Theorem f1ghm0to0 13608

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 13585	. . . . 5 |- ( F e. ( R GrpHom S ) -> ( F ` N ) = .0. )
4	3	3ad2ant1 1021	. . . 4 |- ( ( F e. ( R GrpHom S ) /\ F : A -1-1-> B /\ X e. A ) -> ( F ` N ) = .0. )
5	4	eqeq2d 2217	. . 3 |- ( ( F e. ( R GrpHom S ) /\ F : A -1-1-> B /\ X e. A ) -> ( ( F ` X ) = ( F ` N ) <-> ( F ` X ) = .0. ) )
6		simp2 1001	. . . 4 |- ( ( F e. ( R GrpHom S ) /\ F : A -1-1-> B /\ X e. A ) -> F : A -1-1-> B )
7		simp3 1002	. . . 4 |- ( ( F e. ( R GrpHom S ) /\ F : A -1-1-> B /\ X e. A ) -> X e. A )
8		ghmgrp1 13581	. . . . . 6 |- ( F e. ( R GrpHom S ) -> R e. Grp )
9		f1ghm0to0.a	. . . . . . 7 |- A = ( Base ` R )
10	9, 1	grpidcl 13361	. . . . . 6 |- ( R e. Grp -> N e. A )
11	8, 10	syl 14	. . . . 5 |- ( F e. ( R GrpHom S ) -> N e. A )
12	11	3ad2ant1 1021	. . . 4 |- ( ( F e. ( R GrpHom S ) /\ F : A -1-1-> B /\ X e. A ) -> N e. A )
13		f1veqaeq 5838	. . . 4 |- ( ( F : A -1-1-> B /\ ( X e. A /\ N e. A ) ) -> ( ( F ` X ) = ( F ` N ) -> X = N ) )
14	6, 7, 12, 13	syl12anc 1248	. . 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 5576	. . . 4 |- ( X = N -> ( F ` X ) = ( F ` N ) )
17	16, 4	sylan9eqr 2260	. . 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 981   = wceq 1373   e. wcel 2176   -1-1->wf1 5268   ` cfv 5271  (class class class)co 5944   Basecbs 12832   0gc0g 13088   Grpcgrp 13332   GrpHom cghm 13576

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 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-coll 4159  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-cnex 8016  ax-resscn 8017  ax-1re 8019  ax-addrcl 8022

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-ral 2489  df-rex 2490  df-reu 2491  df-rmo 2492  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-br 4045  df-opab 4106  df-mpt 4107  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-riota 5899  df-ov 5947  df-oprab 5948  df-mpo 5949  df-inn 9037  df-2 9095  df-ndx 12835  df-slot 12836  df-base 12838  df-plusg 12922  df-0g 13090  df-mgm 13188  df-sgrp 13234  df-mnd 13249  df-grp 13335  df-ghm 13577

This theorem is referenced by:  ghmf1  13609  kerf1ghm  13610