MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  grprcan Structured version   Visualization version   GIF version

Theorem grprcan 17376
Description: Right cancellation law for groups. (Contributed by NM, 24-Aug-2011.) (Proof shortened by Mario Carneiro, 6-Jan-2015.)
Hypotheses
Ref Expression
grprcan.b 𝐵 = (Base‘𝐺)
grprcan.p + = (+g𝐺)
Assertion
Ref Expression
grprcan ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 + 𝑍) = (𝑌 + 𝑍) ↔ 𝑋 = 𝑌))

Proof of Theorem grprcan
Dummy variables 𝑣 𝑢 𝑤 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 grprcan.b . . . . 5 𝐵 = (Base‘𝐺)
2 grprcan.p . . . . 5 + = (+g𝐺)
3 eqid 2621 . . . . 5 (0g𝐺) = (0g𝐺)
41, 2, 3grpinvex 17353 . . . 4 ((𝐺 ∈ Grp ∧ 𝑍𝐵) → ∃𝑦𝐵 (𝑦 + 𝑍) = (0g𝐺))
543ad2antr3 1226 . . 3 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ∃𝑦𝐵 (𝑦 + 𝑍) = (0g𝐺))
6 simprr 795 . . . . . . . 8 (((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ ((𝑦𝐵 ∧ (𝑦 + 𝑍) = (0g𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) → (𝑋 + 𝑍) = (𝑌 + 𝑍))
76oveq1d 6619 . . . . . . 7 (((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ ((𝑦𝐵 ∧ (𝑦 + 𝑍) = (0g𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) → ((𝑋 + 𝑍) + 𝑦) = ((𝑌 + 𝑍) + 𝑦))
8 simpll 789 . . . . . . . . 9 (((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ ((𝑦𝐵 ∧ (𝑦 + 𝑍) = (0g𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) → 𝐺 ∈ Grp)
91, 2grpass 17352 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ (𝑢𝐵𝑣𝐵𝑤𝐵)) → ((𝑢 + 𝑣) + 𝑤) = (𝑢 + (𝑣 + 𝑤)))
108, 9sylan 488 . . . . . . . 8 ((((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ ((𝑦𝐵 ∧ (𝑦 + 𝑍) = (0g𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) ∧ (𝑢𝐵𝑣𝐵𝑤𝐵)) → ((𝑢 + 𝑣) + 𝑤) = (𝑢 + (𝑣 + 𝑤)))
11 simplr1 1101 . . . . . . . 8 (((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ ((𝑦𝐵 ∧ (𝑦 + 𝑍) = (0g𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) → 𝑋𝐵)
12 simplr3 1103 . . . . . . . 8 (((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ ((𝑦𝐵 ∧ (𝑦 + 𝑍) = (0g𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) → 𝑍𝐵)
13 simprll 801 . . . . . . . 8 (((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ ((𝑦𝐵 ∧ (𝑦 + 𝑍) = (0g𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) → 𝑦𝐵)
1410, 11, 12, 13caovassd 6786 . . . . . . 7 (((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ ((𝑦𝐵 ∧ (𝑦 + 𝑍) = (0g𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) → ((𝑋 + 𝑍) + 𝑦) = (𝑋 + (𝑍 + 𝑦)))
15 simplr2 1102 . . . . . . . 8 (((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ ((𝑦𝐵 ∧ (𝑦 + 𝑍) = (0g𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) → 𝑌𝐵)
1610, 15, 12, 13caovassd 6786 . . . . . . 7 (((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ ((𝑦𝐵 ∧ (𝑦 + 𝑍) = (0g𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) → ((𝑌 + 𝑍) + 𝑦) = (𝑌 + (𝑍 + 𝑦)))
177, 14, 163eqtr3d 2663 . . . . . 6 (((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ ((𝑦𝐵 ∧ (𝑦 + 𝑍) = (0g𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) → (𝑋 + (𝑍 + 𝑦)) = (𝑌 + (𝑍 + 𝑦)))
181, 2grpcl 17351 . . . . . . . . . 10 ((𝐺 ∈ Grp ∧ 𝑢𝐵𝑣𝐵) → (𝑢 + 𝑣) ∈ 𝐵)
198, 18syl3an1 1356 . . . . . . . . 9 ((((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ ((𝑦𝐵 ∧ (𝑦 + 𝑍) = (0g𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) ∧ 𝑢𝐵𝑣𝐵) → (𝑢 + 𝑣) ∈ 𝐵)
201, 3grpidcl 17371 . . . . . . . . . 10 (𝐺 ∈ Grp → (0g𝐺) ∈ 𝐵)
218, 20syl 17 . . . . . . . . 9 (((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ ((𝑦𝐵 ∧ (𝑦 + 𝑍) = (0g𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) → (0g𝐺) ∈ 𝐵)
221, 2, 3grplid 17373 . . . . . . . . . 10 ((𝐺 ∈ Grp ∧ 𝑢𝐵) → ((0g𝐺) + 𝑢) = 𝑢)
238, 22sylan 488 . . . . . . . . 9 ((((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ ((𝑦𝐵 ∧ (𝑦 + 𝑍) = (0g𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) ∧ 𝑢𝐵) → ((0g𝐺) + 𝑢) = 𝑢)
241, 2, 3grpinvex 17353 . . . . . . . . . 10 ((𝐺 ∈ Grp ∧ 𝑢𝐵) → ∃𝑣𝐵 (𝑣 + 𝑢) = (0g𝐺))
258, 24sylan 488 . . . . . . . . 9 ((((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ ((𝑦𝐵 ∧ (𝑦 + 𝑍) = (0g𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) ∧ 𝑢𝐵) → ∃𝑣𝐵 (𝑣 + 𝑢) = (0g𝐺))
26 simpr 477 . . . . . . . . 9 ((((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ ((𝑦𝐵 ∧ (𝑦 + 𝑍) = (0g𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) ∧ 𝑍𝐵) → 𝑍𝐵)
2713adantr 481 . . . . . . . . 9 ((((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ ((𝑦𝐵 ∧ (𝑦 + 𝑍) = (0g𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) ∧ 𝑍𝐵) → 𝑦𝐵)
28 simprlr 802 . . . . . . . . . 10 (((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ ((𝑦𝐵 ∧ (𝑦 + 𝑍) = (0g𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) → (𝑦 + 𝑍) = (0g𝐺))
2928adantr 481 . . . . . . . . 9 ((((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ ((𝑦𝐵 ∧ (𝑦 + 𝑍) = (0g𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) ∧ 𝑍𝐵) → (𝑦 + 𝑍) = (0g𝐺))
3019, 21, 23, 10, 25, 26, 27, 29grprinvd 6826 . . . . . . . 8 ((((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ ((𝑦𝐵 ∧ (𝑦 + 𝑍) = (0g𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) ∧ 𝑍𝐵) → (𝑍 + 𝑦) = (0g𝐺))
3112, 30mpdan 701 . . . . . . 7 (((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ ((𝑦𝐵 ∧ (𝑦 + 𝑍) = (0g𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) → (𝑍 + 𝑦) = (0g𝐺))
3231oveq2d 6620 . . . . . 6 (((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ ((𝑦𝐵 ∧ (𝑦 + 𝑍) = (0g𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) → (𝑋 + (𝑍 + 𝑦)) = (𝑋 + (0g𝐺)))
3331oveq2d 6620 . . . . . 6 (((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ ((𝑦𝐵 ∧ (𝑦 + 𝑍) = (0g𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) → (𝑌 + (𝑍 + 𝑦)) = (𝑌 + (0g𝐺)))
3417, 32, 333eqtr3d 2663 . . . . 5 (((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ ((𝑦𝐵 ∧ (𝑦 + 𝑍) = (0g𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) → (𝑋 + (0g𝐺)) = (𝑌 + (0g𝐺)))
351, 2, 3grprid 17374 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑋 + (0g𝐺)) = 𝑋)
368, 11, 35syl2anc 692 . . . . 5 (((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ ((𝑦𝐵 ∧ (𝑦 + 𝑍) = (0g𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) → (𝑋 + (0g𝐺)) = 𝑋)
371, 2, 3grprid 17374 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑌𝐵) → (𝑌 + (0g𝐺)) = 𝑌)
388, 15, 37syl2anc 692 . . . . 5 (((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ ((𝑦𝐵 ∧ (𝑦 + 𝑍) = (0g𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) → (𝑌 + (0g𝐺)) = 𝑌)
3934, 36, 383eqtr3d 2663 . . . 4 (((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ ((𝑦𝐵 ∧ (𝑦 + 𝑍) = (0g𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) → 𝑋 = 𝑌)
4039expr 642 . . 3 (((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝑦𝐵 ∧ (𝑦 + 𝑍) = (0g𝐺))) → ((𝑋 + 𝑍) = (𝑌 + 𝑍) → 𝑋 = 𝑌))
415, 40rexlimddv 3028 . 2 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 + 𝑍) = (𝑌 + 𝑍) → 𝑋 = 𝑌))
42 oveq1 6611 . 2 (𝑋 = 𝑌 → (𝑋 + 𝑍) = (𝑌 + 𝑍))
4341, 42impbid1 215 1 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 + 𝑍) = (𝑌 + 𝑍) ↔ 𝑋 = 𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wrex 2908  cfv 5847  (class class class)co 6604  Basecbs 15781  +gcplusg 15862  0gc0g 16021  Grpcgrp 17343
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-iota 5810  df-fun 5849  df-fv 5855  df-riota 6565  df-ov 6607  df-0g 16023  df-mgm 17163  df-sgrp 17205  df-mnd 17216  df-grp 17346
This theorem is referenced by:  grpinveu  17377  grpid  17378  grpidlcan  17402  grpinvssd  17413  grpsubrcan  17417  grpsubadd  17424  sylow1lem4  17937  ringcom  18500  ringrz  18509  lmodcom  18830  ogrpaddlt  29503  rhmunitinv  29607  isnumbasgrplem2  37155
  Copyright terms: Public domain W3C validator