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Theorem grpasscan1 19032
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 2735 . . . . 5 (0g𝐺) = (0g𝐺)
4 grpasscan1.n . . . . 5 𝑁 = (invg𝐺)
51, 2, 3, 4grprinv 19021 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑋 + (𝑁𝑋)) = (0g𝐺))
653adant3 1131 . . 3 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑋 + (𝑁𝑋)) = (0g𝐺))
76oveq1d 7446 . 2 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 + (𝑁𝑋)) + 𝑌) = ((0g𝐺) + 𝑌))
81, 4grpinvcl 19018 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑁𝑋) ∈ 𝐵)
91, 2grpass 18973 . . . . . 6 ((𝐺 ∈ Grp ∧ (𝑋𝐵 ∧ (𝑁𝑋) ∈ 𝐵𝑌𝐵)) → ((𝑋 + (𝑁𝑋)) + 𝑌) = (𝑋 + ((𝑁𝑋) + 𝑌)))
1093exp2 1353 . . . . 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 1116 . 2 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 + (𝑁𝑋)) + 𝑌) = (𝑋 + ((𝑁𝑋) + 𝑌)))
141, 2, 3grplid 18998 . . 3 ((𝐺 ∈ Grp ∧ 𝑌𝐵) → ((0g𝐺) + 𝑌) = 𝑌)
15143adant2 1130 . 2 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → ((0g𝐺) + 𝑌) = 𝑌)
167, 13, 153eqtr3d 2783 1 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑋 + ((𝑁𝑋) + 𝑌)) = 𝑌)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1537  wcel 2106  cfv 6563  (class class class)co 7431  Basecbs 17245  +gcplusg 17298  0gc0g 17486  Grpcgrp 18964  invgcminusg 18965
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-riota 7388  df-ov 7434  df-0g 17488  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-grp 18967  df-minusg 18968
This theorem is referenced by:  mulgaddcomlem  19128  ghmqusnsglem1  19311  ghmquskerlem1  19314  grplsmid  33412  nsgqusf1olem3  33423  qsdrnglem2  33504
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