Theorem ghminv 19198

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 19193	. . . . . 6 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑆 ∈ Grp)
2		ghminv.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝑆)
3		eqid 2736	. . . . . . 7 ⊢ (+g‘𝑆) = (+g‘𝑆)
4		eqid 2736	. . . . . . 7 ⊢ (0g‘𝑆) = (0g‘𝑆)
5		ghminv.y	. . . . . . 7 ⊢ 𝑀 = (invg‘𝑆)
6	2, 3, 4, 5	grprinv 18966	. . . . . 6 ⊢ ((𝑆 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑋(+g‘𝑆)(𝑀‘𝑋)) = (0g‘𝑆))
7	1, 6	sylan 581	. . . . 5 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ 𝐵) → (𝑋(+g‘𝑆)(𝑀‘𝑋)) = (0g‘𝑆))
8	7	fveq2d 6844	. . . 4 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ 𝐵) → (𝐹‘(𝑋(+g‘𝑆)(𝑀‘𝑋))) = (𝐹‘(0g‘𝑆)))
9	2, 5	grpinvcl 18963	. . . . . 6 ⊢ ((𝑆 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑀‘𝑋) ∈ 𝐵)
10	1, 9	sylan 581	. . . . 5 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ 𝐵) → (𝑀‘𝑋) ∈ 𝐵)
11		eqid 2736	. . . . . 6 ⊢ (+g‘𝑇) = (+g‘𝑇)
12	2, 3, 11	ghmlin 19196	. . . . 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 1465	. . . 4 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ 𝐵) → (𝐹‘(𝑋(+g‘𝑆)(𝑀‘𝑋))) = ((𝐹‘𝑋)(+g‘𝑇)(𝐹‘(𝑀‘𝑋))))
14		eqid 2736	. . . . . 6 ⊢ (0g‘𝑇) = (0g‘𝑇)
15	4, 14	ghmid 19197	. . . . 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 2779	. . 3 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ 𝐵) → ((𝐹‘𝑋)(+g‘𝑇)(𝐹‘(𝑀‘𝑋))) = (0g‘𝑇))
18		ghmgrp2 19194	. . . . 5 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑇 ∈ Grp)
19	18	adantr 480	. . . 4 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ 𝐵) → 𝑇 ∈ Grp)
20		eqid 2736	. . . . . 6 ⊢ (Base‘𝑇) = (Base‘𝑇)
21	2, 20	ghmf 19195	. . . . 5 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹:𝐵⟶(Base‘𝑇))
22	21	ffvelcdmda 7036	. . . 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 7037	. . . 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 18967	. . . 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 1374	. . 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 2742	1 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ 𝐵) → (𝐹‘(𝑀‘𝑋)) = (𝑁‘(𝐹‘𝑋)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ⟶wf 6494  ‘cfv 6498  (class class class)co 7367  Basecbs 17179  +_gcplusg 17220  0_gc0g 17402  Grpcgrp 18909  inv_gcminusg 18910   GrpHom cghm 19187

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-fv 6506  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-1st 7942  df-2nd 7943  df-map 8775  df-0g 17404  df-mgm 18608  df-sgrp 18687  df-mnd 18703  df-grp 18912  df-minusg 18913  df-ghm 19188

This theorem is referenced by:  ghmsub  19199  ghmmulg  19203  ghmrn  19204  ghmpreima  19213  ghmeql  19214  ghmqusnsglem1  19255  ghmquskerlem1  19258  frgpup3lem  19752  psgninv  21562  zrhpsgnodpm  21572  asclinvg  21869  mplind  22048  cpmatinvcl  22682  sum2dchr  27237  fxpsubg  33234  zrhneg  34122  zrhcntr  34123  fldhmf1  42529