Theorem nvs 29025

Description: Proportionality property of the norm of a scalar product in a normed complex vector space. (Contributed by NM, 11-Nov-2006.) (New usage is discouraged.)

Hypotheses

Ref	Expression
nvs.1	⊢ 𝑋 = (BaseSet‘𝑈)
nvs.4	⊢ 𝑆 = ( ·𝑠OLD ‘𝑈)
nvs.6	⊢ 𝑁 = (normCV‘𝑈)

Assertion

Ref	Expression
nvs	⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ 𝑋) → (𝑁‘(𝐴𝑆𝐵)) = ((abs‘𝐴) · (𝑁‘𝐵)))

Proof of Theorem nvs

Dummy variables 𝑦 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nvs.1	. . . . . . 7 ⊢ 𝑋 = (BaseSet‘𝑈)
2		eqid 2738	. . . . . . 7 ⊢ ( +𝑣 ‘𝑈) = ( +𝑣 ‘𝑈)
3		nvs.4	. . . . . . 7 ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈)
4		eqid 2738	. . . . . . 7 ⊢ (0vec‘𝑈) = (0vec‘𝑈)
5		nvs.6	. . . . . . 7 ⊢ 𝑁 = (normCV‘𝑈)
6	1, 2, 3, 4, 5	nvi 28976	. . . . . 6 ⊢ (𝑈 ∈ NrmCVec → (⟨( +𝑣 ‘𝑈), 𝑆⟩ ∈ CVecOLD ∧ 𝑁:𝑋⟶ℝ ∧ ∀𝑥 ∈ 𝑋 (((𝑁‘𝑥) = 0 → 𝑥 = (0vec‘𝑈)) ∧ ∀𝑦 ∈ ℂ (𝑁‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑁‘𝑥)) ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥( +𝑣 ‘𝑈)𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦)))))
7	6	simp3d 1143	. . . . 5 ⊢ (𝑈 ∈ NrmCVec → ∀𝑥 ∈ 𝑋 (((𝑁‘𝑥) = 0 → 𝑥 = (0vec‘𝑈)) ∧ ∀𝑦 ∈ ℂ (𝑁‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑁‘𝑥)) ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥( +𝑣 ‘𝑈)𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦))))
8		simp2 1136	. . . . . 6 ⊢ ((((𝑁‘𝑥) = 0 → 𝑥 = (0vec‘𝑈)) ∧ ∀𝑦 ∈ ℂ (𝑁‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑁‘𝑥)) ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥( +𝑣 ‘𝑈)𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦))) → ∀𝑦 ∈ ℂ (𝑁‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑁‘𝑥)))
9	8	ralimi 3087	. . . . 5 ⊢ (∀𝑥 ∈ 𝑋 (((𝑁‘𝑥) = 0 → 𝑥 = (0vec‘𝑈)) ∧ ∀𝑦 ∈ ℂ (𝑁‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑁‘𝑥)) ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥( +𝑣 ‘𝑈)𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦))) → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ ℂ (𝑁‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑁‘𝑥)))
10	7, 9	syl 17	. . . 4 ⊢ (𝑈 ∈ NrmCVec → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ ℂ (𝑁‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑁‘𝑥)))
11		oveq2 7283	. . . . . . 7 ⊢ (𝑥 = 𝐵 → (𝑦𝑆𝑥) = (𝑦𝑆𝐵))
12	11	fveq2d 6778	. . . . . 6 ⊢ (𝑥 = 𝐵 → (𝑁‘(𝑦𝑆𝑥)) = (𝑁‘(𝑦𝑆𝐵)))
13		fveq2 6774	. . . . . . 7 ⊢ (𝑥 = 𝐵 → (𝑁‘𝑥) = (𝑁‘𝐵))
14	13	oveq2d 7291	. . . . . 6 ⊢ (𝑥 = 𝐵 → ((abs‘𝑦) · (𝑁‘𝑥)) = ((abs‘𝑦) · (𝑁‘𝐵)))
15	12, 14	eqeq12d 2754	. . . . 5 ⊢ (𝑥 = 𝐵 → ((𝑁‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑁‘𝑥)) ↔ (𝑁‘(𝑦𝑆𝐵)) = ((abs‘𝑦) · (𝑁‘𝐵))))
16		fvoveq1 7298	. . . . . 6 ⊢ (𝑦 = 𝐴 → (𝑁‘(𝑦𝑆𝐵)) = (𝑁‘(𝐴𝑆𝐵)))
17		fveq2 6774	. . . . . . 7 ⊢ (𝑦 = 𝐴 → (abs‘𝑦) = (abs‘𝐴))
18	17	oveq1d 7290	. . . . . 6 ⊢ (𝑦 = 𝐴 → ((abs‘𝑦) · (𝑁‘𝐵)) = ((abs‘𝐴) · (𝑁‘𝐵)))
19	16, 18	eqeq12d 2754	. . . . 5 ⊢ (𝑦 = 𝐴 → ((𝑁‘(𝑦𝑆𝐵)) = ((abs‘𝑦) · (𝑁‘𝐵)) ↔ (𝑁‘(𝐴𝑆𝐵)) = ((abs‘𝐴) · (𝑁‘𝐵))))
20	15, 19	rspc2v 3570	. . . 4 ⊢ ((𝐵 ∈ 𝑋 ∧ 𝐴 ∈ ℂ) → (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ ℂ (𝑁‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑁‘𝑥)) → (𝑁‘(𝐴𝑆𝐵)) = ((abs‘𝐴) · (𝑁‘𝐵))))
21	10, 20	syl5 34	. . 3 ⊢ ((𝐵 ∈ 𝑋 ∧ 𝐴 ∈ ℂ) → (𝑈 ∈ NrmCVec → (𝑁‘(𝐴𝑆𝐵)) = ((abs‘𝐴) · (𝑁‘𝐵))))
22	21	3impia 1116	. 2 ⊢ ((𝐵 ∈ 𝑋 ∧ 𝐴 ∈ ℂ ∧ 𝑈 ∈ NrmCVec) → (𝑁‘(𝐴𝑆𝐵)) = ((abs‘𝐴) · (𝑁‘𝐵)))
23	22	3com13 1123	1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ 𝑋) → (𝑁‘(𝐴𝑆𝐵)) = ((abs‘𝐴) · (𝑁‘𝐵)))

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106  ∀wral 3064  ⟨cop 4567   class class class wbr 5074  ⟶wf 6429  ‘cfv 6433  (class class class)co 7275  ℂcc 10869  ℝcr 10870  0cc0 10871   + caddc 10874   · cmul 10876   ≤ cle 11010  abscabs 14945  CVec_OLDcvc 28920  NrmCVeccnv 28946   +_𝑣 cpv 28947  BaseSetcba 28948   ·_𝑠OLD cns 28949  0_veccn0v 28950  norm_CVcnmcv 28952

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-1st 7831  df-2nd 7832  df-vc 28921  df-nv 28954  df-va 28957  df-ba 28958  df-sm 28959  df-0v 28960  df-nmcv 28962

This theorem is referenced by:  nvsge0  29026  nvm1  29027  nvpi  29029  nvmtri  29033  smcnlem  29059  ipidsq  29072  minvecolem2  29237