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Theorem grpcominv1 41084
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 18872 . . . . 5 (𝜑 → (𝑁𝑌) ∈ 𝐵)
7 grpcominv.x . . . . 5 (𝜑𝑋𝐵)
81, 2, 3, 6, 5, 7grpassd 18830 . . . 4 (𝜑 → (((𝑁𝑌) + 𝑌) + 𝑋) = ((𝑁𝑌) + (𝑌 + 𝑋)))
9 eqid 2732 . . . . . . 7 (0g𝐺) = (0g𝐺)
101, 2, 9, 4, 3, 5grplinvd 18878 . . . . . 6 (𝜑 → ((𝑁𝑌) + 𝑌) = (0g𝐺))
1110oveq1d 7423 . . . . 5 (𝜑 → (((𝑁𝑌) + 𝑌) + 𝑋) = ((0g𝐺) + 𝑋))
121, 2, 9, 3, 7grplidd 18853 . . . . 5 (𝜑 → ((0g𝐺) + 𝑋) = 𝑋)
1311, 12eqtr2d 2773 . . . 4 (𝜑𝑋 = (((𝑁𝑌) + 𝑌) + 𝑋))
14 grpcominv.1 . . . . 5 (𝜑 → (𝑋 + 𝑌) = (𝑌 + 𝑋))
1514oveq2d 7424 . . . 4 (𝜑 → ((𝑁𝑌) + (𝑋 + 𝑌)) = ((𝑁𝑌) + (𝑌 + 𝑋)))
168, 13, 153eqtr4rd 2783 . . 3 (𝜑 → ((𝑁𝑌) + (𝑋 + 𝑌)) = 𝑋)
171, 2, 3, 6, 7, 5grpassd 18830 . . 3 (𝜑 → (((𝑁𝑌) + 𝑋) + 𝑌) = ((𝑁𝑌) + (𝑋 + 𝑌)))
181, 2, 4, 3, 7, 5grpasscan2d 41083 . . 3 (𝜑 → ((𝑋 + (𝑁𝑌)) + 𝑌) = 𝑋)
1916, 17, 183eqtr4rd 2783 . 2 (𝜑 → ((𝑋 + (𝑁𝑌)) + 𝑌) = (((𝑁𝑌) + 𝑋) + 𝑌))
201, 2, 3, 7, 6grpcld 18832 . . 3 (𝜑 → (𝑋 + (𝑁𝑌)) ∈ 𝐵)
211, 2, 3, 6, 7grpcld 18832 . . 3 (𝜑 → ((𝑁𝑌) + 𝑋) ∈ 𝐵)
221, 2grprcan 18857 . . 3 ((𝐺 ∈ Grp ∧ ((𝑋 + (𝑁𝑌)) ∈ 𝐵 ∧ ((𝑁𝑌) + 𝑋) ∈ 𝐵𝑌𝐵)) → (((𝑋 + (𝑁𝑌)) + 𝑌) = (((𝑁𝑌) + 𝑋) + 𝑌) ↔ (𝑋 + (𝑁𝑌)) = ((𝑁𝑌) + 𝑋)))
233, 20, 21, 5, 22syl13anc 1372 . 2 (𝜑 → (((𝑋 + (𝑁𝑌)) + 𝑌) = (((𝑁𝑌) + 𝑋) + 𝑌) ↔ (𝑋 + (𝑁𝑌)) = ((𝑁𝑌) + 𝑋)))
2419, 23mpbid 231 1 (𝜑 → (𝑋 + (𝑁𝑌)) = ((𝑁𝑌) + 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1541  wcel 2106  cfv 6543  (class class class)co 7408  Basecbs 17143  +gcplusg 17196  0gc0g 17384  Grpcgrp 18818  invgcminusg 18819
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 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724
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 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-riota 7364  df-ov 7411  df-0g 17386  df-mgm 18560  df-sgrp 18609  df-mnd 18625  df-grp 18821  df-minusg 18822
This theorem is referenced by:  grpcominv2  41085
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