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Theorem grpcominv1 42494
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 19018 . . . . 5 (𝜑 → (𝑁𝑌) ∈ 𝐵)
7 grpcominv.x . . . . 5 (𝜑𝑋𝐵)
81, 2, 3, 6, 5, 7grpassd 18975 . . . 4 (𝜑 → (((𝑁𝑌) + 𝑌) + 𝑋) = ((𝑁𝑌) + (𝑌 + 𝑋)))
9 eqid 2734 . . . . . . 7 (0g𝐺) = (0g𝐺)
101, 2, 9, 4, 3, 5grplinvd 19024 . . . . . 6 (𝜑 → ((𝑁𝑌) + 𝑌) = (0g𝐺))
1110oveq1d 7445 . . . . 5 (𝜑 → (((𝑁𝑌) + 𝑌) + 𝑋) = ((0g𝐺) + 𝑋))
121, 2, 9, 3, 7grplidd 18999 . . . . 5 (𝜑 → ((0g𝐺) + 𝑋) = 𝑋)
1311, 12eqtr2d 2775 . . . 4 (𝜑𝑋 = (((𝑁𝑌) + 𝑌) + 𝑋))
14 grpcominv.1 . . . . 5 (𝜑 → (𝑋 + 𝑌) = (𝑌 + 𝑋))
1514oveq2d 7446 . . . 4 (𝜑 → ((𝑁𝑌) + (𝑋 + 𝑌)) = ((𝑁𝑌) + (𝑌 + 𝑋)))
168, 13, 153eqtr4rd 2785 . . 3 (𝜑 → ((𝑁𝑌) + (𝑋 + 𝑌)) = 𝑋)
171, 2, 3, 6, 7, 5grpassd 18975 . . 3 (𝜑 → (((𝑁𝑌) + 𝑋) + 𝑌) = ((𝑁𝑌) + (𝑋 + 𝑌)))
181, 2, 4, 3, 7, 5grpasscan2d 42493 . . 3 (𝜑 → ((𝑋 + (𝑁𝑌)) + 𝑌) = 𝑋)
1916, 17, 183eqtr4rd 2785 . 2 (𝜑 → ((𝑋 + (𝑁𝑌)) + 𝑌) = (((𝑁𝑌) + 𝑋) + 𝑌))
201, 2, 3, 7, 6grpcld 18977 . . 3 (𝜑 → (𝑋 + (𝑁𝑌)) ∈ 𝐵)
211, 2, 3, 6, 7grpcld 18977 . . 3 (𝜑 → ((𝑁𝑌) + 𝑋) ∈ 𝐵)
221, 2grprcan 19003 . . 3 ((𝐺 ∈ Grp ∧ ((𝑋 + (𝑁𝑌)) ∈ 𝐵 ∧ ((𝑁𝑌) + 𝑋) ∈ 𝐵𝑌𝐵)) → (((𝑋 + (𝑁𝑌)) + 𝑌) = (((𝑁𝑌) + 𝑋) + 𝑌) ↔ (𝑋 + (𝑁𝑌)) = ((𝑁𝑌) + 𝑋)))
233, 20, 21, 5, 22syl13anc 1371 . 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 1536  wcel 2105  cfv 6562  (class class class)co 7430  Basecbs 17244  +gcplusg 17297  0gc0g 17485  Grpcgrp 18963  invgcminusg 18964
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-fv 6570  df-riota 7387  df-ov 7433  df-0g 17487  df-mgm 18665  df-sgrp 18744  df-mnd 18760  df-grp 18966  df-minusg 18967
This theorem is referenced by:  grpcominv2  42495
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