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| Mirrors > Home > MPE Home > Th. List > ngplcan | Structured version Visualization version GIF version | ||
| Description: Cancel left 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 |
|---|---|
| ngplcan | β’ (((πΊ β NrmGrp β§ πΊ β Abel) β§ (π΄ β π β§ π΅ β π β§ πΆ β π)) β ((πΆ + π΄)π·(πΆ + π΅)) = (π΄π·π΅)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | simplr 768 | . . . 4 β’ (((πΊ β NrmGrp β§ πΊ β Abel) β§ (π΄ β π β§ π΅ β π β§ πΆ β π)) β πΊ β Abel) | |
| 2 | simpr3 1194 | . . . 4 β’ (((πΊ β NrmGrp β§ πΊ β Abel) β§ (π΄ β π β§ π΅ β π β§ πΆ β π)) β πΆ β π) | |
| 3 | simpr1 1192 | . . . 4 β’ (((πΊ β NrmGrp β§ πΊ β Abel) β§ (π΄ β π β§ π΅ β π β§ πΆ β π)) β π΄ β π) | |
| 4 | ngprcan.x | . . . . 5 β’ π = (BaseβπΊ) | |
| 5 | ngprcan.p | . . . . 5 β’ + = (+gβπΊ) | |
| 6 | 4, 5 | ablcom 19753 | . . . 4 β’ ((πΊ β Abel β§ πΆ β π β§ π΄ β π) β (πΆ + π΄) = (π΄ + πΆ)) |
| 7 | 1, 2, 3, 6 | syl3anc 1369 | . . 3 β’ (((πΊ β NrmGrp β§ πΊ β Abel) β§ (π΄ β π β§ π΅ β π β§ πΆ β π)) β (πΆ + π΄) = (π΄ + πΆ)) |
| 8 | simpr2 1193 | . . . 4 β’ (((πΊ β NrmGrp β§ πΊ β Abel) β§ (π΄ β π β§ π΅ β π β§ πΆ β π)) β π΅ β π) | |
| 9 | 4, 5 | ablcom 19753 | . . . 4 β’ ((πΊ β Abel β§ πΆ β π β§ π΅ β π) β (πΆ + π΅) = (π΅ + πΆ)) |
| 10 | 1, 2, 8, 9 | syl3anc 1369 | . . 3 β’ (((πΊ β NrmGrp β§ πΊ β Abel) β§ (π΄ β π β§ π΅ β π β§ πΆ β π)) β (πΆ + π΅) = (π΅ + πΆ)) |
| 11 | 7, 10 | oveq12d 7438 | . 2 β’ (((πΊ β NrmGrp β§ πΊ β Abel) β§ (π΄ β π β§ π΅ β π β§ πΆ β π)) β ((πΆ + π΄)π·(πΆ + π΅)) = ((π΄ + πΆ)π·(π΅ + πΆ))) |
| 12 | ngprcan.d | . . . 4 β’ π· = (distβπΊ) | |
| 13 | 4, 5, 12 | ngprcan 24518 | . . 3 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΅)) |
| 14 | 13 | adantlr 714 | . 2 β’ (((πΊ β NrmGrp β§ πΊ β Abel) β§ (π΄ β π β§ π΅ β π β§ πΆ β π)) β ((π΄ + πΆ)π·(π΅ + πΆ)) = (π΄π·π΅)) |
| 15 | 11, 14 | eqtrd 2768 | 1 β’ (((πΊ β NrmGrp β§ πΊ β Abel) β§ (π΄ β π β§ π΅ β π β§ πΆ β π)) β ((πΆ + π΄)π·(πΆ + π΅)) = (π΄π·π΅)) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ wa 395 β§ w3a 1085 = wceq 1534 β wcel 2099 βcfv 6548 (class class class)co 7420 Basecbs 17179 +gcplusg 17232 distcds 17241 Abelcabl 19735 NrmGrpcngp 24485 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 ax-pre-sup 11216 |
| This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 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 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-1st 7993 df-2nd 7994 df-frecs 8286 df-wrecs 8317 df-recs 8391 df-rdg 8430 df-er 8724 df-map 8846 df-en 8964 df-dom 8965 df-sdom 8966 df-sup 9465 df-inf 9466 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-div 11902 df-nn 12243 df-2 12305 df-n0 12503 df-z 12589 df-uz 12853 df-q 12963 df-rp 13007 df-xneg 13124 df-xadd 13125 df-xmul 13126 df-0g 17422 df-topgen 17424 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-grp 18892 df-minusg 18893 df-sbg 18894 df-cmn 19736 df-abl 19737 df-psmet 21270 df-xmet 21271 df-met 21272 df-bl 21273 df-mopn 21274 df-top 22795 df-topon 22812 df-topsp 22834 df-bases 22848 df-xms 24225 df-ms 24226 df-nm 24490 df-ngp 24491 |
| This theorem is referenced by: ngptgp 24544 |
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