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Theorem grpasscan1 18931
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 2726 . . . . 5 (0g𝐺) = (0g𝐺)
4 grpasscan1.n . . . . 5 𝑁 = (invg𝐺)
51, 2, 3, 4grprinv 18920 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑋 + (𝑁𝑋)) = (0g𝐺))
653adant3 1129 . . 3 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑋 + (𝑁𝑋)) = (0g𝐺))
76oveq1d 7420 . 2 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 + (𝑁𝑋)) + 𝑌) = ((0g𝐺) + 𝑌))
81, 4grpinvcl 18917 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑁𝑋) ∈ 𝐵)
91, 2grpass 18872 . . . . . 6 ((𝐺 ∈ Grp ∧ (𝑋𝐵 ∧ (𝑁𝑋) ∈ 𝐵𝑌𝐵)) → ((𝑋 + (𝑁𝑋)) + 𝑌) = (𝑋 + ((𝑁𝑋) + 𝑌)))
1093exp2 1351 . . . . 5 (𝐺 ∈ Grp → (𝑋𝐵 → ((𝑁𝑋) ∈ 𝐵 → (𝑌𝐵 → ((𝑋 + (𝑁𝑋)) + 𝑌) = (𝑋 + ((𝑁𝑋) + 𝑌))))))
1110imp 406 . . . 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 1114 . 2 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 + (𝑁𝑋)) + 𝑌) = (𝑋 + ((𝑁𝑋) + 𝑌)))
141, 2, 3grplid 18897 . . 3 ((𝐺 ∈ Grp ∧ 𝑌𝐵) → ((0g𝐺) + 𝑌) = 𝑌)
15143adant2 1128 . 2 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → ((0g𝐺) + 𝑌) = 𝑌)
167, 13, 153eqtr3d 2774 1 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑋 + ((𝑁𝑋) + 𝑌)) = 𝑌)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1084   = wceq 1533  wcel 2098  cfv 6537  (class class class)co 7405  Basecbs 17153  +gcplusg 17206  0gc0g 17394  Grpcgrp 18863  invgcminusg 18864
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-fv 6545  df-riota 7361  df-ov 7408  df-0g 17396  df-mgm 18573  df-sgrp 18652  df-mnd 18668  df-grp 18866  df-minusg 18867
This theorem is referenced by:  mulgaddcomlem  19024  grplsmid  33020  nsgqusf1olem3  33032  ghmquskerlem1  33034  qsdrnglem2  33116
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