Theorem kerf1hrm 18743

Description: A ring 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.)

Hypotheses

Ref	Expression
kerf1hrm.a	|- A = ( Base ` R )
kerf1hrm.b	|- B = ( Base ` S )
kerf1hrm.n	|- N = ( 0g ` R )
kerf1hrm.0	|- .0. = ( 0g ` S )

Assertion

Ref	Expression
kerf1hrm	|- ( F e. ( R RingHom S ) -> ( F : A -1-1-> B <-> ( `' F " { .0. } ) = { N } ) )

Proof of Theorem kerf1hrm

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

Step	Hyp	Ref	Expression
1		simpl 473	. . . . . . 7 |- ( ( ( F e. ( R RingHom S ) /\ F : A -1-1-> B ) /\ x e. ( `' F " { .0. } ) ) -> ( F e. ( R RingHom S ) /\ F : A -1-1-> B ) )
2		f1fn 6102	. . . . . . . . . . 11 |- ( F : A -1-1-> B -> F Fn A )
3	2	adantl 482	. . . . . . . . . 10 |- ( ( F e. ( R RingHom S ) /\ F : A -1-1-> B ) -> F Fn A )
4		elpreima 6337	. . . . . . . . . 10 |- ( F Fn A -> ( x e. ( `' F " { .0. } ) <-> ( x e. A /\ ( F ` x ) e. { .0. } ) ) )
5	3, 4	syl 17	. . . . . . . . 9 |- ( ( F e. ( R RingHom S ) /\ F : A -1-1-> B ) -> ( x e. ( `' F " { .0. } ) <-> ( x e. A /\ ( F ` x ) e. { .0. } ) ) )
6	5	biimpa 501	. . . . . . . 8 |- ( ( ( F e. ( R RingHom S ) /\ F : A -1-1-> B ) /\ x e. ( `' F " { .0. } ) ) -> ( x e. A /\ ( F ` x ) e. { .0. } ) )
7	6	simpld 475	. . . . . . 7 |- ( ( ( F e. ( R RingHom S ) /\ F : A -1-1-> B ) /\ x e. ( `' F " { .0. } ) ) -> x e. A )
8	6	simprd 479	. . . . . . . 8 |- ( ( ( F e. ( R RingHom S ) /\ F : A -1-1-> B ) /\ x e. ( `' F " { .0. } ) ) -> ( F ` x ) e. { .0. } )
9		fvex 6201	. . . . . . . . 9 |- ( F ` x ) e. _V
10	9	elsn 4192	. . . . . . . 8 |- ( ( F ` x ) e. { .0. } <-> ( F ` x ) = .0. )
11	8, 10	sylib 208	. . . . . . 7 |- ( ( ( F e. ( R RingHom S ) /\ F : A -1-1-> B ) /\ x e. ( `' F " { .0. } ) ) -> ( F ` x ) = .0. )
12		kerf1hrm.a	. . . . . . . . . . 11 |- A = ( Base ` R )
13		kerf1hrm.b	. . . . . . . . . . 11 |- B = ( Base ` S )
14		kerf1hrm.0	. . . . . . . . . . 11 |- .0. = ( 0g ` S )
15		kerf1hrm.n	. . . . . . . . . . 11 |- N = ( 0g ` R )
16	12, 13, 14, 15	f1rhm0to0 18740	. . . . . . . . . 10 |- ( ( F e. ( R RingHom S ) /\ F : A -1-1-> B /\ x e. A ) -> ( ( F ` x ) = .0. <-> x = N ) )
17	16	biimpd 219	. . . . . . . . 9 |- ( ( F e. ( R RingHom S ) /\ F : A -1-1-> B /\ x e. A ) -> ( ( F ` x ) = .0. -> x = N ) )
18	17	3expa 1265	. . . . . . . 8 |- ( ( ( F e. ( R RingHom S ) /\ F : A -1-1-> B ) /\ x e. A ) -> ( ( F ` x ) = .0. -> x = N ) )
19	18	imp 445	. . . . . . 7 |- ( ( ( ( F e. ( R RingHom S ) /\ F : A -1-1-> B ) /\ x e. A ) /\ ( F ` x ) = .0. ) -> x = N )
20	1, 7, 11, 19	syl21anc 1325	. . . . . 6 |- ( ( ( F e. ( R RingHom S ) /\ F : A -1-1-> B ) /\ x e. ( `' F " { .0. } ) ) -> x = N )
21	20	ex 450	. . . . 5 |- ( ( F e. ( R RingHom S ) /\ F : A -1-1-> B ) -> ( x e. ( `' F " { .0. } ) -> x = N ) )
22		velsn 4193	. . . . 5 |- ( x e. { N } <-> x = N )
23	21, 22	syl6ibr 242	. . . 4 |- ( ( F e. ( R RingHom S ) /\ F : A -1-1-> B ) -> ( x e. ( `' F " { .0. } ) -> x e. { N } ) )
24	23	ssrdv 3609	. . 3 |- ( ( F e. ( R RingHom S ) /\ F : A -1-1-> B ) -> ( `' F " { .0. } ) C_ { N } )
25		rhmrcl1 18719	. . . . . . 7 |- ( F e. ( R RingHom S ) -> R e. Ring )
26		ringgrp 18552	. . . . . . 7 |- ( R e. Ring -> R e. Grp )
27	12, 15	grpidcl 17450	. . . . . . 7 |- ( R e. Grp -> N e. A )
28	25, 26, 27	3syl 18	. . . . . 6 |- ( F e. ( R RingHom S ) -> N e. A )
29		rhmghm 18725	. . . . . . . 8 |- ( F e. ( R RingHom S ) -> F e. ( R GrpHom S ) )
30	15, 14	ghmid 17666	. . . . . . . 8 |- ( F e. ( R GrpHom S ) -> ( F ` N ) = .0. )
31	29, 30	syl 17	. . . . . . 7 |- ( F e. ( R RingHom S ) -> ( F ` N ) = .0. )
32		fvex 6201	. . . . . . . 8 |- ( F ` N ) e. _V
33	32	elsn 4192	. . . . . . 7 |- ( ( F ` N ) e. { .0. } <-> ( F ` N ) = .0. )
34	31, 33	sylibr 224	. . . . . 6 |- ( F e. ( R RingHom S ) -> ( F ` N ) e. { .0. } )
35	12, 13	rhmf 18726	. . . . . . 7 |- ( F e. ( R RingHom S ) -> F : A --> B )
36		ffn 6045	. . . . . . 7 |- ( F : A --> B -> F Fn A )
37		elpreima 6337	. . . . . . 7 |- ( F Fn A -> ( N e. ( `' F " { .0. } ) <-> ( N e. A /\ ( F ` N ) e. { .0. } ) ) )
38	35, 36, 37	3syl 18	. . . . . 6 |- ( F e. ( R RingHom S ) -> ( N e. ( `' F " { .0. } ) <-> ( N e. A /\ ( F ` N ) e. { .0. } ) ) )
39	28, 34, 38	mpbir2and 957	. . . . 5 |- ( F e. ( R RingHom S ) -> N e. ( `' F " { .0. } ) )
40	39	snssd 4340	. . . 4 |- ( F e. ( R RingHom S ) -> { N } C_ ( `' F " { .0. } ) )
41	40	adantr 481	. . 3 |- ( ( F e. ( R RingHom S ) /\ F : A -1-1-> B ) -> { N } C_ ( `' F " { .0. } ) )
42	24, 41	eqssd 3620	. 2 |- ( ( F e. ( R RingHom S ) /\ F : A -1-1-> B ) -> ( `' F " { .0. } ) = { N } )
43	35	adantr 481	. . 3 |- ( ( F e. ( R RingHom S ) /\ ( `' F " { .0. } ) = { N } ) -> F : A --> B )
44	29	adantr 481	. . . . . . . . . 10 |- ( ( F e. ( R RingHom S ) /\ ( ( `' F " { .0. } ) = { N } /\ ( x e. A /\ y e. A ) /\ ( F ` x ) = ( F ` y ) ) ) -> F e. ( R GrpHom S ) )
45		simpr2l 1120	. . . . . . . . . 10 |- ( ( F e. ( R RingHom S ) /\ ( ( `' F " { .0. } ) = { N } /\ ( x e. A /\ y e. A ) /\ ( F ` x ) = ( F ` y ) ) ) -> x e. A )
46		simpr2r 1121	. . . . . . . . . 10 |- ( ( F e. ( R RingHom S ) /\ ( ( `' F " { .0. } ) = { N } /\ ( x e. A /\ y e. A ) /\ ( F ` x ) = ( F ` y ) ) ) -> y e. A )
47		simpr3 1069	. . . . . . . . . 10 |- ( ( F e. ( R RingHom S ) /\ ( ( `' F " { .0. } ) = { N } /\ ( x e. A /\ y e. A ) /\ ( F ` x ) = ( F ` y ) ) ) -> ( F ` x ) = ( F ` y ) )
48		eqid 2622	. . . . . . . . . . . 12 |- ( `' F " { .0. } ) = ( `' F " { .0. } )
49		eqid 2622	. . . . . . . . . . . 12 |- ( -g ` R ) = ( -g ` R )
50	12, 14, 48, 49	ghmeqker 17687	. . . . . . . . . . 11 |- ( ( F e. ( R GrpHom S ) /\ x e. A /\ y e. A ) -> ( ( F ` x ) = ( F ` y ) <-> ( x ( -g ` R ) y ) e. ( `' F " { .0. } ) ) )
51	50	biimpa 501	. . . . . . . . . 10 |- ( ( ( F e. ( R GrpHom S ) /\ x e. A /\ y e. A ) /\ ( F ` x ) = ( F ` y ) ) -> ( x ( -g ` R ) y ) e. ( `' F " { .0. } ) )
52	44, 45, 46, 47, 51	syl31anc 1329	. . . . . . . . 9 |- ( ( F e. ( R RingHom S ) /\ ( ( `' F " { .0. } ) = { N } /\ ( x e. A /\ y e. A ) /\ ( F ` x ) = ( F ` y ) ) ) -> ( x ( -g ` R ) y ) e. ( `' F " { .0. } ) )
53		simpr1 1067	. . . . . . . . 9 |- ( ( F e. ( R RingHom S ) /\ ( ( `' F " { .0. } ) = { N } /\ ( x e. A /\ y e. A ) /\ ( F ` x ) = ( F ` y ) ) ) -> ( `' F " { .0. } ) = { N } )
54	52, 53	eleqtrd 2703	. . . . . . . 8 |- ( ( F e. ( R RingHom S ) /\ ( ( `' F " { .0. } ) = { N } /\ ( x e. A /\ y e. A ) /\ ( F ` x ) = ( F ` y ) ) ) -> ( x ( -g ` R ) y ) e. { N } )
55		ovex 6678	. . . . . . . . 9 |- ( x ( -g ` R ) y ) e. _V
56	55	elsn 4192	. . . . . . . 8 |- ( ( x ( -g ` R ) y ) e. { N } <-> ( x ( -g ` R ) y ) = N )
57	54, 56	sylib 208	. . . . . . 7 |- ( ( F e. ( R RingHom S ) /\ ( ( `' F " { .0. } ) = { N } /\ ( x e. A /\ y e. A ) /\ ( F ` x ) = ( F ` y ) ) ) -> ( x ( -g ` R ) y ) = N )
58	25	adantr 481	. . . . . . . . 9 |- ( ( F e. ( R RingHom S ) /\ ( ( `' F " { .0. } ) = { N } /\ ( x e. A /\ y e. A ) /\ ( F ` x ) = ( F ` y ) ) ) -> R e. Ring )
59	58, 26	syl 17	. . . . . . . 8 |- ( ( F e. ( R RingHom S ) /\ ( ( `' F " { .0. } ) = { N } /\ ( x e. A /\ y e. A ) /\ ( F ` x ) = ( F ` y ) ) ) -> R e. Grp )
60	12, 15, 49	grpsubeq0 17501	. . . . . . . 8 |- ( ( R e. Grp /\ x e. A /\ y e. A ) -> ( ( x ( -g ` R ) y ) = N <-> x = y ) )
61	59, 45, 46, 60	syl3anc 1326	. . . . . . 7 |- ( ( F e. ( R RingHom S ) /\ ( ( `' F " { .0. } ) = { N } /\ ( x e. A /\ y e. A ) /\ ( F ` x ) = ( F ` y ) ) ) -> ( ( x ( -g ` R ) y ) = N <-> x = y ) )
62	57, 61	mpbid 222	. . . . . 6 |- ( ( F e. ( R RingHom S ) /\ ( ( `' F " { .0. } ) = { N } /\ ( x e. A /\ y e. A ) /\ ( F ` x ) = ( F ` y ) ) ) -> x = y )
63	62	3anassrs 1290	. . . . 5 |- ( ( ( ( F e. ( R RingHom S ) /\ ( `' F " { .0. } ) = { N } ) /\ ( x e. A /\ y e. A ) ) /\ ( F ` x ) = ( F ` y ) ) -> x = y )
64	63	ex 450	. . . 4 |- ( ( ( F e. ( R RingHom S ) /\ ( `' F " { .0. } ) = { N } ) /\ ( x e. A /\ y e. A ) ) -> ( ( F ` x ) = ( F ` y ) -> x = y ) )
65	64	ralrimivva 2971	. . 3 |- ( ( F e. ( R RingHom S ) /\ ( `' F " { .0. } ) = { N } ) -> A. x e. A A. y e. A ( ( F ` x ) = ( F ` y ) -> x = y ) )
66		dff13 6512	. . 3 |- ( F : A -1-1-> B <-> ( F : A --> B /\ A. x e. A A. y e. A ( ( F ` x ) = ( F ` y ) -> x = y ) ) )
67	43, 65, 66	sylanbrc 698	. 2 |- ( ( F e. ( R RingHom S ) /\ ( `' F " { .0. } ) = { N } ) -> F : A -1-1-> B )
68	42, 67	impbida 877	1 |- ( F e. ( R RingHom S ) -> ( F : A -1-1-> B <-> ( `' F " { .0. } ) = { N } ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   C_ wss 3574   {csn 4177   `'ccnv 5113   "cima 5117   Fn wfn 5883   -->wf 5884   -1-1->wf1 5885   ` cfv 5888  (class class class)co 6650   Basecbs 15857   0gc0g 16100   Grpcgrp 17422   -gcsg 17424   GrpHom cghm 17657   Ringcrg 18547   RingHom crh 18712

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-plusg 15954  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-mhm 17335  df-grp 17425  df-minusg 17426  df-sbg 17427  df-ghm 17658  df-mgp 18490  df-ur 18502  df-ring 18549  df-rnghom 18715

This theorem is referenced by:  zrhf1ker  30019