nmfval2 - Metamath Proof Explorer

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Theorem nmfval2 23653

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 18522	. 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 23652	. 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 1539 ∈ wcel 2108 ↦ cmpt 5153 × cxp 5578 ↾ cres 5582 ‘cfv 6418 (class class class)co 7255 Basecbs 16840 distcds 16897 0_gc0g 17067 Grpcgrp 18492 normcnm 23638

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-op 4565 df-uni 4837 df-br 5071 df-opab 5133 df-mpt 5154 df-id 5480 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-fv 6426 df-riota 7212 df-ov 7258 df-0g 17069 df-mgm 18241 df-sgrp 18290 df-mnd 18301 df-grp 18495 df-nm 23644

This theorem is referenced by: nmf2 23655 nmpropd2 23657

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