Theorem f1ghm0to0 13809

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 13786	. . . . 5 |- ( F e. ( R GrpHom S ) -> ( F ` N ) = .0. )
4	3	3ad2ant1 1042	. . . 4 |- ( ( F e. ( R GrpHom S ) /\ F : A -1-1-> B /\ X e. A ) -> ( F ` N ) = .0. )
5	4	eqeq2d 2241	. . 3 |- ( ( F e. ( R GrpHom S ) /\ F : A -1-1-> B /\ X e. A ) -> ( ( F ` X ) = ( F ` N ) <-> ( F ` X ) = .0. ) )
6		simp2 1022	. . . 4 |- ( ( F e. ( R GrpHom S ) /\ F : A -1-1-> B /\ X e. A ) -> F : A -1-1-> B )
7		simp3 1023	. . . 4 |- ( ( F e. ( R GrpHom S ) /\ F : A -1-1-> B /\ X e. A ) -> X e. A )
8		ghmgrp1 13782	. . . . . 6 |- ( F e. ( R GrpHom S ) -> R e. Grp )
9		f1ghm0to0.a	. . . . . . 7 |- A = ( Base ` R )
10	9, 1	grpidcl 13562	. . . . . 6 |- ( R e. Grp -> N e. A )
11	8, 10	syl 14	. . . . 5 |- ( F e. ( R GrpHom S ) -> N e. A )
12	11	3ad2ant1 1042	. . . 4 |- ( ( F e. ( R GrpHom S ) /\ F : A -1-1-> B /\ X e. A ) -> N e. A )
13		f1veqaeq 5893	. . . 4 |- ( ( F : A -1-1-> B /\ ( X e. A /\ N e. A ) ) -> ( ( F ` X ) = ( F ` N ) -> X = N ) )
14	6, 7, 12, 13	syl12anc 1269	. . 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 5627	. . . 4 |- ( X = N -> ( F ` X ) = ( F ` N ) )
17	16, 4	sylan9eqr 2284	. . 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 1002   = wceq 1395   e. wcel 2200   -1-1->wf1 5315   ` cfv 5318  (class class class)co 6001   Basecbs 13032   0gc0g 13289   Grpcgrp 13533   GrpHom cghm 13777

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8090  ax-resscn 8091  ax-1re 8093  ax-addrcl 8096

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-riota 5954  df-ov 6004  df-oprab 6005  df-mpo 6006  df-inn 9111  df-2 9169  df-ndx 13035  df-slot 13036  df-base 13038  df-plusg 13123  df-0g 13291  df-mgm 13389  df-sgrp 13435  df-mnd 13450  df-grp 13536  df-ghm 13778

This theorem is referenced by:  ghmf1  13810  kerf1ghm  13811