keridl - Mathbox for Jeff Madsen

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Theorem keridl 33831

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

**Hypotheses**
Ref	Expression
keridl.1
keridl.2	GId

**Assertion**
Ref	Expression
keridl	$/\$ $/\$

Proof of Theorem keridl Dummy variables

are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2622	. . . 4
2		eqid 2622	. . . 4
3		keridl.1	. . . 4
4		eqid 2622	. . . 4
5	1, 2, 3, 4	rngohomf 33765	. . 3 $/\$ $/\$
6		cnvimass 5485	. . . 4
7		fdm 6051	. . . 4
8	6, 7	syl5sseq 3653	. . 3
9	5, 8	syl 17	. 2 $/\$ $/\$
10		eqid 2622	. . . . 5 GId GId
11	1, 2, 10	rngo0cl 33718	. . . 4 GId
12	11	3ad2ant1 1082	. . 3 $/\$ $/\$ GId
13		keridl.2	. . . . 5 GId
14	1, 10, 3, 13	rngohom0 33771	. . . 4 $/\$ $/\$ GId
15		fvex 6201	. . . . 5 GId
16	15	elsn 4192	. . . 4 GId GId
17	14, 16	sylibr 224	. . 3 $/\$ $/\$ GId
18		ffn 6045	. . . 4
19		elpreima 6337	. . . 4 GId GId $/\$ GId
20	5, 18, 19	3syl 18	. . 3 $/\$ $/\$ GId GId $/\$ GId
21	12, 17, 20	mpbir2and 957	. 2 $/\$ $/\$ GId
22		an4 865	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
23	1, 2, 3	rngohomadd 33768	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
24	23	adantr 481	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
25		oveq12 6659	. . . . . . . . . . . . . 14 $/\$
26	25	adantl 482	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
27	3	rngogrpo 33709	. . . . . . . . . . . . . . . 16
28	4, 13	grpoidcl 27368	. . . . . . . . . . . . . . . . 17
29	4, 13	grpolid 27370	. . . . . . . . . . . . . . . . 17 $/\$
30	28, 29	mpdan 702	. . . . . . . . . . . . . . . 16
31	27, 30	syl 17	. . . . . . . . . . . . . . 15
32	31	3ad2ant2 1083	. . . . . . . . . . . . . 14 $/\$ $/\$
33	32	ad2antrr 762	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
34	24, 26, 33	3eqtrd 2660	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
35	34	ex 450	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$
36		fvex 6201	. . . . . . . . . . . . 13
37	36	elsn 4192	. . . . . . . . . . . 12
38		fvex 6201	. . . . . . . . . . . . 13
39	38	elsn 4192	. . . . . . . . . . . 12
40	37, 39	anbi12i 733	. . . . . . . . . . 11 $/\$ $/\$
41		fvex 6201	. . . . . . . . . . . 12
42	41	elsn 4192	. . . . . . . . . . 11
43	35, 40, 42	3imtr4g 285	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
44	43	imdistanda 729	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
45	1, 2	rngogcl 33711	. . . . . . . . . . . 12 $/\$ $/\$
46	45	3expib 1268	. . . . . . . . . . 11 $/\$
47	46	3ad2ant1 1082	. . . . . . . . . 10 $/\$ $/\$ $/\$
48	47	anim1d 588	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
49	44, 48	syld 47	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
50	22, 49	syl5bi 232	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
51		elpreima 6337	. . . . . . . . 9 $/\$
52	5, 18, 51	3syl 18	. . . . . . . 8 $/\$ $/\$ $/\$
53		elpreima 6337	. . . . . . . . 9 $/\$
54	5, 18, 53	3syl 18	. . . . . . . 8 $/\$ $/\$ $/\$
55	52, 54	anbi12d 747	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
56		elpreima 6337	. . . . . . . 8 $/\$
57	5, 18, 56	3syl 18	. . . . . . 7 $/\$ $/\$ $/\$
58	50, 55, 57	3imtr4d 283	. . . . . 6 $/\$ $/\$ $/\$
59	58	impl 650	. . . . 5 $/\$ $/\$ $/\$ $/\$
60	59	ralrimiva 2966	. . . 4 $/\$ $/\$ $/\$
61	37	anbi2i 730	. . . . . . 7 $/\$ $/\$
62		eqid 2622	. . . . . . . . . . . . . . . 16
63	1, 62, 2	rngocl 33700	. . . . . . . . . . . . . . 15 $/\$ $/\$
64	63	3expb 1266	. . . . . . . . . . . . . 14 $/\$ $/\$
65	64	3ad2antl1 1223	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
66	65	anass1rs 849	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
67	66	adantlrr 757	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$
68		eqid 2622	. . . . . . . . . . . . . . . 16
69	1, 2, 62, 68	rngohommul 33769	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$
70	69	anass1rs 849	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
71	70	adantlrr 757	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$
72		oveq2 6658	. . . . . . . . . . . . . . 15
73	72	adantl 482	. . . . . . . . . . . . . 14 $/\$
74	73	ad2antlr 763	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$
75	1, 2, 3, 4	rngohomcl 33766	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$
76	13, 4, 3, 68	rngorz 33722	. . . . . . . . . . . . . . . 16 $/\$
77	76	3ad2antl2 1224	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$
78	75, 77	syldan 487	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$
79	78	adantlr 751	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$
80	71, 74, 79	3eqtrd 2660	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$
81		fvex 6201	. . . . . . . . . . . . 13
82	81	elsn 4192	. . . . . . . . . . . 12
83	80, 82	sylibr 224	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$
84		elpreima 6337	. . . . . . . . . . . . 13 $/\$
85	5, 18, 84	3syl 18	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
86	85	ad2antrr 762	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
87	67, 83, 86	mpbir2and 957	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
88	1, 62, 2	rngocl 33700	. . . . . . . . . . . . . . 15 $/\$ $/\$
89	88	3expb 1266	. . . . . . . . . . . . . 14 $/\$ $/\$
90	89	3ad2antl1 1223	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
91	90	anassrs 680	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
92	91	adantlrr 757	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$
93	1, 2, 62, 68	rngohommul 33769	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$
94	93	anassrs 680	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
95	94	adantlrr 757	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$
96		oveq1 6657	. . . . . . . . . . . . . . 15
97	96	adantl 482	. . . . . . . . . . . . . 14 $/\$
98	97	ad2antlr 763	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$
99	13, 4, 3, 68	rngolz 33721	. . . . . . . . . . . . . . . 16 $/\$
100	99	3ad2antl2 1224	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$
101	75, 100	syldan 487	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$
102	101	adantlr 751	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$
103	95, 98, 102	3eqtrd 2660	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$
104		fvex 6201	. . . . . . . . . . . . 13
105	104	elsn 4192	. . . . . . . . . . . 12
106	103, 105	sylibr 224	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$
107		elpreima 6337	. . . . . . . . . . . . 13 $/\$
108	5, 18, 107	3syl 18	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
109	108	ad2antrr 762	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
110	92, 106, 109	mpbir2and 957	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
111	87, 110	jca 554	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
112	111	ralrimiva 2966	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
113	112	ex 450	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
114	61, 113	syl5bi 232	. . . . . 6 $/\$ $/\$ $/\$ $/\$
115	52, 114	sylbid 230	. . . . 5 $/\$ $/\$ $/\$
116	115	imp 445	. . . 4 $/\$ $/\$ $/\$ $/\$
117	60, 116	jca 554	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$
118	117	ralrimiva 2966	. 2 $/\$ $/\$ $/\$ $/\$
119	1, 62, 2, 10	isidl 33813	. . 3 $/\$ GId $/\$ $/\$ $/\$
120	119	3ad2ant1 1082	. 2 $/\$ $/\$ $/\$ GId $/\$ $/\$ $/\$
121	9, 21, 118, 120	mpbir3and 1245	1 $/\$ $/\$

Colors of variables: wff setvar class

Syntax hints:

wi 4

wb 196 $/\$ wa 384 $/\$ w3a 1037

wceq 1483

wcel 1990

wral 2912

wss 3574

csn 4177

ccnv 5113

cdm 5114

crn 5115

cima 5117

wfn 5883

wf 5884

cfv 5888 (class class class)co 6650

c1st 7166

c2nd 7167

cgr 27343 GIdcgi 27344

crngo 33693

crnghom 33759

cidl 33806

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

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 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-ral 2917 df-rex 2918 df-reu 2919 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-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 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-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-1st 7168 df-2nd 7169 df-map 7859 df-grpo 27347 df-gid 27348 df-ginv 27349 df-ablo 27399 df-ghomOLD 33683 df-rngo 33694 df-rngohom 33762 df-idl 33809

This theorem is referenced by: (None)

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