ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  grprcan GIF version

Theorem grprcan 12787
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 2177 . . . . 5 (0g𝐺) = (0g𝐺)
41, 2, 3grpinvex 12764 . . . 4 ((𝐺 ∈ Grp ∧ 𝑍𝐵) → ∃𝑦𝐵 (𝑦 + 𝑍) = (0g𝐺))
543ad2antr3 1164 . . 3 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ∃𝑦𝐵 (𝑦 + 𝑍) = (0g𝐺))
6 simprr 531 . . . . . . . 8 (((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ ((𝑦𝐵 ∧ (𝑦 + 𝑍) = (0g𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) → (𝑋 + 𝑍) = (𝑌 + 𝑍))
76oveq1d 5883 . . . . . . 7 (((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ ((𝑦𝐵 ∧ (𝑦 + 𝑍) = (0g𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) → ((𝑋 + 𝑍) + 𝑦) = ((𝑌 + 𝑍) + 𝑦))
8 simpll 527 . . . . . . . . 9 (((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ ((𝑦𝐵 ∧ (𝑦 + 𝑍) = (0g𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) → 𝐺 ∈ Grp)
91, 2grpass 12763 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ (𝑢𝐵𝑣𝐵𝑤𝐵)) → ((𝑢 + 𝑣) + 𝑤) = (𝑢 + (𝑣 + 𝑤)))
108, 9sylan 283 . . . . . . . 8 ((((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ ((𝑦𝐵 ∧ (𝑦 + 𝑍) = (0g𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) ∧ (𝑢𝐵𝑣𝐵𝑤𝐵)) → ((𝑢 + 𝑣) + 𝑤) = (𝑢 + (𝑣 + 𝑤)))
11 simplr1 1039 . . . . . . . 8 (((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ ((𝑦𝐵 ∧ (𝑦 + 𝑍) = (0g𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) → 𝑋𝐵)
12 simplr3 1041 . . . . . . . 8 (((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ ((𝑦𝐵 ∧ (𝑦 + 𝑍) = (0g𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) → 𝑍𝐵)
13 simprll 537 . . . . . . . 8 (((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ ((𝑦𝐵 ∧ (𝑦 + 𝑍) = (0g𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) → 𝑦𝐵)
1410, 11, 12, 13caovassd 6027 . . . . . . 7 (((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ ((𝑦𝐵 ∧ (𝑦 + 𝑍) = (0g𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) → ((𝑋 + 𝑍) + 𝑦) = (𝑋 + (𝑍 + 𝑦)))
15 simplr2 1040 . . . . . . . 8 (((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ ((𝑦𝐵 ∧ (𝑦 + 𝑍) = (0g𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) → 𝑌𝐵)
1610, 15, 12, 13caovassd 6027 . . . . . . 7 (((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ ((𝑦𝐵 ∧ (𝑦 + 𝑍) = (0g𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) → ((𝑌 + 𝑍) + 𝑦) = (𝑌 + (𝑍 + 𝑦)))
177, 14, 163eqtr3d 2218 . . . . . 6 (((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ ((𝑦𝐵 ∧ (𝑦 + 𝑍) = (0g𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) → (𝑋 + (𝑍 + 𝑦)) = (𝑌 + (𝑍 + 𝑦)))
181, 2grpcl 12762 . . . . . . . . . 10 ((𝐺 ∈ Grp ∧ 𝑢𝐵𝑣𝐵) → (𝑢 + 𝑣) ∈ 𝐵)
198, 18syl3an1 1271 . . . . . . . . 9 ((((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ ((𝑦𝐵 ∧ (𝑦 + 𝑍) = (0g𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) ∧ 𝑢𝐵𝑣𝐵) → (𝑢 + 𝑣) ∈ 𝐵)
201, 3grpidcl 12781 . . . . . . . . . 10 (𝐺 ∈ Grp → (0g𝐺) ∈ 𝐵)
218, 20syl 14 . . . . . . . . 9 (((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ ((𝑦𝐵 ∧ (𝑦 + 𝑍) = (0g𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) → (0g𝐺) ∈ 𝐵)
221, 2, 3grplid 12783 . . . . . . . . . 10 ((𝐺 ∈ Grp ∧ 𝑢𝐵) → ((0g𝐺) + 𝑢) = 𝑢)
238, 22sylan 283 . . . . . . . . 9 ((((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ ((𝑦𝐵 ∧ (𝑦 + 𝑍) = (0g𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) ∧ 𝑢𝐵) → ((0g𝐺) + 𝑢) = 𝑢)
241, 2, 3grpinvex 12764 . . . . . . . . . 10 ((𝐺 ∈ Grp ∧ 𝑢𝐵) → ∃𝑣𝐵 (𝑣 + 𝑢) = (0g𝐺))
258, 24sylan 283 . . . . . . . . 9 ((((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ ((𝑦𝐵 ∧ (𝑦 + 𝑍) = (0g𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) ∧ 𝑢𝐵) → ∃𝑣𝐵 (𝑣 + 𝑢) = (0g𝐺))
26 simpr 110 . . . . . . . . 9 ((((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ ((𝑦𝐵 ∧ (𝑦 + 𝑍) = (0g𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) ∧ 𝑍𝐵) → 𝑍𝐵)
2713adantr 276 . . . . . . . . 9 ((((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ ((𝑦𝐵 ∧ (𝑦 + 𝑍) = (0g𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) ∧ 𝑍𝐵) → 𝑦𝐵)
28 simprlr 538 . . . . . . . . . 10 (((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ ((𝑦𝐵 ∧ (𝑦 + 𝑍) = (0g𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) → (𝑦 + 𝑍) = (0g𝐺))
2928adantr 276 . . . . . . . . 9 ((((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ ((𝑦𝐵 ∧ (𝑦 + 𝑍) = (0g𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) ∧ 𝑍𝐵) → (𝑦 + 𝑍) = (0g𝐺))
3019, 21, 23, 10, 25, 26, 27, 29grprinvd 12684 . . . . . . . 8 ((((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ ((𝑦𝐵 ∧ (𝑦 + 𝑍) = (0g𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) ∧ 𝑍𝐵) → (𝑍 + 𝑦) = (0g𝐺))
3112, 30mpdan 421 . . . . . . 7 (((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ ((𝑦𝐵 ∧ (𝑦 + 𝑍) = (0g𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) → (𝑍 + 𝑦) = (0g𝐺))
3231oveq2d 5884 . . . . . 6 (((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ ((𝑦𝐵 ∧ (𝑦 + 𝑍) = (0g𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) → (𝑋 + (𝑍 + 𝑦)) = (𝑋 + (0g𝐺)))
3331oveq2d 5884 . . . . . 6 (((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ ((𝑦𝐵 ∧ (𝑦 + 𝑍) = (0g𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) → (𝑌 + (𝑍 + 𝑦)) = (𝑌 + (0g𝐺)))
3417, 32, 333eqtr3d 2218 . . . . 5 (((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ ((𝑦𝐵 ∧ (𝑦 + 𝑍) = (0g𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) → (𝑋 + (0g𝐺)) = (𝑌 + (0g𝐺)))
351, 2, 3grprid 12784 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑋 + (0g𝐺)) = 𝑋)
368, 11, 35syl2anc 411 . . . . 5 (((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ ((𝑦𝐵 ∧ (𝑦 + 𝑍) = (0g𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) → (𝑋 + (0g𝐺)) = 𝑋)
371, 2, 3grprid 12784 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑌𝐵) → (𝑌 + (0g𝐺)) = 𝑌)
388, 15, 37syl2anc 411 . . . . 5 (((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ ((𝑦𝐵 ∧ (𝑦 + 𝑍) = (0g𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) → (𝑌 + (0g𝐺)) = 𝑌)
3934, 36, 383eqtr3d 2218 . . . 4 (((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ ((𝑦𝐵 ∧ (𝑦 + 𝑍) = (0g𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) → 𝑋 = 𝑌)
4039expr 375 . . 3 (((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝑦𝐵 ∧ (𝑦 + 𝑍) = (0g𝐺))) → ((𝑋 + 𝑍) = (𝑌 + 𝑍) → 𝑋 = 𝑌))
415, 40rexlimddv 2599 . 2 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 + 𝑍) = (𝑌 + 𝑍) → 𝑋 = 𝑌))
42 oveq1 5875 . 2 (𝑋 = 𝑌 → (𝑋 + 𝑍) = (𝑌 + 𝑍))
4341, 42impbid1 142 1 ((𝐺 ∈ Grp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 + 𝑍) = (𝑌 + 𝑍) ↔ 𝑋 = 𝑌))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105  w3a 978   = wceq 1353  wcel 2148  wrex 2456  cfv 5211  (class class class)co 5868  Basecbs 12432  +gcplusg 12505  0gc0g 12640  Grpcgrp 12754
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4205  ax-un 4429  ax-cnex 7880  ax-resscn 7881  ax-1re 7883  ax-addrcl 7886
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4289  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-rn 4633  df-res 4634  df-iota 5173  df-fun 5213  df-fn 5214  df-fv 5219  df-riota 5824  df-ov 5871  df-inn 8896  df-2 8954  df-ndx 12435  df-slot 12436  df-base 12438  df-plusg 12518  df-0g 12642  df-mgm 12654  df-sgrp 12687  df-mnd 12697  df-grp 12757
This theorem is referenced by:  grpinveu  12788  grpid  12789  grpidlcan  12812  grpinvssd  12823  grpsubrcan  12827  grpsubadd  12834  ringcom  13027  ringrz  13036
  Copyright terms: Public domain W3C validator