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Theorem grpasscan2 17302
 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 1054 . . 3 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → 𝐺 ∈ Grp)
2 simp2 1055 . . 3 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → 𝑋𝐵)
3 grplcan.b . . . . 5 𝐵 = (Base‘𝐺)
4 grpasscan1.n . . . . 5 𝑁 = (invg𝐺)
53, 4grpinvcl 17290 . . . 4 ((𝐺 ∈ Grp ∧ 𝑌𝐵) → (𝑁𝑌) ∈ 𝐵)
653adant2 1073 . . 3 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑁𝑌) ∈ 𝐵)
7 simp3 1056 . . 3 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → 𝑌𝐵)
8 grplcan.p . . . 4 + = (+g𝐺)
93, 8grpass 17254 . . 3 ((𝐺 ∈ Grp ∧ (𝑋𝐵 ∧ (𝑁𝑌) ∈ 𝐵𝑌𝐵)) → ((𝑋 + (𝑁𝑌)) + 𝑌) = (𝑋 + ((𝑁𝑌) + 𝑌)))
101, 2, 6, 7, 9syl13anc 1320 . 2 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 + (𝑁𝑌)) + 𝑌) = (𝑋 + ((𝑁𝑌) + 𝑌)))
11 eqid 2610 . . . . 5 (0g𝐺) = (0g𝐺)
123, 8, 11, 4grplinv 17291 . . . 4 ((𝐺 ∈ Grp ∧ 𝑌𝐵) → ((𝑁𝑌) + 𝑌) = (0g𝐺))
13123adant2 1073 . . 3 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → ((𝑁𝑌) + 𝑌) = (0g𝐺))
1413oveq2d 6565 . 2 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑋 + ((𝑁𝑌) + 𝑌)) = (𝑋 + (0g𝐺)))
153, 8, 11grprid 17276 . . 3 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑋 + (0g𝐺)) = 𝑋)
16153adant3 1074 . 2 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑋 + (0g𝐺)) = 𝑋)
1710, 14, 163eqtrd 2648 1 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 + (𝑁𝑌)) + 𝑌) = 𝑋)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  +gcplusg 15768  0gc0g 15923  Grpcgrp 17245  invgcminusg 17246 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-rep 4699  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-csb 3500  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-iun 4457  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-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-0g 15925  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-grp 17248  df-minusg 17249 This theorem is referenced by:  mulgaddcomlem  17386
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