nmfval2 - Metamath Proof Explorer

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Theorem nmfval2 24748

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 19027	. 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 24747	. 2 ⊢ ( 0 ∈ 𝑋 → 𝑁 = (𝑥 ∈ 𝑋 ↦ (𝑥𝐸 0 )))
8	3, 7	syl 18	1 ⊢ (𝑊 ∈ Grp → 𝑁 = (𝑥 ∈ 𝑋 ↦ (𝑥𝐸 0 )))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1570 ∈ wcel 2143 ↦ cmpt 5192 × cxp 5659 ↾ cres 5663 ‘cfv 6536 (class class class)co 7410 Basecbs 17264 distcds 17314 0_gc0g 17487 Grpcgrp 18995 normcnm 24733

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-sep 5257 ax-nul 5269 ax-pow 5336 ax-pr 5404 ax-un 7732

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-ral 3080 df-rex 3090 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3745 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-nul 4287 df-if 4488 df-pw 4564 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4873 df-br 5110 df-opab 5174 df-mpt 5193 df-id 5556 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-fv 6544 df-riota 7367 df-ov 7413 df-0g 17489 df-mgm 18693 df-sgrp 18772 df-mnd 18788 df-grp 18998 df-nm 24739

This theorem is referenced by: nmf2 24750 nmpropd2 24752

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