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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 24459 | . . . 4 β’ (πΊ β NrmGrp β πΊ β Grp) | |
| 2 | ngprcan.x | . . . . 5 β’ π = (BaseβπΊ) | |
| 3 | ngprcan.p | . . . . 5 β’ + = (+gβπΊ) | |
| 4 | eqid 2726 | . . . . 5 β’ (-gβπΊ) = (-gβπΊ) | |
| 5 | 2, 3, 4 | grppnpcan2 18960 | . . . 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 579 | . . 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 6888 | . 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 482 | . . 3 β’ ((πΊ β NrmGrp β§ (π΄ β π β§ π΅ β π β§ πΆ β π)) β πΊ β NrmGrp) | |
| 9 | 1 | adantr 480 | . . . 4 β’ ((πΊ β NrmGrp β§ (π΄ β π β§ π΅ β π β§ πΆ β π)) β πΊ β Grp) |
| 10 | simpr1 1191 | . . . 4 β’ ((πΊ β NrmGrp β§ (π΄ β π β§ π΅ β π β§ πΆ β π)) β π΄ β π) | |
| 11 | simpr3 1193 | . . . 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 18869 | . . . 4 β’ ((πΊ β Grp β§ π΄ β π β§ πΆ β π) β (π΄ + πΆ) β π) |
| 13 | 9, 10, 11, 12 | syl3anc 1368 | . . 3 β’ ((πΊ β NrmGrp β§ (π΄ β π β§ π΅ β π β§ πΆ β π)) β (π΄ + πΆ) β π) |
| 14 | simpr2 1192 | . . . 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 18869 | . . . 4 β’ ((πΊ β Grp β§ π΅ β π β§ πΆ β π) β (π΅ + πΆ) β π) |
| 16 | 9, 14, 11, 15 | syl3anc 1368 | . . 3 β’ ((πΊ β NrmGrp β§ (π΄ β π β§ π΅ β π β§ πΆ β π)) β (π΅ + πΆ) β π) |
| 17 | eqid 2726 | . . . 4 β’ (normβπΊ) = (normβπΊ) | |
| 18 | ngprcan.d | . . . 4 β’ π· = (distβπΊ) | |
| 19 | 17, 2, 4, 18 | ngpds 24464 | . . 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 1368 | . 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 24464 | . . 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 1368 | . 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 2776 | 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 395 β§ w3a 1084 = wceq 1533 β wcel 2098 βcfv 6536 (class class class)co 7404 Basecbs 17151 +gcplusg 17204 distcds 17213 Grpcgrp 18861 -gcsg 18863 normcnm 24436 NrmGrpcngp 24437 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 ax-pre-sup 11187 |
| This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-er 8702 df-map 8821 df-en 8939 df-dom 8940 df-sdom 8941 df-sup 9436 df-inf 9437 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-div 11873 df-nn 12214 df-2 12276 df-n0 12474 df-z 12560 df-uz 12824 df-q 12934 df-rp 12978 df-xneg 13095 df-xadd 13096 df-xmul 13097 df-0g 17394 df-topgen 17396 df-mgm 18571 df-sgrp 18650 df-mnd 18666 df-grp 18864 df-minusg 18865 df-sbg 18866 df-psmet 21228 df-xmet 21229 df-met 21230 df-bl 21231 df-mopn 21232 df-top 22747 df-topon 22764 df-topsp 22786 df-bases 22800 df-xms 24177 df-ms 24178 df-nm 24442 df-ngp 24443 |
| This theorem is referenced by: ngplcan 24471 isngp4 24472 ngpsubcan 24474 |
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