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Mirrors > Home > MPE Home > Th. List > ngprcan | Structured version Visualization version GIF version |
Description: Cancel right addition inside a distance calculation. (Contributed by Mario Carneiro, 2-Oct-2015.) |
Ref | Expression |
---|---|
ngprcan.x | β’ π = (BaseβπΊ) |
ngprcan.p | β’ + = (+gβπΊ) |
ngprcan.d | β’ π· = (distβπΊ) |
Ref | Expression |
---|---|
ngprcan | β’ ((πΊ β NrmGrp β§ (π΄ β π β§ π΅ β π β§ πΆ β π)) β ((π΄ + πΆ)π·(π΅ + πΆ)) = (π΄π·π΅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ngpgrp 23978 | . . . 4 β’ (πΊ β NrmGrp β πΊ β Grp) | |
2 | ngprcan.x | . . . . 5 β’ π = (BaseβπΊ) | |
3 | ngprcan.p | . . . . 5 β’ + = (+gβπΊ) | |
4 | eqid 2733 | . . . . 5 β’ (-gβπΊ) = (-gβπΊ) | |
5 | 2, 3, 4 | grppnpcan2 18849 | . . . 4 β’ ((πΊ β Grp β§ (π΄ β π β§ π΅ β π β§ πΆ β π)) β ((π΄ + πΆ)(-gβπΊ)(π΅ + πΆ)) = (π΄(-gβπΊ)π΅)) |
6 | 1, 5 | sylan 581 | . . 3 β’ ((πΊ β NrmGrp β§ (π΄ β π β§ π΅ β π β§ πΆ β π)) β ((π΄ + πΆ)(-gβπΊ)(π΅ + πΆ)) = (π΄(-gβπΊ)π΅)) |
7 | 6 | fveq2d 6850 | . 2 β’ ((πΊ β NrmGrp β§ (π΄ β π β§ π΅ β π β§ πΆ β π)) β ((normβπΊ)β((π΄ + πΆ)(-gβπΊ)(π΅ + πΆ))) = ((normβπΊ)β(π΄(-gβπΊ)π΅))) |
8 | simpl 484 | . . 3 β’ ((πΊ β NrmGrp β§ (π΄ β π β§ π΅ β π β§ πΆ β π)) β πΊ β NrmGrp) | |
9 | 1 | adantr 482 | . . . 4 β’ ((πΊ β NrmGrp β§ (π΄ β π β§ π΅ β π β§ πΆ β π)) β πΊ β Grp) |
10 | simpr1 1195 | . . . 4 β’ ((πΊ β NrmGrp β§ (π΄ β π β§ π΅ β π β§ πΆ β π)) β π΄ β π) | |
11 | simpr3 1197 | . . . 4 β’ ((πΊ β NrmGrp β§ (π΄ β π β§ π΅ β π β§ πΆ β π)) β πΆ β π) | |
12 | 2, 3 | grpcl 18764 | . . . 4 β’ ((πΊ β Grp β§ π΄ β π β§ πΆ β π) β (π΄ + πΆ) β π) |
13 | 9, 10, 11, 12 | syl3anc 1372 | . . 3 β’ ((πΊ β NrmGrp β§ (π΄ β π β§ π΅ β π β§ πΆ β π)) β (π΄ + πΆ) β π) |
14 | simpr2 1196 | . . . 4 β’ ((πΊ β NrmGrp β§ (π΄ β π β§ π΅ β π β§ πΆ β π)) β π΅ β π) | |
15 | 2, 3 | grpcl 18764 | . . . 4 β’ ((πΊ β Grp β§ π΅ β π β§ πΆ β π) β (π΅ + πΆ) β π) |
16 | 9, 14, 11, 15 | syl3anc 1372 | . . 3 β’ ((πΊ β NrmGrp β§ (π΄ β π β§ π΅ β π β§ πΆ β π)) β (π΅ + πΆ) β π) |
17 | eqid 2733 | . . . 4 β’ (normβπΊ) = (normβπΊ) | |
18 | ngprcan.d | . . . 4 β’ π· = (distβπΊ) | |
19 | 17, 2, 4, 18 | ngpds 23983 | . . 3 β’ ((πΊ β NrmGrp β§ (π΄ + πΆ) β π β§ (π΅ + πΆ) β π) β ((π΄ + πΆ)π·(π΅ + πΆ)) = ((normβπΊ)β((π΄ + πΆ)(-gβπΊ)(π΅ + πΆ)))) |
20 | 8, 13, 16, 19 | syl3anc 1372 | . 2 β’ ((πΊ β NrmGrp β§ (π΄ β π β§ π΅ β π β§ πΆ β π)) β ((π΄ + πΆ)π·(π΅ + πΆ)) = ((normβπΊ)β((π΄ + πΆ)(-gβπΊ)(π΅ + πΆ)))) |
21 | 17, 2, 4, 18 | ngpds 23983 | . . 3 β’ ((πΊ β NrmGrp β§ π΄ β π β§ π΅ β π) β (π΄π·π΅) = ((normβπΊ)β(π΄(-gβπΊ)π΅))) |
22 | 8, 10, 14, 21 | syl3anc 1372 | . 2 β’ ((πΊ β NrmGrp β§ (π΄ β π β§ π΅ β π β§ πΆ β π)) β (π΄π·π΅) = ((normβπΊ)β(π΄(-gβπΊ)π΅))) |
23 | 7, 20, 22 | 3eqtr4d 2783 | 1 β’ ((πΊ β NrmGrp β§ (π΄ β π β§ π΅ β π β§ πΆ β π)) β ((π΄ + πΆ)π·(π΅ + πΆ)) = (π΄π·π΅)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 β§ w3a 1088 = wceq 1542 β wcel 2107 βcfv 6500 (class class class)co 7361 Basecbs 17091 +gcplusg 17141 distcds 17150 Grpcgrp 18756 -gcsg 18758 normcnm 23955 NrmGrpcngp 23956 |
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 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 ax-pre-sup 11137 |
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 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7807 df-1st 7925 df-2nd 7926 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-er 8654 df-map 8773 df-en 8890 df-dom 8891 df-sdom 8892 df-sup 9386 df-inf 9387 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 df-sub 11395 df-neg 11396 df-div 11821 df-nn 12162 df-2 12224 df-n0 12422 df-z 12508 df-uz 12772 df-q 12882 df-rp 12924 df-xneg 13041 df-xadd 13042 df-xmul 13043 df-0g 17331 df-topgen 17333 df-mgm 18505 df-sgrp 18554 df-mnd 18565 df-grp 18759 df-minusg 18760 df-sbg 18761 df-psmet 20811 df-xmet 20812 df-met 20813 df-bl 20814 df-mopn 20815 df-top 22266 df-topon 22283 df-topsp 22305 df-bases 22319 df-xms 23696 df-ms 23697 df-nm 23961 df-ngp 23962 |
This theorem is referenced by: ngplcan 23990 isngp4 23991 ngpsubcan 23993 |
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