nmfval2 - Metamath Proof Explorer

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Theorem nmfval2 22395

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 = (0_g‘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 = (0_g‘𝑊)
4		nmfval.d	. . 3 ⊢ 𝐷 = (dist‘𝑊)
5	1, 2, 3, 4	nmfval 22393	. 2 ⊢ 𝑁 = (𝑥 ∈ 𝑋 ↦ (𝑥𝐷 0 ))
6		nmfval.e	. . . . 5 ⊢ 𝐸 = (𝐷 ↾ (𝑋 × 𝑋))
7	6	oveqi 6663	. . . 4 ⊢ (𝑥𝐸 0 ) = (𝑥(𝐷 ↾ (𝑋 × 𝑋)) 0 )
8		id 22	. . . . 5 ⊢ (𝑥 ∈ 𝑋 → 𝑥 ∈ 𝑋)
9	2, 3	grpidcl 17450	. . . . 5 ⊢ (𝑊 ∈ Grp → 0 ∈ 𝑋)
10		ovres 6800	. . . . 5 ⊢ ((𝑥 ∈ 𝑋 ∧ 0 ∈ 𝑋) → (𝑥(𝐷 ↾ (𝑋 × 𝑋)) 0 ) = (𝑥𝐷 0 ))
11	8, 9, 10	syl2anr 495	. . . 4 ⊢ ((𝑊 ∈ Grp ∧ 𝑥 ∈ 𝑋) → (𝑥(𝐷 ↾ (𝑋 × 𝑋)) 0 ) = (𝑥𝐷 0 ))
12	7, 11	syl5req 2669	. . 3 ⊢ ((𝑊 ∈ Grp ∧ 𝑥 ∈ 𝑋) → (𝑥𝐷 0 ) = (𝑥𝐸 0 ))
13	12	mpteq2dva 4744	. 2 ⊢ (𝑊 ∈ Grp → (𝑥 ∈ 𝑋 ↦ (𝑥𝐷 0 )) = (𝑥 ∈ 𝑋 ↦ (𝑥𝐸 0 )))
14	5, 13	syl5eq 2668	1 ⊢ (𝑊 ∈ Grp → 𝑁 = (𝑥 ∈ 𝑋 ↦ (𝑥𝐸 0 )))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ↦ cmpt 4729 × cxp 5112 ↾ cres 5116 ‘cfv 5888 (class class class)co 6650 Basecbs 15857 distcds 15950 0_gc0g 16100 Grpcgrp 17422 normcnm 22381

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-fv 5896 df-riota 6611 df-ov 6653 df-0g 16102 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-grp 17425 df-nm 22387

This theorem is referenced by: nmf2 22397 nmpropd2 22399

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