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Theorem grpasscan1 19019
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 2737 . . . . 5 (0g𝐺) = (0g𝐺)
4 grpasscan1.n . . . . 5 𝑁 = (invg𝐺)
51, 2, 3, 4grprinv 19008 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑋 + (𝑁𝑋)) = (0g𝐺))
653adant3 1133 . . 3 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑋 + (𝑁𝑋)) = (0g𝐺))
76oveq1d 7446 . 2 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 + (𝑁𝑋)) + 𝑌) = ((0g𝐺) + 𝑌))
81, 4grpinvcl 19005 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑁𝑋) ∈ 𝐵)
91, 2grpass 18960 . . . . . 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 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 1118 . 2 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 + (𝑁𝑋)) + 𝑌) = (𝑋 + ((𝑁𝑋) + 𝑌)))
141, 2, 3grplid 18985 . . 3 ((𝐺 ∈ Grp ∧ 𝑌𝐵) → ((0g𝐺) + 𝑌) = 𝑌)
15143adant2 1132 . 2 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → ((0g𝐺) + 𝑌) = 𝑌)
167, 13, 153eqtr3d 2785 1 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑋 + ((𝑁𝑋) + 𝑌)) = 𝑌)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1087   = wceq 1540  wcel 2108  cfv 6561  (class class class)co 7431  Basecbs 17247  +gcplusg 17297  0gc0g 17484  Grpcgrp 18951  invgcminusg 18952
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-riota 7388  df-ov 7434  df-0g 17486  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-grp 18954  df-minusg 18955
This theorem is referenced by:  mulgaddcomlem  19115  ghmqusnsglem1  19298  ghmquskerlem1  19301  grplsmid  33432  nsgqusf1olem3  33443  qsdrnglem2  33524
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