Theorem ghmid 19154

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

Hypotheses

Ref	Expression
ghmid.y	⊢ 𝑌 = (0g‘𝑆)
ghmid.z	⊢ 0 = (0g‘𝑇)

Assertion

Ref	Expression
ghmid	⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝐹‘𝑌) = 0 )

Proof of Theorem ghmid

Step	Hyp	Ref	Expression
1		ghmgrp1 19150	. . . . . 6 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑆 ∈ Grp)
2		eqid 2729	. . . . . . 7 ⊢ (Base‘𝑆) = (Base‘𝑆)
3		ghmid.y	. . . . . . 7 ⊢ 𝑌 = (0g‘𝑆)
4	2, 3	grpidcl 18897	. . . . . 6 ⊢ (𝑆 ∈ Grp → 𝑌 ∈ (Base‘𝑆))
5	1, 4	syl 17	. . . . 5 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑌 ∈ (Base‘𝑆))
6		eqid 2729	. . . . . 6 ⊢ (+g‘𝑆) = (+g‘𝑆)
7		eqid 2729	. . . . . 6 ⊢ (+g‘𝑇) = (+g‘𝑇)
8	2, 6, 7	ghmlin 19153	. . . . 5 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑌 ∈ (Base‘𝑆) ∧ 𝑌 ∈ (Base‘𝑆)) → (𝐹‘(𝑌(+g‘𝑆)𝑌)) = ((𝐹‘𝑌)(+g‘𝑇)(𝐹‘𝑌)))
9	5, 5, 8	mpd3an23 1465	. . . 4 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝐹‘(𝑌(+g‘𝑆)𝑌)) = ((𝐹‘𝑌)(+g‘𝑇)(𝐹‘𝑌)))
10	2, 6, 3	grplid 18899	. . . . . 6 ⊢ ((𝑆 ∈ Grp ∧ 𝑌 ∈ (Base‘𝑆)) → (𝑌(+g‘𝑆)𝑌) = 𝑌)
11	1, 5, 10	syl2anc 584	. . . . 5 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝑌(+g‘𝑆)𝑌) = 𝑌)
12	11	fveq2d 6862	. . . 4 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝐹‘(𝑌(+g‘𝑆)𝑌)) = (𝐹‘𝑌))
13	9, 12	eqtr3d 2766	. . 3 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → ((𝐹‘𝑌)(+g‘𝑇)(𝐹‘𝑌)) = (𝐹‘𝑌))
14		ghmgrp2 19151	. . . 4 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑇 ∈ Grp)
15		eqid 2729	. . . . . 6 ⊢ (Base‘𝑇) = (Base‘𝑇)
16	2, 15	ghmf 19152	. . . . 5 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
17	16, 5	ffvelcdmd 7057	. . . 4 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝐹‘𝑌) ∈ (Base‘𝑇))
18		ghmid.z	. . . . 5 ⊢ 0 = (0g‘𝑇)
19	15, 7, 18	grpid 18907	. . . 4 ⊢ ((𝑇 ∈ Grp ∧ (𝐹‘𝑌) ∈ (Base‘𝑇)) → (((𝐹‘𝑌)(+g‘𝑇)(𝐹‘𝑌)) = (𝐹‘𝑌) ↔ 0 = (𝐹‘𝑌)))
20	14, 17, 19	syl2anc 584	. . 3 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → (((𝐹‘𝑌)(+g‘𝑇)(𝐹‘𝑌)) = (𝐹‘𝑌) ↔ 0 = (𝐹‘𝑌)))
21	13, 20	mpbid 232	. 2 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 0 = (𝐹‘𝑌))
22	21	eqcomd 2735	1 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝐹‘𝑌) = 0 )

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1540   ∈ wcel 2109  ‘cfv 6511  (class class class)co 7387  Basecbs 17179  +_gcplusg 17220  0_gc0g 17402  Grpcgrp 18865   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-ghm 19145

This theorem is referenced by:  ghminv  19155  ghmmhm  19158  ghmpreima  19170  f1ghm0to0  19177  kerf1ghm  19179  ghmqusker  19219  lactghmga  19335  nrhmzr  20446  zrinitorngc  20551  imadrhmcl  20706  srng0  20763  islmhm2  20945  zrh0  21423  chrrhm  21441  zndvds0  21460  ip0l  21545  evlslem2  21986  evlslem3  21987  evlslem6  21988  rhmmpl  22270  rhmply1vr1  22274  0mat2pmat  22623  nmolb2d  24606  nmoi  24616  nmoix  24617  nmoleub  24619  nmoleub2lem2  25016  nmhmcn  25020  dchrptlem2  27176  psgnid  33054  dimkerim  33623  lvecendof1f1o  33629  ricdrng1  42516  rhmpsr  42540