Theorem grpcominv1 43005

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 18962	. . . . 5 ⊢ (𝜑 → (𝑁‘𝑌) ∈ 𝐵)
7		grpcominv.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
8	1, 2, 3, 6, 5, 7	grpassd 18919	. . . 4 ⊢ (𝜑 → (((𝑁‘𝑌) + 𝑌) + 𝑋) = ((𝑁‘𝑌) + (𝑌 + 𝑋)))
9		eqid 2740	. . . . . . 7 ⊢ (0g‘𝐺) = (0g‘𝐺)
10	1, 2, 9, 4, 3, 5	grplinvd 18968	. . . . . 6 ⊢ (𝜑 → ((𝑁‘𝑌) + 𝑌) = (0g‘𝐺))
11	10	oveq1d 7378	. . . . 5 ⊢ (𝜑 → (((𝑁‘𝑌) + 𝑌) + 𝑋) = ((0g‘𝐺) + 𝑋))
12	1, 2, 9, 3, 7	grplidd 18943	. . . . 5 ⊢ (𝜑 → ((0g‘𝐺) + 𝑋) = 𝑋)
13	11, 12	eqtr2d 2776	. . . 4 ⊢ (𝜑 → 𝑋 = (((𝑁‘𝑌) + 𝑌) + 𝑋))
14		grpcominv.1	. . . . 5 ⊢ (𝜑 → (𝑋 + 𝑌) = (𝑌 + 𝑋))
15	14	oveq2d 7379	. . . 4 ⊢ (𝜑 → ((𝑁‘𝑌) + (𝑋 + 𝑌)) = ((𝑁‘𝑌) + (𝑌 + 𝑋)))
16	8, 13, 15	3eqtr4rd 2786	. . 3 ⊢ (𝜑 → ((𝑁‘𝑌) + (𝑋 + 𝑌)) = 𝑋)
17	1, 2, 3, 6, 7, 5	grpassd 18919	. . 3 ⊢ (𝜑 → (((𝑁‘𝑌) + 𝑋) + 𝑌) = ((𝑁‘𝑌) + (𝑋 + 𝑌)))
18	1, 2, 4, 3, 7, 5	grpasscan2d 43004	. . 3 ⊢ (𝜑 → ((𝑋 + (𝑁‘𝑌)) + 𝑌) = 𝑋)
19	16, 17, 18	3eqtr4rd 2786	. 2 ⊢ (𝜑 → ((𝑋 + (𝑁‘𝑌)) + 𝑌) = (((𝑁‘𝑌) + 𝑋) + 𝑌))
20	1, 2, 3, 7, 6	grpcld 18921	. . 3 ⊢ (𝜑 → (𝑋 + (𝑁‘𝑌)) ∈ 𝐵)
21	1, 2, 3, 6, 7	grpcld 18921	. . 3 ⊢ (𝜑 → ((𝑁‘𝑌) + 𝑋) ∈ 𝐵)
22	1, 2	grprcan 18947	. . 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 1380	. 2 ⊢ (𝜑 → (((𝑋 + (𝑁‘𝑌)) + 𝑌) = (((𝑁‘𝑌) + 𝑋) + 𝑌) ↔ (𝑋 + (𝑁‘𝑌)) = ((𝑁‘𝑌) + 𝑋)))
24	19, 23	mpbid 233	1 ⊢ (𝜑 → (𝑋 + (𝑁‘𝑌)) = ((𝑁‘𝑌) + 𝑋))

Syntax hints:   → wi 4   ↔ wb 207   = wceq 1547   ∈ wcel 2119  ‘cfv 6492  (class class class)co 7363  Basecbs 17177  +_gcplusg 17218  0_gc0g 17400  Grpcgrp 18907  inv_gcminusg 18908

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-rmo 3345  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-br 5080  df-opab 5142  df-mpt 5161  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-fv 6500  df-riota 7320  df-ov 7366  df-0g 17402  df-mgm 18606  df-sgrp 18685  df-mnd 18701  df-grp 18910  df-minusg 18911

This theorem is referenced by:  grpcominv2  43006