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Mirrors > Home > MPE Home > Th. List > nmfval0 | Structured version Visualization version GIF version |
Description: The value of the norm function on a structure containing a zero as the distance restricted to the elements of the base set to zero. Examples of structures containing a "zero" are groups (see nmfval2 23443 proved from this theorem and grpidcl 18349) or more generally monoids (see mndidcl 18142), or pointed sets). (Contributed by Mario Carneiro, 2-Oct-2015.) Extract this result from the proof of nmfval2 23443. (Revised by BJ, 27-Aug-2024.) |
Ref | Expression |
---|---|
nmfval0.n | ⊢ 𝑁 = (norm‘𝑊) |
nmfval0.x | ⊢ 𝑋 = (Base‘𝑊) |
nmfval0.z | ⊢ 0 = (0g‘𝑊) |
nmfval0.d | ⊢ 𝐷 = (dist‘𝑊) |
nmfval0.e | ⊢ 𝐸 = (𝐷 ↾ (𝑋 × 𝑋)) |
Ref | Expression |
---|---|
nmfval0 | ⊢ ( 0 ∈ 𝑋 → 𝑁 = (𝑥 ∈ 𝑋 ↦ (𝑥𝐸 0 ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nmfval0.n | . . 3 ⊢ 𝑁 = (norm‘𝑊) | |
2 | nmfval0.x | . . 3 ⊢ 𝑋 = (Base‘𝑊) | |
3 | nmfval0.z | . . 3 ⊢ 0 = (0g‘𝑊) | |
4 | nmfval0.d | . . 3 ⊢ 𝐷 = (dist‘𝑊) | |
5 | 1, 2, 3, 4 | nmfval 23440 | . 2 ⊢ 𝑁 = (𝑥 ∈ 𝑋 ↦ (𝑥𝐷 0 )) |
6 | nmfval0.e | . . . . 5 ⊢ 𝐸 = (𝐷 ↾ (𝑋 × 𝑋)) | |
7 | 6 | oveqi 7204 | . . . 4 ⊢ (𝑥𝐸 0 ) = (𝑥(𝐷 ↾ (𝑋 × 𝑋)) 0 ) |
8 | ovres 7352 | . . . . 5 ⊢ ((𝑥 ∈ 𝑋 ∧ 0 ∈ 𝑋) → (𝑥(𝐷 ↾ (𝑋 × 𝑋)) 0 ) = (𝑥𝐷 0 )) | |
9 | 8 | ancoms 462 | . . . 4 ⊢ (( 0 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋) → (𝑥(𝐷 ↾ (𝑋 × 𝑋)) 0 ) = (𝑥𝐷 0 )) |
10 | 7, 9 | eqtr2id 2784 | . . 3 ⊢ (( 0 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋) → (𝑥𝐷 0 ) = (𝑥𝐸 0 )) |
11 | 10 | mpteq2dva 5135 | . 2 ⊢ ( 0 ∈ 𝑋 → (𝑥 ∈ 𝑋 ↦ (𝑥𝐷 0 )) = (𝑥 ∈ 𝑋 ↦ (𝑥𝐸 0 ))) |
12 | 5, 11 | syl5eq 2783 | 1 ⊢ ( 0 ∈ 𝑋 → 𝑁 = (𝑥 ∈ 𝑋 ↦ (𝑥𝐸 0 ))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2112 ↦ cmpt 5120 × cxp 5534 ↾ cres 5538 ‘cfv 6358 (class class class)co 7191 Basecbs 16666 distcds 16758 0gc0g 16898 normcnm 23428 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-sbc 3684 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-fv 6366 df-ov 7194 df-nm 23434 |
This theorem is referenced by: nmfval2 23443 |
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