imsdval - Metamath Proof Explorer

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Theorem imsdval 29044

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 29043	. . . . 5 ⊢ (𝑈 ∈ NrmCVec → 𝐷 = (𝑁 ∘ 𝑀))
5	4	3ad2ant1 1132	. . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → 𝐷 = (𝑁 ∘ 𝑀))
6	5	fveq1d 6773	. . 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 29003	. . . . 5 ⊢ (𝑈 ∈ NrmCVec → 𝑀:(𝑋 × 𝑋)⟶𝑋)
9		opelxpi 5627	. . . . 5 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ⟨𝐴, 𝐵⟩ ∈ (𝑋 × 𝑋))
10		fvco3 6864	. . . . 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 2780	. 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐷‘⟨𝐴, 𝐵⟩) = (𝑁‘(𝑀‘⟨𝐴, 𝐵⟩)))
14		df-ov 7274	. 2 ⊢ (𝐴𝐷𝐵) = (𝐷‘⟨𝐴, 𝐵⟩)
15		df-ov 7274	. . 3 ⊢ (𝐴𝑀𝐵) = (𝑀‘⟨𝐴, 𝐵⟩)
16	15	fveq2i 6774	. 2 ⊢ (𝑁‘(𝐴𝑀𝐵)) = (𝑁‘(𝑀‘⟨𝐴, 𝐵⟩))
17	13, 14, 16	3eqtr4g 2805	1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷𝐵) = (𝑁‘(𝐴𝑀𝐵)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1086 = wceq 1542 ∈ wcel 2110 ⟨cop 4573 × cxp 5588 ∘ ccom 5594 ⟶wf 6428 ‘cfv 6432 (class class class)co 7271 NrmCVeccnv 28942 BaseSetcba 28944 −_𝑣 cnsb 28947 norm_CVcnmcv 28948 IndMetcims 28949

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7582 ax-resscn 10929 ax-1cn 10930 ax-icn 10931 ax-addcl 10932 ax-addrcl 10933 ax-mulcl 10934 ax-mulrcl 10935 ax-mulcom 10936 ax-addass 10937 ax-mulass 10938 ax-distr 10939 ax-i2m1 10940 ax-1ne0 10941 ax-1rid 10942 ax-rnegex 10943 ax-rrecex 10944 ax-cnre 10945 ax-pre-lttri 10946 ax-pre-lttrn 10947 ax-pre-ltadd 10948

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-id 5490 df-po 5504 df-so 5505 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-riota 7228 df-ov 7274 df-oprab 7275 df-mpo 7276 df-1st 7824 df-2nd 7825 df-er 8481 df-en 8717 df-dom 8718 df-sdom 8719 df-pnf 11012 df-mnf 11013 df-ltxr 11015 df-sub 11207 df-neg 11208 df-grpo 28851 df-gid 28852 df-ginv 28853 df-gdiv 28854 df-ablo 28903 df-vc 28917 df-nv 28950 df-va 28953 df-ba 28954 df-sm 28955 df-0v 28956 df-vs 28957 df-nmcv 28958 df-ims 28959

This theorem is referenced by: imsdval2 29045 nvnd 29046 vacn 29052 smcnlem 29055 sspimsval 29096 blometi 29161 blocnilem 29162 ubthlem2 29229 minvecolem2 29233 minvecolem4 29238 minvecolem5 29239 minvecolem6 29240 h2hmetdval 29336 hhssmetdval 29635

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