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Mirrors > Home > MPE Home > Th. List > tngngpim | Structured version Visualization version GIF version |
Description: The induced metric of a normed group is a function. (Contributed by AV, 19-Oct-2021.) |
Ref | Expression |
---|---|
tngngpim.t | ⊢ 𝑇 = (𝐺 toNrmGrp 𝑁) |
tngngpim.n | ⊢ 𝑁 = (norm‘𝐺) |
tngngpim.x | ⊢ 𝑋 = (Base‘𝐺) |
tngngpim.d | ⊢ 𝐷 = (dist‘𝑇) |
Ref | Expression |
---|---|
tngngpim | ⊢ (𝐺 ∈ NrmGrp → 𝐷:(𝑋 × 𝑋)⟶ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tngngpim.x | . . 3 ⊢ 𝑋 = (Base‘𝐺) | |
2 | tngngpim.n | . . 3 ⊢ 𝑁 = (norm‘𝐺) | |
3 | 1, 2 | nmf 23923 | . 2 ⊢ (𝐺 ∈ NrmGrp → 𝑁:𝑋⟶ℝ) |
4 | tngngpim.t | . . . . . 6 ⊢ 𝑇 = (𝐺 toNrmGrp 𝑁) | |
5 | 2 | oveq2i 7363 | . . . . . 6 ⊢ (𝐺 toNrmGrp 𝑁) = (𝐺 toNrmGrp (norm‘𝐺)) |
6 | 4, 5 | eqtri 2766 | . . . . 5 ⊢ 𝑇 = (𝐺 toNrmGrp (norm‘𝐺)) |
7 | 6 | nrmtngnrm 23974 | . . . 4 ⊢ (𝐺 ∈ NrmGrp → (𝑇 ∈ NrmGrp ∧ (norm‘𝑇) = (norm‘𝐺))) |
8 | tngngpim.d | . . . . . . . 8 ⊢ 𝐷 = (dist‘𝑇) | |
9 | 4, 1, 8 | tngngp2 23968 | . . . . . . 7 ⊢ (𝑁:𝑋⟶ℝ → (𝑇 ∈ NrmGrp ↔ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋)))) |
10 | simpr 486 | . . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋)) → 𝐷 ∈ (Met‘𝑋)) | |
11 | 9, 10 | syl6bi 253 | . . . . . 6 ⊢ (𝑁:𝑋⟶ℝ → (𝑇 ∈ NrmGrp → 𝐷 ∈ (Met‘𝑋))) |
12 | 11 | com12 32 | . . . . 5 ⊢ (𝑇 ∈ NrmGrp → (𝑁:𝑋⟶ℝ → 𝐷 ∈ (Met‘𝑋))) |
13 | 12 | adantr 482 | . . . 4 ⊢ ((𝑇 ∈ NrmGrp ∧ (norm‘𝑇) = (norm‘𝐺)) → (𝑁:𝑋⟶ℝ → 𝐷 ∈ (Met‘𝑋))) |
14 | 7, 13 | syl 17 | . . 3 ⊢ (𝐺 ∈ NrmGrp → (𝑁:𝑋⟶ℝ → 𝐷 ∈ (Met‘𝑋))) |
15 | metf 23635 | . . 3 ⊢ (𝐷 ∈ (Met‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ) | |
16 | 14, 15 | syl6 35 | . 2 ⊢ (𝐺 ∈ NrmGrp → (𝑁:𝑋⟶ℝ → 𝐷:(𝑋 × 𝑋)⟶ℝ)) |
17 | 3, 16 | mpd 15 | 1 ⊢ (𝐺 ∈ NrmGrp → 𝐷:(𝑋 × 𝑋)⟶ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 × cxp 5630 ⟶wf 6490 ‘cfv 6494 (class class class)co 7352 ℝcr 11009 Basecbs 17043 distcds 17102 Grpcgrp 18708 Metcmet 20735 normcnm 23884 NrmGrpcngp 23885 toNrmGrp ctng 23886 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2709 ax-rep 5241 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7665 ax-cnex 11066 ax-resscn 11067 ax-1cn 11068 ax-icn 11069 ax-addcl 11070 ax-addrcl 11071 ax-mulcl 11072 ax-mulrcl 11073 ax-mulcom 11074 ax-addass 11075 ax-mulass 11076 ax-distr 11077 ax-i2m1 11078 ax-1ne0 11079 ax-1rid 11080 ax-rnegex 11081 ax-rrecex 11082 ax-cnre 11083 ax-pre-lttri 11084 ax-pre-lttrn 11085 ax-pre-ltadd 11086 ax-pre-mulgt0 11087 ax-pre-sup 11088 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-pss 3928 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-iun 4955 df-br 5105 df-opab 5167 df-mpt 5188 df-tr 5222 df-id 5530 df-eprel 5536 df-po 5544 df-so 5545 df-fr 5587 df-we 5589 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6252 df-ord 6319 df-on 6320 df-lim 6321 df-suc 6322 df-iota 6446 df-fun 6496 df-fn 6497 df-f 6498 df-f1 6499 df-fo 6500 df-f1o 6501 df-fv 6502 df-riota 7308 df-ov 7355 df-oprab 7356 df-mpo 7357 df-om 7796 df-1st 7914 df-2nd 7915 df-frecs 8205 df-wrecs 8236 df-recs 8310 df-rdg 8349 df-er 8607 df-map 8726 df-en 8843 df-dom 8844 df-sdom 8845 df-sup 9337 df-inf 9338 df-pnf 11150 df-mnf 11151 df-xr 11152 df-ltxr 11153 df-le 11154 df-sub 11346 df-neg 11347 df-div 11772 df-nn 12113 df-2 12175 df-3 12176 df-4 12177 df-5 12178 df-6 12179 df-7 12180 df-8 12181 df-9 12182 df-n0 12373 df-z 12459 df-dec 12578 df-uz 12723 df-q 12829 df-rp 12871 df-xneg 12988 df-xadd 12989 df-xmul 12990 df-sets 16996 df-slot 17014 df-ndx 17026 df-base 17044 df-plusg 17106 df-tset 17112 df-ds 17115 df-rest 17264 df-topn 17265 df-0g 17283 df-topgen 17285 df-mgm 18457 df-sgrp 18506 df-mnd 18517 df-grp 18711 df-minusg 18712 df-sbg 18713 df-psmet 20741 df-xmet 20742 df-met 20743 df-bl 20744 df-mopn 20745 df-top 22195 df-topon 22212 df-topsp 22234 df-bases 22248 df-xms 23625 df-ms 23626 df-nm 23890 df-ngp 23891 df-tng 23892 |
This theorem is referenced by: (None) |
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