imsdval - Metamath Proof Explorer

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Theorem imsdval 30668

Description: Value of the induced metric (distance function) of a normed complex vector space. Equation 1 of [Kreyszig] p. 59. (Contributed by NM, 11-Sep-2007.) (Revised by Mario Carneiro, 27-Dec-2014.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
imsdval.1	⊢ 𝑋 = (BaseSet‘𝑈)
imsdval.3	⊢ 𝑀 = ( −_𝑣 ‘𝑈)
imsdval.6	⊢ 𝑁 = (norm_CV‘𝑈)
imsdval.8	⊢ 𝐷 = (IndMet‘𝑈)

**Assertion**
Ref	Expression
imsdval	⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷𝐵) = (𝑁‘(𝐴𝑀𝐵)))

**Proof of Theorem imsdval**
Step	Hyp	Ref	Expression
1		imsdval.3	. . . . . 6 ⊢ 𝑀 = ( −_𝑣 ‘𝑈)
2		imsdval.6	. . . . . 6 ⊢ 𝑁 = (norm_CV‘𝑈)
3		imsdval.8	. . . . . 6 ⊢ 𝐷 = (IndMet‘𝑈)
4	1, 2, 3	imsval 30667	. . . . 5 ⊢ (𝑈 ∈ NrmCVec → 𝐷 = (𝑁 ∘ 𝑀))
5	4	3ad2ant1 1133	. . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → 𝐷 = (𝑁 ∘ 𝑀))
6	5	fveq1d 6830	. . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐷‘⟨𝐴, 𝐵⟩) = ((𝑁 ∘ 𝑀)‘⟨𝐴, 𝐵⟩))
7		imsdval.1	. . . . . 6 ⊢ 𝑋 = (BaseSet‘𝑈)
8	7, 1	nvmf 30627	. . . . 5 ⊢ (𝑈 ∈ NrmCVec → 𝑀:(𝑋 × 𝑋)⟶𝑋)
9		opelxpi 5656	. . . . 5 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ⟨𝐴, 𝐵⟩ ∈ (𝑋 × 𝑋))
10		fvco3 6927	. . . . 5 ⊢ ((𝑀:(𝑋 × 𝑋)⟶𝑋 ∧ ⟨𝐴, 𝐵⟩ ∈ (𝑋 × 𝑋)) → ((𝑁 ∘ 𝑀)‘⟨𝐴, 𝐵⟩) = (𝑁‘(𝑀‘⟨𝐴, 𝐵⟩)))
11	8, 9, 10	syl2an 596	. . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → ((𝑁 ∘ 𝑀)‘⟨𝐴, 𝐵⟩) = (𝑁‘(𝑀‘⟨𝐴, 𝐵⟩)))
12	11	3impb 1114	. . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝑁 ∘ 𝑀)‘⟨𝐴, 𝐵⟩) = (𝑁‘(𝑀‘⟨𝐴, 𝐵⟩)))
13	6, 12	eqtrd 2768	. 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐷‘⟨𝐴, 𝐵⟩) = (𝑁‘(𝑀‘⟨𝐴, 𝐵⟩)))
14		df-ov 7355	. 2 ⊢ (𝐴𝐷𝐵) = (𝐷‘⟨𝐴, 𝐵⟩)
15		df-ov 7355	. . 3 ⊢ (𝐴𝑀𝐵) = (𝑀‘⟨𝐴, 𝐵⟩)
16	15	fveq2i 6831	. 2 ⊢ (𝑁‘(𝐴𝑀𝐵)) = (𝑁‘(𝑀‘⟨𝐴, 𝐵⟩))
17	13, 14, 16	3eqtr4g 2793	1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷𝐵) = (𝑁‘(𝐴𝑀𝐵)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1541 ∈ wcel 2113 ⟨cop 4581 × cxp 5617 ∘ ccom 5623 ⟶wf 6482 ‘cfv 6486 (class class class)co 7352 NrmCVeccnv 30566 BaseSetcba 30568 −_𝑣 cnsb 30571 norm_CVcnmcv 30572 IndMetcims 30573

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 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2705 ax-rep 5219 ax-sep 5236 ax-nul 5246 ax-pow 5305 ax-pr 5372 ax-un 7674 ax-resscn 11070 ax-1cn 11071 ax-icn 11072 ax-addcl 11073 ax-addrcl 11074 ax-mulcl 11075 ax-mulrcl 11076 ax-mulcom 11077 ax-addass 11078 ax-mulass 11079 ax-distr 11080 ax-i2m1 11081 ax-1ne0 11082 ax-1rid 11083 ax-rnegex 11084 ax-rrecex 11085 ax-cnre 11086 ax-pre-lttri 11087 ax-pre-lttrn 11088 ax-pre-ltadd 11089

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2725 df-clel 2808 df-nfc 2882 df-ne 2930 df-nel 3034 df-ral 3049 df-rex 3058 df-reu 3348 df-rab 3397 df-v 3439 df-sbc 3738 df-csb 3847 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-nul 4283 df-if 4475 df-pw 4551 df-sn 4576 df-pr 4578 df-op 4582 df-uni 4859 df-iun 4943 df-br 5094 df-opab 5156 df-mpt 5175 df-id 5514 df-po 5527 df-so 5528 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 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-riota 7309 df-ov 7355 df-oprab 7356 df-mpo 7357 df-1st 7927 df-2nd 7928 df-er 8628 df-en 8876 df-dom 8877 df-sdom 8878 df-pnf 11155 df-mnf 11156 df-ltxr 11158 df-sub 11353 df-neg 11354 df-grpo 30475 df-gid 30476 df-ginv 30477 df-gdiv 30478 df-ablo 30527 df-vc 30541 df-nv 30574 df-va 30577 df-ba 30578 df-sm 30579 df-0v 30580 df-vs 30581 df-nmcv 30582 df-ims 30583

This theorem is referenced by: imsdval2 30669 nvnd 30670 vacn 30676 smcnlem 30679 sspimsval 30720 blometi 30785 blocnilem 30786 ubthlem2 30853 minvecolem2 30857 minvecolem4 30862 minvecolem5 30863 minvecolem6 30864 h2hmetdval 30960 hhssmetdval 31259

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