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Theorem grpasscan1 12822
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 2177 . . . . 5 (0g𝐺) = (0g𝐺)
4 grpasscan1.n . . . . 5 𝑁 = (invg𝐺)
51, 2, 3, 4grprinv 12813 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑋 + (𝑁𝑋)) = (0g𝐺))
653adant3 1017 . . 3 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑋 + (𝑁𝑋)) = (0g𝐺))
76oveq1d 5884 . 2 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 + (𝑁𝑋)) + 𝑌) = ((0g𝐺) + 𝑌))
81, 4grpinvcl 12811 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑁𝑋) ∈ 𝐵)
91, 2grpass 12776 . . . . . 6 ((𝐺 ∈ Grp ∧ (𝑋𝐵 ∧ (𝑁𝑋) ∈ 𝐵𝑌𝐵)) → ((𝑋 + (𝑁𝑋)) + 𝑌) = (𝑋 + ((𝑁𝑋) + 𝑌)))
1093exp2 1225 . . . . 5 (𝐺 ∈ Grp → (𝑋𝐵 → ((𝑁𝑋) ∈ 𝐵 → (𝑌𝐵 → ((𝑋 + (𝑁𝑋)) + 𝑌) = (𝑋 + ((𝑁𝑋) + 𝑌))))))
1110imp 124 . . . 4 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → ((𝑁𝑋) ∈ 𝐵 → (𝑌𝐵 → ((𝑋 + (𝑁𝑋)) + 𝑌) = (𝑋 + ((𝑁𝑋) + 𝑌)))))
128, 11mpd 13 . . 3 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑌𝐵 → ((𝑋 + (𝑁𝑋)) + 𝑌) = (𝑋 + ((𝑁𝑋) + 𝑌))))
13123impia 1200 . 2 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 + (𝑁𝑋)) + 𝑌) = (𝑋 + ((𝑁𝑋) + 𝑌)))
141, 2, 3grplid 12796 . . 3 ((𝐺 ∈ Grp ∧ 𝑌𝐵) → ((0g𝐺) + 𝑌) = 𝑌)
15143adant2 1016 . 2 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → ((0g𝐺) + 𝑌) = 𝑌)
167, 13, 153eqtr3d 2218 1 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑋 + ((𝑁𝑋) + 𝑌)) = 𝑌)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  w3a 978   = wceq 1353  wcel 2148  cfv 5212  (class class class)co 5869  Basecbs 12445  +gcplusg 12518  0gc0g 12653  Grpcgrp 12767  invgcminusg 12768
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-cnex 7893  ax-resscn 7894  ax-1re 7896  ax-addrcl 7899
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4290  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-riota 5825  df-ov 5872  df-inn 8909  df-2 8967  df-ndx 12448  df-slot 12449  df-base 12451  df-plusg 12531  df-0g 12655  df-mgm 12667  df-sgrp 12700  df-mnd 12710  df-grp 12770  df-minusg 12771
This theorem is referenced by:  mulgaddcomlem  12894
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