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Theorem grprcan 18128
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 2822 . . . . 5 (0g𝐺) = (0g𝐺)
41, 2, 3grpinvex 18104 . . . 4 ((𝐺 ∈ Grp ∧ 𝑍𝐵) → ∃𝑦𝐵 (𝑦 + 𝑍) = (0g𝐺))
543ad2antr3 1187 . . 3 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ∃𝑦𝐵 (𝑦 + 𝑍) = (0g𝐺))
6 simprr 772 . . . . . . . 8 (((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ ((𝑦𝐵 ∧ (𝑦 + 𝑍) = (0g𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) → (𝑋 + 𝑍) = (𝑌 + 𝑍))
76oveq1d 7155 . . . . . . 7 (((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ ((𝑦𝐵 ∧ (𝑦 + 𝑍) = (0g𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) → ((𝑋 + 𝑍) + 𝑦) = ((𝑌 + 𝑍) + 𝑦))
8 simpll 766 . . . . . . . . 9 (((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ ((𝑦𝐵 ∧ (𝑦 + 𝑍) = (0g𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) → 𝐺 ∈ Grp)
91, 2grpass 18103 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ (𝑢𝐵𝑣𝐵𝑤𝐵)) → ((𝑢 + 𝑣) + 𝑤) = (𝑢 + (𝑣 + 𝑤)))
108, 9sylan 583 . . . . . . . 8 ((((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ ((𝑦𝐵 ∧ (𝑦 + 𝑍) = (0g𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) ∧ (𝑢𝐵𝑣𝐵𝑤𝐵)) → ((𝑢 + 𝑣) + 𝑤) = (𝑢 + (𝑣 + 𝑤)))
11 simplr1 1212 . . . . . . . 8 (((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ ((𝑦𝐵 ∧ (𝑦 + 𝑍) = (0g𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) → 𝑋𝐵)
12 simplr3 1214 . . . . . . . 8 (((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ ((𝑦𝐵 ∧ (𝑦 + 𝑍) = (0g𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) → 𝑍𝐵)
13 simprll 778 . . . . . . . 8 (((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ ((𝑦𝐵 ∧ (𝑦 + 𝑍) = (0g𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) → 𝑦𝐵)
1410, 11, 12, 13caovassd 7332 . . . . . . 7 (((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ ((𝑦𝐵 ∧ (𝑦 + 𝑍) = (0g𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) → ((𝑋 + 𝑍) + 𝑦) = (𝑋 + (𝑍 + 𝑦)))
15 simplr2 1213 . . . . . . . 8 (((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ ((𝑦𝐵 ∧ (𝑦 + 𝑍) = (0g𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) → 𝑌𝐵)
1610, 15, 12, 13caovassd 7332 . . . . . . 7 (((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ ((𝑦𝐵 ∧ (𝑦 + 𝑍) = (0g𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) → ((𝑌 + 𝑍) + 𝑦) = (𝑌 + (𝑍 + 𝑦)))
177, 14, 163eqtr3d 2865 . . . . . 6 (((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ ((𝑦𝐵 ∧ (𝑦 + 𝑍) = (0g𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) → (𝑋 + (𝑍 + 𝑦)) = (𝑌 + (𝑍 + 𝑦)))
181, 2grpcl 18102 . . . . . . . . . 10 ((𝐺 ∈ Grp ∧ 𝑢𝐵𝑣𝐵) → (𝑢 + 𝑣) ∈ 𝐵)
198, 18syl3an1 1160 . . . . . . . . 9 ((((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ ((𝑦𝐵 ∧ (𝑦 + 𝑍) = (0g𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) ∧ 𝑢𝐵𝑣𝐵) → (𝑢 + 𝑣) ∈ 𝐵)
201, 3grpidcl 18122 . . . . . . . . . 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 18124 . . . . . . . . . 10 ((𝐺 ∈ Grp ∧ 𝑢𝐵) → ((0g𝐺) + 𝑢) = 𝑢)
238, 22sylan 583 . . . . . . . . 9 ((((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ ((𝑦𝐵 ∧ (𝑦 + 𝑍) = (0g𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) ∧ 𝑢𝐵) → ((0g𝐺) + 𝑢) = 𝑢)
241, 2, 3grpinvex 18104 . . . . . . . . . 10 ((𝐺 ∈ Grp ∧ 𝑢𝐵) → ∃𝑣𝐵 (𝑣 + 𝑢) = (0g𝐺))
258, 24sylan 583 . . . . . . . . 9 ((((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ ((𝑦𝐵 ∧ (𝑦 + 𝑍) = (0g𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) ∧ 𝑢𝐵) → ∃𝑣𝐵 (𝑣 + 𝑢) = (0g𝐺))
26 simpr 488 . . . . . . . . 9 ((((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ ((𝑦𝐵 ∧ (𝑦 + 𝑍) = (0g𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) ∧ 𝑍𝐵) → 𝑍𝐵)
2713adantr 484 . . . . . . . . 9 ((((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ ((𝑦𝐵 ∧ (𝑦 + 𝑍) = (0g𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) ∧ 𝑍𝐵) → 𝑦𝐵)
28 simprlr 779 . . . . . . . . . 10 (((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ ((𝑦𝐵 ∧ (𝑦 + 𝑍) = (0g𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) → (𝑦 + 𝑍) = (0g𝐺))
2928adantr 484 . . . . . . . . 9 ((((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ ((𝑦𝐵 ∧ (𝑦 + 𝑍) = (0g𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) ∧ 𝑍𝐵) → (𝑦 + 𝑍) = (0g𝐺))
3019, 21, 23, 10, 25, 26, 27, 29grprinvd 17875 . . . . . . . 8 ((((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ ((𝑦𝐵 ∧ (𝑦 + 𝑍) = (0g𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) ∧ 𝑍𝐵) → (𝑍 + 𝑦) = (0g𝐺))
3112, 30mpdan 686 . . . . . . 7 (((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ ((𝑦𝐵 ∧ (𝑦 + 𝑍) = (0g𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) → (𝑍 + 𝑦) = (0g𝐺))
3231oveq2d 7156 . . . . . 6 (((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ ((𝑦𝐵 ∧ (𝑦 + 𝑍) = (0g𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) → (𝑋 + (𝑍 + 𝑦)) = (𝑋 + (0g𝐺)))
3331oveq2d 7156 . . . . . 6 (((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ ((𝑦𝐵 ∧ (𝑦 + 𝑍) = (0g𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) → (𝑌 + (𝑍 + 𝑦)) = (𝑌 + (0g𝐺)))
3417, 32, 333eqtr3d 2865 . . . . 5 (((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ ((𝑦𝐵 ∧ (𝑦 + 𝑍) = (0g𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) → (𝑋 + (0g𝐺)) = (𝑌 + (0g𝐺)))
351, 2, 3grprid 18125 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑋 + (0g𝐺)) = 𝑋)
368, 11, 35syl2anc 587 . . . . 5 (((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ ((𝑦𝐵 ∧ (𝑦 + 𝑍) = (0g𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) → (𝑋 + (0g𝐺)) = 𝑋)
371, 2, 3grprid 18125 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑌𝐵) → (𝑌 + (0g𝐺)) = 𝑌)
388, 15, 37syl2anc 587 . . . . 5 (((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ ((𝑦𝐵 ∧ (𝑦 + 𝑍) = (0g𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) → (𝑌 + (0g𝐺)) = 𝑌)
3934, 36, 383eqtr3d 2865 . . . 4 (((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ ((𝑦𝐵 ∧ (𝑦 + 𝑍) = (0g𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) → 𝑋 = 𝑌)
4039expr 460 . . 3 (((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝑦𝐵 ∧ (𝑦 + 𝑍) = (0g𝐺))) → ((𝑋 + 𝑍) = (𝑌 + 𝑍) → 𝑋 = 𝑌))
415, 40rexlimddv 3277 . 2 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 + 𝑍) = (𝑌 + 𝑍) → 𝑋 = 𝑌))
42 oveq1 7147 . 2 (𝑋 = 𝑌 → (𝑋 + 𝑍) = (𝑌 + 𝑍))
4341, 42impbid1 228 1 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 + 𝑍) = (𝑌 + 𝑍) ↔ 𝑋 = 𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wcel 2114  wrex 3131  cfv 6334  (class class class)co 7140  Basecbs 16474  +gcplusg 16556  0gc0g 16704  Grpcgrp 18094
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-reu 3137  df-rmo 3138  df-rab 3139  df-v 3471  df-sbc 3748  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-iota 6293  df-fun 6336  df-fv 6342  df-riota 7098  df-ov 7143  df-0g 16706  df-mgm 17843  df-sgrp 17892  df-mnd 17903  df-grp 18097
This theorem is referenced by:  grpinveu  18129  grpid  18130  grpidlcan  18156  grpinvssd  18167  grpsubrcan  18171  grpsubadd  18178  sylow1lem4  18717  ringcom  19323  ringrz  19332  lmodcom  19671  ogrpaddlt  30749  rhmunitinv  30927  isnumbasgrplem2  39978
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