Theorem kerf1ghm 13860

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 5544	. . . . . . . . . . 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 5766	. . . . . . . . . 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 3684	. . . . . . . . 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 13858	. . . . . . . . . 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 1229	. . . . . . . 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 1272	. . . . . 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 3686	. . . . 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 3233	. . 3 |- ( ( F e. ( R GrpHom S ) /\ F : A -1-1-> B ) -> ( `' F " { .0. } ) C_ { N } )
25		ghmgrp1 13831	. . . . . . 7 |- ( F e. ( R GrpHom S ) -> R e. Grp )
26	12, 14	grpidcl 13611	. . . . . . 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 13835	. . . . . . 7 |- ( F e. ( R GrpHom S ) -> ( F ` N ) = .0. )
29	12, 13	ghmf 13833	. . . . . . . . 9 |- ( F e. ( R GrpHom S ) -> F : A --> B )
30	29, 27	ffvelcdmd 5783	. . . . . . . 8 |- ( F e. ( R GrpHom S ) -> ( F ` N ) e. B )
31		elsng 3684	. . . . . . . 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 5482	. . . . . . 7 |- ( F : A --> B -> F Fn A )
35		elpreima 5766	. . . . . . 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 952	. . . . 5 |- ( F e. ( R GrpHom S ) -> N e. ( `' F " { .0. } ) )
38	37	snssd 3818	. . . 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 3244	. 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 1082	. . . . . . . . . 10 |- ( ( F e. ( R GrpHom S ) /\ ( ( `' F " { .0. } ) = { N } /\ ( x e. A /\ y e. A ) /\ ( F ` x ) = ( F ` y ) ) ) -> x e. A )
44		simpr2r 1083	. . . . . . . . . 10 |- ( ( F e. ( R GrpHom S ) /\ ( ( `' F " { .0. } ) = { N } /\ ( x e. A /\ y e. A ) /\ ( F ` x ) = ( F ` y ) ) ) -> y e. A )
45		simpr3 1031	. . . . . . . . . 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 2231	. . . . . . . . . . . 12 |- ( `' F " { .0. } ) = ( `' F " { .0. } )
47		eqid 2231	. . . . . . . . . . . 12 |- ( -g ` R ) = ( -g ` R )
48	12, 15, 46, 47	ghmeqker 13857	. . . . . . . . . . 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 1276	. . . . . . . . 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 1029	. . . . . . . . 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 2310	. . . . . . . 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 1024	. . . . . . . . . 10 |- ( ( ( `' F " { .0. } ) = { N } /\ ( x e. A /\ y e. A ) /\ ( F ` x ) = ( F ` y ) ) -> ( x e. A /\ y e. A ) )
54	12, 47	grpsubcl 13662	. . . . . . . . . . 11 |- ( ( R e. Grp /\ x e. A /\ y e. A ) -> ( x ( -g ` R ) y ) e. A )
55	54	3expb 1230	. . . . . . . . . 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 3684	. . . . . . . . 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 13668	. . . . . . . 8 |- ( ( R e. Grp /\ x e. A /\ y e. A ) -> ( ( x ( -g ` R ) y ) = N <-> x = y ) )
62	60, 43, 44, 61	syl3anc 1273	. . . . . . 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 1255	. . . . 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 2614	. . 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 5908	. . 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 600	1 |- ( F e. ( R GrpHom S ) -> ( F : A -1-1-> B <-> ( `' F " { .0. } ) = { N } ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1004   = wceq 1397   e. wcel 2202   A.wral 2510   C_ wss 3200   {csn 3669   `'ccnv 4724   "cima 4728   Fn wfn 5321   -->wf 5322   -1-1->wf1 5323   ` cfv 5326  (class class class)co 6017   Basecbs 13081   0gc0g 13338   Grpcgrp 13582   -gcsg 13584   GrpHom cghm 13826

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8122  ax-resscn 8123  ax-1re 8125  ax-addrcl 8128

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-inn 9143  df-2 9201  df-ndx 13084  df-slot 13085  df-base 13087  df-plusg 13172  df-0g 13340  df-mgm 13438  df-sgrp 13484  df-mnd 13499  df-grp 13585  df-minusg 13586  df-sbg 13587  df-ghm 13827

This theorem is referenced by: (None)