Theorem ghminv 19241

Description: A homomorphism of groups preserves inverses. (Contributed by Stefan O'Rear, 31-Dec-2014.)

Hypotheses

Ref	Expression
ghminv.b	⊢ 𝐵 = (Base‘𝑆)
ghminv.y	⊢ 𝑀 = (invg‘𝑆)
ghminv.z	⊢ 𝑁 = (invg‘𝑇)

Assertion

Ref	Expression
ghminv	⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ 𝐵) → (𝐹‘(𝑀‘𝑋)) = (𝑁‘(𝐹‘𝑋)))

Proof of Theorem ghminv

Step	Hyp	Ref	Expression
1		ghmgrp1 19236	. . . . . 6 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑆 ∈ Grp)
2		ghminv.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝑆)
3		eqid 2737	. . . . . . 7 ⊢ (+g‘𝑆) = (+g‘𝑆)
4		eqid 2737	. . . . . . 7 ⊢ (0g‘𝑆) = (0g‘𝑆)
5		ghminv.y	. . . . . . 7 ⊢ 𝑀 = (invg‘𝑆)
6	2, 3, 4, 5	grprinv 19008	. . . . . 6 ⊢ ((𝑆 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑋(+g‘𝑆)(𝑀‘𝑋)) = (0g‘𝑆))
7	1, 6	sylan 580	. . . . 5 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ 𝐵) → (𝑋(+g‘𝑆)(𝑀‘𝑋)) = (0g‘𝑆))
8	7	fveq2d 6910	. . . 4 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ 𝐵) → (𝐹‘(𝑋(+g‘𝑆)(𝑀‘𝑋))) = (𝐹‘(0g‘𝑆)))
9	2, 5	grpinvcl 19005	. . . . . 6 ⊢ ((𝑆 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑀‘𝑋) ∈ 𝐵)
10	1, 9	sylan 580	. . . . 5 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ 𝐵) → (𝑀‘𝑋) ∈ 𝐵)
11		eqid 2737	. . . . . 6 ⊢ (+g‘𝑇) = (+g‘𝑇)
12	2, 3, 11	ghmlin 19239	. . . . 5 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ 𝐵 ∧ (𝑀‘𝑋) ∈ 𝐵) → (𝐹‘(𝑋(+g‘𝑆)(𝑀‘𝑋))) = ((𝐹‘𝑋)(+g‘𝑇)(𝐹‘(𝑀‘𝑋))))
13	10, 12	mpd3an3 1464	. . . 4 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ 𝐵) → (𝐹‘(𝑋(+g‘𝑆)(𝑀‘𝑋))) = ((𝐹‘𝑋)(+g‘𝑇)(𝐹‘(𝑀‘𝑋))))
14		eqid 2737	. . . . . 6 ⊢ (0g‘𝑇) = (0g‘𝑇)
15	4, 14	ghmid 19240	. . . . 5 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝐹‘(0g‘𝑆)) = (0g‘𝑇))
16	15	adantr 480	. . . 4 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ 𝐵) → (𝐹‘(0g‘𝑆)) = (0g‘𝑇))
17	8, 13, 16	3eqtr3d 2785	. . 3 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ 𝐵) → ((𝐹‘𝑋)(+g‘𝑇)(𝐹‘(𝑀‘𝑋))) = (0g‘𝑇))
18		ghmgrp2 19237	. . . . 5 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑇 ∈ Grp)
19	18	adantr 480	. . . 4 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ 𝐵) → 𝑇 ∈ Grp)
20		eqid 2737	. . . . . 6 ⊢ (Base‘𝑇) = (Base‘𝑇)
21	2, 20	ghmf 19238	. . . . 5 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹:𝐵⟶(Base‘𝑇))
22	21	ffvelcdmda 7104	. . . 4 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ 𝐵) → (𝐹‘𝑋) ∈ (Base‘𝑇))
23	21	adantr 480	. . . . 5 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ 𝐵) → 𝐹:𝐵⟶(Base‘𝑇))
24	23, 10	ffvelcdmd 7105	. . . 4 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ 𝐵) → (𝐹‘(𝑀‘𝑋)) ∈ (Base‘𝑇))
25		ghminv.z	. . . . 5 ⊢ 𝑁 = (invg‘𝑇)
26	20, 11, 14, 25	grpinvid1 19009	. . . 4 ⊢ ((𝑇 ∈ Grp ∧ (𝐹‘𝑋) ∈ (Base‘𝑇) ∧ (𝐹‘(𝑀‘𝑋)) ∈ (Base‘𝑇)) → ((𝑁‘(𝐹‘𝑋)) = (𝐹‘(𝑀‘𝑋)) ↔ ((𝐹‘𝑋)(+g‘𝑇)(𝐹‘(𝑀‘𝑋))) = (0g‘𝑇)))
27	19, 22, 24, 26	syl3anc 1373	. . 3 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ 𝐵) → ((𝑁‘(𝐹‘𝑋)) = (𝐹‘(𝑀‘𝑋)) ↔ ((𝐹‘𝑋)(+g‘𝑇)(𝐹‘(𝑀‘𝑋))) = (0g‘𝑇)))
28	17, 27	mpbird 257	. 2 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ 𝐵) → (𝑁‘(𝐹‘𝑋)) = (𝐹‘(𝑀‘𝑋)))
29	28	eqcomd 2743	1 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ 𝐵) → (𝐹‘(𝑀‘𝑋)) = (𝑁‘(𝐹‘𝑋)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2108  ⟶wf 6557  ‘cfv 6561  (class class class)co 7431  Basecbs 17247  +_gcplusg 17297  0_gc0g 17484  Grpcgrp 18951  inv_gcminusg 18952   GrpHom cghm 19230

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-map 8868  df-0g 17486  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-grp 18954  df-minusg 18955  df-ghm 19231

This theorem is referenced by:  ghmsub  19242  ghmmulg  19246  ghmrn  19247  ghmpreima  19256  ghmeql  19257  ghmqusnsglem1  19298  ghmquskerlem1  19301  frgpup3lem  19795  psgninv  21600  zrhpsgnodpm  21610  asclinvg  21909  mplind  22094  cpmatinvcl  22723  sum2dchr  27318  zrhneg  33979  zrhcntr  33980  fldhmf1  42091