Theorem kerf1ghm 13811

Description: A group homomorphism F is injective if and only if its kernel is the singleton { N }. (Contributed by Thierry Arnoux, 27-Oct-2017.) (Proof shortened 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
kerf1ghm	|- ( F e. ( R GrpHom S ) -> ( F : A -1-1-> B <-> ( `' F " { .0. } ) = { N } ) )

Proof of Theorem kerf1ghm

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl 109	. . . . . . 7 |- ( ( ( F e. ( R GrpHom S ) /\ F : A -1-1-> B ) /\ x e. ( `' F " { .0. } ) ) -> ( F e. ( R GrpHom S ) /\ F : A -1-1-> B ) )
2		f1fn 5533	. . . . . . . . . . 11 |- ( F : A -1-1-> B -> F Fn A )
3	2	adantl 277	. . . . . . . . . 10 |- ( ( F e. ( R GrpHom S ) /\ F : A -1-1-> B ) -> F Fn A )
4		elpreima 5754	. . . . . . . . . 10 |- ( F Fn A -> ( x e. ( `' F " { .0. } ) <-> ( x e. A /\ ( F ` x ) e. { .0. } ) ) )
5	3, 4	syl 14	. . . . . . . . 9 |- ( ( F e. ( R GrpHom S ) /\ F : A -1-1-> B ) -> ( x e. ( `' F " { .0. } ) <-> ( x e. A /\ ( F ` x ) e. { .0. } ) ) )
6	5	biimpa 296	. . . . . . . 8 |- ( ( ( F e. ( R GrpHom S ) /\ F : A -1-1-> B ) /\ x e. ( `' F " { .0. } ) ) -> ( x e. A /\ ( F ` x ) e. { .0. } ) )
7	6	simpld 112	. . . . . . 7 |- ( ( ( F e. ( R GrpHom S ) /\ F : A -1-1-> B ) /\ x e. ( `' F " { .0. } ) ) -> x e. A )
8	6	simprd 114	. . . . . . . 8 |- ( ( ( F e. ( R GrpHom S ) /\ F : A -1-1-> B ) /\ x e. ( `' F " { .0. } ) ) -> ( F ` x ) e. { .0. } )
9		elsng 3681	. . . . . . . . 9 |- ( ( F ` x ) e. { .0. } -> ( ( F ` x ) e. { .0. } <-> ( F ` x ) = .0. ) )
10	8, 9	syl 14	. . . . . . . 8 |- ( ( ( F e. ( R GrpHom S ) /\ F : A -1-1-> B ) /\ x e. ( `' F " { .0. } ) ) -> ( ( F ` x ) e. { .0. } <-> ( F ` x ) = .0. ) )
11	8, 10	mpbid 147	. . . . . . 7 |- ( ( ( F e. ( R GrpHom S ) /\ F : A -1-1-> B ) /\ x e. ( `' F " { .0. } ) ) -> ( F ` x ) = .0. )
12		f1ghm0to0.a	. . . . . . . . . . 11 |- A = ( Base ` R )
13		f1ghm0to0.b	. . . . . . . . . . 11 |- B = ( Base ` S )
14		f1ghm0to0.n	. . . . . . . . . . 11 |- N = ( 0g ` R )
15		f1ghm0to0.0	. . . . . . . . . . 11 |- .0. = ( 0g ` S )
16	12, 13, 14, 15	f1ghm0to0 13809	. . . . . . . . . 10 |- ( ( F e. ( R GrpHom S ) /\ F : A -1-1-> B /\ x e. A ) -> ( ( F ` x ) = .0. <-> x = N ) )
17	16	biimpd 144	. . . . . . . . 9 |- ( ( F e. ( R GrpHom S ) /\ F : A -1-1-> B /\ x e. A ) -> ( ( F ` x ) = .0. -> x = N ) )
18	17	3expa 1227	. . . . . . . 8 |- ( ( ( F e. ( R GrpHom S ) /\ F : A -1-1-> B ) /\ x e. A ) -> ( ( F ` x ) = .0. -> x = N ) )
19	18	imp 124	. . . . . . 7 |- ( ( ( ( F e. ( R GrpHom S ) /\ F : A -1-1-> B ) /\ x e. A ) /\ ( F ` x ) = .0. ) -> x = N )
20	1, 7, 11, 19	syl21anc 1270	. . . . . 6 |- ( ( ( F e. ( R GrpHom S ) /\ F : A -1-1-> B ) /\ x e. ( `' F " { .0. } ) ) -> x = N )
21	20	ex 115	. . . . 5 |- ( ( F e. ( R GrpHom S ) /\ F : A -1-1-> B ) -> ( x e. ( `' F " { .0. } ) -> x = N ) )
22		velsn 3683	. . . . 5 |- ( x e. { N } <-> x = N )
23	21, 22	imbitrrdi 162	. . . 4 |- ( ( F e. ( R GrpHom S ) /\ F : A -1-1-> B ) -> ( x e. ( `' F " { .0. } ) -> x e. { N } ) )
24	23	ssrdv 3230	. . 3 |- ( ( F e. ( R GrpHom S ) /\ F : A -1-1-> B ) -> ( `' F " { .0. } ) C_ { N } )
25		ghmgrp1 13782	. . . . . . 7 |- ( F e. ( R GrpHom S ) -> R e. Grp )
26	12, 14	grpidcl 13562	. . . . . . 7 |- ( R e. Grp -> N e. A )
27	25, 26	syl 14	. . . . . 6 |- ( F e. ( R GrpHom S ) -> N e. A )
28	14, 15	ghmid 13786	. . . . . . 7 |- ( F e. ( R GrpHom S ) -> ( F ` N ) = .0. )
29	12, 13	ghmf 13784	. . . . . . . . 9 |- ( F e. ( R GrpHom S ) -> F : A --> B )
30	29, 27	ffvelcdmd 5771	. . . . . . . 8 |- ( F e. ( R GrpHom S ) -> ( F ` N ) e. B )
31		elsng 3681	. . . . . . . 8 |- ( ( F ` N ) e. B -> ( ( F ` N ) e. { .0. } <-> ( F ` N ) = .0. ) )
32	30, 31	syl 14	. . . . . . 7 |- ( F e. ( R GrpHom S ) -> ( ( F ` N ) e. { .0. } <-> ( F ` N ) = .0. ) )
33	28, 32	mpbird 167	. . . . . 6 |- ( F e. ( R GrpHom S ) -> ( F ` N ) e. { .0. } )
34		ffn 5473	. . . . . . 7 |- ( F : A --> B -> F Fn A )
35		elpreima 5754	. . . . . . 7 |- ( F Fn A -> ( N e. ( `' F " { .0. } ) <-> ( N e. A /\ ( F ` N ) e. { .0. } ) ) )
36	29, 34, 35	3syl 17	. . . . . 6 |- ( F e. ( R GrpHom S ) -> ( N e. ( `' F " { .0. } ) <-> ( N e. A /\ ( F ` N ) e. { .0. } ) ) )
37	27, 33, 36	mpbir2and 950	. . . . 5 |- ( F e. ( R GrpHom S ) -> N e. ( `' F " { .0. } ) )
38	37	snssd 3813	. . . 4 |- ( F e. ( R GrpHom S ) -> { N } C_ ( `' F " { .0. } ) )
39	38	adantr 276	. . 3 |- ( ( F e. ( R GrpHom S ) /\ F : A -1-1-> B ) -> { N } C_ ( `' F " { .0. } ) )
40	24, 39	eqssd 3241	. 2 |- ( ( F e. ( R GrpHom S ) /\ F : A -1-1-> B ) -> ( `' F " { .0. } ) = { N } )
41	29	adantr 276	. . 3 |- ( ( F e. ( R GrpHom S ) /\ ( `' F " { .0. } ) = { N } ) -> F : A --> B )
42		simpl 109	. . . . . . . . . 10 |- ( ( F e. ( R GrpHom S ) /\ ( ( `' F " { .0. } ) = { N } /\ ( x e. A /\ y e. A ) /\ ( F ` x ) = ( F ` y ) ) ) -> F e. ( R GrpHom S ) )
43		simpr2l 1080	. . . . . . . . . 10 |- ( ( F e. ( R GrpHom S ) /\ ( ( `' F " { .0. } ) = { N } /\ ( x e. A /\ y e. A ) /\ ( F ` x ) = ( F ` y ) ) ) -> x e. A )
44		simpr2r 1081	. . . . . . . . . 10 |- ( ( F e. ( R GrpHom S ) /\ ( ( `' F " { .0. } ) = { N } /\ ( x e. A /\ y e. A ) /\ ( F ` x ) = ( F ` y ) ) ) -> y e. A )
45		simpr3 1029	. . . . . . . . . 10 |- ( ( F e. ( R GrpHom S ) /\ ( ( `' F " { .0. } ) = { N } /\ ( x e. A /\ y e. A ) /\ ( F ` x ) = ( F ` y ) ) ) -> ( F ` x ) = ( F ` y ) )
46		eqid 2229	. . . . . . . . . . . 12 |- ( `' F " { .0. } ) = ( `' F " { .0. } )
47		eqid 2229	. . . . . . . . . . . 12 |- ( -g ` R ) = ( -g ` R )
48	12, 15, 46, 47	ghmeqker 13808	. . . . . . . . . . 11 |- ( ( F e. ( R GrpHom S ) /\ x e. A /\ y e. A ) -> ( ( F ` x ) = ( F ` y ) <-> ( x ( -g ` R ) y ) e. ( `' F " { .0. } ) ) )
49	48	biimpa 296	. . . . . . . . . 10 |- ( ( ( F e. ( R GrpHom S ) /\ x e. A /\ y e. A ) /\ ( F ` x ) = ( F ` y ) ) -> ( x ( -g ` R ) y ) e. ( `' F " { .0. } ) )
50	42, 43, 44, 45, 49	syl31anc 1274	. . . . . . . . 9 |- ( ( F e. ( R GrpHom S ) /\ ( ( `' F " { .0. } ) = { N } /\ ( x e. A /\ y e. A ) /\ ( F ` x ) = ( F ` y ) ) ) -> ( x ( -g ` R ) y ) e. ( `' F " { .0. } ) )
51		simpr1 1027	. . . . . . . . 9 |- ( ( F e. ( R GrpHom S ) /\ ( ( `' F " { .0. } ) = { N } /\ ( x e. A /\ y e. A ) /\ ( F ` x ) = ( F ` y ) ) ) -> ( `' F " { .0. } ) = { N } )
52	50, 51	eleqtrd 2308	. . . . . . . 8 |- ( ( F e. ( R GrpHom S ) /\ ( ( `' F " { .0. } ) = { N } /\ ( x e. A /\ y e. A ) /\ ( F ` x ) = ( F ` y ) ) ) -> ( x ( -g ` R ) y ) e. { N } )
53		simp2 1022	. . . . . . . . . 10 |- ( ( ( `' F " { .0. } ) = { N } /\ ( x e. A /\ y e. A ) /\ ( F ` x ) = ( F ` y ) ) -> ( x e. A /\ y e. A ) )
54	12, 47	grpsubcl 13613	. . . . . . . . . . 11 |- ( ( R e. Grp /\ x e. A /\ y e. A ) -> ( x ( -g ` R ) y ) e. A )
55	54	3expb 1228	. . . . . . . . . 10 |- ( ( R e. Grp /\ ( x e. A /\ y e. A ) ) -> ( x ( -g ` R ) y ) e. A )
56	25, 53, 55	syl2an 289	. . . . . . . . 9 |- ( ( F e. ( R GrpHom S ) /\ ( ( `' F " { .0. } ) = { N } /\ ( x e. A /\ y e. A ) /\ ( F ` x ) = ( F ` y ) ) ) -> ( x ( -g ` R ) y ) e. A )
57		elsng 3681	. . . . . . . . 9 |- ( ( x ( -g ` R ) y ) e. A -> ( ( x ( -g ` R ) y ) e. { N } <-> ( x ( -g ` R ) y ) = N ) )
58	56, 57	syl 14	. . . . . . . 8 |- ( ( F e. ( R GrpHom S ) /\ ( ( `' F " { .0. } ) = { N } /\ ( x e. A /\ y e. A ) /\ ( F ` x ) = ( F ` y ) ) ) -> ( ( x ( -g ` R ) y ) e. { N } <-> ( x ( -g ` R ) y ) = N ) )
59	52, 58	mpbid 147	. . . . . . 7 |- ( ( F e. ( R GrpHom S ) /\ ( ( `' F " { .0. } ) = { N } /\ ( x e. A /\ y e. A ) /\ ( F ` x ) = ( F ` y ) ) ) -> ( x ( -g ` R ) y ) = N )
60	25	adantr 276	. . . . . . . 8 |- ( ( F e. ( R GrpHom S ) /\ ( ( `' F " { .0. } ) = { N } /\ ( x e. A /\ y e. A ) /\ ( F ` x ) = ( F ` y ) ) ) -> R e. Grp )
61	12, 14, 47	grpsubeq0 13619	. . . . . . . 8 |- ( ( R e. Grp /\ x e. A /\ y e. A ) -> ( ( x ( -g ` R ) y ) = N <-> x = y ) )
62	60, 43, 44, 61	syl3anc 1271	. . . . . . 7 |- ( ( F e. ( R GrpHom S ) /\ ( ( `' F " { .0. } ) = { N } /\ ( x e. A /\ y e. A ) /\ ( F ` x ) = ( F ` y ) ) ) -> ( ( x ( -g ` R ) y ) = N <-> x = y ) )
63	59, 62	mpbid 147	. . . . . 6 |- ( ( F e. ( R GrpHom S ) /\ ( ( `' F " { .0. } ) = { N } /\ ( x e. A /\ y e. A ) /\ ( F ` x ) = ( F ` y ) ) ) -> x = y )
64	63	3anassrs 1253	. . . . 5 |- ( ( ( ( F e. ( R GrpHom S ) /\ ( `' F " { .0. } ) = { N } ) /\ ( x e. A /\ y e. A ) ) /\ ( F ` x ) = ( F ` y ) ) -> x = y )
65	64	ex 115	. . . 4 |- ( ( ( F e. ( R GrpHom S ) /\ ( `' F " { .0. } ) = { N } ) /\ ( x e. A /\ y e. A ) ) -> ( ( F ` x ) = ( F ` y ) -> x = y ) )
66	65	ralrimivva 2612	. . 3 |- ( ( F e. ( R GrpHom S ) /\ ( `' F " { .0. } ) = { N } ) -> A. x e. A A. y e. A ( ( F ` x ) = ( F ` y ) -> x = y ) )
67		dff13 5892	. . 3 |- ( F : A -1-1-> B <-> ( F : A --> B /\ A. x e. A A. y e. A ( ( F ` x ) = ( F ` y ) -> x = y ) ) )
68	41, 66, 67	sylanbrc 417	. 2 |- ( ( F e. ( R GrpHom S ) /\ ( `' F " { .0. } ) = { N } ) -> F : A -1-1-> B )
69	40, 68	impbida 598	1 |- ( F e. ( R GrpHom S ) -> ( F : A -1-1-> B <-> ( `' F " { .0. } ) = { N } ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1002   = wceq 1395   e. wcel 2200   A.wral 2508   C_ wss 3197   {csn 3666   `'ccnv 4718   "cima 4722   Fn wfn 5313   -->wf 5314   -1-1->wf1 5315   ` cfv 5318  (class class class)co 6001   Basecbs 13032   0gc0g 13289   Grpcgrp 13533   -gcsg 13535   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-1st 6286  df-2nd 6287  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-minusg 13537  df-sbg 13538  df-ghm 13778

This theorem is referenced by: (None)