Theorem kerf1ghm 13725

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 5505	. . . . . . . . . . 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 5722	. . . . . . . . . 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 3658	. . . . . . . . 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 13723	. . . . . . . . . 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 1206	. . . . . . . 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 1249	. . . . . 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 3660	. . . . 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 3207	. . 3 |- ( ( F e. ( R GrpHom S ) /\ F : A -1-1-> B ) -> ( `' F " { .0. } ) C_ { N } )
25		ghmgrp1 13696	. . . . . . 7 |- ( F e. ( R GrpHom S ) -> R e. Grp )
26	12, 14	grpidcl 13476	. . . . . . 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 13700	. . . . . . 7 |- ( F e. ( R GrpHom S ) -> ( F ` N ) = .0. )
29	12, 13	ghmf 13698	. . . . . . . . 9 |- ( F e. ( R GrpHom S ) -> F : A --> B )
30	29, 27	ffvelcdmd 5739	. . . . . . . 8 |- ( F e. ( R GrpHom S ) -> ( F ` N ) e. B )
31		elsng 3658	. . . . . . . 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 5445	. . . . . . 7 |- ( F : A --> B -> F Fn A )
35		elpreima 5722	. . . . . . 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 947	. . . . 5 |- ( F e. ( R GrpHom S ) -> N e. ( `' F " { .0. } ) )
38	37	snssd 3789	. . . 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 3218	. 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 1059	. . . . . . . . . 10 |- ( ( F e. ( R GrpHom S ) /\ ( ( `' F " { .0. } ) = { N } /\ ( x e. A /\ y e. A ) /\ ( F ` x ) = ( F ` y ) ) ) -> x e. A )
44		simpr2r 1060	. . . . . . . . . 10 |- ( ( F e. ( R GrpHom S ) /\ ( ( `' F " { .0. } ) = { N } /\ ( x e. A /\ y e. A ) /\ ( F ` x ) = ( F ` y ) ) ) -> y e. A )
45		simpr3 1008	. . . . . . . . . 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 2207	. . . . . . . . . . . 12 |- ( `' F " { .0. } ) = ( `' F " { .0. } )
47		eqid 2207	. . . . . . . . . . . 12 |- ( -g ` R ) = ( -g ` R )
48	12, 15, 46, 47	ghmeqker 13722	. . . . . . . . . . 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 1253	. . . . . . . . 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 1006	. . . . . . . . 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 2286	. . . . . . . 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 1001	. . . . . . . . . 10 |- ( ( ( `' F " { .0. } ) = { N } /\ ( x e. A /\ y e. A ) /\ ( F ` x ) = ( F ` y ) ) -> ( x e. A /\ y e. A ) )
54	12, 47	grpsubcl 13527	. . . . . . . . . . 11 |- ( ( R e. Grp /\ x e. A /\ y e. A ) -> ( x ( -g ` R ) y ) e. A )
55	54	3expb 1207	. . . . . . . . . 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 3658	. . . . . . . . 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 13533	. . . . . . . 8 |- ( ( R e. Grp /\ x e. A /\ y e. A ) -> ( ( x ( -g ` R ) y ) = N <-> x = y ) )
62	60, 43, 44, 61	syl3anc 1250	. . . . . . 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 1232	. . . . 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 2590	. . 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 5860	. . 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 596	1 |- ( F e. ( R GrpHom S ) -> ( F : A -1-1-> B <-> ( `' F " { .0. } ) = { N } ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 981   = wceq 1373   e. wcel 2178   A.wral 2486   C_ wss 3174   {csn 3643   `'ccnv 4692   "cima 4696   Fn wfn 5285   -->wf 5286   -1-1->wf1 5287   ` cfv 5290  (class class class)co 5967   Basecbs 12947   0gc0g 13203   Grpcgrp 13447   -gcsg 13449   GrpHom cghm 13691

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-cnex 8051  ax-resscn 8052  ax-1re 8054  ax-addrcl 8057

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-ral 2491  df-rex 2492  df-reu 2493  df-rmo 2494  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-riota 5922  df-ov 5970  df-oprab 5971  df-mpo 5972  df-1st 6249  df-2nd 6250  df-inn 9072  df-2 9130  df-ndx 12950  df-slot 12951  df-base 12953  df-plusg 13037  df-0g 13205  df-mgm 13303  df-sgrp 13349  df-mnd 13364  df-grp 13450  df-minusg 13451  df-sbg 13452  df-ghm 13692

This theorem is referenced by: (None)