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Theorem grprcan 17278
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 2610 . . . . 5 (0g𝐺) = (0g𝐺)
41, 2, 3grpinvex 17255 . . . 4 ((𝐺 ∈ Grp ∧ 𝑍𝐵) → ∃𝑦𝐵 (𝑦 + 𝑍) = (0g𝐺))
543ad2antr3 1221 . . 3 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ∃𝑦𝐵 (𝑦 + 𝑍) = (0g𝐺))
6 simprr 792 . . . . . . . 8 (((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ ((𝑦𝐵 ∧ (𝑦 + 𝑍) = (0g𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) → (𝑋 + 𝑍) = (𝑌 + 𝑍))
76oveq1d 6564 . . . . . . 7 (((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ ((𝑦𝐵 ∧ (𝑦 + 𝑍) = (0g𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) → ((𝑋 + 𝑍) + 𝑦) = ((𝑌 + 𝑍) + 𝑦))
8 simpll 786 . . . . . . . . 9 (((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ ((𝑦𝐵 ∧ (𝑦 + 𝑍) = (0g𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) → 𝐺 ∈ Grp)
91, 2grpass 17254 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ (𝑢𝐵𝑣𝐵𝑤𝐵)) → ((𝑢 + 𝑣) + 𝑤) = (𝑢 + (𝑣 + 𝑤)))
108, 9sylan 487 . . . . . . . 8 ((((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ ((𝑦𝐵 ∧ (𝑦 + 𝑍) = (0g𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) ∧ (𝑢𝐵𝑣𝐵𝑤𝐵)) → ((𝑢 + 𝑣) + 𝑤) = (𝑢 + (𝑣 + 𝑤)))
11 simplr1 1096 . . . . . . . 8 (((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ ((𝑦𝐵 ∧ (𝑦 + 𝑍) = (0g𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) → 𝑋𝐵)
12 simplr3 1098 . . . . . . . 8 (((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ ((𝑦𝐵 ∧ (𝑦 + 𝑍) = (0g𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) → 𝑍𝐵)
13 simprll 798 . . . . . . . 8 (((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ ((𝑦𝐵 ∧ (𝑦 + 𝑍) = (0g𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) → 𝑦𝐵)
1410, 11, 12, 13caovassd 6731 . . . . . . 7 (((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ ((𝑦𝐵 ∧ (𝑦 + 𝑍) = (0g𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) → ((𝑋 + 𝑍) + 𝑦) = (𝑋 + (𝑍 + 𝑦)))
15 simplr2 1097 . . . . . . . 8 (((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ ((𝑦𝐵 ∧ (𝑦 + 𝑍) = (0g𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) → 𝑌𝐵)
1610, 15, 12, 13caovassd 6731 . . . . . . 7 (((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ ((𝑦𝐵 ∧ (𝑦 + 𝑍) = (0g𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) → ((𝑌 + 𝑍) + 𝑦) = (𝑌 + (𝑍 + 𝑦)))
177, 14, 163eqtr3d 2652 . . . . . 6 (((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ ((𝑦𝐵 ∧ (𝑦 + 𝑍) = (0g𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) → (𝑋 + (𝑍 + 𝑦)) = (𝑌 + (𝑍 + 𝑦)))
181, 2grpcl 17253 . . . . . . . . . 10 ((𝐺 ∈ Grp ∧ 𝑢𝐵𝑣𝐵) → (𝑢 + 𝑣) ∈ 𝐵)
198, 18syl3an1 1351 . . . . . . . . 9 ((((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ ((𝑦𝐵 ∧ (𝑦 + 𝑍) = (0g𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) ∧ 𝑢𝐵𝑣𝐵) → (𝑢 + 𝑣) ∈ 𝐵)
201, 3grpidcl 17273 . . . . . . . . . 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 17275 . . . . . . . . . 10 ((𝐺 ∈ Grp ∧ 𝑢𝐵) → ((0g𝐺) + 𝑢) = 𝑢)
238, 22sylan 487 . . . . . . . . 9 ((((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ ((𝑦𝐵 ∧ (𝑦 + 𝑍) = (0g𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) ∧ 𝑢𝐵) → ((0g𝐺) + 𝑢) = 𝑢)
241, 2, 3grpinvex 17255 . . . . . . . . . 10 ((𝐺 ∈ Grp ∧ 𝑢𝐵) → ∃𝑣𝐵 (𝑣 + 𝑢) = (0g𝐺))
258, 24sylan 487 . . . . . . . . 9 ((((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ ((𝑦𝐵 ∧ (𝑦 + 𝑍) = (0g𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) ∧ 𝑢𝐵) → ∃𝑣𝐵 (𝑣 + 𝑢) = (0g𝐺))
26 simpr 476 . . . . . . . . 9 ((((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ ((𝑦𝐵 ∧ (𝑦 + 𝑍) = (0g𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) ∧ 𝑍𝐵) → 𝑍𝐵)
2713adantr 480 . . . . . . . . 9 ((((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ ((𝑦𝐵 ∧ (𝑦 + 𝑍) = (0g𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) ∧ 𝑍𝐵) → 𝑦𝐵)
28 simprlr 799 . . . . . . . . . 10 (((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ ((𝑦𝐵 ∧ (𝑦 + 𝑍) = (0g𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) → (𝑦 + 𝑍) = (0g𝐺))
2928adantr 480 . . . . . . . . 9 ((((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ ((𝑦𝐵 ∧ (𝑦 + 𝑍) = (0g𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) ∧ 𝑍𝐵) → (𝑦 + 𝑍) = (0g𝐺))
3019, 21, 23, 10, 25, 26, 27, 29grprinvd 6771 . . . . . . . 8 ((((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ ((𝑦𝐵 ∧ (𝑦 + 𝑍) = (0g𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) ∧ 𝑍𝐵) → (𝑍 + 𝑦) = (0g𝐺))
3112, 30mpdan 699 . . . . . . 7 (((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ ((𝑦𝐵 ∧ (𝑦 + 𝑍) = (0g𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) → (𝑍 + 𝑦) = (0g𝐺))
3231oveq2d 6565 . . . . . 6 (((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ ((𝑦𝐵 ∧ (𝑦 + 𝑍) = (0g𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) → (𝑋 + (𝑍 + 𝑦)) = (𝑋 + (0g𝐺)))
3331oveq2d 6565 . . . . . 6 (((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ ((𝑦𝐵 ∧ (𝑦 + 𝑍) = (0g𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) → (𝑌 + (𝑍 + 𝑦)) = (𝑌 + (0g𝐺)))
3417, 32, 333eqtr3d 2652 . . . . 5 (((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ ((𝑦𝐵 ∧ (𝑦 + 𝑍) = (0g𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) → (𝑋 + (0g𝐺)) = (𝑌 + (0g𝐺)))
351, 2, 3grprid 17276 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑋 + (0g𝐺)) = 𝑋)
368, 11, 35syl2anc 691 . . . . 5 (((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ ((𝑦𝐵 ∧ (𝑦 + 𝑍) = (0g𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) → (𝑋 + (0g𝐺)) = 𝑋)
371, 2, 3grprid 17276 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑌𝐵) → (𝑌 + (0g𝐺)) = 𝑌)
388, 15, 37syl2anc 691 . . . . 5 (((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ ((𝑦𝐵 ∧ (𝑦 + 𝑍) = (0g𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) → (𝑌 + (0g𝐺)) = 𝑌)
3934, 36, 383eqtr3d 2652 . . . 4 (((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ ((𝑦𝐵 ∧ (𝑦 + 𝑍) = (0g𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) → 𝑋 = 𝑌)
4039expr 641 . . 3 (((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝑦𝐵 ∧ (𝑦 + 𝑍) = (0g𝐺))) → ((𝑋 + 𝑍) = (𝑌 + 𝑍) → 𝑋 = 𝑌))
415, 40rexlimddv 3017 . 2 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 + 𝑍) = (𝑌 + 𝑍) → 𝑋 = 𝑌))
42 oveq1 6556 . 2 (𝑋 = 𝑌 → (𝑋 + 𝑍) = (𝑌 + 𝑍))
4341, 42impbid1 214 1 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 + 𝑍) = (𝑌 + 𝑍) ↔ 𝑋 = 𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383  w3a 1031   = wceq 1475  wcel 1977  wrex 2897  cfv 5804  (class class class)co 6549  Basecbs 15695  +gcplusg 15768  0gc0g 15923  Grpcgrp 17245
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-riota 6511  df-ov 6552  df-0g 15925  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-grp 17248
This theorem is referenced by:  grpinveu  17279  grpid  17280  grpidlcan  17304  grpinvssd  17315  grpsubrcan  17319  grpsubadd  17326  sylow1lem4  17839  ringcom  18402  ringrz  18411  lmodcom  18732  ogrpaddlt  29049  rhmunitinv  29153  isnumbasgrplem2  36693
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