Theorem ghminv 19130

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 19125	. . . . . 6 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑆 ∈ Grp)
2		ghminv.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝑆)
3		eqid 2731	. . . . . . 7 ⊢ (+g‘𝑆) = (+g‘𝑆)
4		eqid 2731	. . . . . . 7 ⊢ (0g‘𝑆) = (0g‘𝑆)
5		ghminv.y	. . . . . . 7 ⊢ 𝑀 = (invg‘𝑆)
6	2, 3, 4, 5	grprinv 18898	. . . . . 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 6821	. . . 4 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ 𝐵) → (𝐹‘(𝑋(+g‘𝑆)(𝑀‘𝑋))) = (𝐹‘(0g‘𝑆)))
9	2, 5	grpinvcl 18895	. . . . . 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 2731	. . . . . 6 ⊢ (+g‘𝑇) = (+g‘𝑇)
12	2, 3, 11	ghmlin 19128	. . . . 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 2731	. . . . . 6 ⊢ (0g‘𝑇) = (0g‘𝑇)
15	4, 14	ghmid 19129	. . . . 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 2774	. . 3 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ 𝐵) → ((𝐹‘𝑋)(+g‘𝑇)(𝐹‘(𝑀‘𝑋))) = (0g‘𝑇))
18		ghmgrp2 19126	. . . . 5 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑇 ∈ Grp)
19	18	adantr 480	. . . 4 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ 𝐵) → 𝑇 ∈ Grp)
20		eqid 2731	. . . . . 6 ⊢ (Base‘𝑇) = (Base‘𝑇)
21	2, 20	ghmf 19127	. . . . 5 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹:𝐵⟶(Base‘𝑇))
22	21	ffvelcdmda 7012	. . . 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 7013	. . . 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 18899	. . . 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 2737	1 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑋 ∈ 𝐵) → (𝐹‘(𝑀‘𝑋)) = (𝑁‘(𝐹‘𝑋)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2111  ⟶wf 6472  ‘cfv 6476  (class class class)co 7341  Basecbs 17115  +_gcplusg 17156  0_gc0g 17338  Grpcgrp 18841  inv_gcminusg 18842   GrpHom cghm 19119

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5229  ax-nul 5239  ax-pow 5298  ax-pr 5365  ax-un 7663

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4279  df-if 4471  df-pw 4547  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-iun 4938  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5506  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-rn 5622  df-res 5623  df-ima 5624  df-iota 6432  df-fun 6478  df-fn 6479  df-f 6480  df-fv 6484  df-riota 7298  df-ov 7344  df-oprab 7345  df-mpo 7346  df-1st 7916  df-2nd 7917  df-map 8747  df-0g 17340  df-mgm 18543  df-sgrp 18622  df-mnd 18638  df-grp 18844  df-minusg 18845  df-ghm 19120

This theorem is referenced by:  ghmsub  19131  ghmmulg  19135  ghmrn  19136  ghmpreima  19145  ghmeql  19146  ghmqusnsglem1  19187  ghmquskerlem1  19190  frgpup3lem  19684  psgninv  21514  zrhpsgnodpm  21524  asclinvg  21821  mplind  22000  cpmatinvcl  22627  sum2dchr  27207  fxpsubg  33134  zrhneg  33983  zrhcntr  33984  fldhmf1  42123