Theorem grpcominv1 43555

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 19192	. . . . 5 ⊢ (𝜑 → (𝑁‘𝑌) ∈ 𝐵)
7		grpcominv.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
8	1, 2, 3, 6, 5, 7	grpassd 19149	. . . 4 ⊢ (𝜑 → (((𝑁‘𝑌) + 𝑌) + 𝑋) = ((𝑁‘𝑌) + (𝑌 + 𝑋)))
9		eqid 2761	. . . . . . 7 ⊢ (0g‘𝐺) = (0g‘𝐺)
10	1, 2, 9, 4, 3, 5	grplinvd 19198	. . . . . 6 ⊢ (𝜑 → ((𝑁‘𝑌) + 𝑌) = (0g‘𝐺))
11	10	oveq1d 7433	. . . . 5 ⊢ (𝜑 → (((𝑁‘𝑌) + 𝑌) + 𝑋) = ((0g‘𝐺) + 𝑋))
12	1, 2, 9, 3, 7	grplidd 19173	. . . . 5 ⊢ (𝜑 → ((0g‘𝐺) + 𝑋) = 𝑋)
13	11, 12	eqtr2d 2797	. . . 4 ⊢ (𝜑 → 𝑋 = (((𝑁‘𝑌) + 𝑌) + 𝑋))
14		grpcominv.1	. . . . 5 ⊢ (𝜑 → (𝑋 + 𝑌) = (𝑌 + 𝑋))
15	14	oveq2d 7434	. . . 4 ⊢ (𝜑 → ((𝑁‘𝑌) + (𝑋 + 𝑌)) = ((𝑁‘𝑌) + (𝑌 + 𝑋)))
16	8, 13, 15	3eqtr4rd 2807	. . 3 ⊢ (𝜑 → ((𝑁‘𝑌) + (𝑋 + 𝑌)) = 𝑋)
17	1, 2, 3, 6, 7, 5	grpassd 19149	. . 3 ⊢ (𝜑 → (((𝑁‘𝑌) + 𝑋) + 𝑌) = ((𝑁‘𝑌) + (𝑋 + 𝑌)))
18	1, 2, 4, 3, 7, 5	grpasscan2d 43554	. . 3 ⊢ (𝜑 → ((𝑋 + (𝑁‘𝑌)) + 𝑌) = 𝑋)
19	16, 17, 18	3eqtr4rd 2807	. 2 ⊢ (𝜑 → ((𝑋 + (𝑁‘𝑌)) + 𝑌) = (((𝑁‘𝑌) + 𝑋) + 𝑌))
20	1, 2, 3, 7, 6	grpcld 19151	. . 3 ⊢ (𝜑 → (𝑋 + (𝑁‘𝑌)) ∈ 𝐵)
21	1, 2, 3, 6, 7	grpcld 19151	. . 3 ⊢ (𝜑 → ((𝑁‘𝑌) + 𝑋) ∈ 𝐵)
22	1, 2	grprcan 19177	. . 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))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1570   ∈ wcel 2145  ‘cfv 6537  (class class class)co 7418  Basecbs 17380  +_gcplusg 17421  0_gc0g 17603  Grpcgrp 19137  inv_gcminusg 19138

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7749

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-fv 6545  df-riota 7375  df-ov 7421  df-0g 17605  df-mgm 18809  df-sgrp 18901  df-mnd 18917  df-grp 19140  df-minusg 19141

This theorem is used by:  grpcominv2  43556