Theorem f1ghm0to0 13989

Description: If a group homomorphism F is injective, it maps the zero of one group (and only the zero) to the zero of the other group. (Contributed by AV, 24-Oct-2019.) (Revised by Thierry Arnoux, 13-May-2023.)

Hypotheses

Ref	Expression
f1ghm0to0.a	|- A = ( Base ` R )
f1ghm0to0.b	|- B = ( Base ` S )
f1ghm0to0.n	|- N = ( 0g ` R )
f1ghm0to0.0	|- .0. = ( 0g ` S )

Assertion

Ref	Expression
f1ghm0to0	|- ( ( F e. ( R GrpHom S ) /\ F : A -1-1-> B /\ X e. A ) -> ( ( F ` X ) = .0. <-> X = N ) )

Proof of Theorem f1ghm0to0

Step	Hyp	Ref	Expression
1		f1ghm0to0.n	. . . . . 6 |- N = ( 0g ` R )
2		f1ghm0to0.0	. . . . . 6 |- .0. = ( 0g ` S )
3	1, 2	ghmid 13966	. . . . 5 |- ( F e. ( R GrpHom S ) -> ( F ` N ) = .0. )
4	3	3ad2ant1 1045	. . . 4 |- ( ( F e. ( R GrpHom S ) /\ F : A -1-1-> B /\ X e. A ) -> ( F ` N ) = .0. )
5	4	eqeq2d 2244	. . 3 |- ( ( F e. ( R GrpHom S ) /\ F : A -1-1-> B /\ X e. A ) -> ( ( F ` X ) = ( F ` N ) <-> ( F ` X ) = .0. ) )
6		simp2 1025	. . . 4 |- ( ( F e. ( R GrpHom S ) /\ F : A -1-1-> B /\ X e. A ) -> F : A -1-1-> B )
7		simp3 1026	. . . 4 |- ( ( F e. ( R GrpHom S ) /\ F : A -1-1-> B /\ X e. A ) -> X e. A )
8		ghmgrp1 13962	. . . . . 6 |- ( F e. ( R GrpHom S ) -> R e. Grp )
9		f1ghm0to0.a	. . . . . . 7 |- A = ( Base ` R )
10	9, 1	grpidcl 13742	. . . . . 6 |- ( R e. Grp -> N e. A )
11	8, 10	syl 14	. . . . 5 |- ( F e. ( R GrpHom S ) -> N e. A )
12	11	3ad2ant1 1045	. . . 4 |- ( ( F e. ( R GrpHom S ) /\ F : A -1-1-> B /\ X e. A ) -> N e. A )
13		f1veqaeq 5942	. . . 4 |- ( ( F : A -1-1-> B /\ ( X e. A /\ N e. A ) ) -> ( ( F ` X ) = ( F ` N ) -> X = N ) )
14	6, 7, 12, 13	syl12anc 1272	. . 3 |- ( ( F e. ( R GrpHom S ) /\ F : A -1-1-> B /\ X e. A ) -> ( ( F ` X ) = ( F ` N ) -> X = N ) )
15	5, 14	sylbird 170	. 2 |- ( ( F e. ( R GrpHom S ) /\ F : A -1-1-> B /\ X e. A ) -> ( ( F ` X ) = .0. -> X = N ) )
16		fveq2 5670	. . . 4 |- ( X = N -> ( F ` X ) = ( F ` N ) )
17	16, 4	sylan9eqr 2287	. . 3 |- ( ( ( F e. ( R GrpHom S ) /\ F : A -1-1-> B /\ X e. A ) /\ X = N ) -> ( F ` X ) = .0. )
18	17	ex 115	. 2 |- ( ( F e. ( R GrpHom S ) /\ F : A -1-1-> B /\ X e. A ) -> ( X = N -> ( F ` X ) = .0. ) )
19	15, 18	impbid 129	1 |- ( ( F e. ( R GrpHom S ) /\ F : A -1-1-> B /\ X e. A ) -> ( ( F ` X ) = .0. <-> X = N ) )

Syntax hints:   -> wi 4   <-> wb 105   /\ w3a 1005   = wceq 1398   e. wcel 2203   -1-1->wf1 5349   ` cfv 5352  (class class class)co 6050   Basecbs 13212   0gc0g 13469   Grpcgrp 13713   GrpHom cghm 13957

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-cnex 8218  ax-resscn 8219  ax-1re 8221  ax-addrcl 8224

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-inn 9238  df-2 9296  df-ndx 13215  df-slot 13216  df-base 13218  df-plusg 13303  df-0g 13471  df-mgm 13569  df-sgrp 13615  df-mnd 13630  df-grp 13716  df-ghm 13958

This theorem is referenced by:  ghmf1  13990  kerf1ghm  13991