Theorem ghminv 19155

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 19150	. . . . . 6 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑆 ∈ Grp)
2		ghminv.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝑆)
3		eqid 2729	. . . . . . 7 ⊢ (+g‘𝑆) = (+g‘𝑆)
4		eqid 2729	. . . . . . 7 ⊢ (0g‘𝑆) = (0g‘𝑆)
5		ghminv.y	. . . . . . 7 ⊢ 𝑀 = (invg‘𝑆)
6	2, 3, 4, 5	grprinv 18922	. . . . . 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 6862	. . . 4 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ 𝐵) → (𝐹‘(𝑋(+g‘𝑆)(𝑀‘𝑋))) = (𝐹‘(0g‘𝑆)))
9	2, 5	grpinvcl 18919	. . . . . 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 2729	. . . . . 6 ⊢ (+g‘𝑇) = (+g‘𝑇)
12	2, 3, 11	ghmlin 19153	. . . . 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 2729	. . . . . 6 ⊢ (0g‘𝑇) = (0g‘𝑇)
15	4, 14	ghmid 19154	. . . . 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 2772	. . 3 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ 𝐵) → ((𝐹‘𝑋)(+g‘𝑇)(𝐹‘(𝑀‘𝑋))) = (0g‘𝑇))
18		ghmgrp2 19151	. . . . 5 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑇 ∈ Grp)
19	18	adantr 480	. . . 4 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ 𝐵) → 𝑇 ∈ Grp)
20		eqid 2729	. . . . . 6 ⊢ (Base‘𝑇) = (Base‘𝑇)
21	2, 20	ghmf 19152	. . . . 5 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹:𝐵⟶(Base‘𝑇))
22	21	ffvelcdmda 7056	. . . 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 7057	. . . 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 18923	. . . 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 2735	1 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ 𝐵) → (𝐹‘(𝑀‘𝑋)) = (𝑁‘(𝐹‘𝑋)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ⟶wf 6507  ‘cfv 6511  (class class class)co 7387  Basecbs 17179  +_gcplusg 17220  0_gc0g 17402  Grpcgrp 18865  inv_gcminusg 18866   GrpHom cghm 19144

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-fv 6519  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-1st 7968  df-2nd 7969  df-map 8801  df-0g 17404  df-mgm 18567  df-sgrp 18646  df-mnd 18662  df-grp 18868  df-minusg 18869  df-ghm 19145

This theorem is referenced by:  ghmsub  19156  ghmmulg  19160  ghmrn  19161  ghmpreima  19170  ghmeql  19171  ghmqusnsglem1  19212  ghmquskerlem1  19215  frgpup3lem  19707  psgninv  21491  zrhpsgnodpm  21501  asclinvg  21798  mplind  21977  cpmatinvcl  22604  sum2dchr  27185  zrhneg  33968  zrhcntr  33969  fldhmf1  42078