imsdval - Metamath Proof Explorer

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Theorem imsdval 30705

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 30704	. . . . 5 ⊢ (𝑈 ∈ NrmCVec → 𝐷 = (𝑁 ∘ 𝑀))
5	4	3ad2ant1 1134	. . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → 𝐷 = (𝑁 ∘ 𝑀))
6	5	fveq1d 6908	. . 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 30664	. . . . 5 ⊢ (𝑈 ∈ NrmCVec → 𝑀:(𝑋 × 𝑋)⟶𝑋)
9		opelxpi 5722	. . . . 5 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ⟨𝐴, 𝐵⟩ ∈ (𝑋 × 𝑋))
10		fvco3 7008	. . . . 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 1115	. . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝑁 ∘ 𝑀)‘⟨𝐴, 𝐵⟩) = (𝑁‘(𝑀‘⟨𝐴, 𝐵⟩)))
13	6, 12	eqtrd 2777	. 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐷‘⟨𝐴, 𝐵⟩) = (𝑁‘(𝑀‘⟨𝐴, 𝐵⟩)))
14		df-ov 7434	. 2 ⊢ (𝐴𝐷𝐵) = (𝐷‘⟨𝐴, 𝐵⟩)
15		df-ov 7434	. . 3 ⊢ (𝐴𝑀𝐵) = (𝑀‘⟨𝐴, 𝐵⟩)
16	15	fveq2i 6909	. 2 ⊢ (𝑁‘(𝐴𝑀𝐵)) = (𝑁‘(𝑀‘⟨𝐴, 𝐵⟩))
17	13, 14, 16	3eqtr4g 2802	1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷𝐵) = (𝑁‘(𝐴𝑀𝐵)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ⟨cop 4632 × cxp 5683 ∘ ccom 5689 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 NrmCVeccnv 30603 BaseSetcba 30605 −_𝑣 cnsb 30608 norm_CVcnmcv 30609 IndMetcims 30610

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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-po 5592 df-so 5593 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-ltxr 11300 df-sub 11494 df-neg 11495 df-grpo 30512 df-gid 30513 df-ginv 30514 df-gdiv 30515 df-ablo 30564 df-vc 30578 df-nv 30611 df-va 30614 df-ba 30615 df-sm 30616 df-0v 30617 df-vs 30618 df-nmcv 30619 df-ims 30620

This theorem is referenced by: imsdval2 30706 nvnd 30707 vacn 30713 smcnlem 30716 sspimsval 30757 blometi 30822 blocnilem 30823 ubthlem2 30890 minvecolem2 30894 minvecolem4 30899 minvecolem5 30900 minvecolem6 30901 h2hmetdval 30997 hhssmetdval 31296

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