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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 2821 | . . 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 23196 | . 2 ⊢ ((𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷𝐵) = ((norm‘𝐺)‘(𝐴 − 𝐵))) |
6 | ngpgrp 23191 | . . . 4 ⊢ (𝐺 ∈ NrmGrp → 𝐺 ∈ Grp) | |
7 | 2, 3 | grpsubcl 18162 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴 − 𝐵) ∈ 𝑋) |
8 | 6, 7 | syl3an1 1159 | . . 3 ⊢ ((𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴 − 𝐵) ∈ 𝑋) |
9 | ngpds2.z | . . . 4 ⊢ 0 = (0g‘𝐺) | |
10 | 1, 2, 9, 4 | nmval 23182 | . . 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 2856 | 1 ⊢ ((𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷𝐵) = ((𝐴 − 𝐵)𝐷 0 )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ‘cfv 6341 (class class class)co 7142 Basecbs 16466 distcds 16557 0gc0g 16696 Grpcgrp 18086 -gcsg 18088 normcnm 23169 NrmGrpcngp 23170 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5189 ax-nul 5196 ax-pow 5252 ax-pr 5316 ax-un 7447 ax-cnex 10579 ax-resscn 10580 ax-1cn 10581 ax-icn 10582 ax-addcl 10583 ax-addrcl 10584 ax-mulcl 10585 ax-mulrcl 10586 ax-mulcom 10587 ax-addass 10588 ax-mulass 10589 ax-distr 10590 ax-i2m1 10591 ax-1ne0 10592 ax-1rid 10593 ax-rnegex 10594 ax-rrecex 10595 ax-cnre 10596 ax-pre-lttri 10597 ax-pre-lttrn 10598 ax-pre-ltadd 10599 ax-pre-mulgt0 10600 ax-pre-sup 10601 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3488 df-sbc 3764 df-csb 3872 df-dif 3927 df-un 3929 df-in 3931 df-ss 3940 df-pss 3942 df-nul 4280 df-if 4454 df-pw 4527 df-sn 4554 df-pr 4556 df-tp 4558 df-op 4560 df-uni 4825 df-iun 4907 df-br 5053 df-opab 5115 df-mpt 5133 df-tr 5159 df-id 5446 df-eprel 5451 df-po 5460 df-so 5461 df-fr 5500 df-we 5502 df-xp 5547 df-rel 5548 df-cnv 5549 df-co 5550 df-dm 5551 df-rn 5552 df-res 5553 df-ima 5554 df-pred 6134 df-ord 6180 df-on 6181 df-lim 6182 df-suc 6183 df-iota 6300 df-fun 6343 df-fn 6344 df-f 6345 df-f1 6346 df-fo 6347 df-f1o 6348 df-fv 6349 df-riota 7100 df-ov 7145 df-oprab 7146 df-mpo 7147 df-om 7567 df-1st 7675 df-2nd 7676 df-wrecs 7933 df-recs 7994 df-rdg 8032 df-er 8275 df-map 8394 df-en 8496 df-dom 8497 df-sdom 8498 df-sup 8892 df-inf 8893 df-pnf 10663 df-mnf 10664 df-xr 10665 df-ltxr 10666 df-le 10667 df-sub 10858 df-neg 10859 df-div 11284 df-nn 11625 df-2 11687 df-n0 11885 df-z 11969 df-uz 12231 df-q 12336 df-rp 12377 df-xneg 12494 df-xadd 12495 df-xmul 12496 df-0g 16698 df-topgen 16700 df-mgm 17835 df-sgrp 17884 df-mnd 17895 df-grp 18089 df-minusg 18090 df-sbg 18091 df-psmet 20520 df-xmet 20521 df-met 20522 df-bl 20523 df-mopn 20524 df-top 21485 df-topon 21502 df-topsp 21524 df-bases 21537 df-xms 22913 df-ms 22914 df-nm 23175 df-ngp 23176 |
This theorem is referenced by: ngpds2r 23199 ngpds3 23200 |
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