Theorem grpcominv1 41648

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 18918	. . . . 5 ⊢ (𝜑 → (𝑁‘𝑌) ∈ 𝐵)
7		grpcominv.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
8	1, 2, 3, 6, 5, 7	grpassd 18875	. . . 4 ⊢ (𝜑 → (((𝑁‘𝑌) + 𝑌) + 𝑋) = ((𝑁‘𝑌) + (𝑌 + 𝑋)))
9		eqid 2726	. . . . . . 7 ⊢ (0g‘𝐺) = (0g‘𝐺)
10	1, 2, 9, 4, 3, 5	grplinvd 18924	. . . . . 6 ⊢ (𝜑 → ((𝑁‘𝑌) + 𝑌) = (0g‘𝐺))
11	10	oveq1d 7420	. . . . 5 ⊢ (𝜑 → (((𝑁‘𝑌) + 𝑌) + 𝑋) = ((0g‘𝐺) + 𝑋))
12	1, 2, 9, 3, 7	grplidd 18899	. . . . 5 ⊢ (𝜑 → ((0g‘𝐺) + 𝑋) = 𝑋)
13	11, 12	eqtr2d 2767	. . . 4 ⊢ (𝜑 → 𝑋 = (((𝑁‘𝑌) + 𝑌) + 𝑋))
14		grpcominv.1	. . . . 5 ⊢ (𝜑 → (𝑋 + 𝑌) = (𝑌 + 𝑋))
15	14	oveq2d 7421	. . . 4 ⊢ (𝜑 → ((𝑁‘𝑌) + (𝑋 + 𝑌)) = ((𝑁‘𝑌) + (𝑌 + 𝑋)))
16	8, 13, 15	3eqtr4rd 2777	. . 3 ⊢ (𝜑 → ((𝑁‘𝑌) + (𝑋 + 𝑌)) = 𝑋)
17	1, 2, 3, 6, 7, 5	grpassd 18875	. . 3 ⊢ (𝜑 → (((𝑁‘𝑌) + 𝑋) + 𝑌) = ((𝑁‘𝑌) + (𝑋 + 𝑌)))
18	1, 2, 4, 3, 7, 5	grpasscan2d 41647	. . 3 ⊢ (𝜑 → ((𝑋 + (𝑁‘𝑌)) + 𝑌) = 𝑋)
19	16, 17, 18	3eqtr4rd 2777	. 2 ⊢ (𝜑 → ((𝑋 + (𝑁‘𝑌)) + 𝑌) = (((𝑁‘𝑌) + 𝑋) + 𝑌))
20	1, 2, 3, 7, 6	grpcld 18877	. . 3 ⊢ (𝜑 → (𝑋 + (𝑁‘𝑌)) ∈ 𝐵)
21	1, 2, 3, 6, 7	grpcld 18877	. . 3 ⊢ (𝜑 → ((𝑁‘𝑌) + 𝑋) ∈ 𝐵)
22	1, 2	grprcan 18903	. . 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 1369	. 2 ⊢ (𝜑 → (((𝑋 + (𝑁‘𝑌)) + 𝑌) = (((𝑁‘𝑌) + 𝑋) + 𝑌) ↔ (𝑋 + (𝑁‘𝑌)) = ((𝑁‘𝑌) + 𝑋)))
24	19, 23	mpbid 231	1 ⊢ (𝜑 → (𝑋 + (𝑁‘𝑌)) = ((𝑁‘𝑌) + 𝑋))

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1533   ∈ wcel 2098  ‘cfv 6537  (class class class)co 7405  Basecbs 17153  +_gcplusg 17206  0_gc0g 17394  Grpcgrp 18863  inv_gcminusg 18864

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-fv 6545  df-riota 7361  df-ov 7408  df-0g 17396  df-mgm 18573  df-sgrp 18652  df-mnd 18668  df-grp 18866  df-minusg 18867

This theorem is referenced by:  grpcominv2  41649