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Mirrors > Home > MPE Home > Th. List > ngpds3 | Structured version Visualization version GIF version |
Description: Write the distance between two points in terms of distance from zero. (Contributed by Mario Carneiro, 2-Oct-2015.) |
Ref | Expression |
---|---|
ngpds2.x | ⊢ 𝑋 = (Base‘𝐺) |
ngpds2.z | ⊢ 0 = (0g‘𝐺) |
ngpds2.m | ⊢ − = (-g‘𝐺) |
ngpds2.d | ⊢ 𝐷 = (dist‘𝐺) |
Ref | Expression |
---|---|
ngpds3 | ⊢ ((𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷𝐵) = ( 0 𝐷(𝐴 − 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ngpds2.x | . . 3 ⊢ 𝑋 = (Base‘𝐺) | |
2 | ngpds2.z | . . 3 ⊢ 0 = (0g‘𝐺) | |
3 | ngpds2.m | . . 3 ⊢ − = (-g‘𝐺) | |
4 | ngpds2.d | . . 3 ⊢ 𝐷 = (dist‘𝐺) | |
5 | 1, 2, 3, 4 | ngpds2 24084 | . 2 ⊢ ((𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷𝐵) = ((𝐴 − 𝐵)𝐷 0 )) |
6 | ngpxms 24079 | . . . 4 ⊢ (𝐺 ∈ NrmGrp → 𝐺 ∈ ∞MetSp) | |
7 | 6 | 3ad2ant1 1134 | . . 3 ⊢ ((𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → 𝐺 ∈ ∞MetSp) |
8 | ngpgrp 24077 | . . . 4 ⊢ (𝐺 ∈ NrmGrp → 𝐺 ∈ Grp) | |
9 | 1, 3 | grpsubcl 18890 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴 − 𝐵) ∈ 𝑋) |
10 | 8, 9 | syl3an1 1164 | . . 3 ⊢ ((𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴 − 𝐵) ∈ 𝑋) |
11 | 8 | 3ad2ant1 1134 | . . . 4 ⊢ ((𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → 𝐺 ∈ Grp) |
12 | 1, 2 | grpidcl 18837 | . . . 4 ⊢ (𝐺 ∈ Grp → 0 ∈ 𝑋) |
13 | 11, 12 | syl 17 | . . 3 ⊢ ((𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → 0 ∈ 𝑋) |
14 | 1, 4 | xmssym 23940 | . . 3 ⊢ ((𝐺 ∈ ∞MetSp ∧ (𝐴 − 𝐵) ∈ 𝑋 ∧ 0 ∈ 𝑋) → ((𝐴 − 𝐵)𝐷 0 ) = ( 0 𝐷(𝐴 − 𝐵))) |
15 | 7, 10, 13, 14 | syl3anc 1372 | . 2 ⊢ ((𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴 − 𝐵)𝐷 0 ) = ( 0 𝐷(𝐴 − 𝐵))) |
16 | 5, 15 | eqtrd 2773 | 1 ⊢ ((𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷𝐵) = ( 0 𝐷(𝐴 − 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ‘cfv 6535 (class class class)co 7396 Basecbs 17131 distcds 17193 0gc0g 17372 Grpcgrp 18806 -gcsg 18808 ∞MetSpcxms 23792 NrmGrpcngp 24055 |
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 2704 ax-sep 5295 ax-nul 5302 ax-pow 5359 ax-pr 5423 ax-un 7712 ax-cnex 11153 ax-resscn 11154 ax-1cn 11155 ax-icn 11156 ax-addcl 11157 ax-addrcl 11158 ax-mulcl 11159 ax-mulrcl 11160 ax-mulcom 11161 ax-addass 11162 ax-mulass 11163 ax-distr 11164 ax-i2m1 11165 ax-1ne0 11166 ax-1rid 11167 ax-rnegex 11168 ax-rrecex 11169 ax-cnre 11170 ax-pre-lttri 11171 ax-pre-lttrn 11172 ax-pre-ltadd 11173 ax-pre-mulgt0 11174 ax-pre-sup 11175 |
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 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3965 df-nul 4321 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4905 df-iun 4995 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6292 df-ord 6359 df-on 6360 df-lim 6361 df-suc 6362 df-iota 6487 df-fun 6537 df-fn 6538 df-f 6539 df-f1 6540 df-fo 6541 df-f1o 6542 df-fv 6543 df-riota 7352 df-ov 7399 df-oprab 7400 df-mpo 7401 df-om 7843 df-1st 7962 df-2nd 7963 df-frecs 8253 df-wrecs 8284 df-recs 8358 df-rdg 8397 df-er 8691 df-map 8810 df-en 8928 df-dom 8929 df-sdom 8930 df-sup 9424 df-inf 9425 df-pnf 11237 df-mnf 11238 df-xr 11239 df-ltxr 11240 df-le 11241 df-sub 11433 df-neg 11434 df-div 11859 df-nn 12200 df-2 12262 df-n0 12460 df-z 12546 df-uz 12810 df-q 12920 df-rp 12962 df-xneg 13079 df-xadd 13080 df-xmul 13081 df-0g 17374 df-topgen 17376 df-mgm 18548 df-sgrp 18597 df-mnd 18613 df-grp 18809 df-minusg 18810 df-sbg 18811 df-psmet 20910 df-xmet 20911 df-met 20912 df-bl 20913 df-mopn 20914 df-top 22365 df-topon 22382 df-topsp 22404 df-bases 22418 df-xms 23795 df-ms 23796 df-nm 24060 df-ngp 24061 |
This theorem is referenced by: ngpds3r 24087 |
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