Theorem keridl 30404

Description: The kernel of a ring homomorphism is an ideal. (Contributed by Jeff Madsen, 3-Jan-2011.)

Hypotheses

Ref	Expression
keridl.1	|- G = ( 1st ` S )
keridl.2	|- Z = (GId ` G )

Assertion

Ref	Expression
keridl	|- ( ( R e. RingOps /\ S e. RingOps /\ F e. ( R RngHom S ) ) -> ( `' F " { Z } ) e. ( Idl ` R ) )

Proof of Theorem keridl

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

Step	Hyp	Ref	Expression
1		eqid 2443	. . . 4 |- ( 1st ` R ) = ( 1st ` R )
2		eqid 2443	. . . 4 |- ran ( 1st ` R ) = ran ( 1st ` R )
3		keridl.1	. . . 4 |- G = ( 1st ` S )
4		eqid 2443	. . . 4 |- ran G = ran G
5	1, 2, 3, 4	rngohomf 30344	. . 3 |- ( ( R e. RingOps /\ S e. RingOps /\ F e. ( R RngHom S ) ) -> F : ran ( 1st ` R ) --> ran G )
6		cnvimass 5347	. . . 4 |- ( `' F " { Z } ) C_ dom F
7		fdm 5725	. . . 4 |- ( F : ran ( 1st ` R ) --> ran G -> dom F = ran ( 1st ` R ) )
8	6, 7	syl5sseq 3537	. . 3 |- ( F : ran ( 1st ` R ) --> ran G -> ( `' F " { Z } ) C_ ran ( 1st ` R ) )
9	5, 8	syl 16	. 2 |- ( ( R e. RingOps /\ S e. RingOps /\ F e. ( R RngHom S ) ) -> ( `' F " { Z } ) C_ ran ( 1st ` R ) )
10		eqid 2443	. . . . 5 |- (GId ` ( 1st ` R ) ) = (GId ` ( 1st ` R ) )
11	1, 2, 10	rngo0cl 25272	. . . 4 |- ( R e. RingOps -> (GId ` ( 1st ` R ) ) e. ran ( 1st ` R ) )
12	11	3ad2ant1 1018	. . 3 |- ( ( R e. RingOps /\ S e. RingOps /\ F e. ( R RngHom S ) ) -> (GId ` ( 1st ` R ) ) e. ran ( 1st ` R ) )
13		keridl.2	. . . . 5 |- Z = (GId ` G )
14	1, 10, 3, 13	rngohom0 30350	. . . 4 |- ( ( R e. RingOps /\ S e. RingOps /\ F e. ( R RngHom S ) ) -> ( F ` (GId ` ( 1st ` R ) ) ) = Z )
15		fvex 5866	. . . . 5 |- ( F ` (GId ` ( 1st ` R ) ) ) e. _V
16	15	elsnc 4038	. . . 4 |- ( ( F ` (GId ` ( 1st ` R ) ) ) e. { Z } <-> ( F ` (GId ` ( 1st ` R ) ) ) = Z )
17	14, 16	sylibr 212	. . 3 |- ( ( R e. RingOps /\ S e. RingOps /\ F e. ( R RngHom S ) ) -> ( F ` (GId ` ( 1st ` R ) ) ) e. { Z } )
18		ffn 5721	. . . 4 |- ( F : ran ( 1st ` R ) --> ran G -> F Fn ran ( 1st ` R ) )
19		elpreima 5992	. . . 4 |- ( F Fn ran ( 1st ` R ) -> ( (GId ` ( 1st ` R ) ) e. ( `' F " { Z } ) <-> ( (GId ` ( 1st ` R ) ) e. ran ( 1st ` R ) /\ ( F ` (GId ` ( 1st ` R ) ) ) e. { Z } ) ) )
20	5, 18, 19	3syl 20	. . 3 |- ( ( R e. RingOps /\ S e. RingOps /\ F e. ( R RngHom S ) ) -> ( (GId ` ( 1st ` R ) ) e. ( `' F " { Z } ) <-> ( (GId ` ( 1st ` R ) ) e. ran ( 1st ` R ) /\ ( F ` (GId ` ( 1st ` R ) ) ) e. { Z } ) ) )
21	12, 17, 20	mpbir2and 922	. 2 |- ( ( R e. RingOps /\ S e. RingOps /\ F e. ( R RngHom S ) ) -> (GId ` ( 1st ` R ) ) e. ( `' F " { Z } ) )
22		an4 824	. . . . . . . 8 |- ( ( ( x e. ran ( 1st ` R ) /\ ( F ` x ) e. { Z } ) /\ ( y e. ran ( 1st ` R ) /\ ( F ` y ) e. { Z } ) ) <-> ( ( x e. ran ( 1st ` R ) /\ y e. ran ( 1st ` R ) ) /\ ( ( F ` x ) e. { Z } /\ ( F ` y ) e. { Z } ) ) )
23	1, 2, 3	rngohomadd 30347	. . . . . . . . . . . . . 14 |- ( ( ( R e. RingOps /\ S e. RingOps /\ F e. ( R RngHom S ) ) /\ ( x e. ran ( 1st ` R ) /\ y e. ran ( 1st ` R ) ) ) -> ( F ` ( x ( 1st ` R ) y ) ) = ( ( F ` x ) G ( F ` y ) ) )
24	23	adantr 465	. . . . . . . . . . . . 13 |- ( ( ( ( R e. RingOps /\ S e. RingOps /\ F e. ( R RngHom S ) ) /\ ( x e. ran ( 1st ` R ) /\ y e. ran ( 1st ` R ) ) ) /\ ( ( F ` x ) = Z /\ ( F ` y ) = Z ) ) -> ( F ` ( x ( 1st ` R ) y ) ) = ( ( F ` x ) G ( F ` y ) ) )
25		oveq12 6290	. . . . . . . . . . . . . 14 |- ( ( ( F ` x ) = Z /\ ( F ` y ) = Z ) -> ( ( F ` x ) G ( F ` y ) ) = ( Z G Z ) )
26	25	adantl 466	. . . . . . . . . . . . 13 |- ( ( ( ( R e. RingOps /\ S e. RingOps /\ F e. ( R RngHom S ) ) /\ ( x e. ran ( 1st ` R ) /\ y e. ran ( 1st ` R ) ) ) /\ ( ( F ` x ) = Z /\ ( F ` y ) = Z ) ) -> ( ( F ` x ) G ( F ` y ) ) = ( Z G Z ) )
27	3	rngogrpo 25264	. . . . . . . . . . . . . . . 16 |- ( S e. RingOps -> G e. GrpOp )
28	4, 13	grpoidcl 25091	. . . . . . . . . . . . . . . . 17 |- ( G e. GrpOp -> Z e. ran G )
29	4, 13	grpolid 25093	. . . . . . . . . . . . . . . . 17 |- ( ( G e. GrpOp /\ Z e. ran G ) -> ( Z G Z ) = Z )
30	28, 29	mpdan 668	. . . . . . . . . . . . . . . 16 |- ( G e. GrpOp -> ( Z G Z ) = Z )
31	27, 30	syl 16	. . . . . . . . . . . . . . 15 |- ( S e. RingOps -> ( Z G Z ) = Z )
32	31	3ad2ant2 1019	. . . . . . . . . . . . . 14 |- ( ( R e. RingOps /\ S e. RingOps /\ F e. ( R RngHom S ) ) -> ( Z G Z ) = Z )
33	32	ad2antrr 725	. . . . . . . . . . . . 13 |- ( ( ( ( R e. RingOps /\ S e. RingOps /\ F e. ( R RngHom S ) ) /\ ( x e. ran ( 1st ` R ) /\ y e. ran ( 1st ` R ) ) ) /\ ( ( F ` x ) = Z /\ ( F ` y ) = Z ) ) -> ( Z G Z ) = Z )
34	24, 26, 33	3eqtrd 2488	. . . . . . . . . . . 12 |- ( ( ( ( R e. RingOps /\ S e. RingOps /\ F e. ( R RngHom S ) ) /\ ( x e. ran ( 1st ` R ) /\ y e. ran ( 1st ` R ) ) ) /\ ( ( F ` x ) = Z /\ ( F ` y ) = Z ) ) -> ( F ` ( x ( 1st ` R ) y ) ) = Z )
35	34	ex 434	. . . . . . . . . . 11 |- ( ( ( R e. RingOps /\ S e. RingOps /\ F e. ( R RngHom S ) ) /\ ( x e. ran ( 1st ` R ) /\ y e. ran ( 1st ` R ) ) ) -> ( ( ( F ` x ) = Z /\ ( F ` y ) = Z ) -> ( F ` ( x ( 1st ` R ) y ) ) = Z ) )
36		fvex 5866	. . . . . . . . . . . . 13 |- ( F ` x ) e. _V
37	36	elsnc 4038	. . . . . . . . . . . 12 |- ( ( F ` x ) e. { Z } <-> ( F ` x ) = Z )
38		fvex 5866	. . . . . . . . . . . . 13 |- ( F ` y ) e. _V
39	38	elsnc 4038	. . . . . . . . . . . 12 |- ( ( F ` y ) e. { Z } <-> ( F ` y ) = Z )
40	37, 39	anbi12i 697	. . . . . . . . . . 11 |- ( ( ( F ` x ) e. { Z } /\ ( F ` y ) e. { Z } ) <-> ( ( F ` x ) = Z /\ ( F ` y ) = Z ) )
41		fvex 5866	. . . . . . . . . . . 12 |- ( F ` ( x ( 1st ` R ) y ) ) e. _V
42	41	elsnc 4038	. . . . . . . . . . 11 |- ( ( F ` ( x ( 1st ` R ) y ) ) e. { Z } <-> ( F ` ( x ( 1st ` R ) y ) ) = Z )
43	35, 40, 42	3imtr4g 270	. . . . . . . . . 10 |- ( ( ( R e. RingOps /\ S e. RingOps /\ F e. ( R RngHom S ) ) /\ ( x e. ran ( 1st ` R ) /\ y e. ran ( 1st ` R ) ) ) -> ( ( ( F ` x ) e. { Z } /\ ( F ` y ) e. { Z } ) -> ( F ` ( x ( 1st ` R ) y ) ) e. { Z } ) )
44	43	imdistanda 693	. . . . . . . . 9 |- ( ( R e. RingOps /\ S e. RingOps /\ F e. ( R RngHom S ) ) -> ( ( ( x e. ran ( 1st ` R ) /\ y e. ran ( 1st ` R ) ) /\ ( ( F ` x ) e. { Z } /\ ( F ` y ) e. { Z } ) ) -> ( ( x e. ran ( 1st ` R ) /\ y e. ran ( 1st ` R ) ) /\ ( F ` ( x ( 1st ` R ) y ) ) e. { Z } ) ) )
45	1, 2	rngogcl 25265	. . . . . . . . . . . 12 |- ( ( R e. RingOps /\ x e. ran ( 1st ` R ) /\ y e. ran ( 1st ` R ) ) -> ( x ( 1st ` R ) y ) e. ran ( 1st ` R ) )
46	45	3expib 1200	. . . . . . . . . . 11 |- ( R e. RingOps -> ( ( x e. ran ( 1st ` R ) /\ y e. ran ( 1st ` R ) ) -> ( x ( 1st ` R ) y ) e. ran ( 1st ` R ) ) )
47	46	3ad2ant1 1018	. . . . . . . . . 10 |- ( ( R e. RingOps /\ S e. RingOps /\ F e. ( R RngHom S ) ) -> ( ( x e. ran ( 1st ` R ) /\ y e. ran ( 1st ` R ) ) -> ( x ( 1st ` R ) y ) e. ran ( 1st ` R ) ) )
48	47	anim1d 564	. . . . . . . . 9 |- ( ( R e. RingOps /\ S e. RingOps /\ F e. ( R RngHom S ) ) -> ( ( ( x e. ran ( 1st ` R ) /\ y e. ran ( 1st ` R ) ) /\ ( F ` ( x ( 1st ` R ) y ) ) e. { Z } ) -> ( ( x ( 1st ` R ) y ) e. ran ( 1st ` R ) /\ ( F ` ( x ( 1st ` R ) y ) ) e. { Z } ) ) )
49	44, 48	syld 44	. . . . . . . 8 |- ( ( R e. RingOps /\ S e. RingOps /\ F e. ( R RngHom S ) ) -> ( ( ( x e. ran ( 1st ` R ) /\ y e. ran ( 1st ` R ) ) /\ ( ( F ` x ) e. { Z } /\ ( F ` y ) e. { Z } ) ) -> ( ( x ( 1st ` R ) y ) e. ran ( 1st ` R ) /\ ( F ` ( x ( 1st ` R ) y ) ) e. { Z } ) ) )
50	22, 49	syl5bi 217	. . . . . . 7 |- ( ( R e. RingOps /\ S e. RingOps /\ F e. ( R RngHom S ) ) -> ( ( ( x e. ran ( 1st ` R ) /\ ( F ` x ) e. { Z } ) /\ ( y e. ran ( 1st ` R ) /\ ( F ` y ) e. { Z } ) ) -> ( ( x ( 1st ` R ) y ) e. ran ( 1st ` R ) /\ ( F ` ( x ( 1st ` R ) y ) ) e. { Z } ) ) )
51		elpreima 5992	. . . . . . . . 9 |- ( F Fn ran ( 1st ` R ) -> ( x e. ( `' F " { Z } ) <-> ( x e. ran ( 1st ` R ) /\ ( F ` x ) e. { Z } ) ) )
52	5, 18, 51	3syl 20	. . . . . . . 8 |- ( ( R e. RingOps /\ S e. RingOps /\ F e. ( R RngHom S ) ) -> ( x e. ( `' F " { Z } ) <-> ( x e. ran ( 1st ` R ) /\ ( F ` x ) e. { Z } ) ) )
53		elpreima 5992	. . . . . . . . 9 |- ( F Fn ran ( 1st ` R ) -> ( y e. ( `' F " { Z } ) <-> ( y e. ran ( 1st ` R ) /\ ( F ` y ) e. { Z } ) ) )
54	5, 18, 53	3syl 20	. . . . . . . 8 |- ( ( R e. RingOps /\ S e. RingOps /\ F e. ( R RngHom S ) ) -> ( y e. ( `' F " { Z } ) <-> ( y e. ran ( 1st ` R ) /\ ( F ` y ) e. { Z } ) ) )
55	52, 54	anbi12d 710	. . . . . . 7 |- ( ( R e. RingOps /\ S e. RingOps /\ F e. ( R RngHom S ) ) -> ( ( x e. ( `' F " { Z } ) /\ y e. ( `' F " { Z } ) ) <-> ( ( x e. ran ( 1st ` R ) /\ ( F ` x ) e. { Z } ) /\ ( y e. ran ( 1st ` R ) /\ ( F ` y ) e. { Z } ) ) ) )
56		elpreima 5992	. . . . . . . 8 |- ( F Fn ran ( 1st ` R ) -> ( ( x ( 1st ` R ) y ) e. ( `' F " { Z } ) <-> ( ( x ( 1st ` R ) y ) e. ran ( 1st ` R ) /\ ( F ` ( x ( 1st ` R ) y ) ) e. { Z } ) ) )
57	5, 18, 56	3syl 20	. . . . . . 7 |- ( ( R e. RingOps /\ S e. RingOps /\ F e. ( R RngHom S ) ) -> ( ( x ( 1st ` R ) y ) e. ( `' F " { Z } ) <-> ( ( x ( 1st ` R ) y ) e. ran ( 1st ` R ) /\ ( F ` ( x ( 1st ` R ) y ) ) e. { Z } ) ) )
58	50, 55, 57	3imtr4d 268	. . . . . 6 |- ( ( R e. RingOps /\ S e. RingOps /\ F e. ( R RngHom S ) ) -> ( ( x e. ( `' F " { Z } ) /\ y e. ( `' F " { Z } ) ) -> ( x ( 1st ` R ) y ) e. ( `' F " { Z } ) ) )
59	58	impl 620	. . . . 5 |- ( ( ( ( R e. RingOps /\ S e. RingOps /\ F e. ( R RngHom S ) ) /\ x e. ( `' F " { Z } ) ) /\ y e. ( `' F " { Z } ) ) -> ( x ( 1st ` R ) y ) e. ( `' F " { Z } ) )
60	59	ralrimiva 2857	. . . 4 |- ( ( ( R e. RingOps /\ S e. RingOps /\ F e. ( R RngHom S ) ) /\ x e. ( `' F " { Z } ) ) -> A. y e. ( `' F " { Z } ) ( x ( 1st ` R ) y ) e. ( `' F " { Z } ) )
61	37	anbi2i 694	. . . . . . 7 |- ( ( x e. ran ( 1st ` R ) /\ ( F ` x ) e. { Z } ) <-> ( x e. ran ( 1st ` R ) /\ ( F ` x ) = Z ) )
62		eqid 2443	. . . . . . . . . . . . . . . 16 |- ( 2nd ` R ) = ( 2nd ` R )
63	1, 62, 2	rngocl 25256	. . . . . . . . . . . . . . 15 |- ( ( R e. RingOps /\ z e. ran ( 1st ` R ) /\ x e. ran ( 1st ` R ) ) -> ( z ( 2nd ` R ) x ) e. ran ( 1st ` R ) )
64	63	3expb 1198	. . . . . . . . . . . . . 14 |- ( ( R e. RingOps /\ ( z e. ran ( 1st ` R ) /\ x e. ran ( 1st ` R ) ) ) -> ( z ( 2nd ` R ) x ) e. ran ( 1st ` R ) )
65	64	3ad2antl1 1159	. . . . . . . . . . . . 13 |- ( ( ( R e. RingOps /\ S e. RingOps /\ F e. ( R RngHom S ) ) /\ ( z e. ran ( 1st ` R ) /\ x e. ran ( 1st ` R ) ) ) -> ( z ( 2nd ` R ) x ) e. ran ( 1st ` R ) )
66	65	anass1rs 807	. . . . . . . . . . . 12 |- ( ( ( ( R e. RingOps /\ S e. RingOps /\ F e. ( R RngHom S ) ) /\ x e. ran ( 1st ` R ) ) /\ z e. ran ( 1st ` R ) ) -> ( z ( 2nd ` R ) x ) e. ran ( 1st ` R ) )
67	66	adantlrr 720	. . . . . . . . . . 11 |- ( ( ( ( R e. RingOps /\ S e. RingOps /\ F e. ( R RngHom S ) ) /\ ( x e. ran ( 1st ` R ) /\ ( F ` x ) = Z ) ) /\ z e. ran ( 1st ` R ) ) -> ( z ( 2nd ` R ) x ) e. ran ( 1st ` R ) )
68		eqid 2443	. . . . . . . . . . . . . . . 16 |- ( 2nd ` S ) = ( 2nd ` S )
69	1, 2, 62, 68	rngohommul 30348	. . . . . . . . . . . . . . 15 |- ( ( ( R e. RingOps /\ S e. RingOps /\ F e. ( R RngHom S ) ) /\ ( z e. ran ( 1st ` R ) /\ x e. ran ( 1st ` R ) ) ) -> ( F ` ( z ( 2nd ` R ) x ) ) = ( ( F ` z ) ( 2nd ` S ) ( F ` x ) ) )
70	69	anass1rs 807	. . . . . . . . . . . . . 14 |- ( ( ( ( R e. RingOps /\ S e. RingOps /\ F e. ( R RngHom S ) ) /\ x e. ran ( 1st ` R ) ) /\ z e. ran ( 1st ` R ) ) -> ( F ` ( z ( 2nd ` R ) x ) ) = ( ( F ` z ) ( 2nd ` S ) ( F ` x ) ) )
71	70	adantlrr 720	. . . . . . . . . . . . 13 |- ( ( ( ( R e. RingOps /\ S e. RingOps /\ F e. ( R RngHom S ) ) /\ ( x e. ran ( 1st ` R ) /\ ( F ` x ) = Z ) ) /\ z e. ran ( 1st ` R ) ) -> ( F ` ( z ( 2nd ` R ) x ) ) = ( ( F ` z ) ( 2nd ` S ) ( F ` x ) ) )
72		oveq2 6289	. . . . . . . . . . . . . . 15 |- ( ( F ` x ) = Z -> ( ( F ` z ) ( 2nd ` S ) ( F ` x ) ) = ( ( F ` z ) ( 2nd ` S ) Z ) )
73	72	adantl 466	. . . . . . . . . . . . . 14 |- ( ( x e. ran ( 1st ` R ) /\ ( F ` x ) = Z ) -> ( ( F ` z ) ( 2nd ` S ) ( F ` x ) ) = ( ( F ` z ) ( 2nd ` S ) Z ) )
74	73	ad2antlr 726	. . . . . . . . . . . . 13 |- ( ( ( ( R e. RingOps /\ S e. RingOps /\ F e. ( R RngHom S ) ) /\ ( x e. ran ( 1st ` R ) /\ ( F ` x ) = Z ) ) /\ z e. ran ( 1st ` R ) ) -> ( ( F ` z ) ( 2nd ` S ) ( F ` x ) ) = ( ( F ` z ) ( 2nd ` S ) Z ) )
75	1, 2, 3, 4	rngohomcl 30345	. . . . . . . . . . . . . . 15 |- ( ( ( R e. RingOps /\ S e. RingOps /\ F e. ( R RngHom S ) ) /\ z e. ran ( 1st ` R ) ) -> ( F ` z ) e. ran G )
76	13, 4, 3, 68	rngorz 25276	. . . . . . . . . . . . . . . 16 |- ( ( S e. RingOps /\ ( F ` z ) e. ran G ) -> ( ( F ` z ) ( 2nd ` S ) Z ) = Z )
77	76	3ad2antl2 1160	. . . . . . . . . . . . . . 15 |- ( ( ( R e. RingOps /\ S e. RingOps /\ F e. ( R RngHom S ) ) /\ ( F ` z ) e. ran G ) -> ( ( F ` z ) ( 2nd ` S ) Z ) = Z )
78	75, 77	syldan 470	. . . . . . . . . . . . . 14 |- ( ( ( R e. RingOps /\ S e. RingOps /\ F e. ( R RngHom S ) ) /\ z e. ran ( 1st ` R ) ) -> ( ( F ` z ) ( 2nd ` S ) Z ) = Z )
79	78	adantlr 714	. . . . . . . . . . . . 13 |- ( ( ( ( R e. RingOps /\ S e. RingOps /\ F e. ( R RngHom S ) ) /\ ( x e. ran ( 1st ` R ) /\ ( F ` x ) = Z ) ) /\ z e. ran ( 1st ` R ) ) -> ( ( F ` z ) ( 2nd ` S ) Z ) = Z )
80	71, 74, 79	3eqtrd 2488	. . . . . . . . . . . 12 |- ( ( ( ( R e. RingOps /\ S e. RingOps /\ F e. ( R RngHom S ) ) /\ ( x e. ran ( 1st ` R ) /\ ( F ` x ) = Z ) ) /\ z e. ran ( 1st ` R ) ) -> ( F ` ( z ( 2nd ` R ) x ) ) = Z )
81		fvex 5866	. . . . . . . . . . . . 13 |- ( F ` ( z ( 2nd ` R ) x ) ) e. _V
82	81	elsnc 4038	. . . . . . . . . . . 12 |- ( ( F ` ( z ( 2nd ` R ) x ) ) e. { Z } <-> ( F ` ( z ( 2nd ` R ) x ) ) = Z )
83	80, 82	sylibr 212	. . . . . . . . . . 11 |- ( ( ( ( R e. RingOps /\ S e. RingOps /\ F e. ( R RngHom S ) ) /\ ( x e. ran ( 1st ` R ) /\ ( F ` x ) = Z ) ) /\ z e. ran ( 1st ` R ) ) -> ( F ` ( z ( 2nd ` R ) x ) ) e. { Z } )
84		elpreima 5992	. . . . . . . . . . . . 13 |- ( F Fn ran ( 1st ` R ) -> ( ( z ( 2nd ` R ) x ) e. ( `' F " { Z } ) <-> ( ( z ( 2nd ` R ) x ) e. ran ( 1st ` R ) /\ ( F ` ( z ( 2nd ` R ) x ) ) e. { Z } ) ) )
85	5, 18, 84	3syl 20	. . . . . . . . . . . 12 |- ( ( R e. RingOps /\ S e. RingOps /\ F e. ( R RngHom S ) ) -> ( ( z ( 2nd ` R ) x ) e. ( `' F " { Z } ) <-> ( ( z ( 2nd ` R ) x ) e. ran ( 1st ` R ) /\ ( F ` ( z ( 2nd ` R ) x ) ) e. { Z } ) ) )
86	85	ad2antrr 725	. . . . . . . . . . 11 |- ( ( ( ( R e. RingOps /\ S e. RingOps /\ F e. ( R RngHom S ) ) /\ ( x e. ran ( 1st ` R ) /\ ( F ` x ) = Z ) ) /\ z e. ran ( 1st ` R ) ) -> ( ( z ( 2nd ` R ) x ) e. ( `' F " { Z } ) <-> ( ( z ( 2nd ` R ) x ) e. ran ( 1st ` R ) /\ ( F ` ( z ( 2nd ` R ) x ) ) e. { Z } ) ) )
87	67, 83, 86	mpbir2and 922	. . . . . . . . . 10 |- ( ( ( ( R e. RingOps /\ S e. RingOps /\ F e. ( R RngHom S ) ) /\ ( x e. ran ( 1st ` R ) /\ ( F ` x ) = Z ) ) /\ z e. ran ( 1st ` R ) ) -> ( z ( 2nd ` R ) x ) e. ( `' F " { Z } ) )
88	1, 62, 2	rngocl 25256	. . . . . . . . . . . . . . 15 |- ( ( R e. RingOps /\ x e. ran ( 1st ` R ) /\ z e. ran ( 1st ` R ) ) -> ( x ( 2nd ` R ) z ) e. ran ( 1st ` R ) )
89	88	3expb 1198	. . . . . . . . . . . . . 14 |- ( ( R e. RingOps /\ ( x e. ran ( 1st ` R ) /\ z e. ran ( 1st ` R ) ) ) -> ( x ( 2nd ` R ) z ) e. ran ( 1st ` R ) )
90	89	3ad2antl1 1159	. . . . . . . . . . . . 13 |- ( ( ( R e. RingOps /\ S e. RingOps /\ F e. ( R RngHom S ) ) /\ ( x e. ran ( 1st ` R ) /\ z e. ran ( 1st ` R ) ) ) -> ( x ( 2nd ` R ) z ) e. ran ( 1st ` R ) )
91	90	anassrs 648	. . . . . . . . . . . 12 |- ( ( ( ( R e. RingOps /\ S e. RingOps /\ F e. ( R RngHom S ) ) /\ x e. ran ( 1st ` R ) ) /\ z e. ran ( 1st ` R ) ) -> ( x ( 2nd ` R ) z ) e. ran ( 1st ` R ) )
92	91	adantlrr 720	. . . . . . . . . . 11 |- ( ( ( ( R e. RingOps /\ S e. RingOps /\ F e. ( R RngHom S ) ) /\ ( x e. ran ( 1st ` R ) /\ ( F ` x ) = Z ) ) /\ z e. ran ( 1st ` R ) ) -> ( x ( 2nd ` R ) z ) e. ran ( 1st ` R ) )
93	1, 2, 62, 68	rngohommul 30348	. . . . . . . . . . . . . . 15 |- ( ( ( R e. RingOps /\ S e. RingOps /\ F e. ( R RngHom S ) ) /\ ( x e. ran ( 1st ` R ) /\ z e. ran ( 1st ` R ) ) ) -> ( F ` ( x ( 2nd ` R ) z ) ) = ( ( F ` x ) ( 2nd ` S ) ( F ` z ) ) )
94	93	anassrs 648	. . . . . . . . . . . . . 14 |- ( ( ( ( R e. RingOps /\ S e. RingOps /\ F e. ( R RngHom S ) ) /\ x e. ran ( 1st ` R ) ) /\ z e. ran ( 1st ` R ) ) -> ( F ` ( x ( 2nd ` R ) z ) ) = ( ( F ` x ) ( 2nd ` S ) ( F ` z ) ) )
95	94	adantlrr 720	. . . . . . . . . . . . 13 |- ( ( ( ( R e. RingOps /\ S e. RingOps /\ F e. ( R RngHom S ) ) /\ ( x e. ran ( 1st ` R ) /\ ( F ` x ) = Z ) ) /\ z e. ran ( 1st ` R ) ) -> ( F ` ( x ( 2nd ` R ) z ) ) = ( ( F ` x ) ( 2nd ` S ) ( F ` z ) ) )
96		oveq1 6288	. . . . . . . . . . . . . . 15 |- ( ( F ` x ) = Z -> ( ( F ` x ) ( 2nd ` S ) ( F ` z ) ) = ( Z ( 2nd ` S ) ( F ` z ) ) )
97	96	adantl 466	. . . . . . . . . . . . . 14 |- ( ( x e. ran ( 1st ` R ) /\ ( F ` x ) = Z ) -> ( ( F ` x ) ( 2nd ` S ) ( F ` z ) ) = ( Z ( 2nd ` S ) ( F ` z ) ) )
98	97	ad2antlr 726	. . . . . . . . . . . . 13 |- ( ( ( ( R e. RingOps /\ S e. RingOps /\ F e. ( R RngHom S ) ) /\ ( x e. ran ( 1st ` R ) /\ ( F ` x ) = Z ) ) /\ z e. ran ( 1st ` R ) ) -> ( ( F ` x ) ( 2nd ` S ) ( F ` z ) ) = ( Z ( 2nd ` S ) ( F ` z ) ) )
99	13, 4, 3, 68	rngolz 25275	. . . . . . . . . . . . . . . 16 |- ( ( S e. RingOps /\ ( F ` z ) e. ran G ) -> ( Z ( 2nd ` S ) ( F ` z ) ) = Z )
100	99	3ad2antl2 1160	. . . . . . . . . . . . . . 15 |- ( ( ( R e. RingOps /\ S e. RingOps /\ F e. ( R RngHom S ) ) /\ ( F ` z ) e. ran G ) -> ( Z ( 2nd ` S ) ( F ` z ) ) = Z )
101	75, 100	syldan 470	. . . . . . . . . . . . . 14 |- ( ( ( R e. RingOps /\ S e. RingOps /\ F e. ( R RngHom S ) ) /\ z e. ran ( 1st ` R ) ) -> ( Z ( 2nd ` S ) ( F ` z ) ) = Z )
102	101	adantlr 714	. . . . . . . . . . . . 13 |- ( ( ( ( R e. RingOps /\ S e. RingOps /\ F e. ( R RngHom S ) ) /\ ( x e. ran ( 1st ` R ) /\ ( F ` x ) = Z ) ) /\ z e. ran ( 1st ` R ) ) -> ( Z ( 2nd ` S ) ( F ` z ) ) = Z )
103	95, 98, 102	3eqtrd 2488	. . . . . . . . . . . 12 |- ( ( ( ( R e. RingOps /\ S e. RingOps /\ F e. ( R RngHom S ) ) /\ ( x e. ran ( 1st ` R ) /\ ( F ` x ) = Z ) ) /\ z e. ran ( 1st ` R ) ) -> ( F ` ( x ( 2nd ` R ) z ) ) = Z )
104		fvex 5866	. . . . . . . . . . . . 13 |- ( F ` ( x ( 2nd ` R ) z ) ) e. _V
105	104	elsnc 4038	. . . . . . . . . . . 12 |- ( ( F ` ( x ( 2nd ` R ) z ) ) e. { Z } <-> ( F ` ( x ( 2nd ` R ) z ) ) = Z )
106	103, 105	sylibr 212	. . . . . . . . . . 11 |- ( ( ( ( R e. RingOps /\ S e. RingOps /\ F e. ( R RngHom S ) ) /\ ( x e. ran ( 1st ` R ) /\ ( F ` x ) = Z ) ) /\ z e. ran ( 1st ` R ) ) -> ( F ` ( x ( 2nd ` R ) z ) ) e. { Z } )
107		elpreima 5992	. . . . . . . . . . . . 13 |- ( F Fn ran ( 1st ` R ) -> ( ( x ( 2nd ` R ) z ) e. ( `' F " { Z } ) <-> ( ( x ( 2nd ` R ) z ) e. ran ( 1st ` R ) /\ ( F ` ( x ( 2nd ` R ) z ) ) e. { Z } ) ) )
108	5, 18, 107	3syl 20	. . . . . . . . . . . 12 |- ( ( R e. RingOps /\ S e. RingOps /\ F e. ( R RngHom S ) ) -> ( ( x ( 2nd ` R ) z ) e. ( `' F " { Z } ) <-> ( ( x ( 2nd ` R ) z ) e. ran ( 1st ` R ) /\ ( F ` ( x ( 2nd ` R ) z ) ) e. { Z } ) ) )
109	108	ad2antrr 725	. . . . . . . . . . 11 |- ( ( ( ( R e. RingOps /\ S e. RingOps /\ F e. ( R RngHom S ) ) /\ ( x e. ran ( 1st ` R ) /\ ( F ` x ) = Z ) ) /\ z e. ran ( 1st ` R ) ) -> ( ( x ( 2nd ` R ) z ) e. ( `' F " { Z } ) <-> ( ( x ( 2nd ` R ) z ) e. ran ( 1st ` R ) /\ ( F ` ( x ( 2nd ` R ) z ) ) e. { Z } ) ) )
110	92, 106, 109	mpbir2and 922	. . . . . . . . . 10 |- ( ( ( ( R e. RingOps /\ S e. RingOps /\ F e. ( R RngHom S ) ) /\ ( x e. ran ( 1st ` R ) /\ ( F ` x ) = Z ) ) /\ z e. ran ( 1st ` R ) ) -> ( x ( 2nd ` R ) z ) e. ( `' F " { Z } ) )
111	87, 110	jca 532	. . . . . . . . 9 |- ( ( ( ( R e. RingOps /\ S e. RingOps /\ F e. ( R RngHom S ) ) /\ ( x e. ran ( 1st ` R ) /\ ( F ` x ) = Z ) ) /\ z e. ran ( 1st ` R ) ) -> ( ( z ( 2nd ` R ) x ) e. ( `' F " { Z } ) /\ ( x ( 2nd ` R ) z ) e. ( `' F " { Z } ) ) )
112	111	ralrimiva 2857	. . . . . . . 8 |- ( ( ( R e. RingOps /\ S e. RingOps /\ F e. ( R RngHom S ) ) /\ ( x e. ran ( 1st ` R ) /\ ( F ` x ) = Z ) ) -> A. z e. ran ( 1st ` R ) ( ( z ( 2nd ` R ) x ) e. ( `' F " { Z } ) /\ ( x ( 2nd ` R ) z ) e. ( `' F " { Z } ) ) )
113	112	ex 434	. . . . . . 7 |- ( ( R e. RingOps /\ S e. RingOps /\ F e. ( R RngHom S ) ) -> ( ( x e. ran ( 1st ` R ) /\ ( F ` x ) = Z ) -> A. z e. ran ( 1st ` R ) ( ( z ( 2nd ` R ) x ) e. ( `' F " { Z } ) /\ ( x ( 2nd ` R ) z ) e. ( `' F " { Z } ) ) ) )
114	61, 113	syl5bi 217	. . . . . 6 |- ( ( R e. RingOps /\ S e. RingOps /\ F e. ( R RngHom S ) ) -> ( ( x e. ran ( 1st ` R ) /\ ( F ` x ) e. { Z } ) -> A. z e. ran ( 1st ` R ) ( ( z ( 2nd ` R ) x ) e. ( `' F " { Z } ) /\ ( x ( 2nd ` R ) z ) e. ( `' F " { Z } ) ) ) )
115	52, 114	sylbid 215	. . . . 5 |- ( ( R e. RingOps /\ S e. RingOps /\ F e. ( R RngHom S ) ) -> ( x e. ( `' F " { Z } ) -> A. z e. ran ( 1st ` R ) ( ( z ( 2nd ` R ) x ) e. ( `' F " { Z } ) /\ ( x ( 2nd ` R ) z ) e. ( `' F " { Z } ) ) ) )
116	115	imp 429	. . . 4 |- ( ( ( R e. RingOps /\ S e. RingOps /\ F e. ( R RngHom S ) ) /\ x e. ( `' F " { Z } ) ) -> A. z e. ran ( 1st ` R ) ( ( z ( 2nd ` R ) x ) e. ( `' F " { Z } ) /\ ( x ( 2nd ` R ) z ) e. ( `' F " { Z } ) ) )
117	60, 116	jca 532	. . 3 |- ( ( ( R e. RingOps /\ S e. RingOps /\ F e. ( R RngHom S ) ) /\ x e. ( `' F " { Z } ) ) -> ( A. y e. ( `' F " { Z } ) ( x ( 1st ` R ) y ) e. ( `' F " { Z } ) /\ A. z e. ran ( 1st ` R ) ( ( z ( 2nd ` R ) x ) e. ( `' F " { Z } ) /\ ( x ( 2nd ` R ) z ) e. ( `' F " { Z } ) ) ) )
118	117	ralrimiva 2857	. 2 |- ( ( R e. RingOps /\ S e. RingOps /\ F e. ( R RngHom S ) ) -> A. x e. ( `' F " { Z } ) ( A. y e. ( `' F " { Z } ) ( x ( 1st ` R ) y ) e. ( `' F " { Z } ) /\ A. z e. ran ( 1st ` R ) ( ( z ( 2nd ` R ) x ) e. ( `' F " { Z } ) /\ ( x ( 2nd ` R ) z ) e. ( `' F " { Z } ) ) ) )
119	1, 62, 2, 10	isidl 30386	. . 3 |- ( R e. RingOps -> ( ( `' F " { Z } ) e. ( Idl ` R ) <-> ( ( `' F " { Z } ) C_ ran ( 1st ` R ) /\ (GId ` ( 1st ` R ) ) e. ( `' F " { Z } ) /\ A. x e. ( `' F " { Z } ) ( A. y e. ( `' F " { Z } ) ( x ( 1st ` R ) y ) e. ( `' F " { Z } ) /\ A. z e. ran ( 1st ` R ) ( ( z ( 2nd ` R ) x ) e. ( `' F " { Z } ) /\ ( x ( 2nd ` R ) z ) e. ( `' F " { Z } ) ) ) ) ) )
120	119	3ad2ant1 1018	. 2 |- ( ( R e. RingOps /\ S e. RingOps /\ F e. ( R RngHom S ) ) -> ( ( `' F " { Z } ) e. ( Idl ` R ) <-> ( ( `' F " { Z } ) C_ ran ( 1st ` R ) /\ (GId ` ( 1st ` R ) ) e. ( `' F " { Z } ) /\ A. x e. ( `' F " { Z } ) ( A. y e. ( `' F " { Z } ) ( x ( 1st ` R ) y ) e. ( `' F " { Z } ) /\ A. z e. ran ( 1st ` R ) ( ( z ( 2nd ` R ) x ) e. ( `' F " { Z } ) /\ ( x ( 2nd ` R ) z ) e. ( `' F " { Z } ) ) ) ) ) )
121	9, 21, 118, 120	mpbir3and 1180	1 |- ( ( R e. RingOps /\ S e. RingOps /\ F e. ( R RngHom S ) ) -> ( `' F " { Z } ) e. ( Idl ` R ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 974   = wceq 1383   e. wcel 1804   A.wral 2793   C_ wss 3461   {csn 4014   `'ccnv 4988   dom cdm 4989   ran crn 4990   "cima 4992   Fn wfn 5573   -->wf 5574   ` cfv 5578  (class class class)co 6281   1stc1st 6783   2ndc2nd 6784   GrpOpcgr 25060  GIdcgi 25061   RingOpscrngo 25249   RngHom crnghom 30338   Idlcidl 30379

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-1st 6785  df-2nd 6786  df-map 7424  df-grpo 25065  df-gid 25066  df-ginv 25067  df-ablo 25156  df-ghomOLD 25232  df-rngo 25250  df-rngohom 30341  df-idl 30382

This theorem is referenced by: (None)