Theorem grpcominv1 43263

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 19056	. . . . 5 ⊢ (𝜑 → (𝑁‘𝑌) ∈ 𝐵)
7		grpcominv.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
8	1, 2, 3, 6, 5, 7	grpassd 19013	. . . 4 ⊢ (𝜑 → (((𝑁‘𝑌) + 𝑌) + 𝑋) = ((𝑁‘𝑌) + (𝑌 + 𝑋)))
9		eqid 2763	. . . . . . 7 ⊢ (0g‘𝐺) = (0g‘𝐺)
10	1, 2, 9, 4, 3, 5	grplinvd 19062	. . . . . 6 ⊢ (𝜑 → ((𝑁‘𝑌) + 𝑌) = (0g‘𝐺))
11	10	oveq1d 7427	. . . . 5 ⊢ (𝜑 → (((𝑁‘𝑌) + 𝑌) + 𝑋) = ((0g‘𝐺) + 𝑋))
12	1, 2, 9, 3, 7	grplidd 19037	. . . . 5 ⊢ (𝜑 → ((0g‘𝐺) + 𝑋) = 𝑋)
13	11, 12	eqtr2d 2799	. . . 4 ⊢ (𝜑 → 𝑋 = (((𝑁‘𝑌) + 𝑌) + 𝑋))
14		grpcominv.1	. . . . 5 ⊢ (𝜑 → (𝑋 + 𝑌) = (𝑌 + 𝑋))
15	14	oveq2d 7428	. . . 4 ⊢ (𝜑 → ((𝑁‘𝑌) + (𝑋 + 𝑌)) = ((𝑁‘𝑌) + (𝑌 + 𝑋)))
16	8, 13, 15	3eqtr4rd 2809	. . 3 ⊢ (𝜑 → ((𝑁‘𝑌) + (𝑋 + 𝑌)) = 𝑋)
17	1, 2, 3, 6, 7, 5	grpassd 19013	. . 3 ⊢ (𝜑 → (((𝑁‘𝑌) + 𝑋) + 𝑌) = ((𝑁‘𝑌) + (𝑋 + 𝑌)))
18	1, 2, 4, 3, 7, 5	grpasscan2d 43262	. . 3 ⊢ (𝜑 → ((𝑋 + (𝑁‘𝑌)) + 𝑌) = 𝑋)
19	16, 17, 18	3eqtr4rd 2809	. 2 ⊢ (𝜑 → ((𝑋 + (𝑁‘𝑌)) + 𝑌) = (((𝑁‘𝑌) + 𝑋) + 𝑌))
20	1, 2, 3, 7, 6	grpcld 19015	. . 3 ⊢ (𝜑 → (𝑋 + (𝑁‘𝑌)) ∈ 𝐵)
21	1, 2, 3, 6, 7	grpcld 19015	. . 3 ⊢ (𝜑 → ((𝑁‘𝑌) + 𝑋) ∈ 𝐵)
22	1, 2	grprcan 19041	. . 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 1399	. 2 ⊢ (𝜑 → (((𝑋 + (𝑁‘𝑌)) + 𝑌) = (((𝑁‘𝑌) + 𝑋) + 𝑌) ↔ (𝑋 + (𝑁‘𝑌)) = ((𝑁‘𝑌) + 𝑋)))
24	19, 23	mpbid 235	1 ⊢ (𝜑 → (𝑋 + (𝑁‘𝑌)) = ((𝑁‘𝑌) + 𝑋))

Syntax hints:   → wi 4   ↔ wb 209   = wceq 1570   ∈ wcel 2143  ‘cfv 6538  (class class class)co 7412  Basecbs 17270  +_gcplusg 17311  0_gc0g 17493  Grpcgrp 19001  inv_gcminusg 19002

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-fv 6546  df-riota 7369  df-ov 7415  df-0g 17495  df-mgm 18699  df-sgrp 18778  df-mnd 18794  df-grp 19004  df-minusg 19005

This theorem is referenced by:  grpcominv2  43264