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Mirrors > Home > MPE Home > Th. List > nmfval2 | Structured version Visualization version GIF version |
Description: The value of the norm function using a restricted metric. (Contributed by Mario Carneiro, 2-Oct-2015.) |
Ref | Expression |
---|---|
nmfval.n | ⊢ 𝑁 = (norm‘𝑊) |
nmfval.x | ⊢ 𝑋 = (Base‘𝑊) |
nmfval.z | ⊢ 0 = (0g‘𝑊) |
nmfval.d | ⊢ 𝐷 = (dist‘𝑊) |
nmfval.e | ⊢ 𝐸 = (𝐷 ↾ (𝑋 × 𝑋)) |
Ref | Expression |
---|---|
nmfval2 | ⊢ (𝑊 ∈ Grp → 𝑁 = (𝑥 ∈ 𝑋 ↦ (𝑥𝐸 0 ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nmfval.n | . . 3 ⊢ 𝑁 = (norm‘𝑊) | |
2 | nmfval.x | . . 3 ⊢ 𝑋 = (Base‘𝑊) | |
3 | nmfval.z | . . 3 ⊢ 0 = (0g‘𝑊) | |
4 | nmfval.d | . . 3 ⊢ 𝐷 = (dist‘𝑊) | |
5 | 1, 2, 3, 4 | nmfval 23195 | . 2 ⊢ 𝑁 = (𝑥 ∈ 𝑋 ↦ (𝑥𝐷 0 )) |
6 | nmfval.e | . . . . 5 ⊢ 𝐸 = (𝐷 ↾ (𝑋 × 𝑋)) | |
7 | 6 | oveqi 7148 | . . . 4 ⊢ (𝑥𝐸 0 ) = (𝑥(𝐷 ↾ (𝑋 × 𝑋)) 0 ) |
8 | id 22 | . . . . 5 ⊢ (𝑥 ∈ 𝑋 → 𝑥 ∈ 𝑋) | |
9 | 2, 3 | grpidcl 18123 | . . . . 5 ⊢ (𝑊 ∈ Grp → 0 ∈ 𝑋) |
10 | ovres 7294 | . . . . 5 ⊢ ((𝑥 ∈ 𝑋 ∧ 0 ∈ 𝑋) → (𝑥(𝐷 ↾ (𝑋 × 𝑋)) 0 ) = (𝑥𝐷 0 )) | |
11 | 8, 9, 10 | syl2anr 599 | . . . 4 ⊢ ((𝑊 ∈ Grp ∧ 𝑥 ∈ 𝑋) → (𝑥(𝐷 ↾ (𝑋 × 𝑋)) 0 ) = (𝑥𝐷 0 )) |
12 | 7, 11 | syl5req 2846 | . . 3 ⊢ ((𝑊 ∈ Grp ∧ 𝑥 ∈ 𝑋) → (𝑥𝐷 0 ) = (𝑥𝐸 0 )) |
13 | 12 | mpteq2dva 5125 | . 2 ⊢ (𝑊 ∈ Grp → (𝑥 ∈ 𝑋 ↦ (𝑥𝐷 0 )) = (𝑥 ∈ 𝑋 ↦ (𝑥𝐸 0 ))) |
14 | 5, 13 | syl5eq 2845 | 1 ⊢ (𝑊 ∈ Grp → 𝑁 = (𝑥 ∈ 𝑋 ↦ (𝑥𝐸 0 ))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ↦ cmpt 5110 × cxp 5517 ↾ cres 5521 ‘cfv 6324 (class class class)co 7135 Basecbs 16475 distcds 16566 0gc0g 16705 Grpcgrp 18095 normcnm 23183 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-fv 6332 df-riota 7093 df-ov 7138 df-0g 16707 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-grp 18098 df-nm 23189 |
This theorem is referenced by: nmf2 23199 nmpropd2 23201 |
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