rngoisocnv - Mathbox for Jeff Madsen

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Theorem rngoisocnv 32950

Description: The inverse of a ring isomorphism is a ring isomorphism. (Contributed by Jeff Madsen, 16-Jun-2011.)

**Assertion**
Ref	Expression
rngoisocnv	⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) → ^◡𝐹 ∈ (𝑆 RngIso 𝑅))

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

Step	Hyp	Ref	Expression
1		f1ocnv 6062	. . . . . . . 8 ⊢ (𝐹:ran (1^st ‘𝑅)–1-1-onto→ran (1^st ‘𝑆) → ^◡𝐹:ran (1^st ‘𝑆)–1-1-onto→ran (1^st ‘𝑅))
2		f1of 6050	. . . . . . . 8 ⊢ (^◡𝐹:ran (1^st ‘𝑆)–1-1-onto→ran (1^st ‘𝑅) → ^◡𝐹:ran (1^st ‘𝑆)⟶ran (1^st ‘𝑅))
3	1, 2	syl 17	. . . . . . 7 ⊢ (𝐹:ran (1^st ‘𝑅)–1-1-onto→ran (1^st ‘𝑆) → ^◡𝐹:ran (1^st ‘𝑆)⟶ran (1^st ‘𝑅))
4	3	ad2antll 761	. . . . . 6 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:ran (1^st ‘𝑅)–1-1-onto→ran (1^st ‘𝑆))) → ^◡𝐹:ran (1^st ‘𝑆)⟶ran (1^st ‘𝑅))
5		eqid 2610	. . . . . . . . . 10 ⊢ (2^nd ‘𝑅) = (2^nd ‘𝑅)
6		eqid 2610	. . . . . . . . . 10 ⊢ (GId‘(2^nd ‘𝑅)) = (GId‘(2^nd ‘𝑅))
7		eqid 2610	. . . . . . . . . 10 ⊢ (2^nd ‘𝑆) = (2^nd ‘𝑆)
8		eqid 2610	. . . . . . . . . 10 ⊢ (GId‘(2^nd ‘𝑆)) = (GId‘(2^nd ‘𝑆))
9	5, 6, 7, 8	rngohom1 32937	. . . . . . . . 9 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (𝐹‘(GId‘(2^nd ‘𝑅))) = (GId‘(2^nd ‘𝑆)))
10	9	3expa 1257	. . . . . . . 8 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (𝐹‘(GId‘(2^nd ‘𝑅))) = (GId‘(2^nd ‘𝑆)))
11	10	adantrr 749	. . . . . . 7 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:ran (1^st ‘𝑅)–1-1-onto→ran (1^st ‘𝑆))) → (𝐹‘(GId‘(2^nd ‘𝑅))) = (GId‘(2^nd ‘𝑆)))
12		eqid 2610	. . . . . . . . . . 11 ⊢ ran (1^st ‘𝑅) = ran (1^st ‘𝑅)
13	12, 5, 6	rngo1cl 32908	. . . . . . . . . 10 ⊢ (𝑅 ∈ RingOps → (GId‘(2^nd ‘𝑅)) ∈ ran (1^st ‘𝑅))
14		f1ocnvfv 6434	. . . . . . . . . 10 ⊢ ((𝐹:ran (1^st ‘𝑅)–1-1-onto→ran (1^st ‘𝑆) ∧ (GId‘(2^nd ‘𝑅)) ∈ ran (1^st ‘𝑅)) → ((𝐹‘(GId‘(2^nd ‘𝑅))) = (GId‘(2^nd ‘𝑆)) → (^◡𝐹‘(GId‘(2^nd ‘𝑆))) = (GId‘(2^nd ‘𝑅))))
15	13, 14	sylan2 490	. . . . . . . . 9 ⊢ ((𝐹:ran (1^st ‘𝑅)–1-1-onto→ran (1^st ‘𝑆) ∧ 𝑅 ∈ RingOps) → ((𝐹‘(GId‘(2^nd ‘𝑅))) = (GId‘(2^nd ‘𝑆)) → (^◡𝐹‘(GId‘(2^nd ‘𝑆))) = (GId‘(2^nd ‘𝑅))))
16	15	ancoms 468	. . . . . . . 8 ⊢ ((𝑅 ∈ RingOps ∧ 𝐹:ran (1^st ‘𝑅)–1-1-onto→ran (1^st ‘𝑆)) → ((𝐹‘(GId‘(2^nd ‘𝑅))) = (GId‘(2^nd ‘𝑆)) → (^◡𝐹‘(GId‘(2^nd ‘𝑆))) = (GId‘(2^nd ‘𝑅))))
17	16	ad2ant2rl 781	. . . . . . 7 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:ran (1^st ‘𝑅)–1-1-onto→ran (1^st ‘𝑆))) → ((𝐹‘(GId‘(2^nd ‘𝑅))) = (GId‘(2^nd ‘𝑆)) → (^◡𝐹‘(GId‘(2^nd ‘𝑆))) = (GId‘(2^nd ‘𝑅))))
18	11, 17	mpd 15	. . . . . 6 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:ran (1^st ‘𝑅)–1-1-onto→ran (1^st ‘𝑆))) → (^◡𝐹‘(GId‘(2^nd ‘𝑆))) = (GId‘(2^nd ‘𝑅)))
19		f1ocnvfv2 6433	. . . . . . . . . . . . . 14 ⊢ ((𝐹:ran (1^st ‘𝑅)–1-1-onto→ran (1^st ‘𝑆) ∧ 𝑥 ∈ ran (1^st ‘𝑆)) → (𝐹‘(^◡𝐹‘𝑥)) = 𝑥)
20		f1ocnvfv2 6433	. . . . . . . . . . . . . 14 ⊢ ((𝐹:ran (1^st ‘𝑅)–1-1-onto→ran (1^st ‘𝑆) ∧ 𝑦 ∈ ran (1^st ‘𝑆)) → (𝐹‘(^◡𝐹‘𝑦)) = 𝑦)
21	19, 20	anim12da 32676	. . . . . . . . . . . . 13 ⊢ ((𝐹:ran (1^st ‘𝑅)–1-1-onto→ran (1^st ‘𝑆) ∧ (𝑥 ∈ ran (1^st ‘𝑆) ∧ 𝑦 ∈ ran (1^st ‘𝑆))) → ((𝐹‘(^◡𝐹‘𝑥)) = 𝑥 ∧ (𝐹‘(^◡𝐹‘𝑦)) = 𝑦))
22		oveq12 6558	. . . . . . . . . . . . 13 ⊢ (((𝐹‘(^◡𝐹‘𝑥)) = 𝑥 ∧ (𝐹‘(^◡𝐹‘𝑦)) = 𝑦) → ((𝐹‘(^◡𝐹‘𝑥))(1^st ‘𝑆)(𝐹‘(^◡𝐹‘𝑦))) = (𝑥(1^st ‘𝑆)𝑦))
23	21, 22	syl 17	. . . . . . . . . . . 12 ⊢ ((𝐹:ran (1^st ‘𝑅)–1-1-onto→ran (1^st ‘𝑆) ∧ (𝑥 ∈ ran (1^st ‘𝑆) ∧ 𝑦 ∈ ran (1^st ‘𝑆))) → ((𝐹‘(^◡𝐹‘𝑥))(1^st ‘𝑆)(𝐹‘(^◡𝐹‘𝑦))) = (𝑥(1^st ‘𝑆)𝑦))
24	23	adantll 746	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:ran (1^st ‘𝑅)–1-1-onto→ran (1^st ‘𝑆)) ∧ (𝑥 ∈ ran (1^st ‘𝑆) ∧ 𝑦 ∈ ran (1^st ‘𝑆))) → ((𝐹‘(^◡𝐹‘𝑥))(1^st ‘𝑆)(𝐹‘(^◡𝐹‘𝑦))) = (𝑥(1^st ‘𝑆)𝑦))
25	24	adantll 746	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:ran (1^st ‘𝑅)–1-1-onto→ran (1^st ‘𝑆))) ∧ (𝑥 ∈ ran (1^st ‘𝑆) ∧ 𝑦 ∈ ran (1^st ‘𝑆))) → ((𝐹‘(^◡𝐹‘𝑥))(1^st ‘𝑆)(𝐹‘(^◡𝐹‘𝑦))) = (𝑥(1^st ‘𝑆)𝑦))
26		f1ocnvdm 6440	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹:ran (1^st ‘𝑅)–1-1-onto→ran (1^st ‘𝑆) ∧ 𝑥 ∈ ran (1^st ‘𝑆)) → (^◡𝐹‘𝑥) ∈ ran (1^st ‘𝑅))
27		f1ocnvdm 6440	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹:ran (1^st ‘𝑅)–1-1-onto→ran (1^st ‘𝑆) ∧ 𝑦 ∈ ran (1^st ‘𝑆)) → (^◡𝐹‘𝑦) ∈ ran (1^st ‘𝑅))
28	26, 27	anim12da 32676	. . . . . . . . . . . . . . 15 ⊢ ((𝐹:ran (1^st ‘𝑅)–1-1-onto→ran (1^st ‘𝑆) ∧ (𝑥 ∈ ran (1^st ‘𝑆) ∧ 𝑦 ∈ ran (1^st ‘𝑆))) → ((^◡𝐹‘𝑥) ∈ ran (1^st ‘𝑅) ∧ (^◡𝐹‘𝑦) ∈ ran (1^st ‘𝑅)))
29		eqid 2610	. . . . . . . . . . . . . . . 16 ⊢ (1^st ‘𝑅) = (1^st ‘𝑅)
30		eqid 2610	. . . . . . . . . . . . . . . 16 ⊢ (1^st ‘𝑆) = (1^st ‘𝑆)
31	29, 12, 30	rngohomadd 32938	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ ((^◡𝐹‘𝑥) ∈ ran (1^st ‘𝑅) ∧ (^◡𝐹‘𝑦) ∈ ran (1^st ‘𝑅))) → (𝐹‘((^◡𝐹‘𝑥)(1^st ‘𝑅)(^◡𝐹‘𝑦))) = ((𝐹‘(^◡𝐹‘𝑥))(1^st ‘𝑆)(𝐹‘(^◡𝐹‘𝑦))))
32	28, 31	sylan2 490	. . . . . . . . . . . . . 14 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝐹:ran (1^st ‘𝑅)–1-1-onto→ran (1^st ‘𝑆) ∧ (𝑥 ∈ ran (1^st ‘𝑆) ∧ 𝑦 ∈ ran (1^st ‘𝑆)))) → (𝐹‘((^◡𝐹‘𝑥)(1^st ‘𝑅)(^◡𝐹‘𝑦))) = ((𝐹‘(^◡𝐹‘𝑥))(1^st ‘𝑆)(𝐹‘(^◡𝐹‘𝑦))))
33	32	exp32 629	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (𝐹:ran (1^st ‘𝑅)–1-1-onto→ran (1^st ‘𝑆) → ((𝑥 ∈ ran (1^st ‘𝑆) ∧ 𝑦 ∈ ran (1^st ‘𝑆)) → (𝐹‘((^◡𝐹‘𝑥)(1^st ‘𝑅)(^◡𝐹‘𝑦))) = ((𝐹‘(^◡𝐹‘𝑥))(1^st ‘𝑆)(𝐹‘(^◡𝐹‘𝑦))))))
34	33	3expa 1257	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (𝐹:ran (1^st ‘𝑅)–1-1-onto→ran (1^st ‘𝑆) → ((𝑥 ∈ ran (1^st ‘𝑆) ∧ 𝑦 ∈ ran (1^st ‘𝑆)) → (𝐹‘((^◡𝐹‘𝑥)(1^st ‘𝑅)(^◡𝐹‘𝑦))) = ((𝐹‘(^◡𝐹‘𝑥))(1^st ‘𝑆)(𝐹‘(^◡𝐹‘𝑦))))))
35	34	impr 647	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:ran (1^st ‘𝑅)–1-1-onto→ran (1^st ‘𝑆))) → ((𝑥 ∈ ran (1^st ‘𝑆) ∧ 𝑦 ∈ ran (1^st ‘𝑆)) → (𝐹‘((^◡𝐹‘𝑥)(1^st ‘𝑅)(^◡𝐹‘𝑦))) = ((𝐹‘(^◡𝐹‘𝑥))(1^st ‘𝑆)(𝐹‘(^◡𝐹‘𝑦)))))
36	35	imp 444	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:ran (1^st ‘𝑅)–1-1-onto→ran (1^st ‘𝑆))) ∧ (𝑥 ∈ ran (1^st ‘𝑆) ∧ 𝑦 ∈ ran (1^st ‘𝑆))) → (𝐹‘((^◡𝐹‘𝑥)(1^st ‘𝑅)(^◡𝐹‘𝑦))) = ((𝐹‘(^◡𝐹‘𝑥))(1^st ‘𝑆)(𝐹‘(^◡𝐹‘𝑦))))
37		eqid 2610	. . . . . . . . . . . . . . . 16 ⊢ ran (1^st ‘𝑆) = ran (1^st ‘𝑆)
38	30, 37	rngogcl 32881	. . . . . . . . . . . . . . 15 ⊢ ((𝑆 ∈ RingOps ∧ 𝑥 ∈ ran (1^st ‘𝑆) ∧ 𝑦 ∈ ran (1^st ‘𝑆)) → (𝑥(1^st ‘𝑆)𝑦) ∈ ran (1^st ‘𝑆))
39	38	3expb 1258	. . . . . . . . . . . . . 14 ⊢ ((𝑆 ∈ RingOps ∧ (𝑥 ∈ ran (1^st ‘𝑆) ∧ 𝑦 ∈ ran (1^st ‘𝑆))) → (𝑥(1^st ‘𝑆)𝑦) ∈ ran (1^st ‘𝑆))
40		f1ocnvfv2 6433	. . . . . . . . . . . . . . 15 ⊢ ((𝐹:ran (1^st ‘𝑅)–1-1-onto→ran (1^st ‘𝑆) ∧ (𝑥(1^st ‘𝑆)𝑦) ∈ ran (1^st ‘𝑆)) → (𝐹‘(^◡𝐹‘(𝑥(1^st ‘𝑆)𝑦))) = (𝑥(1^st ‘𝑆)𝑦))
41	40	ancoms 468	. . . . . . . . . . . . . 14 ⊢ (((𝑥(1^st ‘𝑆)𝑦) ∈ ran (1^st ‘𝑆) ∧ 𝐹:ran (1^st ‘𝑅)–1-1-onto→ran (1^st ‘𝑆)) → (𝐹‘(^◡𝐹‘(𝑥(1^st ‘𝑆)𝑦))) = (𝑥(1^st ‘𝑆)𝑦))
42	39, 41	sylan 487	. . . . . . . . . . . . 13 ⊢ (((𝑆 ∈ RingOps ∧ (𝑥 ∈ ran (1^st ‘𝑆) ∧ 𝑦 ∈ ran (1^st ‘𝑆))) ∧ 𝐹:ran (1^st ‘𝑅)–1-1-onto→ran (1^st ‘𝑆)) → (𝐹‘(^◡𝐹‘(𝑥(1^st ‘𝑆)𝑦))) = (𝑥(1^st ‘𝑆)𝑦))
43	42	an32s 842	. . . . . . . . . . . 12 ⊢ (((𝑆 ∈ RingOps ∧ 𝐹:ran (1^st ‘𝑅)–1-1-onto→ran (1^st ‘𝑆)) ∧ (𝑥 ∈ ran (1^st ‘𝑆) ∧ 𝑦 ∈ ran (1^st ‘𝑆))) → (𝐹‘(^◡𝐹‘(𝑥(1^st ‘𝑆)𝑦))) = (𝑥(1^st ‘𝑆)𝑦))
44	43	adantlll 750	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹:ran (1^st ‘𝑅)–1-1-onto→ran (1^st ‘𝑆)) ∧ (𝑥 ∈ ran (1^st ‘𝑆) ∧ 𝑦 ∈ ran (1^st ‘𝑆))) → (𝐹‘(^◡𝐹‘(𝑥(1^st ‘𝑆)𝑦))) = (𝑥(1^st ‘𝑆)𝑦))
45	44	adantlrl 752	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:ran (1^st ‘𝑅)–1-1-onto→ran (1^st ‘𝑆))) ∧ (𝑥 ∈ ran (1^st ‘𝑆) ∧ 𝑦 ∈ ran (1^st ‘𝑆))) → (𝐹‘(^◡𝐹‘(𝑥(1^st ‘𝑆)𝑦))) = (𝑥(1^st ‘𝑆)𝑦))
46	25, 36, 45	3eqtr4rd 2655	. . . . . . . . 9 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:ran (1^st ‘𝑅)–1-1-onto→ran (1^st ‘𝑆))) ∧ (𝑥 ∈ ran (1^st ‘𝑆) ∧ 𝑦 ∈ ran (1^st ‘𝑆))) → (𝐹‘(^◡𝐹‘(𝑥(1^st ‘𝑆)𝑦))) = (𝐹‘((^◡𝐹‘𝑥)(1^st ‘𝑅)(^◡𝐹‘𝑦))))
47		f1of1 6049	. . . . . . . . . . . 12 ⊢ (𝐹:ran (1^st ‘𝑅)–1-1-onto→ran (1^st ‘𝑆) → 𝐹:ran (1^st ‘𝑅)–1-1→ran (1^st ‘𝑆))
48	47	ad2antlr 759	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹:ran (1^st ‘𝑅)–1-1-onto→ran (1^st ‘𝑆)) ∧ (𝑥 ∈ ran (1^st ‘𝑆) ∧ 𝑦 ∈ ran (1^st ‘𝑆))) → 𝐹:ran (1^st ‘𝑅)–1-1→ran (1^st ‘𝑆))
49		f1ocnvdm 6440	. . . . . . . . . . . . . . 15 ⊢ ((𝐹:ran (1^st ‘𝑅)–1-1-onto→ran (1^st ‘𝑆) ∧ (𝑥(1^st ‘𝑆)𝑦) ∈ ran (1^st ‘𝑆)) → (^◡𝐹‘(𝑥(1^st ‘𝑆)𝑦)) ∈ ran (1^st ‘𝑅))
50	49	ancoms 468	. . . . . . . . . . . . . 14 ⊢ (((𝑥(1^st ‘𝑆)𝑦) ∈ ran (1^st ‘𝑆) ∧ 𝐹:ran (1^st ‘𝑅)–1-1-onto→ran (1^st ‘𝑆)) → (^◡𝐹‘(𝑥(1^st ‘𝑆)𝑦)) ∈ ran (1^st ‘𝑅))
51	39, 50	sylan 487	. . . . . . . . . . . . 13 ⊢ (((𝑆 ∈ RingOps ∧ (𝑥 ∈ ran (1^st ‘𝑆) ∧ 𝑦 ∈ ran (1^st ‘𝑆))) ∧ 𝐹:ran (1^st ‘𝑅)–1-1-onto→ran (1^st ‘𝑆)) → (^◡𝐹‘(𝑥(1^st ‘𝑆)𝑦)) ∈ ran (1^st ‘𝑅))
52	51	an32s 842	. . . . . . . . . . . 12 ⊢ (((𝑆 ∈ RingOps ∧ 𝐹:ran (1^st ‘𝑅)–1-1-onto→ran (1^st ‘𝑆)) ∧ (𝑥 ∈ ran (1^st ‘𝑆) ∧ 𝑦 ∈ ran (1^st ‘𝑆))) → (^◡𝐹‘(𝑥(1^st ‘𝑆)𝑦)) ∈ ran (1^st ‘𝑅))
53	52	adantlll 750	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹:ran (1^st ‘𝑅)–1-1-onto→ran (1^st ‘𝑆)) ∧ (𝑥 ∈ ran (1^st ‘𝑆) ∧ 𝑦 ∈ ran (1^st ‘𝑆))) → (^◡𝐹‘(𝑥(1^st ‘𝑆)𝑦)) ∈ ran (1^st ‘𝑅))
54	29, 12	rngogcl 32881	. . . . . . . . . . . . . . 15 ⊢ ((𝑅 ∈ RingOps ∧ (^◡𝐹‘𝑥) ∈ ran (1^st ‘𝑅) ∧ (^◡𝐹‘𝑦) ∈ ran (1^st ‘𝑅)) → ((^◡𝐹‘𝑥)(1^st ‘𝑅)(^◡𝐹‘𝑦)) ∈ ran (1^st ‘𝑅))
55	54	3expb 1258	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ RingOps ∧ ((^◡𝐹‘𝑥) ∈ ran (1^st ‘𝑅) ∧ (^◡𝐹‘𝑦) ∈ ran (1^st ‘𝑅))) → ((^◡𝐹‘𝑥)(1^st ‘𝑅)(^◡𝐹‘𝑦)) ∈ ran (1^st ‘𝑅))
56	28, 55	sylan2 490	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ RingOps ∧ (𝐹:ran (1^st ‘𝑅)–1-1-onto→ran (1^st ‘𝑆) ∧ (𝑥 ∈ ran (1^st ‘𝑆) ∧ 𝑦 ∈ ran (1^st ‘𝑆)))) → ((^◡𝐹‘𝑥)(1^st ‘𝑅)(^◡𝐹‘𝑦)) ∈ ran (1^st ‘𝑅))
57	56	anassrs 678	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ RingOps ∧ 𝐹:ran (1^st ‘𝑅)–1-1-onto→ran (1^st ‘𝑆)) ∧ (𝑥 ∈ ran (1^st ‘𝑆) ∧ 𝑦 ∈ ran (1^st ‘𝑆))) → ((^◡𝐹‘𝑥)(1^st ‘𝑅)(^◡𝐹‘𝑦)) ∈ ran (1^st ‘𝑅))
58	57	adantllr 751	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹:ran (1^st ‘𝑅)–1-1-onto→ran (1^st ‘𝑆)) ∧ (𝑥 ∈ ran (1^st ‘𝑆) ∧ 𝑦 ∈ ran (1^st ‘𝑆))) → ((^◡𝐹‘𝑥)(1^st ‘𝑅)(^◡𝐹‘𝑦)) ∈ ran (1^st ‘𝑅))
59		f1fveq 6420	. . . . . . . . . . 11 ⊢ ((𝐹:ran (1^st ‘𝑅)–1-1→ran (1^st ‘𝑆) ∧ ((^◡𝐹‘(𝑥(1^st ‘𝑆)𝑦)) ∈ ran (1^st ‘𝑅) ∧ ((^◡𝐹‘𝑥)(1^st ‘𝑅)(^◡𝐹‘𝑦)) ∈ ran (1^st ‘𝑅))) → ((𝐹‘(^◡𝐹‘(𝑥(1^st ‘𝑆)𝑦))) = (𝐹‘((^◡𝐹‘𝑥)(1^st ‘𝑅)(^◡𝐹‘𝑦))) ↔ (^◡𝐹‘(𝑥(1^st ‘𝑆)𝑦)) = ((^◡𝐹‘𝑥)(1^st ‘𝑅)(^◡𝐹‘𝑦))))
60	48, 53, 58, 59	syl12anc 1316	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹:ran (1^st ‘𝑅)–1-1-onto→ran (1^st ‘𝑆)) ∧ (𝑥 ∈ ran (1^st ‘𝑆) ∧ 𝑦 ∈ ran (1^st ‘𝑆))) → ((𝐹‘(^◡𝐹‘(𝑥(1^st ‘𝑆)𝑦))) = (𝐹‘((^◡𝐹‘𝑥)(1^st ‘𝑅)(^◡𝐹‘𝑦))) ↔ (^◡𝐹‘(𝑥(1^st ‘𝑆)𝑦)) = ((^◡𝐹‘𝑥)(1^st ‘𝑅)(^◡𝐹‘𝑦))))
61	60	adantlrl 752	. . . . . . . . 9 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:ran (1^st ‘𝑅)–1-1-onto→ran (1^st ‘𝑆))) ∧ (𝑥 ∈ ran (1^st ‘𝑆) ∧ 𝑦 ∈ ran (1^st ‘𝑆))) → ((𝐹‘(^◡𝐹‘(𝑥(1^st ‘𝑆)𝑦))) = (𝐹‘((^◡𝐹‘𝑥)(1^st ‘𝑅)(^◡𝐹‘𝑦))) ↔ (^◡𝐹‘(𝑥(1^st ‘𝑆)𝑦)) = ((^◡𝐹‘𝑥)(1^st ‘𝑅)(^◡𝐹‘𝑦))))
62	46, 61	mpbid 221	. . . . . . . 8 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:ran (1^st ‘𝑅)–1-1-onto→ran (1^st ‘𝑆))) ∧ (𝑥 ∈ ran (1^st ‘𝑆) ∧ 𝑦 ∈ ran (1^st ‘𝑆))) → (^◡𝐹‘(𝑥(1^st ‘𝑆)𝑦)) = ((^◡𝐹‘𝑥)(1^st ‘𝑅)(^◡𝐹‘𝑦)))
63		oveq12 6558	. . . . . . . . . . . . 13 ⊢ (((𝐹‘(^◡𝐹‘𝑥)) = 𝑥 ∧ (𝐹‘(^◡𝐹‘𝑦)) = 𝑦) → ((𝐹‘(^◡𝐹‘𝑥))(2^nd ‘𝑆)(𝐹‘(^◡𝐹‘𝑦))) = (𝑥(2^nd ‘𝑆)𝑦))
64	21, 63	syl 17	. . . . . . . . . . . 12 ⊢ ((𝐹:ran (1^st ‘𝑅)–1-1-onto→ran (1^st ‘𝑆) ∧ (𝑥 ∈ ran (1^st ‘𝑆) ∧ 𝑦 ∈ ran (1^st ‘𝑆))) → ((𝐹‘(^◡𝐹‘𝑥))(2^nd ‘𝑆)(𝐹‘(^◡𝐹‘𝑦))) = (𝑥(2^nd ‘𝑆)𝑦))
65	64	adantll 746	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:ran (1^st ‘𝑅)–1-1-onto→ran (1^st ‘𝑆)) ∧ (𝑥 ∈ ran (1^st ‘𝑆) ∧ 𝑦 ∈ ran (1^st ‘𝑆))) → ((𝐹‘(^◡𝐹‘𝑥))(2^nd ‘𝑆)(𝐹‘(^◡𝐹‘𝑦))) = (𝑥(2^nd ‘𝑆)𝑦))
66	65	adantll 746	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:ran (1^st ‘𝑅)–1-1-onto→ran (1^st ‘𝑆))) ∧ (𝑥 ∈ ran (1^st ‘𝑆) ∧ 𝑦 ∈ ran (1^st ‘𝑆))) → ((𝐹‘(^◡𝐹‘𝑥))(2^nd ‘𝑆)(𝐹‘(^◡𝐹‘𝑦))) = (𝑥(2^nd ‘𝑆)𝑦))
67	29, 12, 5, 7	rngohommul 32939	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ ((^◡𝐹‘𝑥) ∈ ran (1^st ‘𝑅) ∧ (^◡𝐹‘𝑦) ∈ ran (1^st ‘𝑅))) → (𝐹‘((^◡𝐹‘𝑥)(2^nd ‘𝑅)(^◡𝐹‘𝑦))) = ((𝐹‘(^◡𝐹‘𝑥))(2^nd ‘𝑆)(𝐹‘(^◡𝐹‘𝑦))))
68	28, 67	sylan2 490	. . . . . . . . . . . . . 14 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝐹:ran (1^st ‘𝑅)–1-1-onto→ran (1^st ‘𝑆) ∧ (𝑥 ∈ ran (1^st ‘𝑆) ∧ 𝑦 ∈ ran (1^st ‘𝑆)))) → (𝐹‘((^◡𝐹‘𝑥)(2^nd ‘𝑅)(^◡𝐹‘𝑦))) = ((𝐹‘(^◡𝐹‘𝑥))(2^nd ‘𝑆)(𝐹‘(^◡𝐹‘𝑦))))
69	68	exp32 629	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (𝐹:ran (1^st ‘𝑅)–1-1-onto→ran (1^st ‘𝑆) → ((𝑥 ∈ ran (1^st ‘𝑆) ∧ 𝑦 ∈ ran (1^st ‘𝑆)) → (𝐹‘((^◡𝐹‘𝑥)(2^nd ‘𝑅)(^◡𝐹‘𝑦))) = ((𝐹‘(^◡𝐹‘𝑥))(2^nd ‘𝑆)(𝐹‘(^◡𝐹‘𝑦))))))
70	69	3expa 1257	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (𝐹:ran (1^st ‘𝑅)–1-1-onto→ran (1^st ‘𝑆) → ((𝑥 ∈ ran (1^st ‘𝑆) ∧ 𝑦 ∈ ran (1^st ‘𝑆)) → (𝐹‘((^◡𝐹‘𝑥)(2^nd ‘𝑅)(^◡𝐹‘𝑦))) = ((𝐹‘(^◡𝐹‘𝑥))(2^nd ‘𝑆)(𝐹‘(^◡𝐹‘𝑦))))))
71	70	impr 647	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:ran (1^st ‘𝑅)–1-1-onto→ran (1^st ‘𝑆))) → ((𝑥 ∈ ran (1^st ‘𝑆) ∧ 𝑦 ∈ ran (1^st ‘𝑆)) → (𝐹‘((^◡𝐹‘𝑥)(2^nd ‘𝑅)(^◡𝐹‘𝑦))) = ((𝐹‘(^◡𝐹‘𝑥))(2^nd ‘𝑆)(𝐹‘(^◡𝐹‘𝑦)))))
72	71	imp 444	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:ran (1^st ‘𝑅)–1-1-onto→ran (1^st ‘𝑆))) ∧ (𝑥 ∈ ran (1^st ‘𝑆) ∧ 𝑦 ∈ ran (1^st ‘𝑆))) → (𝐹‘((^◡𝐹‘𝑥)(2^nd ‘𝑅)(^◡𝐹‘𝑦))) = ((𝐹‘(^◡𝐹‘𝑥))(2^nd ‘𝑆)(𝐹‘(^◡𝐹‘𝑦))))
73	30, 7, 37	rngocl 32870	. . . . . . . . . . . . . . 15 ⊢ ((𝑆 ∈ RingOps ∧ 𝑥 ∈ ran (1^st ‘𝑆) ∧ 𝑦 ∈ ran (1^st ‘𝑆)) → (𝑥(2^nd ‘𝑆)𝑦) ∈ ran (1^st ‘𝑆))
74	73	3expb 1258	. . . . . . . . . . . . . 14 ⊢ ((𝑆 ∈ RingOps ∧ (𝑥 ∈ ran (1^st ‘𝑆) ∧ 𝑦 ∈ ran (1^st ‘𝑆))) → (𝑥(2^nd ‘𝑆)𝑦) ∈ ran (1^st ‘𝑆))
75		f1ocnvfv2 6433	. . . . . . . . . . . . . . 15 ⊢ ((𝐹:ran (1^st ‘𝑅)–1-1-onto→ran (1^st ‘𝑆) ∧ (𝑥(2^nd ‘𝑆)𝑦) ∈ ran (1^st ‘𝑆)) → (𝐹‘(^◡𝐹‘(𝑥(2^nd ‘𝑆)𝑦))) = (𝑥(2^nd ‘𝑆)𝑦))
76	75	ancoms 468	. . . . . . . . . . . . . 14 ⊢ (((𝑥(2^nd ‘𝑆)𝑦) ∈ ran (1^st ‘𝑆) ∧ 𝐹:ran (1^st ‘𝑅)–1-1-onto→ran (1^st ‘𝑆)) → (𝐹‘(^◡𝐹‘(𝑥(2^nd ‘𝑆)𝑦))) = (𝑥(2^nd ‘𝑆)𝑦))
77	74, 76	sylan 487	. . . . . . . . . . . . 13 ⊢ (((𝑆 ∈ RingOps ∧ (𝑥 ∈ ran (1^st ‘𝑆) ∧ 𝑦 ∈ ran (1^st ‘𝑆))) ∧ 𝐹:ran (1^st ‘𝑅)–1-1-onto→ran (1^st ‘𝑆)) → (𝐹‘(^◡𝐹‘(𝑥(2^nd ‘𝑆)𝑦))) = (𝑥(2^nd ‘𝑆)𝑦))
78	77	an32s 842	. . . . . . . . . . . 12 ⊢ (((𝑆 ∈ RingOps ∧ 𝐹:ran (1^st ‘𝑅)–1-1-onto→ran (1^st ‘𝑆)) ∧ (𝑥 ∈ ran (1^st ‘𝑆) ∧ 𝑦 ∈ ran (1^st ‘𝑆))) → (𝐹‘(^◡𝐹‘(𝑥(2^nd ‘𝑆)𝑦))) = (𝑥(2^nd ‘𝑆)𝑦))
79	78	adantlll 750	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹:ran (1^st ‘𝑅)–1-1-onto→ran (1^st ‘𝑆)) ∧ (𝑥 ∈ ran (1^st ‘𝑆) ∧ 𝑦 ∈ ran (1^st ‘𝑆))) → (𝐹‘(^◡𝐹‘(𝑥(2^nd ‘𝑆)𝑦))) = (𝑥(2^nd ‘𝑆)𝑦))
80	79	adantlrl 752	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:ran (1^st ‘𝑅)–1-1-onto→ran (1^st ‘𝑆))) ∧ (𝑥 ∈ ran (1^st ‘𝑆) ∧ 𝑦 ∈ ran (1^st ‘𝑆))) → (𝐹‘(^◡𝐹‘(𝑥(2^nd ‘𝑆)𝑦))) = (𝑥(2^nd ‘𝑆)𝑦))
81	66, 72, 80	3eqtr4rd 2655	. . . . . . . . 9 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:ran (1^st ‘𝑅)–1-1-onto→ran (1^st ‘𝑆))) ∧ (𝑥 ∈ ran (1^st ‘𝑆) ∧ 𝑦 ∈ ran (1^st ‘𝑆))) → (𝐹‘(^◡𝐹‘(𝑥(2^nd ‘𝑆)𝑦))) = (𝐹‘((^◡𝐹‘𝑥)(2^nd ‘𝑅)(^◡𝐹‘𝑦))))
82		f1ocnvdm 6440	. . . . . . . . . . . . . . 15 ⊢ ((𝐹:ran (1^st ‘𝑅)–1-1-onto→ran (1^st ‘𝑆) ∧ (𝑥(2^nd ‘𝑆)𝑦) ∈ ran (1^st ‘𝑆)) → (^◡𝐹‘(𝑥(2^nd ‘𝑆)𝑦)) ∈ ran (1^st ‘𝑅))
83	82	ancoms 468	. . . . . . . . . . . . . 14 ⊢ (((𝑥(2^nd ‘𝑆)𝑦) ∈ ran (1^st ‘𝑆) ∧ 𝐹:ran (1^st ‘𝑅)–1-1-onto→ran (1^st ‘𝑆)) → (^◡𝐹‘(𝑥(2^nd ‘𝑆)𝑦)) ∈ ran (1^st ‘𝑅))
84	74, 83	sylan 487	. . . . . . . . . . . . 13 ⊢ (((𝑆 ∈ RingOps ∧ (𝑥 ∈ ran (1^st ‘𝑆) ∧ 𝑦 ∈ ran (1^st ‘𝑆))) ∧ 𝐹:ran (1^st ‘𝑅)–1-1-onto→ran (1^st ‘𝑆)) → (^◡𝐹‘(𝑥(2^nd ‘𝑆)𝑦)) ∈ ran (1^st ‘𝑅))
85	84	an32s 842	. . . . . . . . . . . 12 ⊢ (((𝑆 ∈ RingOps ∧ 𝐹:ran (1^st ‘𝑅)–1-1-onto→ran (1^st ‘𝑆)) ∧ (𝑥 ∈ ran (1^st ‘𝑆) ∧ 𝑦 ∈ ran (1^st ‘𝑆))) → (^◡𝐹‘(𝑥(2^nd ‘𝑆)𝑦)) ∈ ran (1^st ‘𝑅))
86	85	adantlll 750	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹:ran (1^st ‘𝑅)–1-1-onto→ran (1^st ‘𝑆)) ∧ (𝑥 ∈ ran (1^st ‘𝑆) ∧ 𝑦 ∈ ran (1^st ‘𝑆))) → (^◡𝐹‘(𝑥(2^nd ‘𝑆)𝑦)) ∈ ran (1^st ‘𝑅))
87	29, 5, 12	rngocl 32870	. . . . . . . . . . . . . . 15 ⊢ ((𝑅 ∈ RingOps ∧ (^◡𝐹‘𝑥) ∈ ran (1^st ‘𝑅) ∧ (^◡𝐹‘𝑦) ∈ ran (1^st ‘𝑅)) → ((^◡𝐹‘𝑥)(2^nd ‘𝑅)(^◡𝐹‘𝑦)) ∈ ran (1^st ‘𝑅))
88	87	3expb 1258	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ RingOps ∧ ((^◡𝐹‘𝑥) ∈ ran (1^st ‘𝑅) ∧ (^◡𝐹‘𝑦) ∈ ran (1^st ‘𝑅))) → ((^◡𝐹‘𝑥)(2^nd ‘𝑅)(^◡𝐹‘𝑦)) ∈ ran (1^st ‘𝑅))
89	28, 88	sylan2 490	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ RingOps ∧ (𝐹:ran (1^st ‘𝑅)–1-1-onto→ran (1^st ‘𝑆) ∧ (𝑥 ∈ ran (1^st ‘𝑆) ∧ 𝑦 ∈ ran (1^st ‘𝑆)))) → ((^◡𝐹‘𝑥)(2^nd ‘𝑅)(^◡𝐹‘𝑦)) ∈ ran (1^st ‘𝑅))
90	89	anassrs 678	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ RingOps ∧ 𝐹:ran (1^st ‘𝑅)–1-1-onto→ran (1^st ‘𝑆)) ∧ (𝑥 ∈ ran (1^st ‘𝑆) ∧ 𝑦 ∈ ran (1^st ‘𝑆))) → ((^◡𝐹‘𝑥)(2^nd ‘𝑅)(^◡𝐹‘𝑦)) ∈ ran (1^st ‘𝑅))
91	90	adantllr 751	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹:ran (1^st ‘𝑅)–1-1-onto→ran (1^st ‘𝑆)) ∧ (𝑥 ∈ ran (1^st ‘𝑆) ∧ 𝑦 ∈ ran (1^st ‘𝑆))) → ((^◡𝐹‘𝑥)(2^nd ‘𝑅)(^◡𝐹‘𝑦)) ∈ ran (1^st ‘𝑅))
92		f1fveq 6420	. . . . . . . . . . 11 ⊢ ((𝐹:ran (1^st ‘𝑅)–1-1→ran (1^st ‘𝑆) ∧ ((^◡𝐹‘(𝑥(2^nd ‘𝑆)𝑦)) ∈ ran (1^st ‘𝑅) ∧ ((^◡𝐹‘𝑥)(2^nd ‘𝑅)(^◡𝐹‘𝑦)) ∈ ran (1^st ‘𝑅))) → ((𝐹‘(^◡𝐹‘(𝑥(2^nd ‘𝑆)𝑦))) = (𝐹‘((^◡𝐹‘𝑥)(2^nd ‘𝑅)(^◡𝐹‘𝑦))) ↔ (^◡𝐹‘(𝑥(2^nd ‘𝑆)𝑦)) = ((^◡𝐹‘𝑥)(2^nd ‘𝑅)(^◡𝐹‘𝑦))))
93	48, 86, 91, 92	syl12anc 1316	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹:ran (1^st ‘𝑅)–1-1-onto→ran (1^st ‘𝑆)) ∧ (𝑥 ∈ ran (1^st ‘𝑆) ∧ 𝑦 ∈ ran (1^st ‘𝑆))) → ((𝐹‘(^◡𝐹‘(𝑥(2^nd ‘𝑆)𝑦))) = (𝐹‘((^◡𝐹‘𝑥)(2^nd ‘𝑅)(^◡𝐹‘𝑦))) ↔ (^◡𝐹‘(𝑥(2^nd ‘𝑆)𝑦)) = ((^◡𝐹‘𝑥)(2^nd ‘𝑅)(^◡𝐹‘𝑦))))
94	93	adantlrl 752	. . . . . . . . 9 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:ran (1^st ‘𝑅)–1-1-onto→ran (1^st ‘𝑆))) ∧ (𝑥 ∈ ran (1^st ‘𝑆) ∧ 𝑦 ∈ ran (1^st ‘𝑆))) → ((𝐹‘(^◡𝐹‘(𝑥(2^nd ‘𝑆)𝑦))) = (𝐹‘((^◡𝐹‘𝑥)(2^nd ‘𝑅)(^◡𝐹‘𝑦))) ↔ (^◡𝐹‘(𝑥(2^nd ‘𝑆)𝑦)) = ((^◡𝐹‘𝑥)(2^nd ‘𝑅)(^◡𝐹‘𝑦))))
95	81, 94	mpbid 221	. . . . . . . 8 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:ran (1^st ‘𝑅)–1-1-onto→ran (1^st ‘𝑆))) ∧ (𝑥 ∈ ran (1^st ‘𝑆) ∧ 𝑦 ∈ ran (1^st ‘𝑆))) → (^◡𝐹‘(𝑥(2^nd ‘𝑆)𝑦)) = ((^◡𝐹‘𝑥)(2^nd ‘𝑅)(^◡𝐹‘𝑦)))
96	62, 95	jca 553	. . . . . . 7 ⊢ ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:ran (1^st ‘𝑅)–1-1-onto→ran (1^st ‘𝑆))) ∧ (𝑥 ∈ ran (1^st ‘𝑆) ∧ 𝑦 ∈ ran (1^st ‘𝑆))) → ((^◡𝐹‘(𝑥(1^st ‘𝑆)𝑦)) = ((^◡𝐹‘𝑥)(1^st ‘𝑅)(^◡𝐹‘𝑦)) ∧ (^◡𝐹‘(𝑥(2^nd ‘𝑆)𝑦)) = ((^◡𝐹‘𝑥)(2^nd ‘𝑅)(^◡𝐹‘𝑦))))
97	96	ralrimivva 2954	. . . . . 6 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:ran (1^st ‘𝑅)–1-1-onto→ran (1^st ‘𝑆))) → ∀𝑥 ∈ ran (1^st ‘𝑆)∀𝑦 ∈ ran (1^st ‘𝑆)((^◡𝐹‘(𝑥(1^st ‘𝑆)𝑦)) = ((^◡𝐹‘𝑥)(1^st ‘𝑅)(^◡𝐹‘𝑦)) ∧ (^◡𝐹‘(𝑥(2^nd ‘𝑆)𝑦)) = ((^◡𝐹‘𝑥)(2^nd ‘𝑅)(^◡𝐹‘𝑦))))
98	30, 7, 37, 8, 29, 5, 12, 6	isrngohom 32934	. . . . . . . 8 ⊢ ((𝑆 ∈ RingOps ∧ 𝑅 ∈ RingOps) → (^◡𝐹 ∈ (𝑆 RngHom 𝑅) ↔ (^◡𝐹:ran (1^st ‘𝑆)⟶ran (1^st ‘𝑅) ∧ (^◡𝐹‘(GId‘(2^nd ‘𝑆))) = (GId‘(2^nd ‘𝑅)) ∧ ∀𝑥 ∈ ran (1^st ‘𝑆)∀𝑦 ∈ ran (1^st ‘𝑆)((^◡𝐹‘(𝑥(1^st ‘𝑆)𝑦)) = ((^◡𝐹‘𝑥)(1^st ‘𝑅)(^◡𝐹‘𝑦)) ∧ (^◡𝐹‘(𝑥(2^nd ‘𝑆)𝑦)) = ((^◡𝐹‘𝑥)(2^nd ‘𝑅)(^◡𝐹‘𝑦))))))
99	98	ancoms 468	. . . . . . 7 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) → (^◡𝐹 ∈ (𝑆 RngHom 𝑅) ↔ (^◡𝐹:ran (1^st ‘𝑆)⟶ran (1^st ‘𝑅) ∧ (^◡𝐹‘(GId‘(2^nd ‘𝑆))) = (GId‘(2^nd ‘𝑅)) ∧ ∀𝑥 ∈ ran (1^st ‘𝑆)∀𝑦 ∈ ran (1^st ‘𝑆)((^◡𝐹‘(𝑥(1^st ‘𝑆)𝑦)) = ((^◡𝐹‘𝑥)(1^st ‘𝑅)(^◡𝐹‘𝑦)) ∧ (^◡𝐹‘(𝑥(2^nd ‘𝑆)𝑦)) = ((^◡𝐹‘𝑥)(2^nd ‘𝑅)(^◡𝐹‘𝑦))))))
100	99	adantr 480	. . . . . 6 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:ran (1^st ‘𝑅)–1-1-onto→ran (1^st ‘𝑆))) → (^◡𝐹 ∈ (𝑆 RngHom 𝑅) ↔ (^◡𝐹:ran (1^st ‘𝑆)⟶ran (1^st ‘𝑅) ∧ (^◡𝐹‘(GId‘(2^nd ‘𝑆))) = (GId‘(2^nd ‘𝑅)) ∧ ∀𝑥 ∈ ran (1^st ‘𝑆)∀𝑦 ∈ ran (1^st ‘𝑆)((^◡𝐹‘(𝑥(1^st ‘𝑆)𝑦)) = ((^◡𝐹‘𝑥)(1^st ‘𝑅)(^◡𝐹‘𝑦)) ∧ (^◡𝐹‘(𝑥(2^nd ‘𝑆)𝑦)) = ((^◡𝐹‘𝑥)(2^nd ‘𝑅)(^◡𝐹‘𝑦))))))
101	4, 18, 97, 100	mpbir3and 1238	. . . . 5 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:ran (1^st ‘𝑅)–1-1-onto→ran (1^st ‘𝑆))) → ^◡𝐹 ∈ (𝑆 RngHom 𝑅))
102	1	ad2antll 761	. . . . 5 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:ran (1^st ‘𝑅)–1-1-onto→ran (1^st ‘𝑆))) → ^◡𝐹:ran (1^st ‘𝑆)–1-1-onto→ran (1^st ‘𝑅))
103	101, 102	jca 553	. . . 4 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:ran (1^st ‘𝑅)–1-1-onto→ran (1^st ‘𝑆))) → (^◡𝐹 ∈ (𝑆 RngHom 𝑅) ∧ ^◡𝐹:ran (1^st ‘𝑆)–1-1-onto→ran (1^st ‘𝑅)))
104	103	ex 449	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) → ((𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:ran (1^st ‘𝑅)–1-1-onto→ran (1^st ‘𝑆)) → (^◡𝐹 ∈ (𝑆 RngHom 𝑅) ∧ ^◡𝐹:ran (1^st ‘𝑆)–1-1-onto→ran (1^st ‘𝑅))))
105	29, 12, 30, 37	isrngoiso 32947	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) → (𝐹 ∈ (𝑅 RngIso 𝑆) ↔ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:ran (1^st ‘𝑅)–1-1-onto→ran (1^st ‘𝑆))))
106	30, 37, 29, 12	isrngoiso 32947	. . . 4 ⊢ ((𝑆 ∈ RingOps ∧ 𝑅 ∈ RingOps) → (^◡𝐹 ∈ (𝑆 RngIso 𝑅) ↔ (^◡𝐹 ∈ (𝑆 RngHom 𝑅) ∧ ^◡𝐹:ran (1^st ‘𝑆)–1-1-onto→ran (1^st ‘𝑅))))
107	106	ancoms 468	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) → (^◡𝐹 ∈ (𝑆 RngIso 𝑅) ↔ (^◡𝐹 ∈ (𝑆 RngHom 𝑅) ∧ ^◡𝐹:ran (1^st ‘𝑆)–1-1-onto→ran (1^st ‘𝑅))))
108	104, 105, 107	3imtr4d 282	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) → (𝐹 ∈ (𝑅 RngIso 𝑆) → ^◡𝐹 ∈ (𝑆 RngIso 𝑅)))
109	108	3impia 1253	1 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) → ^◡𝐹 ∈ (𝑆 RngIso 𝑅))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ^◡ccnv 5037 ran crn 5039 ⟶wf 5800 –1-1→wf1 5801 –1-1-onto→wf1o 5803 ‘cfv 5804 (class class class)co 6549 1^st c1st 7057 2^nd c2nd 7058 GIdcgi 26728 RingOpscrngo 32863 RngHom crnghom 32929 RngIso crngiso 32930

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-1st 7059 df-2nd 7060 df-map 7746 df-grpo 26731 df-gid 26732 df-ablo 26783 df-ass 32812 df-exid 32814 df-mgmOLD 32818 df-sgrOLD 32830 df-mndo 32836 df-rngo 32864 df-rngohom 32932 df-rngoiso 32945

This theorem is referenced by: riscer 32957

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