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Theorem grprcan 18709
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 2736 . . . . 5 (0g𝐺) = (0g𝐺)
41, 2, 3grpinvex 18683 . . . 4 ((𝐺 ∈ Grp ∧ 𝑍𝐵) → ∃𝑦𝐵 (𝑦 + 𝑍) = (0g𝐺))
543ad2antr3 1189 . . 3 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ∃𝑦𝐵 (𝑦 + 𝑍) = (0g𝐺))
6 simprr 770 . . . . . . . 8 (((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ ((𝑦𝐵 ∧ (𝑦 + 𝑍) = (0g𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) → (𝑋 + 𝑍) = (𝑌 + 𝑍))
76oveq1d 7352 . . . . . . 7 (((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ ((𝑦𝐵 ∧ (𝑦 + 𝑍) = (0g𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) → ((𝑋 + 𝑍) + 𝑦) = ((𝑌 + 𝑍) + 𝑦))
8 simpll 764 . . . . . . . . 9 (((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ ((𝑦𝐵 ∧ (𝑦 + 𝑍) = (0g𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) → 𝐺 ∈ Grp)
91, 2grpass 18682 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ (𝑢𝐵𝑣𝐵𝑤𝐵)) → ((𝑢 + 𝑣) + 𝑤) = (𝑢 + (𝑣 + 𝑤)))
108, 9sylan 580 . . . . . . . 8 ((((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ ((𝑦𝐵 ∧ (𝑦 + 𝑍) = (0g𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) ∧ (𝑢𝐵𝑣𝐵𝑤𝐵)) → ((𝑢 + 𝑣) + 𝑤) = (𝑢 + (𝑣 + 𝑤)))
11 simplr1 1214 . . . . . . . 8 (((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ ((𝑦𝐵 ∧ (𝑦 + 𝑍) = (0g𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) → 𝑋𝐵)
12 simplr3 1216 . . . . . . . 8 (((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ ((𝑦𝐵 ∧ (𝑦 + 𝑍) = (0g𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) → 𝑍𝐵)
13 simprll 776 . . . . . . . 8 (((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ ((𝑦𝐵 ∧ (𝑦 + 𝑍) = (0g𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) → 𝑦𝐵)
1410, 11, 12, 13caovassd 7533 . . . . . . 7 (((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ ((𝑦𝐵 ∧ (𝑦 + 𝑍) = (0g𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) → ((𝑋 + 𝑍) + 𝑦) = (𝑋 + (𝑍 + 𝑦)))
15 simplr2 1215 . . . . . . . 8 (((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ ((𝑦𝐵 ∧ (𝑦 + 𝑍) = (0g𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) → 𝑌𝐵)
1610, 15, 12, 13caovassd 7533 . . . . . . 7 (((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ ((𝑦𝐵 ∧ (𝑦 + 𝑍) = (0g𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) → ((𝑌 + 𝑍) + 𝑦) = (𝑌 + (𝑍 + 𝑦)))
177, 14, 163eqtr3d 2784 . . . . . 6 (((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ ((𝑦𝐵 ∧ (𝑦 + 𝑍) = (0g𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) → (𝑋 + (𝑍 + 𝑦)) = (𝑌 + (𝑍 + 𝑦)))
181, 2grpcl 18681 . . . . . . . . . 10 ((𝐺 ∈ Grp ∧ 𝑢𝐵𝑣𝐵) → (𝑢 + 𝑣) ∈ 𝐵)
198, 18syl3an1 1162 . . . . . . . . 9 ((((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ ((𝑦𝐵 ∧ (𝑦 + 𝑍) = (0g𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) ∧ 𝑢𝐵𝑣𝐵) → (𝑢 + 𝑣) ∈ 𝐵)
201, 3grpidcl 18703 . . . . . . . . . 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 18705 . . . . . . . . . 10 ((𝐺 ∈ Grp ∧ 𝑢𝐵) → ((0g𝐺) + 𝑢) = 𝑢)
238, 22sylan 580 . . . . . . . . 9 ((((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ ((𝑦𝐵 ∧ (𝑦 + 𝑍) = (0g𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) ∧ 𝑢𝐵) → ((0g𝐺) + 𝑢) = 𝑢)
241, 2, 3grpinvex 18683 . . . . . . . . . 10 ((𝐺 ∈ Grp ∧ 𝑢𝐵) → ∃𝑣𝐵 (𝑣 + 𝑢) = (0g𝐺))
258, 24sylan 580 . . . . . . . . 9 ((((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ ((𝑦𝐵 ∧ (𝑦 + 𝑍) = (0g𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) ∧ 𝑢𝐵) → ∃𝑣𝐵 (𝑣 + 𝑢) = (0g𝐺))
26 simpr 485 . . . . . . . . 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 777 . . . . . . . . . 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 18455 . . . . . . . 8 ((((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ ((𝑦𝐵 ∧ (𝑦 + 𝑍) = (0g𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) ∧ 𝑍𝐵) → (𝑍 + 𝑦) = (0g𝐺))
3112, 30mpdan 684 . . . . . . 7 (((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ ((𝑦𝐵 ∧ (𝑦 + 𝑍) = (0g𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) → (𝑍 + 𝑦) = (0g𝐺))
3231oveq2d 7353 . . . . . 6 (((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ ((𝑦𝐵 ∧ (𝑦 + 𝑍) = (0g𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) → (𝑋 + (𝑍 + 𝑦)) = (𝑋 + (0g𝐺)))
3331oveq2d 7353 . . . . . 6 (((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ ((𝑦𝐵 ∧ (𝑦 + 𝑍) = (0g𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) → (𝑌 + (𝑍 + 𝑦)) = (𝑌 + (0g𝐺)))
3417, 32, 333eqtr3d 2784 . . . . 5 (((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ ((𝑦𝐵 ∧ (𝑦 + 𝑍) = (0g𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) → (𝑋 + (0g𝐺)) = (𝑌 + (0g𝐺)))
351, 2, 3grprid 18706 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑋 + (0g𝐺)) = 𝑋)
368, 11, 35syl2anc 584 . . . . 5 (((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ ((𝑦𝐵 ∧ (𝑦 + 𝑍) = (0g𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) → (𝑋 + (0g𝐺)) = 𝑋)
371, 2, 3grprid 18706 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑌𝐵) → (𝑌 + (0g𝐺)) = 𝑌)
388, 15, 37syl2anc 584 . . . . 5 (((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ ((𝑦𝐵 ∧ (𝑦 + 𝑍) = (0g𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) → (𝑌 + (0g𝐺)) = 𝑌)
3934, 36, 383eqtr3d 2784 . . . 4 (((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ ((𝑦𝐵 ∧ (𝑦 + 𝑍) = (0g𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) → 𝑋 = 𝑌)
4039expr 457 . . 3 (((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝑦𝐵 ∧ (𝑦 + 𝑍) = (0g𝐺))) → ((𝑋 + 𝑍) = (𝑌 + 𝑍) → 𝑋 = 𝑌))
415, 40rexlimddv 3154 . 2 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 + 𝑍) = (𝑌 + 𝑍) → 𝑋 = 𝑌))
42 oveq1 7344 . 2 (𝑋 = 𝑌 → (𝑋 + 𝑍) = (𝑌 + 𝑍))
4341, 42impbid1 224 1 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 + 𝑍) = (𝑌 + 𝑍) ↔ 𝑋 = 𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1086   = wceq 1540  wcel 2105  wrex 3070  cfv 6479  (class class class)co 7337  Basecbs 17009  +gcplusg 17059  0gc0g 17247  Grpcgrp 18673
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-sep 5243  ax-nul 5250  ax-pr 5372
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3349  df-reu 3350  df-rab 3404  df-v 3443  df-sbc 3728  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4270  df-if 4474  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4853  df-br 5093  df-opab 5155  df-mpt 5176  df-id 5518  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-iota 6431  df-fun 6481  df-fv 6487  df-riota 7293  df-ov 7340  df-0g 17249  df-mgm 18423  df-sgrp 18472  df-mnd 18483  df-grp 18676
This theorem is referenced by:  grpinveu  18710  grpid  18711  grpidlcan  18737  grpinvssd  18748  grpsubrcan  18752  grpsubadd  18759  sylow1lem4  19302  ringcom  19913  ringrz  19922  rhmunitinv  20092  lmodcom  20275  ogrpaddlt  31630  isnumbasgrplem2  41200  grptcepi  46738
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