imsdval - Metamath Proof Explorer

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Theorem imsdval 28466

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 28465	. . . . 5 ⊢ (𝑈 ∈ NrmCVec → 𝐷 = (𝑁 ∘ 𝑀))
5	4	3ad2ant1 1129	. . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → 𝐷 = (𝑁 ∘ 𝑀))
6	5	fveq1d 6675	. . 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 28425	. . . . 5 ⊢ (𝑈 ∈ NrmCVec → 𝑀:(𝑋 × 𝑋)⟶𝑋)
9		opelxpi 5595	. . . . 5 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ⟨𝐴, 𝐵⟩ ∈ (𝑋 × 𝑋))
10		fvco3 6763	. . . . 5 ⊢ ((𝑀:(𝑋 × 𝑋)⟶𝑋 ∧ ⟨𝐴, 𝐵⟩ ∈ (𝑋 × 𝑋)) → ((𝑁 ∘ 𝑀)‘⟨𝐴, 𝐵⟩) = (𝑁‘(𝑀‘⟨𝐴, 𝐵⟩)))
11	8, 9, 10	syl2an 597	. . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → ((𝑁 ∘ 𝑀)‘⟨𝐴, 𝐵⟩) = (𝑁‘(𝑀‘⟨𝐴, 𝐵⟩)))
12	11	3impb 1111	. . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝑁 ∘ 𝑀)‘⟨𝐴, 𝐵⟩) = (𝑁‘(𝑀‘⟨𝐴, 𝐵⟩)))
13	6, 12	eqtrd 2859	. 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐷‘⟨𝐴, 𝐵⟩) = (𝑁‘(𝑀‘⟨𝐴, 𝐵⟩)))
14		df-ov 7162	. 2 ⊢ (𝐴𝐷𝐵) = (𝐷‘⟨𝐴, 𝐵⟩)
15		df-ov 7162	. . 3 ⊢ (𝐴𝑀𝐵) = (𝑀‘⟨𝐴, 𝐵⟩)
16	15	fveq2i 6676	. 2 ⊢ (𝑁‘(𝐴𝑀𝐵)) = (𝑁‘(𝑀‘⟨𝐴, 𝐵⟩))
17	13, 14, 16	3eqtr4g 2884	1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷𝐵) = (𝑁‘(𝐴𝑀𝐵)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1536 ∈ wcel 2113 ⟨cop 4576 × cxp 5556 ∘ ccom 5562 ⟶wf 6354 ‘cfv 6358 (class class class)co 7159 NrmCVeccnv 28364 BaseSetcba 28366 −_𝑣 cnsb 28369 norm_CVcnmcv 28370 IndMetcims 28371

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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-po 5477 df-so 5478 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-1st 7692 df-2nd 7693 df-er 8292 df-en 8513 df-dom 8514 df-sdom 8515 df-pnf 10680 df-mnf 10681 df-ltxr 10683 df-sub 10875 df-neg 10876 df-grpo 28273 df-gid 28274 df-ginv 28275 df-gdiv 28276 df-ablo 28325 df-vc 28339 df-nv 28372 df-va 28375 df-ba 28376 df-sm 28377 df-0v 28378 df-vs 28379 df-nmcv 28380 df-ims 28381

This theorem is referenced by: imsdval2 28467 nvnd 28468 vacn 28474 smcnlem 28477 sspimsval 28518 blometi 28583 blocnilem 28584 ubthlem2 28651 minvecolem2 28655 minvecolem4 28660 minvecolem5 28661 minvecolem6 28662 h2hmetdval 28758 hhssmetdval 29057

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