Theorem isnvi 30575

Description: Properties that determine a normed complex vector space. (Contributed by NM, 15-Apr-2007.) (New usage is discouraged.)

Hypotheses

Ref	Expression
isnvi.5	⊢ 𝑋 = ran 𝐺
isnvi.6	⊢ 𝑍 = (GId‘𝐺)
isnvi.7	⊢ ⟨𝐺, 𝑆⟩ ∈ CVecOLD
isnvi.8	⊢ 𝑁:𝑋⟶ℝ
isnvi.9	⊢ ((𝑥 ∈ 𝑋 ∧ (𝑁‘𝑥) = 0) → 𝑥 = 𝑍)
isnvi.10	⊢ ((𝑦 ∈ ℂ ∧ 𝑥 ∈ 𝑋) → (𝑁‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑁‘𝑥)))
isnvi.11	⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑁‘(𝑥𝐺𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦)))
isnvi.12	⊢ 𝑈 = ⟨⟨𝐺, 𝑆⟩, 𝑁⟩

Assertion

Ref	Expression
isnvi	⊢ 𝑈 ∈ NrmCVec

Distinct variable groups:   𝑥,𝑦,𝐺   𝑥,𝑁,𝑦   𝑥,𝑆,𝑦   𝑥,𝑋,𝑦

Allowed substitution hints:   𝑈(𝑥,𝑦)   𝑍(𝑥,𝑦)

Proof of Theorem isnvi

Step	Hyp	Ref	Expression
1		isnvi.12	. 2 ⊢ 𝑈 = ⟨⟨𝐺, 𝑆⟩, 𝑁⟩
2		isnvi.7	. . 3 ⊢ ⟨𝐺, 𝑆⟩ ∈ CVecOLD
3		isnvi.8	. . 3 ⊢ 𝑁:𝑋⟶ℝ
4		isnvi.9	. . . . . 6 ⊢ ((𝑥 ∈ 𝑋 ∧ (𝑁‘𝑥) = 0) → 𝑥 = 𝑍)
5	4	ex 412	. . . . 5 ⊢ (𝑥 ∈ 𝑋 → ((𝑁‘𝑥) = 0 → 𝑥 = 𝑍))
6		isnvi.10	. . . . . . 7 ⊢ ((𝑦 ∈ ℂ ∧ 𝑥 ∈ 𝑋) → (𝑁‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑁‘𝑥)))
7	6	ancoms 458	. . . . . 6 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ ℂ) → (𝑁‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑁‘𝑥)))
8	7	ralrimiva 3121	. . . . 5 ⊢ (𝑥 ∈ 𝑋 → ∀𝑦 ∈ ℂ (𝑁‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑁‘𝑥)))
9		isnvi.11	. . . . . 6 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑁‘(𝑥𝐺𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦)))
10	9	ralrimiva 3121	. . . . 5 ⊢ (𝑥 ∈ 𝑋 → ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥𝐺𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦)))
11	5, 8, 10	3jca 1128	. . . 4 ⊢ (𝑥 ∈ 𝑋 → (((𝑁‘𝑥) = 0 → 𝑥 = 𝑍) ∧ ∀𝑦 ∈ ℂ (𝑁‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑁‘𝑥)) ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥𝐺𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦))))
12	11	rgen 3046	. . 3 ⊢ ∀𝑥 ∈ 𝑋 (((𝑁‘𝑥) = 0 → 𝑥 = 𝑍) ∧ ∀𝑦 ∈ ℂ (𝑁‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑁‘𝑥)) ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥𝐺𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦)))
13		isnvi.5	. . . 4 ⊢ 𝑋 = ran 𝐺
14		isnvi.6	. . . 4 ⊢ 𝑍 = (GId‘𝐺)
15	13, 14	isnv 30574	. . 3 ⊢ (⟨⟨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)))))
16	2, 3, 12, 15	mpbir3an 1342	. 2 ⊢ ⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ NrmCVec
17	1, 16	eqeltri 2824	1 ⊢ 𝑈 ∈ NrmCVec

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3044  ⟨cop 4585   class class class wbr 5095  ran crn 5624  ⟶wf 6482  ‘cfv 6486  (class class class)co 7353  ℂcc 11026  ℝcr 11027  0cc0 11028   + caddc 11031   · cmul 11033   ≤ cle 11169  abscabs 15159  GIdcgi 30452  CVec_OLDcvc 30520  NrmCVeccnv 30546

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pr 5374  ax-un 7675

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-ov 7356  df-oprab 7357  df-vc 30521  df-nv 30554

This theorem is referenced by:  cnnv  30639  hhnv  31127  hhssnv  31226