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Theorem grpasscan2 17832
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 1172 . . 3 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → 𝐺 ∈ Grp)
2 simp2 1173 . . 3 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → 𝑋𝐵)
3 grplcan.b . . . . 5 𝐵 = (Base‘𝐺)
4 grpasscan1.n . . . . 5 𝑁 = (invg𝐺)
53, 4grpinvcl 17820 . . . 4 ((𝐺 ∈ Grp ∧ 𝑌𝐵) → (𝑁𝑌) ∈ 𝐵)
653adant2 1167 . . 3 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑁𝑌) ∈ 𝐵)
7 simp3 1174 . . 3 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → 𝑌𝐵)
8 grplcan.p . . . 4 + = (+g𝐺)
93, 8grpass 17784 . . 3 ((𝐺 ∈ Grp ∧ (𝑋𝐵 ∧ (𝑁𝑌) ∈ 𝐵𝑌𝐵)) → ((𝑋 + (𝑁𝑌)) + 𝑌) = (𝑋 + ((𝑁𝑌) + 𝑌)))
101, 2, 6, 7, 9syl13anc 1497 . 2 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 + (𝑁𝑌)) + 𝑌) = (𝑋 + ((𝑁𝑌) + 𝑌)))
11 eqid 2824 . . . . 5 (0g𝐺) = (0g𝐺)
123, 8, 11, 4grplinv 17821 . . . 4 ((𝐺 ∈ Grp ∧ 𝑌𝐵) → ((𝑁𝑌) + 𝑌) = (0g𝐺))
13123adant2 1167 . . 3 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → ((𝑁𝑌) + 𝑌) = (0g𝐺))
1413oveq2d 6920 . 2 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑋 + ((𝑁𝑌) + 𝑌)) = (𝑋 + (0g𝐺)))
153, 8, 11grprid 17806 . . 3 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑋 + (0g𝐺)) = 𝑋)
16153adant3 1168 . 2 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑋 + (0g𝐺)) = 𝑋)
1710, 14, 163eqtrd 2864 1 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 + (𝑁𝑌)) + 𝑌) = 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1113   = wceq 1658  wcel 2166  cfv 6122  (class class class)co 6904  Basecbs 16221  +gcplusg 16304  0gc0g 16452  Grpcgrp 17775  invgcminusg 17776
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2390  ax-ext 2802  ax-rep 4993  ax-sep 5004  ax-nul 5012  ax-pow 5064  ax-pr 5126
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2604  df-eu 2639  df-clab 2811  df-cleq 2817  df-clel 2820  df-nfc 2957  df-ne 2999  df-ral 3121  df-rex 3122  df-reu 3123  df-rmo 3124  df-rab 3125  df-v 3415  df-sbc 3662  df-csb 3757  df-dif 3800  df-un 3802  df-in 3804  df-ss 3811  df-nul 4144  df-if 4306  df-sn 4397  df-pr 4399  df-op 4403  df-uni 4658  df-iun 4741  df-br 4873  df-opab 4935  df-mpt 4952  df-id 5249  df-xp 5347  df-rel 5348  df-cnv 5349  df-co 5350  df-dm 5351  df-rn 5352  df-res 5353  df-ima 5354  df-iota 6085  df-fun 6124  df-fn 6125  df-f 6126  df-f1 6127  df-fo 6128  df-f1o 6129  df-fv 6130  df-riota 6865  df-ov 6907  df-0g 16454  df-mgm 17594  df-sgrp 17636  df-mnd 17647  df-grp 17778  df-minusg 17779
This theorem is referenced by:  mulgaddcomlem  17915
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