Theorem nmfval2 23195

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 = (0g‘𝑊)
nmfval.d	⊢ 𝐷 = (dist‘𝑊)
nmfval.e	⊢ 𝐸 = (𝐷 ↾ (𝑋 × 𝑋))

Assertion

Ref	Expression
nmfval2	⊢ (𝑊 ∈ Grp → 𝑁 = (𝑥 ∈ 𝑋 ↦ (𝑥𝐸 0 )))

Distinct variable groups:   𝑥,𝐷   𝑥,𝑊   𝑥,𝑋   𝑥, 0

Allowed substitution hints:   𝐸(𝑥)   𝑁(𝑥)

Proof of Theorem nmfval2

Step	Hyp	Ref	Expression
1		nmfval.n	. . 3 ⊢ 𝑁 = (norm‘𝑊)
2		nmfval.x	. . 3 ⊢ 𝑋 = (Base‘𝑊)
3		nmfval.z	. . 3 ⊢ 0 = (0g‘𝑊)
4		nmfval.d	. . 3 ⊢ 𝐷 = (dist‘𝑊)
5	1, 2, 3, 4	nmfval 23193	. 2 ⊢ 𝑁 = (𝑥 ∈ 𝑋 ↦ (𝑥𝐷 0 ))
6		nmfval.e	. . . . 5 ⊢ 𝐸 = (𝐷 ↾ (𝑋 × 𝑋))
7	6	oveqi 7162	. . . 4 ⊢ (𝑥𝐸 0 ) = (𝑥(𝐷 ↾ (𝑋 × 𝑋)) 0 )
8		id 22	. . . . 5 ⊢ (𝑥 ∈ 𝑋 → 𝑥 ∈ 𝑋)
9	2, 3	grpidcl 18126	. . . . 5 ⊢ (𝑊 ∈ Grp → 0 ∈ 𝑋)
10		ovres 7307	. . . . 5 ⊢ ((𝑥 ∈ 𝑋 ∧ 0 ∈ 𝑋) → (𝑥(𝐷 ↾ (𝑋 × 𝑋)) 0 ) = (𝑥𝐷 0 ))
11	8, 9, 10	syl2anr 598	. . . 4 ⊢ ((𝑊 ∈ Grp ∧ 𝑥 ∈ 𝑋) → (𝑥(𝐷 ↾ (𝑋 × 𝑋)) 0 ) = (𝑥𝐷 0 ))
12	7, 11	syl5req 2868	. . 3 ⊢ ((𝑊 ∈ Grp ∧ 𝑥 ∈ 𝑋) → (𝑥𝐷 0 ) = (𝑥𝐸 0 ))
13	12	mpteq2dva 5154	. 2 ⊢ (𝑊 ∈ Grp → (𝑥 ∈ 𝑋 ↦ (𝑥𝐷 0 )) = (𝑥 ∈ 𝑋 ↦ (𝑥𝐸 0 )))
14	5, 13	syl5eq 2867	1 ⊢ (𝑊 ∈ Grp → 𝑁 = (𝑥 ∈ 𝑋 ↦ (𝑥𝐸 0 )))

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1536   ∈ wcel 2113   ↦ cmpt 5139   × cxp 5546   ↾ cres 5550  ‘cfv 6348  (class class class)co 7149  Basecbs 16478  distcds 16569  0_gc0g 16708  Grpcgrp 18098  normcnm 23181

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 2792  ax-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5323  ax-un 7454

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2892  df-nfc 2962  df-ne 3016  df-ral 3142  df-rex 3143  df-reu 3144  df-rmo 3145  df-rab 3146  df-v 3493  df-sbc 3769  df-dif 3932  df-un 3934  df-in 3936  df-ss 3945  df-nul 4285  df-if 4461  df-pw 4534  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-br 5060  df-opab 5122  df-mpt 5140  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fv 6356  df-riota 7107  df-ov 7152  df-0g 16710  df-mgm 17847  df-sgrp 17896  df-mnd 17907  df-grp 18101  df-nm 23187

This theorem is referenced by:  nmf2  23197  nmpropd2  23199