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Theorem grpasscan2d 40624
Description: An associative cancellation law for groups. (Contributed by SN, 29-Jan-2025.)
Hypotheses
Ref Expression
grpasscan2d.b 𝐵 = (Base‘𝐺)
grpasscan2d.p + = (+g𝐺)
grpasscan2d.n 𝑁 = (invg𝐺)
grpasscan2d.g (𝜑𝐺 ∈ Grp)
grpasscan2d.1 (𝜑𝑋𝐵)
grpasscan2d.2 (𝜑𝑌𝐵)
Assertion
Ref Expression
grpasscan2d (𝜑 → ((𝑋 + (𝑁𝑌)) + 𝑌) = 𝑋)

Proof of Theorem grpasscan2d
StepHypRef Expression
1 grpasscan2d.g . 2 (𝜑𝐺 ∈ Grp)
2 grpasscan2d.1 . 2 (𝜑𝑋𝐵)
3 grpasscan2d.2 . 2 (𝜑𝑌𝐵)
4 grpasscan2d.b . . 3 𝐵 = (Base‘𝐺)
5 grpasscan2d.p . . 3 + = (+g𝐺)
6 grpasscan2d.n . . 3 𝑁 = (invg𝐺)
74, 5, 6grpasscan2 18769 . 2 ((𝐺 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 + (𝑁𝑌)) + 𝑌) = 𝑋)
81, 2, 3, 7syl3anc 1371 1 (𝜑 → ((𝑋 + (𝑁𝑌)) + 𝑌) = 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1541  wcel 2106  cfv 6493  (class class class)co 7351  Basecbs 17042  +gcplusg 17092  Grpcgrp 18707  invgcminusg 18708
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2708  ax-sep 5254  ax-nul 5261  ax-pow 5318  ax-pr 5382  ax-un 7664
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3351  df-reu 3352  df-rab 3406  df-v 3445  df-sbc 3738  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-pw 4560  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-br 5104  df-opab 5166  df-mpt 5187  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6445  df-fun 6495  df-fn 6496  df-f 6497  df-fv 6501  df-riota 7307  df-ov 7354  df-0g 17282  df-mgm 18456  df-sgrp 18505  df-mnd 18516  df-grp 18710  df-minusg 18711
This theorem is referenced by:  grpcominv1  40625
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