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Mirrors > Home > MPE Home > Th. List > ngpds2 | 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 |
---|---|
ngpds2 | ⊢ ((𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷𝐵) = ((𝐴 − 𝐵)𝐷 0 )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2736 | . . 3 ⊢ (norm‘𝐺) = (norm‘𝐺) | |
2 | ngpds2.x | . . 3 ⊢ 𝑋 = (Base‘𝐺) | |
3 | ngpds2.m | . . 3 ⊢ − = (-g‘𝐺) | |
4 | ngpds2.d | . . 3 ⊢ 𝐷 = (dist‘𝐺) | |
5 | 1, 2, 3, 4 | ngpds 23944 | . 2 ⊢ ((𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷𝐵) = ((norm‘𝐺)‘(𝐴 − 𝐵))) |
6 | ngpgrp 23939 | . . . 4 ⊢ (𝐺 ∈ NrmGrp → 𝐺 ∈ Grp) | |
7 | 2, 3 | grpsubcl 18818 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴 − 𝐵) ∈ 𝑋) |
8 | 6, 7 | syl3an1 1163 | . . 3 ⊢ ((𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴 − 𝐵) ∈ 𝑋) |
9 | ngpds2.z | . . . 4 ⊢ 0 = (0g‘𝐺) | |
10 | 1, 2, 9, 4 | nmval 23929 | . . 3 ⊢ ((𝐴 − 𝐵) ∈ 𝑋 → ((norm‘𝐺)‘(𝐴 − 𝐵)) = ((𝐴 − 𝐵)𝐷 0 )) |
11 | 8, 10 | syl 17 | . 2 ⊢ ((𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((norm‘𝐺)‘(𝐴 − 𝐵)) = ((𝐴 − 𝐵)𝐷 0 )) |
12 | 5, 11 | eqtrd 2776 | 1 ⊢ ((𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷𝐵) = ((𝐴 − 𝐵)𝐷 0 )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ‘cfv 6493 (class class class)co 7353 Basecbs 17075 distcds 17134 0gc0g 17313 Grpcgrp 18740 -gcsg 18742 normcnm 23916 NrmGrpcngp 23917 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7668 ax-cnex 11103 ax-resscn 11104 ax-1cn 11105 ax-icn 11106 ax-addcl 11107 ax-addrcl 11108 ax-mulcl 11109 ax-mulrcl 11110 ax-mulcom 11111 ax-addass 11112 ax-mulass 11113 ax-distr 11114 ax-i2m1 11115 ax-1ne0 11116 ax-1rid 11117 ax-rnegex 11118 ax-rrecex 11119 ax-cnre 11120 ax-pre-lttri 11121 ax-pre-lttrn 11122 ax-pre-ltadd 11123 ax-pre-mulgt0 11124 ax-pre-sup 11125 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7309 df-ov 7356 df-oprab 7357 df-mpo 7358 df-om 7799 df-1st 7917 df-2nd 7918 df-frecs 8208 df-wrecs 8239 df-recs 8313 df-rdg 8352 df-er 8644 df-map 8763 df-en 8880 df-dom 8881 df-sdom 8882 df-sup 9374 df-inf 9375 df-pnf 11187 df-mnf 11188 df-xr 11189 df-ltxr 11190 df-le 11191 df-sub 11383 df-neg 11384 df-div 11809 df-nn 12150 df-2 12212 df-n0 12410 df-z 12496 df-uz 12760 df-q 12866 df-rp 12908 df-xneg 13025 df-xadd 13026 df-xmul 13027 df-0g 17315 df-topgen 17317 df-mgm 18489 df-sgrp 18538 df-mnd 18549 df-grp 18743 df-minusg 18744 df-sbg 18745 df-psmet 20773 df-xmet 20774 df-met 20775 df-bl 20776 df-mopn 20777 df-top 22227 df-topon 22244 df-topsp 22266 df-bases 22280 df-xms 23657 df-ms 23658 df-nm 23922 df-ngp 23923 |
This theorem is referenced by: ngpds2r 23947 ngpds3 23948 |
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