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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 2797 | . . . 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 22887 | . . 3 ⊢ (𝑊 ∈ Grp → 𝑁 = (𝑥 ∈ 𝑋 ↦ (𝑥𝐸(0g‘𝑊)))) |
7 | 6 | adantr 481 | . 2 ⊢ ((𝑊 ∈ Grp ∧ 𝐸 ∈ (Met‘𝑋)) → 𝑁 = (𝑥 ∈ 𝑋 ↦ (𝑥𝐸(0g‘𝑊)))) |
8 | 2, 3 | grpidcl 17893 | . . . 4 ⊢ (𝑊 ∈ Grp → (0g‘𝑊) ∈ 𝑋) |
9 | metcl 22629 | . . . . 5 ⊢ ((𝐸 ∈ (Met‘𝑋) ∧ 𝑥 ∈ 𝑋 ∧ (0g‘𝑊) ∈ 𝑋) → (𝑥𝐸(0g‘𝑊)) ∈ ℝ) | |
10 | 9 | 3comr 1118 | . . . 4 ⊢ (((0g‘𝑊) ∈ 𝑋 ∧ 𝐸 ∈ (Met‘𝑋) ∧ 𝑥 ∈ 𝑋) → (𝑥𝐸(0g‘𝑊)) ∈ ℝ) |
11 | 8, 10 | syl3an1 1156 | . . 3 ⊢ ((𝑊 ∈ Grp ∧ 𝐸 ∈ (Met‘𝑋) ∧ 𝑥 ∈ 𝑋) → (𝑥𝐸(0g‘𝑊)) ∈ ℝ) |
12 | 11 | 3expa 1111 | . 2 ⊢ (((𝑊 ∈ Grp ∧ 𝐸 ∈ (Met‘𝑋)) ∧ 𝑥 ∈ 𝑋) → (𝑥𝐸(0g‘𝑊)) ∈ ℝ) |
13 | 7, 12 | fmpt3d 6750 | 1 ⊢ ((𝑊 ∈ Grp ∧ 𝐸 ∈ (Met‘𝑋)) → 𝑁:𝑋⟶ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1525 ∈ wcel 2083 ↦ cmpt 5047 × cxp 5448 ↾ cres 5452 ⟶wf 6228 ‘cfv 6232 (class class class)co 7023 ℝcr 10389 Basecbs 16316 distcds 16407 0gc0g 16546 Grpcgrp 17865 Metcmet 20217 normcnm 22873 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1781 ax-4 1795 ax-5 1892 ax-6 1951 ax-7 1996 ax-8 2085 ax-9 2093 ax-10 2114 ax-11 2128 ax-12 2143 ax-13 2346 ax-ext 2771 ax-sep 5101 ax-nul 5108 ax-pow 5164 ax-pr 5228 ax-un 7326 ax-cnex 10446 ax-resscn 10447 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1082 df-tru 1528 df-ex 1766 df-nf 1770 df-sb 2045 df-mo 2578 df-eu 2614 df-clab 2778 df-cleq 2790 df-clel 2865 df-nfc 2937 df-ne 2987 df-ral 3112 df-rex 3113 df-reu 3114 df-rmo 3115 df-rab 3116 df-v 3442 df-sbc 3712 df-dif 3868 df-un 3870 df-in 3872 df-ss 3880 df-nul 4218 df-if 4388 df-pw 4461 df-sn 4479 df-pr 4481 df-op 4485 df-uni 4752 df-br 4969 df-opab 5031 df-mpt 5048 df-id 5355 df-xp 5456 df-rel 5457 df-cnv 5458 df-co 5459 df-dm 5460 df-rn 5461 df-res 5462 df-ima 5463 df-iota 6196 df-fun 6234 df-fn 6235 df-f 6236 df-fv 6240 df-riota 6984 df-ov 7026 df-oprab 7027 df-mpo 7028 df-map 8265 df-0g 16548 df-mgm 17685 df-sgrp 17727 df-mnd 17738 df-grp 17868 df-met 20225 df-nm 22879 |
This theorem is referenced by: isngp2 22893 isngp3 22894 nmf 22911 |
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