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Mirrors > Home > MPE Home > Th. List > nmf2 | Structured version Visualization version GIF version |
Description: The norm on a metric group 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 2725 | . . . 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 24561 | . . 3 ⊢ (𝑊 ∈ Grp → 𝑁 = (𝑥 ∈ 𝑋 ↦ (𝑥𝐸(0g‘𝑊)))) |
7 | 6 | adantr 479 | . 2 ⊢ ((𝑊 ∈ Grp ∧ 𝐸 ∈ (Met‘𝑋)) → 𝑁 = (𝑥 ∈ 𝑋 ↦ (𝑥𝐸(0g‘𝑊)))) |
8 | 2, 3 | grpidcl 18946 | . . . 4 ⊢ (𝑊 ∈ Grp → (0g‘𝑊) ∈ 𝑋) |
9 | metcl 24299 | . . . . 5 ⊢ ((𝐸 ∈ (Met‘𝑋) ∧ 𝑥 ∈ 𝑋 ∧ (0g‘𝑊) ∈ 𝑋) → (𝑥𝐸(0g‘𝑊)) ∈ ℝ) | |
10 | 9 | 3comr 1122 | . . . 4 ⊢ (((0g‘𝑊) ∈ 𝑋 ∧ 𝐸 ∈ (Met‘𝑋) ∧ 𝑥 ∈ 𝑋) → (𝑥𝐸(0g‘𝑊)) ∈ ℝ) |
11 | 8, 10 | syl3an1 1160 | . . 3 ⊢ ((𝑊 ∈ Grp ∧ 𝐸 ∈ (Met‘𝑋) ∧ 𝑥 ∈ 𝑋) → (𝑥𝐸(0g‘𝑊)) ∈ ℝ) |
12 | 11 | 3expa 1115 | . 2 ⊢ (((𝑊 ∈ Grp ∧ 𝐸 ∈ (Met‘𝑋)) ∧ 𝑥 ∈ 𝑋) → (𝑥𝐸(0g‘𝑊)) ∈ ℝ) |
13 | 7, 12 | fmpt3d 7125 | 1 ⊢ ((𝑊 ∈ Grp ∧ 𝐸 ∈ (Met‘𝑋)) → 𝑁:𝑋⟶ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ↦ cmpt 5232 × cxp 5676 ↾ cres 5680 ⟶wf 6545 ‘cfv 6549 (class class class)co 7419 ℝcr 11144 Basecbs 17199 distcds 17261 0gc0g 17440 Grpcgrp 18914 Metcmet 21299 normcnm 24546 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11201 ax-resscn 11202 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-map 8847 df-0g 17442 df-mgm 18619 df-sgrp 18698 df-mnd 18714 df-grp 18917 df-met 21307 df-nm 24552 |
This theorem is referenced by: isngp2 24567 isngp3 24568 nmf 24585 |
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