nmfval2 - Metamath Proof Explorer

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Theorem nmfval2 24477

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 18844	. 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 24476	. 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 5173 × cxp 5617 ↾ cres 5621 ‘cfv 6482 (class class class)co 7349 Basecbs 17120 distcds 17170 0_gc0g 17343 Grpcgrp 18812 normcnm 24462

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 2701 ax-sep 5235 ax-nul 5245 ax-pow 5304 ax-pr 5371 ax-un 7671

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 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-ral 3045 df-rex 3054 df-rmo 3343 df-reu 3344 df-rab 3395 df-v 3438 df-sbc 3743 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-nul 4285 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-op 4584 df-uni 4859 df-br 5093 df-opab 5155 df-mpt 5174 df-id 5514 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-iota 6438 df-fun 6484 df-fn 6485 df-f 6486 df-fv 6490 df-riota 7306 df-ov 7352 df-0g 17345 df-mgm 18514 df-sgrp 18593 df-mnd 18609 df-grp 18815 df-nm 24468

This theorem is referenced by: nmf2 24479 nmpropd2 24481

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