nmfval2 - Metamath Proof Explorer

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Theorem nmfval2 24485

Description: The value of the norm function on a group as the distance restricted to the elements of the base set to zero. (Contributed by Mario Carneiro, 2-Oct-2015.)

**Hypotheses**
Ref	Expression
nmfval2.n	⊢ 𝑁 = (norm‘𝑊)
nmfval2.x	⊢ 𝑋 = (Base‘𝑊)
nmfval2.z	⊢ 0 = (0_g‘𝑊)
nmfval2.d	⊢ 𝐷 = (dist‘𝑊)
nmfval2.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		nmfval2.x	. . 3 ⊢ 𝑋 = (Base‘𝑊)
2		nmfval2.z	. . 3 ⊢ 0 = (0_g‘𝑊)
3	1, 2	grpidcl 18903	. 2 ⊢ (𝑊 ∈ Grp → 0 ∈ 𝑋)
4		nmfval2.n	. . 3 ⊢ 𝑁 = (norm‘𝑊)
5		nmfval2.d	. . 3 ⊢ 𝐷 = (dist‘𝑊)
6		nmfval2.e	. . 3 ⊢ 𝐸 = (𝐷 ↾ (𝑋 × 𝑋))
7	4, 1, 2, 5, 6	nmfval0 24484	. 2 ⊢ ( 0 ∈ 𝑋 → 𝑁 = (𝑥 ∈ 𝑋 ↦ (𝑥𝐸 0 )))
8	3, 7	syl 17	1 ⊢ (𝑊 ∈ Grp → 𝑁 = (𝑥 ∈ 𝑋 ↦ (𝑥𝐸 0 )))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2109 ↦ cmpt 5190 × cxp 5638 ↾ cres 5642 ‘cfv 6513 (class class class)co 7389 Basecbs 17185 distcds 17235 0_gc0g 17408 Grpcgrp 18871 normcnm 24470

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 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-sep 5253 ax-nul 5263 ax-pow 5322 ax-pr 5389 ax-un 7713

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-ral 3046 df-rex 3055 df-rmo 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3756 df-dif 3919 df-un 3921 df-in 3923 df-ss 3933 df-nul 4299 df-if 4491 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4874 df-br 5110 df-opab 5172 df-mpt 5191 df-id 5535 df-xp 5646 df-rel 5647 df-cnv 5648 df-co 5649 df-dm 5650 df-rn 5651 df-res 5652 df-ima 5653 df-iota 6466 df-fun 6515 df-fn 6516 df-f 6517 df-fv 6521 df-riota 7346 df-ov 7392 df-0g 17410 df-mgm 18573 df-sgrp 18652 df-mnd 18668 df-grp 18874 df-nm 24476

This theorem is referenced by: nmf2 24487 nmpropd2 24489

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