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Theorem grpcominv1 42518
Description: If two elements commute, then they commute with each other's inverses (case of the first element commuting with the inverse of the second element). (Contributed by SN, 29-Jan-2025.)
Hypotheses
Ref Expression
grpcominv.b 𝐵 = (Base‘𝐺)
grpcominv.p + = (+g𝐺)
grpcominv.n 𝑁 = (invg𝐺)
grpcominv.g (𝜑𝐺 ∈ Grp)
grpcominv.x (𝜑𝑋𝐵)
grpcominv.y (𝜑𝑌𝐵)
grpcominv.1 (𝜑 → (𝑋 + 𝑌) = (𝑌 + 𝑋))
Assertion
Ref Expression
grpcominv1 (𝜑 → (𝑋 + (𝑁𝑌)) = ((𝑁𝑌) + 𝑋))

Proof of Theorem grpcominv1
StepHypRef Expression
1 grpcominv.b . . . . 5 𝐵 = (Base‘𝐺)
2 grpcominv.p . . . . 5 + = (+g𝐺)
3 grpcominv.g . . . . 5 (𝜑𝐺 ∈ Grp)
4 grpcominv.n . . . . . 6 𝑁 = (invg𝐺)
5 grpcominv.y . . . . . 6 (𝜑𝑌𝐵)
61, 4, 3, 5grpinvcld 19006 . . . . 5 (𝜑 → (𝑁𝑌) ∈ 𝐵)
7 grpcominv.x . . . . 5 (𝜑𝑋𝐵)
81, 2, 3, 6, 5, 7grpassd 18963 . . . 4 (𝜑 → (((𝑁𝑌) + 𝑌) + 𝑋) = ((𝑁𝑌) + (𝑌 + 𝑋)))
9 eqid 2737 . . . . . . 7 (0g𝐺) = (0g𝐺)
101, 2, 9, 4, 3, 5grplinvd 19012 . . . . . 6 (𝜑 → ((𝑁𝑌) + 𝑌) = (0g𝐺))
1110oveq1d 7446 . . . . 5 (𝜑 → (((𝑁𝑌) + 𝑌) + 𝑋) = ((0g𝐺) + 𝑋))
121, 2, 9, 3, 7grplidd 18987 . . . . 5 (𝜑 → ((0g𝐺) + 𝑋) = 𝑋)
1311, 12eqtr2d 2778 . . . 4 (𝜑𝑋 = (((𝑁𝑌) + 𝑌) + 𝑋))
14 grpcominv.1 . . . . 5 (𝜑 → (𝑋 + 𝑌) = (𝑌 + 𝑋))
1514oveq2d 7447 . . . 4 (𝜑 → ((𝑁𝑌) + (𝑋 + 𝑌)) = ((𝑁𝑌) + (𝑌 + 𝑋)))
168, 13, 153eqtr4rd 2788 . . 3 (𝜑 → ((𝑁𝑌) + (𝑋 + 𝑌)) = 𝑋)
171, 2, 3, 6, 7, 5grpassd 18963 . . 3 (𝜑 → (((𝑁𝑌) + 𝑋) + 𝑌) = ((𝑁𝑌) + (𝑋 + 𝑌)))
181, 2, 4, 3, 7, 5grpasscan2d 42517 . . 3 (𝜑 → ((𝑋 + (𝑁𝑌)) + 𝑌) = 𝑋)
1916, 17, 183eqtr4rd 2788 . 2 (𝜑 → ((𝑋 + (𝑁𝑌)) + 𝑌) = (((𝑁𝑌) + 𝑋) + 𝑌))
201, 2, 3, 7, 6grpcld 18965 . . 3 (𝜑 → (𝑋 + (𝑁𝑌)) ∈ 𝐵)
211, 2, 3, 6, 7grpcld 18965 . . 3 (𝜑 → ((𝑁𝑌) + 𝑋) ∈ 𝐵)
221, 2grprcan 18991 . . 3 ((𝐺 ∈ Grp ∧ ((𝑋 + (𝑁𝑌)) ∈ 𝐵 ∧ ((𝑁𝑌) + 𝑋) ∈ 𝐵𝑌𝐵)) → (((𝑋 + (𝑁𝑌)) + 𝑌) = (((𝑁𝑌) + 𝑋) + 𝑌) ↔ (𝑋 + (𝑁𝑌)) = ((𝑁𝑌) + 𝑋)))
233, 20, 21, 5, 22syl13anc 1374 . 2 (𝜑 → (((𝑋 + (𝑁𝑌)) + 𝑌) = (((𝑁𝑌) + 𝑋) + 𝑌) ↔ (𝑋 + (𝑁𝑌)) = ((𝑁𝑌) + 𝑋)))
2419, 23mpbid 232 1 (𝜑 → (𝑋 + (𝑁𝑌)) = ((𝑁𝑌) + 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206   = 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:  grpcominv2  42519
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