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Mirrors > Home > MPE Home > Th. List > nmf2 | Structured version Visualization version GIF version |
Description: The norm is a function from the base set into the reals. (Contributed by Mario Carneiro, 2-Oct-2015.) |
Ref | Expression |
---|---|
nmf2.n | ⊢ 𝑁 = (norm‘𝑊) |
nmf2.x | ⊢ 𝑋 = (Base‘𝑊) |
nmf2.d | ⊢ 𝐷 = (dist‘𝑊) |
nmf2.e | ⊢ 𝐸 = (𝐷 ↾ (𝑋 × 𝑋)) |
Ref | Expression |
---|---|
nmf2 | ⊢ ((𝑊 ∈ Grp ∧ 𝐸 ∈ (Met‘𝑋)) → 𝑁:𝑋⟶ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nmf2.n | . . . 4 ⊢ 𝑁 = (norm‘𝑊) | |
2 | nmf2.x | . . . 4 ⊢ 𝑋 = (Base‘𝑊) | |
3 | eqid 2821 | . . . 4 ⊢ (0g‘𝑊) = (0g‘𝑊) | |
4 | nmf2.d | . . . 4 ⊢ 𝐷 = (dist‘𝑊) | |
5 | nmf2.e | . . . 4 ⊢ 𝐸 = (𝐷 ↾ (𝑋 × 𝑋)) | |
6 | 1, 2, 3, 4, 5 | nmfval2 23194 | . . 3 ⊢ (𝑊 ∈ Grp → 𝑁 = (𝑥 ∈ 𝑋 ↦ (𝑥𝐸(0g‘𝑊)))) |
7 | 6 | adantr 483 | . 2 ⊢ ((𝑊 ∈ Grp ∧ 𝐸 ∈ (Met‘𝑋)) → 𝑁 = (𝑥 ∈ 𝑋 ↦ (𝑥𝐸(0g‘𝑊)))) |
8 | 2, 3 | grpidcl 18125 | . . . 4 ⊢ (𝑊 ∈ Grp → (0g‘𝑊) ∈ 𝑋) |
9 | metcl 22936 | . . . . 5 ⊢ ((𝐸 ∈ (Met‘𝑋) ∧ 𝑥 ∈ 𝑋 ∧ (0g‘𝑊) ∈ 𝑋) → (𝑥𝐸(0g‘𝑊)) ∈ ℝ) | |
10 | 9 | 3comr 1121 | . . . 4 ⊢ (((0g‘𝑊) ∈ 𝑋 ∧ 𝐸 ∈ (Met‘𝑋) ∧ 𝑥 ∈ 𝑋) → (𝑥𝐸(0g‘𝑊)) ∈ ℝ) |
11 | 8, 10 | syl3an1 1159 | . . 3 ⊢ ((𝑊 ∈ Grp ∧ 𝐸 ∈ (Met‘𝑋) ∧ 𝑥 ∈ 𝑋) → (𝑥𝐸(0g‘𝑊)) ∈ ℝ) |
12 | 11 | 3expa 1114 | . 2 ⊢ (((𝑊 ∈ Grp ∧ 𝐸 ∈ (Met‘𝑋)) ∧ 𝑥 ∈ 𝑋) → (𝑥𝐸(0g‘𝑊)) ∈ ℝ) |
13 | 7, 12 | fmpt3d 6874 | 1 ⊢ ((𝑊 ∈ Grp ∧ 𝐸 ∈ (Met‘𝑋)) → 𝑁:𝑋⟶ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ↦ cmpt 5138 × cxp 5547 ↾ cres 5551 ⟶wf 6345 ‘cfv 6349 (class class class)co 7150 ℝcr 10530 Basecbs 16477 distcds 16568 0gc0g 16707 Grpcgrp 18097 Metcmet 20525 normcnm 23180 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-map 8402 df-0g 16709 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-grp 18100 df-met 20533 df-nm 23186 |
This theorem is referenced by: isngp2 23200 isngp3 23201 nmf 23218 |
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