𝑁 ∈ V)
12		df-3an 1105	. . . 4 ⊢ ((𝐺 ∈ V ∧ 𝑆 ∈ V ∧ 𝑁 ∈ V) ↔ ((𝐺 ∈ V ∧ 𝑆 ∈ V) ∧ 𝑁 ∈ V))
13	3, 11, 12	sylanbrc 595	. . 3 ⊢ ((⟨𝐺, 𝑆⟩ ∈ CVecOLD ∧ 𝑁:𝑋⟶ℝ) → (𝐺 ∈ V ∧ 𝑆 ∈ V ∧ 𝑁 ∈ V))
14	13	3adant3 1150	. 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 31091	. 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 381	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)))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∀wral 3076  Vcvv 3450  ⟨cop 4590   class class class wbr 5103  ran crn 5656  ⟶wf 6529  ‘cfv 6533  (class class class)co 7413  ℂcc 11122  ℝcr 11123  0cc0 11124   + caddc 11127   · cmul 11129   ≤ cle 11268  abscabs 15321  GIdcgi 30971  CVec_OLDcvc 31039  NrmCVeccnv 31065

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pr 5398  ax-un 7736

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7416  df-oprab 7417  df-vc 31040  df-nv 31073

This theorem is used by:  isnvi  31094  nvi  31095