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Theorem nmval2 23198

Description: The value of the norm function using a restricted metric. (Contributed by Mario Carneiro, 2-Oct-2015.)

**Hypotheses**
Ref	Expression
nmfval.n	⊢ 𝑁 = (norm‘𝑊)
nmfval.x	⊢ 𝑋 = (Base‘𝑊)
nmfval.z	⊢ 0 = (0_g‘𝑊)
nmfval.d	⊢ 𝐷 = (dist‘𝑊)
nmfval.e	⊢ 𝐸 = (𝐷 ↾ (𝑋 × 𝑋))

**Assertion**
Ref	Expression
nmval2	⊢ ((𝑊 ∈ Grp ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝐴) = (𝐴𝐸 0 ))

**Proof of Theorem nmval2**
Step	Hyp	Ref	Expression
1		nmfval.n	. . . 4 ⊢ 𝑁 = (norm‘𝑊)
2		nmfval.x	. . . 4 ⊢ 𝑋 = (Base‘𝑊)
3		nmfval.z	. . . 4 ⊢ 0 = (0_g‘𝑊)
4		nmfval.d	. . . 4 ⊢ 𝐷 = (dist‘𝑊)
5	1, 2, 3, 4	nmval 23196	. . 3 ⊢ (𝐴 ∈ 𝑋 → (𝑁‘𝐴) = (𝐴𝐷 0 ))
6	5	adantl 485	. 2 ⊢ ((𝑊 ∈ Grp ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝐴) = (𝐴𝐷 0 ))
7		nmfval.e	. . . 4 ⊢ 𝐸 = (𝐷 ↾ (𝑋 × 𝑋))
8	7	oveqi 7148	. . 3 ⊢ (𝐴𝐸 0 ) = (𝐴(𝐷 ↾ (𝑋 × 𝑋)) 0 )
9		id 22	. . . 4 ⊢ (𝐴 ∈ 𝑋 → 𝐴 ∈ 𝑋)
10	2, 3	grpidcl 18123	. . . 4 ⊢ (𝑊 ∈ Grp → 0 ∈ 𝑋)
11		ovres 7294	. . . 4 ⊢ ((𝐴 ∈ 𝑋 ∧ 0 ∈ 𝑋) → (𝐴(𝐷 ↾ (𝑋 × 𝑋)) 0 ) = (𝐴𝐷 0 ))
12	9, 10, 11	syl2anr 599	. . 3 ⊢ ((𝑊 ∈ Grp ∧ 𝐴 ∈ 𝑋) → (𝐴(𝐷 ↾ (𝑋 × 𝑋)) 0 ) = (𝐴𝐷 0 ))
13	8, 12	syl5req 2846	. 2 ⊢ ((𝑊 ∈ Grp ∧ 𝐴 ∈ 𝑋) → (𝐴𝐷 0 ) = (𝐴𝐸 0 ))
14	6, 13	eqtrd 2833	1 ⊢ ((𝑊 ∈ Grp ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝐴) = (𝐴𝐸 0 ))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 × cxp 5517 ↾ cres 5521 ‘cfv 6324 (class class class)co 7135 Basecbs 16475 distcds 16566 0_gc0g 16705 Grpcgrp 18095 normcnm 23183

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-fv 6332 df-riota 7093 df-ov 7138 df-0g 16707 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-grp 18098 df-nm 23189

This theorem is referenced by: nmhmcn 23725 nglmle 23906

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