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Theorem grpasscan2 18095
Description: An associative cancellation law for groups. (Contributed by Paul Chapman, 17-Apr-2009.) (Revised by AV, 30-Aug-2021.)
Hypotheses
Ref Expression
grplcan.b 𝐵 = (Base‘𝐺)
grplcan.p + = (+g𝐺)
grpasscan1.n 𝑁 = (invg𝐺)
Assertion
Ref Expression
grpasscan2 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 + (𝑁𝑌)) + 𝑌) = 𝑋)

Proof of Theorem grpasscan2
StepHypRef Expression
1 simp1 1130 . . 3 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → 𝐺 ∈ Grp)
2 simp2 1131 . . 3 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → 𝑋𝐵)
3 grplcan.b . . . . 5 𝐵 = (Base‘𝐺)
4 grpasscan1.n . . . . 5 𝑁 = (invg𝐺)
53, 4grpinvcl 18083 . . . 4 ((𝐺 ∈ Grp ∧ 𝑌𝐵) → (𝑁𝑌) ∈ 𝐵)
653adant2 1125 . . 3 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑁𝑌) ∈ 𝐵)
7 simp3 1132 . . 3 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → 𝑌𝐵)
8 grplcan.p . . . 4 + = (+g𝐺)
93, 8grpass 18044 . . 3 ((𝐺 ∈ Grp ∧ (𝑋𝐵 ∧ (𝑁𝑌) ∈ 𝐵𝑌𝐵)) → ((𝑋 + (𝑁𝑌)) + 𝑌) = (𝑋 + ((𝑁𝑌) + 𝑌)))
101, 2, 6, 7, 9syl13anc 1366 . 2 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 + (𝑁𝑌)) + 𝑌) = (𝑋 + ((𝑁𝑌) + 𝑌)))
11 eqid 2824 . . . . 5 (0g𝐺) = (0g𝐺)
123, 8, 11, 4grplinv 18084 . . . 4 ((𝐺 ∈ Grp ∧ 𝑌𝐵) → ((𝑁𝑌) + 𝑌) = (0g𝐺))
13123adant2 1125 . . 3 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → ((𝑁𝑌) + 𝑌) = (0g𝐺))
1413oveq2d 7167 . 2 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑋 + ((𝑁𝑌) + 𝑌)) = (𝑋 + (0g𝐺)))
153, 8, 11grprid 18066 . . 3 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑋 + (0g𝐺)) = 𝑋)
16153adant3 1126 . 2 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑋 + (0g𝐺)) = 𝑋)
1710, 14, 163eqtrd 2864 1 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 + (𝑁𝑌)) + 𝑌) = 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1081   = wceq 1530  wcel 2106  cfv 6351  (class class class)co 7151  Basecbs 16475  +gcplusg 16557  0gc0g 16705  Grpcgrp 18035  invgcminusg 18036
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2152  ax-12 2167  ax-ext 2796  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2615  df-eu 2649  df-clab 2803  df-cleq 2817  df-clel 2897  df-nfc 2967  df-ne 3021  df-ral 3147  df-rex 3148  df-reu 3149  df-rmo 3150  df-rab 3151  df-v 3501  df-sbc 3776  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-br 5063  df-opab 5125  df-mpt 5143  df-id 5458  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-fv 6359  df-riota 7109  df-ov 7154  df-0g 16707  df-mgm 17844  df-sgrp 17892  df-mnd 17903  df-grp 18038  df-minusg 18039
This theorem is referenced by:  mulgaddcomlem  18182
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