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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 23199 | . . 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 18130 | . . . 4 ⊢ (𝑊 ∈ Grp → (0g‘𝑊) ∈ 𝑋) |
9 | metcl 22941 | . . . . 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 6879 | 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 5145 × cxp 5552 ↾ cres 5556 ⟶wf 6350 ‘cfv 6354 (class class class)co 7155 ℝcr 10535 Basecbs 16482 distcds 16573 0gc0g 16712 Grpcgrp 18102 Metcmet 20530 normcnm 23185 |
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 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 |
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 4567 df-pr 4569 df-op 4573 df-uni 4838 df-br 5066 df-opab 5128 df-mpt 5146 df-id 5459 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-fv 6362 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-map 8407 df-0g 16714 df-mgm 17851 df-sgrp 17900 df-mnd 17911 df-grp 18105 df-met 20538 df-nm 23191 |
This theorem is referenced by: isngp2 23205 isngp3 23206 nmf 23223 |
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