Theorem f1ghm0to0 14052

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 14029	. . . . 5 |- ( F e. ( R GrpHom S ) -> ( F ` N ) = .0. )
4	3	3ad2ant1 1049	. . . 4 |- ( ( F e. ( R GrpHom S ) /\ F : A -1-1-> B /\ X e. A ) -> ( F ` N ) = .0. )
5	4	eqeq2d 2250	. . 3 |- ( ( F e. ( R GrpHom S ) /\ F : A -1-1-> B /\ X e. A ) -> ( ( F ` X ) = ( F ` N ) <-> ( F ` X ) = .0. ) )
6		simp2 1029	. . . 4 |- ( ( F e. ( R GrpHom S ) /\ F : A -1-1-> B /\ X e. A ) -> F : A -1-1-> B )
7		simp3 1030	. . . 4 |- ( ( F e. ( R GrpHom S ) /\ F : A -1-1-> B /\ X e. A ) -> X e. A )
8		ghmgrp1 14025	. . . . . 6 |- ( F e. ( R GrpHom S ) -> R e. Grp )
9		f1ghm0to0.a	. . . . . . 7 |- A = ( Base ` R )
10	9, 1	grpidcl 13811	. . . . . 6 |- ( R e. Grp -> N e. A )
11	8, 10	syl 14	. . . . 5 |- ( F e. ( R GrpHom S ) -> N e. A )
12	11	3ad2ant1 1049	. . . 4 |- ( ( F e. ( R GrpHom S ) /\ F : A -1-1-> B /\ X e. A ) -> N e. A )
13		f1veqaeq 5965	. . . 4 |- ( ( F : A -1-1-> B /\ ( X e. A /\ N e. A ) ) -> ( ( F ` X ) = ( F ` N ) -> X = N ) )
14	6, 7, 12, 13	syl12anc 1276	. . 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 5690	. . . 4 |- ( X = N -> ( F ` X ) = ( F ` N ) )
17	16, 4	sylan9eqr 2293	. . 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 1009   = wceq 1402   e. wcel 2209   -1-1->wf1 5369   ` cfv 5372  (class class class)co 6075   Basecbs 13330   0gc0g 13587   Grpcgrp 13782   GrpHom cghm 14020

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-1re 8263  ax-addrcl 8266

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-inn 9284  df-2 9342  df-ndx 13333  df-slot 13334  df-base 13336  df-plusg 13421  df-0g 13589  df-mgm 13653  df-sgrp 13694  df-mnd 13707  df-grp 13785  df-ghm 14021

This theorem is referenced by:  ghmf1  14053  kerf1ghm  14054