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Mirrors > Home > MPE Home > Th. List > nmval2 | 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 |
---|---|
nmval2 | ⊢ ((𝑊 ∈ Grp ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝐴) = (𝐴𝐸 0 )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nmfval.n | . . . 4 ⊢ 𝑁 = (norm‘𝑊) | |
2 | nmfval.x | . . . 4 ⊢ 𝑋 = (Base‘𝑊) | |
3 | nmfval.z | . . . 4 ⊢ 0 = (0g‘𝑊) | |
4 | nmfval.d | . . . 4 ⊢ 𝐷 = (dist‘𝑊) | |
5 | 1, 2, 3, 4 | nmval 22204 | . . 3 ⊢ (𝐴 ∈ 𝑋 → (𝑁‘𝐴) = (𝐴𝐷 0 )) |
6 | 5 | adantl 481 | . 2 ⊢ ((𝑊 ∈ Grp ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝐴) = (𝐴𝐷 0 )) |
7 | nmfval.e | . . . 4 ⊢ 𝐸 = (𝐷 ↾ (𝑋 × 𝑋)) | |
8 | 7 | oveqi 6562 | . . 3 ⊢ (𝐴𝐸 0 ) = (𝐴(𝐷 ↾ (𝑋 × 𝑋)) 0 ) |
9 | id 22 | . . . 4 ⊢ (𝐴 ∈ 𝑋 → 𝐴 ∈ 𝑋) | |
10 | 2, 3 | grpidcl 17273 | . . . 4 ⊢ (𝑊 ∈ Grp → 0 ∈ 𝑋) |
11 | ovres 6698 | . . . 4 ⊢ ((𝐴 ∈ 𝑋 ∧ 0 ∈ 𝑋) → (𝐴(𝐷 ↾ (𝑋 × 𝑋)) 0 ) = (𝐴𝐷 0 )) | |
12 | 9, 10, 11 | syl2anr 494 | . . 3 ⊢ ((𝑊 ∈ Grp ∧ 𝐴 ∈ 𝑋) → (𝐴(𝐷 ↾ (𝑋 × 𝑋)) 0 ) = (𝐴𝐷 0 )) |
13 | 8, 12 | syl5req 2657 | . 2 ⊢ ((𝑊 ∈ Grp ∧ 𝐴 ∈ 𝑋) → (𝐴𝐷 0 ) = (𝐴𝐸 0 )) |
14 | 6, 13 | eqtrd 2644 | 1 ⊢ ((𝑊 ∈ Grp ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝐴) = (𝐴𝐸 0 )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 × cxp 5036 ↾ cres 5040 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 distcds 15777 0gc0g 15923 Grpcgrp 17245 normcnm 22191 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-fv 5812 df-riota 6511 df-ov 6552 df-0g 15925 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-grp 17248 df-nm 22197 |
This theorem is referenced by: nmhmcn 22728 nglmle 22908 |
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