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Theorem grpasscan1 18886
Description: An associative cancellation law for groups. (Contributed by Paul Chapman, 25-Feb-2008.) (Revised by AV, 30-Aug-2021.)
Hypotheses
Ref Expression
grplcan.b 𝐵 = (Base‘𝐺)
grplcan.p + = (+g𝐺)
grpasscan1.n 𝑁 = (invg𝐺)
Assertion
Ref Expression
grpasscan1 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑋 + ((𝑁𝑋) + 𝑌)) = 𝑌)

Proof of Theorem grpasscan1
StepHypRef Expression
1 grplcan.b . . . . 5 𝐵 = (Base‘𝐺)
2 grplcan.p . . . . 5 + = (+g𝐺)
3 eqid 2733 . . . . 5 (0g𝐺) = (0g𝐺)
4 grpasscan1.n . . . . 5 𝑁 = (invg𝐺)
51, 2, 3, 4grprinv 18875 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑋 + (𝑁𝑋)) = (0g𝐺))
653adant3 1133 . . 3 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑋 + (𝑁𝑋)) = (0g𝐺))
76oveq1d 7424 . 2 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 + (𝑁𝑋)) + 𝑌) = ((0g𝐺) + 𝑌))
81, 4grpinvcl 18872 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑁𝑋) ∈ 𝐵)
91, 2grpass 18828 . . . . . 6 ((𝐺 ∈ Grp ∧ (𝑋𝐵 ∧ (𝑁𝑋) ∈ 𝐵𝑌𝐵)) → ((𝑋 + (𝑁𝑋)) + 𝑌) = (𝑋 + ((𝑁𝑋) + 𝑌)))
1093exp2 1355 . . . . 5 (𝐺 ∈ Grp → (𝑋𝐵 → ((𝑁𝑋) ∈ 𝐵 → (𝑌𝐵 → ((𝑋 + (𝑁𝑋)) + 𝑌) = (𝑋 + ((𝑁𝑋) + 𝑌))))))
1110imp 408 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → ((𝑁𝑋) ∈ 𝐵 → (𝑌𝐵 → ((𝑋 + (𝑁𝑋)) + 𝑌) = (𝑋 + ((𝑁𝑋) + 𝑌)))))
128, 11mpd 15 . . 3 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑌𝐵 → ((𝑋 + (𝑁𝑋)) + 𝑌) = (𝑋 + ((𝑁𝑋) + 𝑌))))
13123impia 1118 . 2 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 + (𝑁𝑋)) + 𝑌) = (𝑋 + ((𝑁𝑋) + 𝑌)))
141, 2, 3grplid 18852 . . 3 ((𝐺 ∈ Grp ∧ 𝑌𝐵) → ((0g𝐺) + 𝑌) = 𝑌)
15143adant2 1132 . 2 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → ((0g𝐺) + 𝑌) = 𝑌)
167, 13, 153eqtr3d 2781 1 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑋 + ((𝑁𝑋) + 𝑌)) = 𝑌)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  w3a 1088   = wceq 1542  wcel 2107  cfv 6544  (class class class)co 7409  Basecbs 17144  +gcplusg 17197  0gc0g 17385  Grpcgrp 18819  invgcminusg 18820
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-fv 6552  df-riota 7365  df-ov 7412  df-0g 17387  df-mgm 18561  df-sgrp 18610  df-mnd 18626  df-grp 18822  df-minusg 18823
This theorem is referenced by:  mulgaddcomlem  18977  grplsmid  32514  nsgqusf1olem3  32526  ghmquskerlem1  32528  qsdrnglem2  32610
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