Theorem ghminv 19430

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 19425	. . . . . 6 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑆 ∈ Grp)
2		ghminv.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝑆)
3		eqid 2761	. . . . . . 7 ⊢ (+g‘𝑆) = (+g‘𝑆)
4		eqid 2761	. . . . . . 7 ⊢ (0g‘𝑆) = (0g‘𝑆)
5		ghminv.y	. . . . . . 7 ⊢ 𝑀 = (invg‘𝑆)
6	2, 3, 4, 5	grprinv 19194	. . . . . 6 ⊢ ((𝑆 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑋(+g‘𝑆)(𝑀‘𝑋)) = (0g‘𝑆))
7	1, 6	sylan 592	. . . . 5 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ 𝐵) → (𝑋(+g‘𝑆)(𝑀‘𝑋)) = (0g‘𝑆))
8	7	fveq2d 6887	. . . 4 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ 𝐵) → (𝐹‘(𝑋(+g‘𝑆)(𝑀‘𝑋))) = (𝐹‘(0g‘𝑆)))
9	2, 5	grpinvcl 19191	. . . . . 6 ⊢ ((𝑆 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑀‘𝑋) ∈ 𝐵)
10	1, 9	sylan 592	. . . . 5 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ 𝐵) → (𝑀‘𝑋) ∈ 𝐵)
11		eqid 2761	. . . . . 6 ⊢ (+g‘𝑇) = (+g‘𝑇)
12	2, 3, 11	ghmlin 19428	. . . . 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 1491	. . . 4 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ 𝐵) → (𝐹‘(𝑋(+g‘𝑆)(𝑀‘𝑋))) = ((𝐹‘𝑋)(+g‘𝑇)(𝐹‘(𝑀‘𝑋))))
14		eqid 2761	. . . . . 6 ⊢ (0g‘𝑇) = (0g‘𝑇)
15	4, 14	ghmid 19429	. . . . 5 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝐹‘(0g‘𝑆)) = (0g‘𝑇))
16	15	adantr 486	. . . 4 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ 𝐵) → (𝐹‘(0g‘𝑆)) = (0g‘𝑇))
17	8, 13, 16	3eqtr3d 2804	. . 3 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ 𝐵) → ((𝐹‘𝑋)(+g‘𝑇)(𝐹‘(𝑀‘𝑋))) = (0g‘𝑇))
18		ghmgrp2 19426	. . . . 5 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑇 ∈ Grp)
19	18	adantr 486	. . . 4 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ 𝐵) → 𝑇 ∈ Grp)
20		eqid 2761	. . . . . 6 ⊢ (Base‘𝑇) = (Base‘𝑇)
21	2, 20	ghmf 19427	. . . . 5 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹:𝐵⟶(Base‘𝑇))
22	21	ffvelcdmda 7082	. . . 4 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ 𝐵) → (𝐹‘𝑋) ∈ (Base‘𝑇))
23	21	adantr 486	. . . . 5 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ 𝐵) → 𝐹:𝐵⟶(Base‘𝑇))
24	23, 10	ffvelcdmd 7083	. . . 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 19195	. . . 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 1398	. . 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 260	. 2 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ 𝐵) → (𝑁‘(𝐹‘𝑋)) = (𝐹‘(𝑀‘𝑋)))
29	28	eqcomd 2767	1 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ 𝐵) → (𝐹‘(𝑀‘𝑋)) = (𝑁‘(𝐹‘𝑋)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ⟶wf 6533  ‘cfv 6537  (class class class)co 7418  Basecbs 17380  +_gcplusg 17421  0_gc0g 17603  Grpcgrp 19137  inv_gcminusg 19138   GrpHom cghm 19420

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-csb 3848  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-iun 4953  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-oprab 7422  df-mpo 7423  df-1st 7999  df-2nd 8000  df-map 8842  df-0g 17605  df-mgm 18809  df-sgrp 18901  df-mnd 18917  df-grp 19140  df-minusg 19141  df-ghm 19421

This theorem is used by:  ghmsub  19431  ghmmulg  19435  ghmrn  19436  ghmpreima  19445  ghmeql  19446  ghmqusnsglem1  19487  ghmquskerlem1  19490  frgpup3lem  19984  psgninv  21881  zrhpsgnodpm  21891  asclinvg  22190  mplind  22372  cpmatinvcl  23028  sum2dchr  27594  fxpsubg  33727  zrhneg  34603  zrhcntr  34604  fldhmf1  43120