Nearby theorems
|
|
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
| 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 24526 | . . . 4 β’ (πΊ β NrmGrp β πΊ β Grp) | |
| 2 | ngprcan.x | . . . . 5 β’ π = (BaseβπΊ) | |
| 3 | ngprcan.p | . . . . 5 β’ + = (+gβπΊ) | |
| 4 | eqid 2727 | . . . . 5 β’ (-gβπΊ) = (-gβπΊ) | |
| 5 | 2, 3, 4 | grppnpcan2 18995 | . . . 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 578 | . . 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 6904 | . 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 481 | . . 3 β’ ((πΊ β NrmGrp β§ (π΄ β π β§ π΅ β π β§ πΆ β π)) β πΊ β NrmGrp) | |
| 9 | 1 | adantr 479 | . . . 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 18903 | . . . 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 18903 | . . . 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 2727 | . . . 4 β’ (normβπΊ) = (normβπΊ) | |
| 18 | ngprcan.d | . . . 4 β’ π· = (distβπΊ) | |
| 19 | 17, 2, 4, 18 | ngpds 24531 | . . 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 24531 | . . 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 2777 | 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 394 β§ w3a 1084 = wceq 1533 β wcel 2098 βcfv 6551 (class class class)co 7424 Basecbs 17185 +gcplusg 17238 distcds 17247 Grpcgrp 18895 -gcsg 18897 normcnm 24503 NrmGrpcngp 24504 |
| 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 2166 ax-ext 2698 ax-sep 5301 ax-nul 5308 ax-pow 5367 ax-pr 5431 ax-un 7744 ax-cnex 11200 ax-resscn 11201 ax-1cn 11202 ax-icn 11203 ax-addcl 11204 ax-addrcl 11205 ax-mulcl 11206 ax-mulrcl 11207 ax-mulcom 11208 ax-addass 11209 ax-mulass 11210 ax-distr 11211 ax-i2m1 11212 ax-1ne0 11213 ax-1rid 11214 ax-rnegex 11215 ax-rrecex 11216 ax-cnre 11217 ax-pre-lttri 11218 ax-pre-lttrn 11219 ax-pre-ltadd 11220 ax-pre-mulgt0 11221 ax-pre-sup 11222 |
| This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4325 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4911 df-iun 5000 df-br 5151 df-opab 5213 df-mpt 5234 df-tr 5268 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5635 df-we 5637 df-xp 5686 df-rel 5687 df-cnv 5688 df-co 5689 df-dm 5690 df-rn 5691 df-res 5692 df-ima 5693 df-pred 6308 df-ord 6375 df-on 6376 df-lim 6377 df-suc 6378 df-iota 6503 df-fun 6553 df-fn 6554 df-f 6555 df-f1 6556 df-fo 6557 df-f1o 6558 df-fv 6559 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7875 df-1st 7997 df-2nd 7998 df-frecs 8291 df-wrecs 8322 df-recs 8396 df-rdg 8435 df-er 8729 df-map 8851 df-en 8969 df-dom 8970 df-sdom 8971 df-sup 9471 df-inf 9472 df-pnf 11286 df-mnf 11287 df-xr 11288 df-ltxr 11289 df-le 11290 df-sub 11482 df-neg 11483 df-div 11908 df-nn 12249 df-2 12311 df-n0 12509 df-z 12595 df-uz 12859 df-q 12969 df-rp 13013 df-xneg 13130 df-xadd 13131 df-xmul 13132 df-0g 17428 df-topgen 17430 df-mgm 18605 df-sgrp 18684 df-mnd 18700 df-grp 18898 df-minusg 18899 df-sbg 18900 df-psmet 21276 df-xmet 21277 df-met 21278 df-bl 21279 df-mopn 21280 df-top 22814 df-topon 22831 df-topsp 22853 df-bases 22867 df-xms 24244 df-ms 24245 df-nm 24509 df-ngp 24510 |
| This theorem is referenced by: ngplcan 24538 isngp4 24539 ngpsubcan 24541 |
| Copyright terms: Public domain | W3C validator |