Theorem grpcominv1 42503

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	⊢ + = (+g‘𝐺)
grpcominv.n	⊢ 𝑁 = (invg‘𝐺)
grpcominv.g	⊢ (𝜑 → 𝐺 ∈ Grp)
grpcominv.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
grpcominv.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)
grpcominv.1	⊢ (𝜑 → (𝑋 + 𝑌) = (𝑌 + 𝑋))

Assertion

Ref	Expression
grpcominv1	⊢ (𝜑 → (𝑋 + (𝑁‘𝑌)) = ((𝑁‘𝑌) + 𝑋))

Proof of Theorem grpcominv1

Step	Hyp	Ref	Expression
1		grpcominv.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐺)
2		grpcominv.p	. . . . 5 ⊢ + = (+g‘𝐺)
3		grpcominv.g	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ Grp)
4		grpcominv.n	. . . . . 6 ⊢ 𝑁 = (invg‘𝐺)
5		grpcominv.y	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
6	1, 4, 3, 5	grpinvcld 18927	. . . . 5 ⊢ (𝜑 → (𝑁‘𝑌) ∈ 𝐵)
7		grpcominv.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
8	1, 2, 3, 6, 5, 7	grpassd 18884	. . . 4 ⊢ (𝜑 → (((𝑁‘𝑌) + 𝑌) + 𝑋) = ((𝑁‘𝑌) + (𝑌 + 𝑋)))
9		eqid 2730	. . . . . . 7 ⊢ (0g‘𝐺) = (0g‘𝐺)
10	1, 2, 9, 4, 3, 5	grplinvd 18933	. . . . . 6 ⊢ (𝜑 → ((𝑁‘𝑌) + 𝑌) = (0g‘𝐺))
11	10	oveq1d 7405	. . . . 5 ⊢ (𝜑 → (((𝑁‘𝑌) + 𝑌) + 𝑋) = ((0g‘𝐺) + 𝑋))
12	1, 2, 9, 3, 7	grplidd 18908	. . . . 5 ⊢ (𝜑 → ((0g‘𝐺) + 𝑋) = 𝑋)
13	11, 12	eqtr2d 2766	. . . 4 ⊢ (𝜑 → 𝑋 = (((𝑁‘𝑌) + 𝑌) + 𝑋))
14		grpcominv.1	. . . . 5 ⊢ (𝜑 → (𝑋 + 𝑌) = (𝑌 + 𝑋))
15	14	oveq2d 7406	. . . 4 ⊢ (𝜑 → ((𝑁‘𝑌) + (𝑋 + 𝑌)) = ((𝑁‘𝑌) + (𝑌 + 𝑋)))
16	8, 13, 15	3eqtr4rd 2776	. . 3 ⊢ (𝜑 → ((𝑁‘𝑌) + (𝑋 + 𝑌)) = 𝑋)
17	1, 2, 3, 6, 7, 5	grpassd 18884	. . 3 ⊢ (𝜑 → (((𝑁‘𝑌) + 𝑋) + 𝑌) = ((𝑁‘𝑌) + (𝑋 + 𝑌)))
18	1, 2, 4, 3, 7, 5	grpasscan2d 42502	. . 3 ⊢ (𝜑 → ((𝑋 + (𝑁‘𝑌)) + 𝑌) = 𝑋)
19	16, 17, 18	3eqtr4rd 2776	. 2 ⊢ (𝜑 → ((𝑋 + (𝑁‘𝑌)) + 𝑌) = (((𝑁‘𝑌) + 𝑋) + 𝑌))
20	1, 2, 3, 7, 6	grpcld 18886	. . 3 ⊢ (𝜑 → (𝑋 + (𝑁‘𝑌)) ∈ 𝐵)
21	1, 2, 3, 6, 7	grpcld 18886	. . 3 ⊢ (𝜑 → ((𝑁‘𝑌) + 𝑋) ∈ 𝐵)
22	1, 2	grprcan 18912	. . 3 ⊢ ((𝐺 ∈ Grp ∧ ((𝑋 + (𝑁‘𝑌)) ∈ 𝐵 ∧ ((𝑁‘𝑌) + 𝑋) ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (((𝑋 + (𝑁‘𝑌)) + 𝑌) = (((𝑁‘𝑌) + 𝑋) + 𝑌) ↔ (𝑋 + (𝑁‘𝑌)) = ((𝑁‘𝑌) + 𝑋)))
23	3, 20, 21, 5, 22	syl13anc 1374	. 2 ⊢ (𝜑 → (((𝑋 + (𝑁‘𝑌)) + 𝑌) = (((𝑁‘𝑌) + 𝑋) + 𝑌) ↔ (𝑋 + (𝑁‘𝑌)) = ((𝑁‘𝑌) + 𝑋)))
24	19, 23	mpbid 232	1 ⊢ (𝜑 → (𝑋 + (𝑁‘𝑌)) = ((𝑁‘𝑌) + 𝑋))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1540   ∈ wcel 2109  ‘cfv 6514  (class class class)co 7390  Basecbs 17186  +_gcplusg 17227  0_gc0g 17409  Grpcgrp 18872  inv_gcminusg 18873

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-fv 6522  df-riota 7347  df-ov 7393  df-0g 17411  df-mgm 18574  df-sgrp 18653  df-mnd 18669  df-grp 18875  df-minusg 18876

This theorem is referenced by:  grpcominv2  42504