rngohomco - Mathbox for Jeff Madsen

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Theorem rngohomco 33773

Description: The composition of two ring homomorphisms is a ring homomorphism. (Contributed by Jeff Madsen, 16-Jun-2011.)

**Assertion**
Ref	Expression
rngohomco	$/\$ $/\$ $/\$ $/\$

Proof of Theorem rngohomco Dummy variables

are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2622	. . . . . . 7
2		eqid 2622	. . . . . . 7
3		eqid 2622	. . . . . . 7
4		eqid 2622	. . . . . . 7
5	1, 2, 3, 4	rngohomf 33765	. . . . . 6 $/\$ $/\$
6	5	3expa 1265	. . . . 5 $/\$ $/\$
7	6	3adantl1 1217	. . . 4 $/\$ $/\$ $/\$
8	7	adantrl 752	. . 3 $/\$ $/\$ $/\$ $/\$
9		eqid 2622	. . . . . . 7
10		eqid 2622	. . . . . . 7
11	9, 10, 1, 2	rngohomf 33765	. . . . . 6 $/\$ $/\$
12	11	3expa 1265	. . . . 5 $/\$ $/\$
13	12	3adantl3 1219	. . . 4 $/\$ $/\$ $/\$
14	13	adantrr 753	. . 3 $/\$ $/\$ $/\$ $/\$
15		fco 6058	. . 3 $/\$
16	8, 14, 15	syl2anc 693	. 2 $/\$ $/\$ $/\$ $/\$
17		eqid 2622	. . . . . . 7
18		eqid 2622	. . . . . . 7 GId GId
19	10, 17, 18	rngo1cl 33738	. . . . . 6 GId
20	19	3ad2ant1 1082	. . . . 5 $/\$ $/\$ GId
21	20	adantr 481	. . . 4 $/\$ $/\$ $/\$ $/\$ GId
22		fvco3 6275	. . . 4 $/\$ GId GId GId
23	14, 21, 22	syl2anc 693	. . 3 $/\$ $/\$ $/\$ $/\$ GId GId
24		eqid 2622	. . . . . . . . 9
25		eqid 2622	. . . . . . . . 9 GId GId
26	17, 18, 24, 25	rngohom1 33767	. . . . . . . 8 $/\$ $/\$ GId GId
27	26	3expa 1265	. . . . . . 7 $/\$ $/\$ GId GId
28	27	3adantl3 1219	. . . . . 6 $/\$ $/\$ $/\$ GId GId
29	28	adantrr 753	. . . . 5 $/\$ $/\$ $/\$ $/\$ GId GId
30	29	fveq2d 6195	. . . 4 $/\$ $/\$ $/\$ $/\$ GId GId
31		eqid 2622	. . . . . . . 8
32		eqid 2622	. . . . . . . 8 GId GId
33	24, 25, 31, 32	rngohom1 33767	. . . . . . 7 $/\$ $/\$ GId GId
34	33	3expa 1265	. . . . . 6 $/\$ $/\$ GId GId
35	34	3adantl1 1217	. . . . 5 $/\$ $/\$ $/\$ GId GId
36	35	adantrl 752	. . . 4 $/\$ $/\$ $/\$ $/\$ GId GId
37	30, 36	eqtrd 2656	. . 3 $/\$ $/\$ $/\$ $/\$ GId GId
38	23, 37	eqtrd 2656	. 2 $/\$ $/\$ $/\$ $/\$ GId GId
39	9, 10, 1	rngohomadd 33768	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
40	39	ex 450	. . . . . . . . . . 11 $/\$ $/\$ $/\$
41	40	3expa 1265	. . . . . . . . . 10 $/\$ $/\$ $/\$
42	41	3adantl3 1219	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
43	42	imp 445	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
44	43	adantlrr 757	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
45	44	fveq2d 6195	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
46	9, 10, 1, 2	rngohomcl 33766	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$
47	9, 10, 1, 2	rngohomcl 33766	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$
48	46, 47	anim12da 33506	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$
49	48	ex 450	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
50	49	3expa 1265	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
51	50	3adantl3 1219	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
52	51	imp 445	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
53	52	adantlrr 757	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
54	1, 2, 3	rngohomadd 33768	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
55	54	ex 450	. . . . . . . . . . 11 $/\$ $/\$ $/\$
56	55	3expa 1265	. . . . . . . . . 10 $/\$ $/\$ $/\$
57	56	3adantl1 1217	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
58	57	imp 445	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
59	58	adantlrl 756	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
60	53, 59	syldan 487	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
61	45, 60	eqtrd 2656	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
62	9, 10	rngogcl 33711	. . . . . . . . 9 $/\$ $/\$
63	62	3expb 1266	. . . . . . . 8 $/\$ $/\$
64	63	3ad2antl1 1223	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
65	64	adantlr 751	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
66		fvco3 6275	. . . . . . 7 $/\$
67	14, 66	sylan 488	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
68	65, 67	syldan 487	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
69		fvco3 6275	. . . . . . . 8 $/\$
70	14, 69	sylan 488	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
71		fvco3 6275	. . . . . . . 8 $/\$
72	14, 71	sylan 488	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
73	70, 72	anim12da 33506	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
74		oveq12 6659	. . . . . 6 $/\$
75	73, 74	syl 17	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
76	61, 68, 75	3eqtr4d 2666	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
77	9, 10, 17, 24	rngohommul 33769	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
78	77	ex 450	. . . . . . . . . . 11 $/\$ $/\$ $/\$
79	78	3expa 1265	. . . . . . . . . 10 $/\$ $/\$ $/\$
80	79	3adantl3 1219	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
81	80	imp 445	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
82	81	adantlrr 757	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
83	82	fveq2d 6195	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
84	1, 2, 24, 31	rngohommul 33769	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
85	84	ex 450	. . . . . . . . . . 11 $/\$ $/\$ $/\$
86	85	3expa 1265	. . . . . . . . . 10 $/\$ $/\$ $/\$
87	86	3adantl1 1217	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
88	87	imp 445	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
89	88	adantlrl 756	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
90	53, 89	syldan 487	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
91	83, 90	eqtrd 2656	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
92	9, 17, 10	rngocl 33700	. . . . . . . . 9 $/\$ $/\$
93	92	3expb 1266	. . . . . . . 8 $/\$ $/\$
94	93	3ad2antl1 1223	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
95	94	adantlr 751	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
96		fvco3 6275	. . . . . . 7 $/\$
97	14, 96	sylan 488	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
98	95, 97	syldan 487	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
99		oveq12 6659	. . . . . 6 $/\$
100	73, 99	syl 17	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
101	91, 98, 100	3eqtr4d 2666	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
102	76, 101	jca 554	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
103	102	ralrimivva 2971	. 2 $/\$ $/\$ $/\$ $/\$ $/\$
104	9, 17, 10, 18, 3, 31, 4, 32	isrngohom 33764	. . . 4 $/\$ $/\$ GId GId $/\$ $/\$
105	104	3adant2 1080	. . 3 $/\$ $/\$ $/\$ GId GId $/\$ $/\$
106	105	adantr 481	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ GId GId $/\$ $/\$
107	16, 38, 103, 106	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

crn 5115

ccom 5118

wf 5884

cfv 5888 (class class class)co 6650

c1st 7166

c2nd 7167 GIdcgi 27344

crngo 33693

crnghom 33759

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-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-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-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-fo 5894 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-ablo 27399 df-ass 33642 df-exid 33644 df-mgmOLD 33648 df-sgrOLD 33660 df-mndo 33666 df-rngo 33694 df-rngohom 33762

This theorem is referenced by: rngoisoco 33781

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