rngohomco - Mathbox for Jeff Madsen

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Theorem rngohomco 36059

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

**Assertion**
Ref	Expression
rngohomco	⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐺 ∈ (𝑆 RngHom 𝑇))) → (𝐺 ∘ 𝐹) ∈ (𝑅 RngHom 𝑇))

Proof of Theorem rngohomco Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2738	. . . . . . 7 ⊢ (1^st ‘𝑆) = (1^st ‘𝑆)
2		eqid 2738	. . . . . . 7 ⊢ ran (1^st ‘𝑆) = ran (1^st ‘𝑆)
3		eqid 2738	. . . . . . 7 ⊢ (1^st ‘𝑇) = (1^st ‘𝑇)
4		eqid 2738	. . . . . . 7 ⊢ ran (1^st ‘𝑇) = ran (1^st ‘𝑇)
5	1, 2, 3, 4	rngohomf 36051	. . . . . 6 ⊢ ((𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps ∧ 𝐺 ∈ (𝑆 RngHom 𝑇)) → 𝐺:ran (1^st ‘𝑆)⟶ran (1^st ‘𝑇))
6	5	3expa 1116	. . . . 5 ⊢ (((𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ 𝐺 ∈ (𝑆 RngHom 𝑇)) → 𝐺:ran (1^st ‘𝑆)⟶ran (1^st ‘𝑇))
7	6	3adantl1 1164	. . . 4 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ 𝐺 ∈ (𝑆 RngHom 𝑇)) → 𝐺:ran (1^st ‘𝑆)⟶ran (1^st ‘𝑇))
8	7	adantrl 712	. . 3 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐺 ∈ (𝑆 RngHom 𝑇))) → 𝐺:ran (1^st ‘𝑆)⟶ran (1^st ‘𝑇))
9		eqid 2738	. . . . . . 7 ⊢ (1^st ‘𝑅) = (1^st ‘𝑅)
10		eqid 2738	. . . . . . 7 ⊢ ran (1^st ‘𝑅) = ran (1^st ‘𝑅)
11	9, 10, 1, 2	rngohomf 36051	. . . . . 6 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → 𝐹:ran (1^st ‘𝑅)⟶ran (1^st ‘𝑆))
12	11	3expa 1116	. . . . 5 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → 𝐹:ran (1^st ‘𝑅)⟶ran (1^st ‘𝑆))
13	12	3adantl3 1166	. . . 4 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → 𝐹:ran (1^st ‘𝑅)⟶ran (1^st ‘𝑆))
14	13	adantrr 713	. . 3 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐺 ∈ (𝑆 RngHom 𝑇))) → 𝐹:ran (1^st ‘𝑅)⟶ran (1^st ‘𝑆))
15		fco 6608	. . 3 ⊢ ((𝐺:ran (1^st ‘𝑆)⟶ran (1^st ‘𝑇) ∧ 𝐹:ran (1^st ‘𝑅)⟶ran (1^st ‘𝑆)) → (𝐺 ∘ 𝐹):ran (1^st ‘𝑅)⟶ran (1^st ‘𝑇))
16	8, 14, 15	syl2anc 583	. 2 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐺 ∈ (𝑆 RngHom 𝑇))) → (𝐺 ∘ 𝐹):ran (1^st ‘𝑅)⟶ran (1^st ‘𝑇))
17		eqid 2738	. . . . . . 7 ⊢ (2^nd ‘𝑅) = (2^nd ‘𝑅)
18		eqid 2738	. . . . . . 7 ⊢ (GId‘(2^nd ‘𝑅)) = (GId‘(2^nd ‘𝑅))
19	10, 17, 18	rngo1cl 36024	. . . . . 6 ⊢ (𝑅 ∈ RingOps → (GId‘(2^nd ‘𝑅)) ∈ ran (1^st ‘𝑅))
20	19	3ad2ant1 1131	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) → (GId‘(2^nd ‘𝑅)) ∈ ran (1^st ‘𝑅))
21	20	adantr 480	. . . 4 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐺 ∈ (𝑆 RngHom 𝑇))) → (GId‘(2^nd ‘𝑅)) ∈ ran (1^st ‘𝑅))
22		fvco3 6849	. . . 4 ⊢ ((𝐹:ran (1^st ‘𝑅)⟶ran (1^st ‘𝑆) ∧ (GId‘(2^nd ‘𝑅)) ∈ ran (1^st ‘𝑅)) → ((𝐺 ∘ 𝐹)‘(GId‘(2^nd ‘𝑅))) = (𝐺‘(𝐹‘(GId‘(2^nd ‘𝑅)))))
23	14, 21, 22	syl2anc 583	. . 3 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐺 ∈ (𝑆 RngHom 𝑇))) → ((𝐺 ∘ 𝐹)‘(GId‘(2^nd ‘𝑅))) = (𝐺‘(𝐹‘(GId‘(2^nd ‘𝑅)))))
24		eqid 2738	. . . . . . . . 9 ⊢ (2^nd ‘𝑆) = (2^nd ‘𝑆)
25		eqid 2738	. . . . . . . . 9 ⊢ (GId‘(2^nd ‘𝑆)) = (GId‘(2^nd ‘𝑆))
26	17, 18, 24, 25	rngohom1 36053	. . . . . . . 8 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (𝐹‘(GId‘(2^nd ‘𝑅))) = (GId‘(2^nd ‘𝑆)))
27	26	3expa 1116	. . . . . . 7 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (𝐹‘(GId‘(2^nd ‘𝑅))) = (GId‘(2^nd ‘𝑆)))
28	27	3adantl3 1166	. . . . . 6 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (𝐹‘(GId‘(2^nd ‘𝑅))) = (GId‘(2^nd ‘𝑆)))
29	28	adantrr 713	. . . . 5 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐺 ∈ (𝑆 RngHom 𝑇))) → (𝐹‘(GId‘(2^nd ‘𝑅))) = (GId‘(2^nd ‘𝑆)))
30	29	fveq2d 6760	. . . 4 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐺 ∈ (𝑆 RngHom 𝑇))) → (𝐺‘(𝐹‘(GId‘(2^nd ‘𝑅)))) = (𝐺‘(GId‘(2^nd ‘𝑆))))
31		eqid 2738	. . . . . . . 8 ⊢ (2^nd ‘𝑇) = (2^nd ‘𝑇)
32		eqid 2738	. . . . . . . 8 ⊢ (GId‘(2^nd ‘𝑇)) = (GId‘(2^nd ‘𝑇))
33	24, 25, 31, 32	rngohom1 36053	. . . . . . 7 ⊢ ((𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps ∧ 𝐺 ∈ (𝑆 RngHom 𝑇)) → (𝐺‘(GId‘(2^nd ‘𝑆))) = (GId‘(2^nd ‘𝑇)))
34	33	3expa 1116	. . . . . 6 ⊢ (((𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ 𝐺 ∈ (𝑆 RngHom 𝑇)) → (𝐺‘(GId‘(2^nd ‘𝑆))) = (GId‘(2^nd ‘𝑇)))
35	34	3adantl1 1164	. . . . 5 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ 𝐺 ∈ (𝑆 RngHom 𝑇)) → (𝐺‘(GId‘(2^nd ‘𝑆))) = (GId‘(2^nd ‘𝑇)))
36	35	adantrl 712	. . . 4 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐺 ∈ (𝑆 RngHom 𝑇))) → (𝐺‘(GId‘(2^nd ‘𝑆))) = (GId‘(2^nd ‘𝑇)))
37	30, 36	eqtrd 2778	. . 3 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐺 ∈ (𝑆 RngHom 𝑇))) → (𝐺‘(𝐹‘(GId‘(2^nd ‘𝑅)))) = (GId‘(2^nd ‘𝑇)))
38	23, 37	eqtrd 2778	. 2 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐺 ∈ (𝑆 RngHom 𝑇))) → ((𝐺 ∘ 𝐹)‘(GId‘(2^nd ‘𝑅))) = (GId‘(2^nd ‘𝑇)))
39	9, 10, 1	rngohomadd 36054	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝑥 ∈ ran (1^st ‘𝑅) ∧ 𝑦 ∈ ran (1^st ‘𝑅))) → (𝐹‘(𝑥(1^st ‘𝑅)𝑦)) = ((𝐹‘𝑥)(1^st ‘𝑆)(𝐹‘𝑦)))
40	39	ex 412	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → ((𝑥 ∈ ran (1^st ‘𝑅) ∧ 𝑦 ∈ ran (1^st ‘𝑅)) → (𝐹‘(𝑥(1^st ‘𝑅)𝑦)) = ((𝐹‘𝑥)(1^st ‘𝑆)(𝐹‘𝑦))))
41	40	3expa 1116	. . . . . . . . . 10 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → ((𝑥 ∈ ran (1^st ‘𝑅) ∧ 𝑦 ∈ ran (1^st ‘𝑅)) → (𝐹‘(𝑥(1^st ‘𝑅)𝑦)) = ((𝐹‘𝑥)(1^st ‘𝑆)(𝐹‘𝑦))))
42	41	3adantl3 1166	. . . . . . . . 9 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → ((𝑥 ∈ ran (1^st ‘𝑅) ∧ 𝑦 ∈ ran (1^st ‘𝑅)) → (𝐹‘(𝑥(1^st ‘𝑅)𝑦)) = ((𝐹‘𝑥)(1^st ‘𝑆)(𝐹‘𝑦))))
43	42	imp 406	. . . . . . . 8 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝑥 ∈ ran (1^st ‘𝑅) ∧ 𝑦 ∈ ran (1^st ‘𝑅))) → (𝐹‘(𝑥(1^st ‘𝑅)𝑦)) = ((𝐹‘𝑥)(1^st ‘𝑆)(𝐹‘𝑦)))
44	43	adantlrr 717	. . . . . . 7 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐺 ∈ (𝑆 RngHom 𝑇))) ∧ (𝑥 ∈ ran (1^st ‘𝑅) ∧ 𝑦 ∈ ran (1^st ‘𝑅))) → (𝐹‘(𝑥(1^st ‘𝑅)𝑦)) = ((𝐹‘𝑥)(1^st ‘𝑆)(𝐹‘𝑦)))
45	44	fveq2d 6760	. . . . . 6 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐺 ∈ (𝑆 RngHom 𝑇))) ∧ (𝑥 ∈ ran (1^st ‘𝑅) ∧ 𝑦 ∈ ran (1^st ‘𝑅))) → (𝐺‘(𝐹‘(𝑥(1^st ‘𝑅)𝑦))) = (𝐺‘((𝐹‘𝑥)(1^st ‘𝑆)(𝐹‘𝑦))))
46	9, 10, 1, 2	rngohomcl 36052	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ 𝑥 ∈ ran (1^st ‘𝑅)) → (𝐹‘𝑥) ∈ ran (1^st ‘𝑆))
47	9, 10, 1, 2	rngohomcl 36052	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ 𝑦 ∈ ran (1^st ‘𝑅)) → (𝐹‘𝑦) ∈ ran (1^st ‘𝑆))
48	46, 47	anim12dan 618	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝑥 ∈ ran (1^st ‘𝑅) ∧ 𝑦 ∈ ran (1^st ‘𝑅))) → ((𝐹‘𝑥) ∈ ran (1^st ‘𝑆) ∧ (𝐹‘𝑦) ∈ ran (1^st ‘𝑆)))
49	48	ex 412	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → ((𝑥 ∈ ran (1^st ‘𝑅) ∧ 𝑦 ∈ ran (1^st ‘𝑅)) → ((𝐹‘𝑥) ∈ ran (1^st ‘𝑆) ∧ (𝐹‘𝑦) ∈ ran (1^st ‘𝑆))))
50	49	3expa 1116	. . . . . . . . . 10 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → ((𝑥 ∈ ran (1^st ‘𝑅) ∧ 𝑦 ∈ ran (1^st ‘𝑅)) → ((𝐹‘𝑥) ∈ ran (1^st ‘𝑆) ∧ (𝐹‘𝑦) ∈ ran (1^st ‘𝑆))))
51	50	3adantl3 1166	. . . . . . . . 9 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → ((𝑥 ∈ ran (1^st ‘𝑅) ∧ 𝑦 ∈ ran (1^st ‘𝑅)) → ((𝐹‘𝑥) ∈ ran (1^st ‘𝑆) ∧ (𝐹‘𝑦) ∈ ran (1^st ‘𝑆))))
52	51	imp 406	. . . . . . . 8 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝑥 ∈ ran (1^st ‘𝑅) ∧ 𝑦 ∈ ran (1^st ‘𝑅))) → ((𝐹‘𝑥) ∈ ran (1^st ‘𝑆) ∧ (𝐹‘𝑦) ∈ ran (1^st ‘𝑆)))
53	52	adantlrr 717	. . . . . . 7 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐺 ∈ (𝑆 RngHom 𝑇))) ∧ (𝑥 ∈ ran (1^st ‘𝑅) ∧ 𝑦 ∈ ran (1^st ‘𝑅))) → ((𝐹‘𝑥) ∈ ran (1^st ‘𝑆) ∧ (𝐹‘𝑦) ∈ ran (1^st ‘𝑆)))
54	1, 2, 3	rngohomadd 36054	. . . . . . . . . . . 12 ⊢ (((𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps ∧ 𝐺 ∈ (𝑆 RngHom 𝑇)) ∧ ((𝐹‘𝑥) ∈ ran (1^st ‘𝑆) ∧ (𝐹‘𝑦) ∈ ran (1^st ‘𝑆))) → (𝐺‘((𝐹‘𝑥)(1^st ‘𝑆)(𝐹‘𝑦))) = ((𝐺‘(𝐹‘𝑥))(1^st ‘𝑇)(𝐺‘(𝐹‘𝑦))))
55	54	ex 412	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps ∧ 𝐺 ∈ (𝑆 RngHom 𝑇)) → (((𝐹‘𝑥) ∈ ran (1^st ‘𝑆) ∧ (𝐹‘𝑦) ∈ ran (1^st ‘𝑆)) → (𝐺‘((𝐹‘𝑥)(1^st ‘𝑆)(𝐹‘𝑦))) = ((𝐺‘(𝐹‘𝑥))(1^st ‘𝑇)(𝐺‘(𝐹‘𝑦)))))
56	55	3expa 1116	. . . . . . . . . 10 ⊢ (((𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ 𝐺 ∈ (𝑆 RngHom 𝑇)) → (((𝐹‘𝑥) ∈ ran (1^st ‘𝑆) ∧ (𝐹‘𝑦) ∈ ran (1^st ‘𝑆)) → (𝐺‘((𝐹‘𝑥)(1^st ‘𝑆)(𝐹‘𝑦))) = ((𝐺‘(𝐹‘𝑥))(1^st ‘𝑇)(𝐺‘(𝐹‘𝑦)))))
57	56	3adantl1 1164	. . . . . . . . 9 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ 𝐺 ∈ (𝑆 RngHom 𝑇)) → (((𝐹‘𝑥) ∈ ran (1^st ‘𝑆) ∧ (𝐹‘𝑦) ∈ ran (1^st ‘𝑆)) → (𝐺‘((𝐹‘𝑥)(1^st ‘𝑆)(𝐹‘𝑦))) = ((𝐺‘(𝐹‘𝑥))(1^st ‘𝑇)(𝐺‘(𝐹‘𝑦)))))
58	57	imp 406	. . . . . . . 8 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ 𝐺 ∈ (𝑆 RngHom 𝑇)) ∧ ((𝐹‘𝑥) ∈ ran (1^st ‘𝑆) ∧ (𝐹‘𝑦) ∈ ran (1^st ‘𝑆))) → (𝐺‘((𝐹‘𝑥)(1^st ‘𝑆)(𝐹‘𝑦))) = ((𝐺‘(𝐹‘𝑥))(1^st ‘𝑇)(𝐺‘(𝐹‘𝑦))))
59	58	adantlrl 716	. . . . . . 7 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐺 ∈ (𝑆 RngHom 𝑇))) ∧ ((𝐹‘𝑥) ∈ ran (1^st ‘𝑆) ∧ (𝐹‘𝑦) ∈ ran (1^st ‘𝑆))) → (𝐺‘((𝐹‘𝑥)(1^st ‘𝑆)(𝐹‘𝑦))) = ((𝐺‘(𝐹‘𝑥))(1^st ‘𝑇)(𝐺‘(𝐹‘𝑦))))
60	53, 59	syldan 590	. . . . . 6 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐺 ∈ (𝑆 RngHom 𝑇))) ∧ (𝑥 ∈ ran (1^st ‘𝑅) ∧ 𝑦 ∈ ran (1^st ‘𝑅))) → (𝐺‘((𝐹‘𝑥)(1^st ‘𝑆)(𝐹‘𝑦))) = ((𝐺‘(𝐹‘𝑥))(1^st ‘𝑇)(𝐺‘(𝐹‘𝑦))))
61	45, 60	eqtrd 2778	. . . . 5 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐺 ∈ (𝑆 RngHom 𝑇))) ∧ (𝑥 ∈ ran (1^st ‘𝑅) ∧ 𝑦 ∈ ran (1^st ‘𝑅))) → (𝐺‘(𝐹‘(𝑥(1^st ‘𝑅)𝑦))) = ((𝐺‘(𝐹‘𝑥))(1^st ‘𝑇)(𝐺‘(𝐹‘𝑦))))
62	9, 10	rngogcl 35997	. . . . . . . . 9 ⊢ ((𝑅 ∈ RingOps ∧ 𝑥 ∈ ran (1^st ‘𝑅) ∧ 𝑦 ∈ ran (1^st ‘𝑅)) → (𝑥(1^st ‘𝑅)𝑦) ∈ ran (1^st ‘𝑅))
63	62	3expb 1118	. . . . . . . 8 ⊢ ((𝑅 ∈ RingOps ∧ (𝑥 ∈ ran (1^st ‘𝑅) ∧ 𝑦 ∈ ran (1^st ‘𝑅))) → (𝑥(1^st ‘𝑅)𝑦) ∈ ran (1^st ‘𝑅))
64	63	3ad2antl1 1183	. . . . . . 7 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝑥 ∈ ran (1^st ‘𝑅) ∧ 𝑦 ∈ ran (1^st ‘𝑅))) → (𝑥(1^st ‘𝑅)𝑦) ∈ ran (1^st ‘𝑅))
65	64	adantlr 711	. . . . . 6 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐺 ∈ (𝑆 RngHom 𝑇))) ∧ (𝑥 ∈ ran (1^st ‘𝑅) ∧ 𝑦 ∈ ran (1^st ‘𝑅))) → (𝑥(1^st ‘𝑅)𝑦) ∈ ran (1^st ‘𝑅))
66		fvco3 6849	. . . . . . 7 ⊢ ((𝐹:ran (1^st ‘𝑅)⟶ran (1^st ‘𝑆) ∧ (𝑥(1^st ‘𝑅)𝑦) ∈ ran (1^st ‘𝑅)) → ((𝐺 ∘ 𝐹)‘(𝑥(1^st ‘𝑅)𝑦)) = (𝐺‘(𝐹‘(𝑥(1^st ‘𝑅)𝑦))))
67	14, 66	sylan 579	. . . . . 6 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐺 ∈ (𝑆 RngHom 𝑇))) ∧ (𝑥(1^st ‘𝑅)𝑦) ∈ ran (1^st ‘𝑅)) → ((𝐺 ∘ 𝐹)‘(𝑥(1^st ‘𝑅)𝑦)) = (𝐺‘(𝐹‘(𝑥(1^st ‘𝑅)𝑦))))
68	65, 67	syldan 590	. . . . 5 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐺 ∈ (𝑆 RngHom 𝑇))) ∧ (𝑥 ∈ ran (1^st ‘𝑅) ∧ 𝑦 ∈ ran (1^st ‘𝑅))) → ((𝐺 ∘ 𝐹)‘(𝑥(1^st ‘𝑅)𝑦)) = (𝐺‘(𝐹‘(𝑥(1^st ‘𝑅)𝑦))))
69		fvco3 6849	. . . . . . . 8 ⊢ ((𝐹:ran (1^st ‘𝑅)⟶ran (1^st ‘𝑆) ∧ 𝑥 ∈ ran (1^st ‘𝑅)) → ((𝐺 ∘ 𝐹)‘𝑥) = (𝐺‘(𝐹‘𝑥)))
70	14, 69	sylan 579	. . . . . . 7 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐺 ∈ (𝑆 RngHom 𝑇))) ∧ 𝑥 ∈ ran (1^st ‘𝑅)) → ((𝐺 ∘ 𝐹)‘𝑥) = (𝐺‘(𝐹‘𝑥)))
71		fvco3 6849	. . . . . . . 8 ⊢ ((𝐹:ran (1^st ‘𝑅)⟶ran (1^st ‘𝑆) ∧ 𝑦 ∈ ran (1^st ‘𝑅)) → ((𝐺 ∘ 𝐹)‘𝑦) = (𝐺‘(𝐹‘𝑦)))
72	14, 71	sylan 579	. . . . . . 7 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐺 ∈ (𝑆 RngHom 𝑇))) ∧ 𝑦 ∈ ran (1^st ‘𝑅)) → ((𝐺 ∘ 𝐹)‘𝑦) = (𝐺‘(𝐹‘𝑦)))
73	70, 72	anim12dan 618	. . . . . 6 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐺 ∈ (𝑆 RngHom 𝑇))) ∧ (𝑥 ∈ ran (1^st ‘𝑅) ∧ 𝑦 ∈ ran (1^st ‘𝑅))) → (((𝐺 ∘ 𝐹)‘𝑥) = (𝐺‘(𝐹‘𝑥)) ∧ ((𝐺 ∘ 𝐹)‘𝑦) = (𝐺‘(𝐹‘𝑦))))
74		oveq12 7264	. . . . . 6 ⊢ ((((𝐺 ∘ 𝐹)‘𝑥) = (𝐺‘(𝐹‘𝑥)) ∧ ((𝐺 ∘ 𝐹)‘𝑦) = (𝐺‘(𝐹‘𝑦))) → (((𝐺 ∘ 𝐹)‘𝑥)(1^st ‘𝑇)((𝐺 ∘ 𝐹)‘𝑦)) = ((𝐺‘(𝐹‘𝑥))(1^st ‘𝑇)(𝐺‘(𝐹‘𝑦))))
75	73, 74	syl 17	. . . . 5 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐺 ∈ (𝑆 RngHom 𝑇))) ∧ (𝑥 ∈ ran (1^st ‘𝑅) ∧ 𝑦 ∈ ran (1^st ‘𝑅))) → (((𝐺 ∘ 𝐹)‘𝑥)(1^st ‘𝑇)((𝐺 ∘ 𝐹)‘𝑦)) = ((𝐺‘(𝐹‘𝑥))(1^st ‘𝑇)(𝐺‘(𝐹‘𝑦))))
76	61, 68, 75	3eqtr4d 2788	. . . 4 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐺 ∈ (𝑆 RngHom 𝑇))) ∧ (𝑥 ∈ ran (1^st ‘𝑅) ∧ 𝑦 ∈ ran (1^st ‘𝑅))) → ((𝐺 ∘ 𝐹)‘(𝑥(1^st ‘𝑅)𝑦)) = (((𝐺 ∘ 𝐹)‘𝑥)(1^st ‘𝑇)((𝐺 ∘ 𝐹)‘𝑦)))
77	9, 10, 17, 24	rngohommul 36055	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝑥 ∈ ran (1^st ‘𝑅) ∧ 𝑦 ∈ ran (1^st ‘𝑅))) → (𝐹‘(𝑥(2^nd ‘𝑅)𝑦)) = ((𝐹‘𝑥)(2^nd ‘𝑆)(𝐹‘𝑦)))
78	77	ex 412	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → ((𝑥 ∈ ran (1^st ‘𝑅) ∧ 𝑦 ∈ ran (1^st ‘𝑅)) → (𝐹‘(𝑥(2^nd ‘𝑅)𝑦)) = ((𝐹‘𝑥)(2^nd ‘𝑆)(𝐹‘𝑦))))
79	78	3expa 1116	. . . . . . . . . 10 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → ((𝑥 ∈ ran (1^st ‘𝑅) ∧ 𝑦 ∈ ran (1^st ‘𝑅)) → (𝐹‘(𝑥(2^nd ‘𝑅)𝑦)) = ((𝐹‘𝑥)(2^nd ‘𝑆)(𝐹‘𝑦))))
80	79	3adantl3 1166	. . . . . . . . 9 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → ((𝑥 ∈ ran (1^st ‘𝑅) ∧ 𝑦 ∈ ran (1^st ‘𝑅)) → (𝐹‘(𝑥(2^nd ‘𝑅)𝑦)) = ((𝐹‘𝑥)(2^nd ‘𝑆)(𝐹‘𝑦))))
81	80	imp 406	. . . . . . . 8 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝑥 ∈ ran (1^st ‘𝑅) ∧ 𝑦 ∈ ran (1^st ‘𝑅))) → (𝐹‘(𝑥(2^nd ‘𝑅)𝑦)) = ((𝐹‘𝑥)(2^nd ‘𝑆)(𝐹‘𝑦)))
82	81	adantlrr 717	. . . . . . 7 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐺 ∈ (𝑆 RngHom 𝑇))) ∧ (𝑥 ∈ ran (1^st ‘𝑅) ∧ 𝑦 ∈ ran (1^st ‘𝑅))) → (𝐹‘(𝑥(2^nd ‘𝑅)𝑦)) = ((𝐹‘𝑥)(2^nd ‘𝑆)(𝐹‘𝑦)))
83	82	fveq2d 6760	. . . . . 6 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐺 ∈ (𝑆 RngHom 𝑇))) ∧ (𝑥 ∈ ran (1^st ‘𝑅) ∧ 𝑦 ∈ ran (1^st ‘𝑅))) → (𝐺‘(𝐹‘(𝑥(2^nd ‘𝑅)𝑦))) = (𝐺‘((𝐹‘𝑥)(2^nd ‘𝑆)(𝐹‘𝑦))))
84	1, 2, 24, 31	rngohommul 36055	. . . . . . . . . . . 12 ⊢ (((𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps ∧ 𝐺 ∈ (𝑆 RngHom 𝑇)) ∧ ((𝐹‘𝑥) ∈ ran (1^st ‘𝑆) ∧ (𝐹‘𝑦) ∈ ran (1^st ‘𝑆))) → (𝐺‘((𝐹‘𝑥)(2^nd ‘𝑆)(𝐹‘𝑦))) = ((𝐺‘(𝐹‘𝑥))(2^nd ‘𝑇)(𝐺‘(𝐹‘𝑦))))
85	84	ex 412	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps ∧ 𝐺 ∈ (𝑆 RngHom 𝑇)) → (((𝐹‘𝑥) ∈ ran (1^st ‘𝑆) ∧ (𝐹‘𝑦) ∈ ran (1^st ‘𝑆)) → (𝐺‘((𝐹‘𝑥)(2^nd ‘𝑆)(𝐹‘𝑦))) = ((𝐺‘(𝐹‘𝑥))(2^nd ‘𝑇)(𝐺‘(𝐹‘𝑦)))))
86	85	3expa 1116	. . . . . . . . . 10 ⊢ (((𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ 𝐺 ∈ (𝑆 RngHom 𝑇)) → (((𝐹‘𝑥) ∈ ran (1^st ‘𝑆) ∧ (𝐹‘𝑦) ∈ ran (1^st ‘𝑆)) → (𝐺‘((𝐹‘𝑥)(2^nd ‘𝑆)(𝐹‘𝑦))) = ((𝐺‘(𝐹‘𝑥))(2^nd ‘𝑇)(𝐺‘(𝐹‘𝑦)))))
87	86	3adantl1 1164	. . . . . . . . 9 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ 𝐺 ∈ (𝑆 RngHom 𝑇)) → (((𝐹‘𝑥) ∈ ran (1^st ‘𝑆) ∧ (𝐹‘𝑦) ∈ ran (1^st ‘𝑆)) → (𝐺‘((𝐹‘𝑥)(2^nd ‘𝑆)(𝐹‘𝑦))) = ((𝐺‘(𝐹‘𝑥))(2^nd ‘𝑇)(𝐺‘(𝐹‘𝑦)))))
88	87	imp 406	. . . . . . . 8 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ 𝐺 ∈ (𝑆 RngHom 𝑇)) ∧ ((𝐹‘𝑥) ∈ ran (1^st ‘𝑆) ∧ (𝐹‘𝑦) ∈ ran (1^st ‘𝑆))) → (𝐺‘((𝐹‘𝑥)(2^nd ‘𝑆)(𝐹‘𝑦))) = ((𝐺‘(𝐹‘𝑥))(2^nd ‘𝑇)(𝐺‘(𝐹‘𝑦))))
89	88	adantlrl 716	. . . . . . 7 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐺 ∈ (𝑆 RngHom 𝑇))) ∧ ((𝐹‘𝑥) ∈ ran (1^st ‘𝑆) ∧ (𝐹‘𝑦) ∈ ran (1^st ‘𝑆))) → (𝐺‘((𝐹‘𝑥)(2^nd ‘𝑆)(𝐹‘𝑦))) = ((𝐺‘(𝐹‘𝑥))(2^nd ‘𝑇)(𝐺‘(𝐹‘𝑦))))
90	53, 89	syldan 590	. . . . . 6 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐺 ∈ (𝑆 RngHom 𝑇))) ∧ (𝑥 ∈ ran (1^st ‘𝑅) ∧ 𝑦 ∈ ran (1^st ‘𝑅))) → (𝐺‘((𝐹‘𝑥)(2^nd ‘𝑆)(𝐹‘𝑦))) = ((𝐺‘(𝐹‘𝑥))(2^nd ‘𝑇)(𝐺‘(𝐹‘𝑦))))
91	83, 90	eqtrd 2778	. . . . 5 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐺 ∈ (𝑆 RngHom 𝑇))) ∧ (𝑥 ∈ ran (1^st ‘𝑅) ∧ 𝑦 ∈ ran (1^st ‘𝑅))) → (𝐺‘(𝐹‘(𝑥(2^nd ‘𝑅)𝑦))) = ((𝐺‘(𝐹‘𝑥))(2^nd ‘𝑇)(𝐺‘(𝐹‘𝑦))))
92	9, 17, 10	rngocl 35986	. . . . . . . . 9 ⊢ ((𝑅 ∈ RingOps ∧ 𝑥 ∈ ran (1^st ‘𝑅) ∧ 𝑦 ∈ ran (1^st ‘𝑅)) → (𝑥(2^nd ‘𝑅)𝑦) ∈ ran (1^st ‘𝑅))
93	92	3expb 1118	. . . . . . . 8 ⊢ ((𝑅 ∈ RingOps ∧ (𝑥 ∈ ran (1^st ‘𝑅) ∧ 𝑦 ∈ ran (1^st ‘𝑅))) → (𝑥(2^nd ‘𝑅)𝑦) ∈ ran (1^st ‘𝑅))
94	93	3ad2antl1 1183	. . . . . . 7 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝑥 ∈ ran (1^st ‘𝑅) ∧ 𝑦 ∈ ran (1^st ‘𝑅))) → (𝑥(2^nd ‘𝑅)𝑦) ∈ ran (1^st ‘𝑅))
95	94	adantlr 711	. . . . . 6 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐺 ∈ (𝑆 RngHom 𝑇))) ∧ (𝑥 ∈ ran (1^st ‘𝑅) ∧ 𝑦 ∈ ran (1^st ‘𝑅))) → (𝑥(2^nd ‘𝑅)𝑦) ∈ ran (1^st ‘𝑅))
96		fvco3 6849	. . . . . . 7 ⊢ ((𝐹:ran (1^st ‘𝑅)⟶ran (1^st ‘𝑆) ∧ (𝑥(2^nd ‘𝑅)𝑦) ∈ ran (1^st ‘𝑅)) → ((𝐺 ∘ 𝐹)‘(𝑥(2^nd ‘𝑅)𝑦)) = (𝐺‘(𝐹‘(𝑥(2^nd ‘𝑅)𝑦))))
97	14, 96	sylan 579	. . . . . 6 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐺 ∈ (𝑆 RngHom 𝑇))) ∧ (𝑥(2^nd ‘𝑅)𝑦) ∈ ran (1^st ‘𝑅)) → ((𝐺 ∘ 𝐹)‘(𝑥(2^nd ‘𝑅)𝑦)) = (𝐺‘(𝐹‘(𝑥(2^nd ‘𝑅)𝑦))))
98	95, 97	syldan 590	. . . . 5 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐺 ∈ (𝑆 RngHom 𝑇))) ∧ (𝑥 ∈ ran (1^st ‘𝑅) ∧ 𝑦 ∈ ran (1^st ‘𝑅))) → ((𝐺 ∘ 𝐹)‘(𝑥(2^nd ‘𝑅)𝑦)) = (𝐺‘(𝐹‘(𝑥(2^nd ‘𝑅)𝑦))))
99		oveq12 7264	. . . . . 6 ⊢ ((((𝐺 ∘ 𝐹)‘𝑥) = (𝐺‘(𝐹‘𝑥)) ∧ ((𝐺 ∘ 𝐹)‘𝑦) = (𝐺‘(𝐹‘𝑦))) → (((𝐺 ∘ 𝐹)‘𝑥)(2^nd ‘𝑇)((𝐺 ∘ 𝐹)‘𝑦)) = ((𝐺‘(𝐹‘𝑥))(2^nd ‘𝑇)(𝐺‘(𝐹‘𝑦))))
100	73, 99	syl 17	. . . . 5 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐺 ∈ (𝑆 RngHom 𝑇))) ∧ (𝑥 ∈ ran (1^st ‘𝑅) ∧ 𝑦 ∈ ran (1^st ‘𝑅))) → (((𝐺 ∘ 𝐹)‘𝑥)(2^nd ‘𝑇)((𝐺 ∘ 𝐹)‘𝑦)) = ((𝐺‘(𝐹‘𝑥))(2^nd ‘𝑇)(𝐺‘(𝐹‘𝑦))))
101	91, 98, 100	3eqtr4d 2788	. . . 4 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐺 ∈ (𝑆 RngHom 𝑇))) ∧ (𝑥 ∈ ran (1^st ‘𝑅) ∧ 𝑦 ∈ ran (1^st ‘𝑅))) → ((𝐺 ∘ 𝐹)‘(𝑥(2^nd ‘𝑅)𝑦)) = (((𝐺 ∘ 𝐹)‘𝑥)(2^nd ‘𝑇)((𝐺 ∘ 𝐹)‘𝑦)))
102	76, 101	jca 511	. . 3 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐺 ∈ (𝑆 RngHom 𝑇))) ∧ (𝑥 ∈ ran (1^st ‘𝑅) ∧ 𝑦 ∈ ran (1^st ‘𝑅))) → (((𝐺 ∘ 𝐹)‘(𝑥(1^st ‘𝑅)𝑦)) = (((𝐺 ∘ 𝐹)‘𝑥)(1^st ‘𝑇)((𝐺 ∘ 𝐹)‘𝑦)) ∧ ((𝐺 ∘ 𝐹)‘(𝑥(2^nd ‘𝑅)𝑦)) = (((𝐺 ∘ 𝐹)‘𝑥)(2^nd ‘𝑇)((𝐺 ∘ 𝐹)‘𝑦))))
103	102	ralrimivva 3114	. 2 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐺 ∈ (𝑆 RngHom 𝑇))) → ∀𝑥 ∈ ran (1^st ‘𝑅)∀𝑦 ∈ ran (1^st ‘𝑅)(((𝐺 ∘ 𝐹)‘(𝑥(1^st ‘𝑅)𝑦)) = (((𝐺 ∘ 𝐹)‘𝑥)(1^st ‘𝑇)((𝐺 ∘ 𝐹)‘𝑦)) ∧ ((𝐺 ∘ 𝐹)‘(𝑥(2^nd ‘𝑅)𝑦)) = (((𝐺 ∘ 𝐹)‘𝑥)(2^nd ‘𝑇)((𝐺 ∘ 𝐹)‘𝑦))))
104	9, 17, 10, 18, 3, 31, 4, 32	isrngohom 36050	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝑇 ∈ RingOps) → ((𝐺 ∘ 𝐹) ∈ (𝑅 RngHom 𝑇) ↔ ((𝐺 ∘ 𝐹):ran (1^st ‘𝑅)⟶ran (1^st ‘𝑇) ∧ ((𝐺 ∘ 𝐹)‘(GId‘(2^nd ‘𝑅))) = (GId‘(2^nd ‘𝑇)) ∧ ∀𝑥 ∈ ran (1^st ‘𝑅)∀𝑦 ∈ ran (1^st ‘𝑅)(((𝐺 ∘ 𝐹)‘(𝑥(1^st ‘𝑅)𝑦)) = (((𝐺 ∘ 𝐹)‘𝑥)(1^st ‘𝑇)((𝐺 ∘ 𝐹)‘𝑦)) ∧ ((𝐺 ∘ 𝐹)‘(𝑥(2^nd ‘𝑅)𝑦)) = (((𝐺 ∘ 𝐹)‘𝑥)(2^nd ‘𝑇)((𝐺 ∘ 𝐹)‘𝑦))))))
105	104	3adant2 1129	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) → ((𝐺 ∘ 𝐹) ∈ (𝑅 RngHom 𝑇) ↔ ((𝐺 ∘ 𝐹):ran (1^st ‘𝑅)⟶ran (1^st ‘𝑇) ∧ ((𝐺 ∘ 𝐹)‘(GId‘(2^nd ‘𝑅))) = (GId‘(2^nd ‘𝑇)) ∧ ∀𝑥 ∈ ran (1^st ‘𝑅)∀𝑦 ∈ ran (1^st ‘𝑅)(((𝐺 ∘ 𝐹)‘(𝑥(1^st ‘𝑅)𝑦)) = (((𝐺 ∘ 𝐹)‘𝑥)(1^st ‘𝑇)((𝐺 ∘ 𝐹)‘𝑦)) ∧ ((𝐺 ∘ 𝐹)‘(𝑥(2^nd ‘𝑅)𝑦)) = (((𝐺 ∘ 𝐹)‘𝑥)(2^nd ‘𝑇)((𝐺 ∘ 𝐹)‘𝑦))))))
106	105	adantr 480	. 2 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐺 ∈ (𝑆 RngHom 𝑇))) → ((𝐺 ∘ 𝐹) ∈ (𝑅 RngHom 𝑇) ↔ ((𝐺 ∘ 𝐹):ran (1^st ‘𝑅)⟶ran (1^st ‘𝑇) ∧ ((𝐺 ∘ 𝐹)‘(GId‘(2^nd ‘𝑅))) = (GId‘(2^nd ‘𝑇)) ∧ ∀𝑥 ∈ ran (1^st ‘𝑅)∀𝑦 ∈ ran (1^st ‘𝑅)(((𝐺 ∘ 𝐹)‘(𝑥(1^st ‘𝑅)𝑦)) = (((𝐺 ∘ 𝐹)‘𝑥)(1^st ‘𝑇)((𝐺 ∘ 𝐹)‘𝑦)) ∧ ((𝐺 ∘ 𝐹)‘(𝑥(2^nd ‘𝑅)𝑦)) = (((𝐺 ∘ 𝐹)‘𝑥)(2^nd ‘𝑇)((𝐺 ∘ 𝐹)‘𝑦))))))
107	16, 38, 103, 106	mpbir3and 1340	1 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐺 ∈ (𝑆 RngHom 𝑇))) → (𝐺 ∘ 𝐹) ∈ (𝑅 RngHom 𝑇))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1085 = wceq 1539 ∈ wcel 2108 ∀wral 3063 ran crn 5581 ∘ ccom 5584 ⟶wf 6414 ‘cfv 6418 (class class class)co 7255 1^st c1st 7802 2^nd c2nd 7803 GIdcgi 28753 RingOpscrngo 35979 RngHom crnghom 36045

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-op 4565 df-uni 4837 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-id 5480 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-fo 6424 df-fv 6426 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-1st 7804 df-2nd 7805 df-map 8575 df-grpo 28756 df-gid 28757 df-ablo 28808 df-ass 35928 df-exid 35930 df-mgmOLD 35934 df-sgrOLD 35946 df-mndo 35952 df-rngo 35980 df-rngohom 36048

This theorem is referenced by: rngoisoco 36067

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