Theorem keridl 32265

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 2451	. . . 4 |- ( 1st ` R ) = ( 1st ` R )
2		eqid 2451	. . . 4 |- ran ( 1st ` R ) = ran ( 1st ` R )
3		keridl.1	. . . 4 |- G = ( 1st ` S )
4		eqid 2451	. . . 4 |- ran G = ran G
5	1, 2, 3, 4	rngohomf 32205	. . 3 |- ( ( R e. RingOps /\ S e. RingOps /\ F e. ( R RngHom S ) ) -> F : ran ( 1st ` R ) --> ran G )
6		cnvimass 5188	. . . 4 |- ( `' F " { Z } ) C_ dom F
7		fdm 5733	. . . 4 |- ( F : ran ( 1st ` R ) --> ran G -> dom F = ran ( 1st ` R ) )
8	6, 7	syl5sseq 3480	. . 3 |- ( F : ran ( 1st ` R ) --> ran G -> ( `' F " { Z } ) C_ ran ( 1st ` R ) )
9	5, 8	syl 17	. 2 |- ( ( R e. RingOps /\ S e. RingOps /\ F e. ( R RngHom S ) ) -> ( `' F " { Z } ) C_ ran ( 1st ` R ) )
10		eqid 2451	. . . . 5 |- (GId ` ( 1st ` R ) ) = (GId ` ( 1st ` R ) )
11	1, 2, 10	rngo0cl 26126	. . . 4 |- ( R e. RingOps -> (GId ` ( 1st ` R ) ) e. ran ( 1st ` R ) )
12	11	3ad2ant1 1029	. . 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 32211	. . . 4 |- ( ( R e. RingOps /\ S e. RingOps /\ F e. ( R RngHom S ) ) -> ( F ` (GId ` ( 1st ` R ) ) ) = Z )
15		fvex 5875	. . . . 5 |- ( F ` (GId ` ( 1st ` R ) ) ) e. _V
16	15	elsnc 3992	. . . 4 |- ( ( F ` (GId ` ( 1st ` R ) ) ) e. { Z } <-> ( F ` (GId ` ( 1st ` R ) ) ) = Z )
17	14, 16	sylibr 216	. . 3 |- ( ( R e. RingOps /\ S e. RingOps /\ F e. ( R RngHom S ) ) -> ( F ` (GId ` ( 1st ` R ) ) ) e. { Z } )
18		ffn 5728	. . . 4 |- ( F : ran ( 1st ` R ) --> ran G -> F Fn ran ( 1st ` R ) )
19		elpreima 6002	. . . 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 18	. . 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 933	. 2 |- ( ( R e. RingOps /\ S e. RingOps /\ F e. ( R RngHom S ) ) -> (GId ` ( 1st ` R ) ) e. ( `' F " { Z } ) )
22		an4 833	. . . . . . . 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 32208	. . . . . . . . . . . . . 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 467	. . . . . . . . . . . . 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 6299	. . . . . . . . . . . . . 14 |- ( ( ( F ` x ) = Z /\ ( F ` y ) = Z ) -> ( ( F ` x ) G ( F ` y ) ) = ( Z G Z ) )
26	25	adantl 468	. . . . . . . . . . . . 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 26118	. . . . . . . . . . . . . . . 16 |- ( S e. RingOps -> G e. GrpOp )
28	4, 13	grpoidcl 25945	. . . . . . . . . . . . . . . . 17 |- ( G e. GrpOp -> Z e. ran G )
29	4, 13	grpolid 25947	. . . . . . . . . . . . . . . . 17 |- ( ( G e. GrpOp /\ Z e. ran G ) -> ( Z G Z ) = Z )
30	28, 29	mpdan 674	. . . . . . . . . . . . . . . 16 |- ( G e. GrpOp -> ( Z G Z ) = Z )
31	27, 30	syl 17	. . . . . . . . . . . . . . 15 |- ( S e. RingOps -> ( Z G Z ) = Z )
32	31	3ad2ant2 1030	. . . . . . . . . . . . . 14 |- ( ( R e. RingOps /\ S e. RingOps /\ F e. ( R RngHom S ) ) -> ( Z G Z ) = Z )
33	32	ad2antrr 732	. . . . . . . . . . . . 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 2489	. . . . . . . . . . . 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 436	. . . . . . . . . . 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 5875	. . . . . . . . . . . . 13 |- ( F ` x ) e. _V
37	36	elsnc 3992	. . . . . . . . . . . 12 |- ( ( F ` x ) e. { Z } <-> ( F ` x ) = Z )
38		fvex 5875	. . . . . . . . . . . . 13 |- ( F ` y ) e. _V
39	38	elsnc 3992	. . . . . . . . . . . 12 |- ( ( F ` y ) e. { Z } <-> ( F ` y ) = Z )
40	37, 39	anbi12i 703	. . . . . . . . . . 11 |- ( ( ( F ` x ) e. { Z } /\ ( F ` y ) e. { Z } ) <-> ( ( F ` x ) = Z /\ ( F ` y ) = Z ) )
41		fvex 5875	. . . . . . . . . . . 12 |- ( F ` ( x ( 1st ` R ) y ) ) e. _V
42	41	elsnc 3992	. . . . . . . . . . 11 |- ( ( F ` ( x ( 1st ` R ) y ) ) e. { Z } <-> ( F ` ( x ( 1st ` R ) y ) ) = Z )
43	35, 40, 42	3imtr4g 274	. . . . . . . . . 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 699	. . . . . . . . 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 26119	. . . . . . . . . . . 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 1211	. . . . . . . . . . 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 1029	. . . . . . . . . 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 568	. . . . . . . . 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 45	. . . . . . . 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 221	. . . . . . 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 6002	. . . . . . . . 9 |- ( F Fn ran ( 1st ` R ) -> ( x e. ( `' F " { Z } ) <-> ( x e. ran ( 1st ` R ) /\ ( F ` x ) e. { Z } ) ) )
52	5, 18, 51	3syl 18	. . . . . . . 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 6002	. . . . . . . . 9 |- ( F Fn ran ( 1st ` R ) -> ( y e. ( `' F " { Z } ) <-> ( y e. ran ( 1st ` R ) /\ ( F ` y ) e. { Z } ) ) )
54	5, 18, 53	3syl 18	. . . . . . . 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 717	. . . . . . 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 6002	. . . . . . . 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 18	. . . . . . 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 272	. . . . . 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 626	. . . . 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 2802	. . . 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 700	. . . . . . 7 |- ( ( x e. ran ( 1st ` R ) /\ ( F ` x ) e. { Z } ) <-> ( x e. ran ( 1st ` R ) /\ ( F ` x ) = Z ) )
62		eqid 2451	. . . . . . . . . . . . . . . 16 |- ( 2nd ` R ) = ( 2nd ` R )
63	1, 62, 2	rngocl 26110	. . . . . . . . . . . . . . 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 1209	. . . . . . . . . . . . . 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 1170	. . . . . . . . . . . . 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 816	. . . . . . . . . . . 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 727	. . . . . . . . . . 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 2451	. . . . . . . . . . . . . . . 16 |- ( 2nd ` S ) = ( 2nd ` S )
69	1, 2, 62, 68	rngohommul 32209	. . . . . . . . . . . . . . 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 816	. . . . . . . . . . . . . 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 727	. . . . . . . . . . . . 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 6298	. . . . . . . . . . . . . . 15 |- ( ( F ` x ) = Z -> ( ( F ` z ) ( 2nd ` S ) ( F ` x ) ) = ( ( F ` z ) ( 2nd ` S ) Z ) )
73	72	adantl 468	. . . . . . . . . . . . . 14 |- ( ( x e. ran ( 1st ` R ) /\ ( F ` x ) = Z ) -> ( ( F ` z ) ( 2nd ` S ) ( F ` x ) ) = ( ( F ` z ) ( 2nd ` S ) Z ) )
74	73	ad2antlr 733	. . . . . . . . . . . . 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 32206	. . . . . . . . . . . . . . 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 26130	. . . . . . . . . . . . . . . 16 |- ( ( S e. RingOps /\ ( F ` z ) e. ran G ) -> ( ( F ` z ) ( 2nd ` S ) Z ) = Z )
77	76	3ad2antl2 1171	. . . . . . . . . . . . . . 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 473	. . . . . . . . . . . . . 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 721	. . . . . . . . . . . . 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 2489	. . . . . . . . . . . 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 5875	. . . . . . . . . . . . 13 |- ( F ` ( z ( 2nd ` R ) x ) ) e. _V
82	81	elsnc 3992	. . . . . . . . . . . 12 |- ( ( F ` ( z ( 2nd ` R ) x ) ) e. { Z } <-> ( F ` ( z ( 2nd ` R ) x ) ) = Z )
83	80, 82	sylibr 216	. . . . . . . . . . 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 6002	. . . . . . . . . . . . 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 18	. . . . . . . . . . . 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 732	. . . . . . . . . . 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 933	. . . . . . . . . 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 26110	. . . . . . . . . . . . . . 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 1209	. . . . . . . . . . . . . 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 1170	. . . . . . . . . . . . 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 654	. . . . . . . . . . . 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 727	. . . . . . . . . . 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 32209	. . . . . . . . . . . . . . 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 654	. . . . . . . . . . . . . 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 727	. . . . . . . . . . . . 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 6297	. . . . . . . . . . . . . . 15 |- ( ( F ` x ) = Z -> ( ( F ` x ) ( 2nd ` S ) ( F ` z ) ) = ( Z ( 2nd ` S ) ( F ` z ) ) )
97	96	adantl 468	. . . . . . . . . . . . . 14 |- ( ( x e. ran ( 1st ` R ) /\ ( F ` x ) = Z ) -> ( ( F ` x ) ( 2nd ` S ) ( F ` z ) ) = ( Z ( 2nd ` S ) ( F ` z ) ) )
98	97	ad2antlr 733	. . . . . . . . . . . . 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 26129	. . . . . . . . . . . . . . . 16 |- ( ( S e. RingOps /\ ( F ` z ) e. ran G ) -> ( Z ( 2nd ` S ) ( F ` z ) ) = Z )
100	99	3ad2antl2 1171	. . . . . . . . . . . . . . 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 473	. . . . . . . . . . . . . 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 721	. . . . . . . . . . . . 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 2489	. . . . . . . . . . . 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 5875	. . . . . . . . . . . . 13 |- ( F ` ( x ( 2nd ` R ) z ) ) e. _V
105	104	elsnc 3992	. . . . . . . . . . . 12 |- ( ( F ` ( x ( 2nd ` R ) z ) ) e. { Z } <-> ( F ` ( x ( 2nd ` R ) z ) ) = Z )
106	103, 105	sylibr 216	. . . . . . . . . . 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 6002	. . . . . . . . . . . . 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 18	. . . . . . . . . . . 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 732	. . . . . . . . . . 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 933	. . . . . . . . . 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 535	. . . . . . . . 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 2802	. . . . . . . 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 436	. . . . . . 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 221	. . . . . 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 219	. . . . 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 431	. . . 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 535	. . 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 2802	. 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 32247	. . 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 1029	. 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 1191	1 |- ( ( R e. RingOps /\ S e. RingOps /\ F e. ( R RngHom S ) ) -> ( `' F " { Z } ) e. ( Idl ` R ) )

Syntax hints:   -> wi 4   <-> wb 188   /\ wa 371   /\ w3a 985   = wceq 1444   e. wcel 1887   A.wral 2737   C_ wss 3404   {csn 3968   `'ccnv 4833   dom cdm 4834   ran crn 4835   "cima 4837   Fn wfn 5577   -->wf 5578   ` cfv 5582  (class class class)co 6290   1stc1st 6791   2ndc2nd 6792   GrpOpcgr 25914  GIdcgi 25915   RingOpscrngo 26103   RngHom crnghom 32199   Idlcidl 32240

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-reu 2744  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-1st 6793  df-2nd 6794  df-map 7474  df-grpo 25919  df-gid 25920  df-ginv 25921  df-ablo 26010  df-ghomOLD 26086  df-rngo 26104  df-rngohom 32202  df-idl 32243

This theorem is referenced by: (None)