rngoisoco - Mathbox for Jeff Madsen

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Theorem rngoisoco 38185

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

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

**Proof of Theorem rngoisoco**
Step	Hyp	Ref	Expression
1		rngoisohom 38183	. . . . . 6 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsIso 𝑆)) → 𝐹 ∈ (𝑅 RingOpsHom 𝑆))
2	1	3expa 1118	. . . . 5 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹 ∈ (𝑅 RingOpsIso 𝑆)) → 𝐹 ∈ (𝑅 RingOpsHom 𝑆))
3	2	3adantl3 1169	. . . 4 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ 𝐹 ∈ (𝑅 RingOpsIso 𝑆)) → 𝐹 ∈ (𝑅 RingOpsHom 𝑆))
4		rngoisohom 38183	. . . . . 6 ⊢ ((𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps ∧ 𝐺 ∈ (𝑆 RingOpsIso 𝑇)) → 𝐺 ∈ (𝑆 RingOpsHom 𝑇))
5	4	3expa 1118	. . . . 5 ⊢ (((𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ 𝐺 ∈ (𝑆 RingOpsIso 𝑇)) → 𝐺 ∈ (𝑆 RingOpsHom 𝑇))
6	5	3adantl1 1167	. . . 4 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ 𝐺 ∈ (𝑆 RingOpsIso 𝑇)) → 𝐺 ∈ (𝑆 RingOpsHom 𝑇))
7	3, 6	anim12dan 619	. . 3 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RingOpsIso 𝑆) ∧ 𝐺 ∈ (𝑆 RingOpsIso 𝑇))) → (𝐹 ∈ (𝑅 RingOpsHom 𝑆) ∧ 𝐺 ∈ (𝑆 RingOpsHom 𝑇)))
8		rngohomco 38177	. . 3 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RingOpsHom 𝑆) ∧ 𝐺 ∈ (𝑆 RingOpsHom 𝑇))) → (𝐺 ∘ 𝐹) ∈ (𝑅 RingOpsHom 𝑇))
9	7, 8	syldan 591	. 2 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RingOpsIso 𝑆) ∧ 𝐺 ∈ (𝑆 RingOpsIso 𝑇))) → (𝐺 ∘ 𝐹) ∈ (𝑅 RingOpsHom 𝑇))
10		eqid 2736	. . . . . . 7 ⊢ (1^st ‘𝑆) = (1^st ‘𝑆)
11		eqid 2736	. . . . . . 7 ⊢ ran (1^st ‘𝑆) = ran (1^st ‘𝑆)
12		eqid 2736	. . . . . . 7 ⊢ (1^st ‘𝑇) = (1^st ‘𝑇)
13		eqid 2736	. . . . . . 7 ⊢ ran (1^st ‘𝑇) = ran (1^st ‘𝑇)
14	10, 11, 12, 13	rngoiso1o 38182	. . . . . 6 ⊢ ((𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps ∧ 𝐺 ∈ (𝑆 RingOpsIso 𝑇)) → 𝐺:ran (1^st ‘𝑆)–1-1-onto→ran (1^st ‘𝑇))
15	14	3expa 1118	. . . . 5 ⊢ (((𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ 𝐺 ∈ (𝑆 RingOpsIso 𝑇)) → 𝐺:ran (1^st ‘𝑆)–1-1-onto→ran (1^st ‘𝑇))
16	15	3adantl1 1167	. . . 4 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ 𝐺 ∈ (𝑆 RingOpsIso 𝑇)) → 𝐺:ran (1^st ‘𝑆)–1-1-onto→ran (1^st ‘𝑇))
17	16	adantrl 716	. . 3 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RingOpsIso 𝑆) ∧ 𝐺 ∈ (𝑆 RingOpsIso 𝑇))) → 𝐺:ran (1^st ‘𝑆)–1-1-onto→ran (1^st ‘𝑇))
18		eqid 2736	. . . . . . 7 ⊢ (1^st ‘𝑅) = (1^st ‘𝑅)
19		eqid 2736	. . . . . . 7 ⊢ ran (1^st ‘𝑅) = ran (1^st ‘𝑅)
20	18, 19, 10, 11	rngoiso1o 38182	. . . . . 6 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsIso 𝑆)) → 𝐹:ran (1^st ‘𝑅)–1-1-onto→ran (1^st ‘𝑆))
21	20	3expa 1118	. . . . 5 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹 ∈ (𝑅 RingOpsIso 𝑆)) → 𝐹:ran (1^st ‘𝑅)–1-1-onto→ran (1^st ‘𝑆))
22	21	3adantl3 1169	. . . 4 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ 𝐹 ∈ (𝑅 RingOpsIso 𝑆)) → 𝐹:ran (1^st ‘𝑅)–1-1-onto→ran (1^st ‘𝑆))
23	22	adantrr 717	. . 3 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RingOpsIso 𝑆) ∧ 𝐺 ∈ (𝑆 RingOpsIso 𝑇))) → 𝐹:ran (1^st ‘𝑅)–1-1-onto→ran (1^st ‘𝑆))
24		f1oco 6797	. . 3 ⊢ ((𝐺:ran (1^st ‘𝑆)–1-1-onto→ran (1^st ‘𝑇) ∧ 𝐹:ran (1^st ‘𝑅)–1-1-onto→ran (1^st ‘𝑆)) → (𝐺 ∘ 𝐹):ran (1^st ‘𝑅)–1-1-onto→ran (1^st ‘𝑇))
25	17, 23, 24	syl2anc 584	. 2 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RingOpsIso 𝑆) ∧ 𝐺 ∈ (𝑆 RingOpsIso 𝑇))) → (𝐺 ∘ 𝐹):ran (1^st ‘𝑅)–1-1-onto→ran (1^st ‘𝑇))
26	18, 19, 12, 13	isrngoiso 38181	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝑇 ∈ RingOps) → ((𝐺 ∘ 𝐹) ∈ (𝑅 RingOpsIso 𝑇) ↔ ((𝐺 ∘ 𝐹) ∈ (𝑅 RingOpsHom 𝑇) ∧ (𝐺 ∘ 𝐹):ran (1^st ‘𝑅)–1-1-onto→ran (1^st ‘𝑇))))
27	26	3adant2 1131	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) → ((𝐺 ∘ 𝐹) ∈ (𝑅 RingOpsIso 𝑇) ↔ ((𝐺 ∘ 𝐹) ∈ (𝑅 RingOpsHom 𝑇) ∧ (𝐺 ∘ 𝐹):ran (1^st ‘𝑅)–1-1-onto→ran (1^st ‘𝑇))))
28	27	adantr 480	. 2 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RingOpsIso 𝑆) ∧ 𝐺 ∈ (𝑆 RingOpsIso 𝑇))) → ((𝐺 ∘ 𝐹) ∈ (𝑅 RingOpsIso 𝑇) ↔ ((𝐺 ∘ 𝐹) ∈ (𝑅 RingOpsHom 𝑇) ∧ (𝐺 ∘ 𝐹):ran (1^st ‘𝑅)–1-1-onto→ran (1^st ‘𝑇))))
29	9, 25, 28	mpbir2and 713	1 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RingOpsIso 𝑆) ∧ 𝐺 ∈ (𝑆 RingOpsIso 𝑇))) → (𝐺 ∘ 𝐹) ∈ (𝑅 RingOpsIso 𝑇))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 ∈ wcel 2113 ran crn 5625 ∘ ccom 5628 –1-1-onto→wf1o 6491 ‘cfv 6492 (class class class)co 7358 1^st c1st 7931 RingOpscrngo 38097 RingOpsHom crngohom 38163 RingOpsIso crngoiso 38164

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-ral 3052 df-rex 3061 df-rmo 3350 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-op 4587 df-uni 4864 df-iun 4948 df-br 5099 df-opab 5161 df-mpt 5180 df-id 5519 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-1st 7933 df-2nd 7934 df-map 8767 df-grpo 30570 df-gid 30571 df-ablo 30622 df-ass 38046 df-exid 38048 df-mgmOLD 38052 df-sgrOLD 38064 df-mndo 38070 df-rngo 38098 df-rngohom 38166 df-rngoiso 38179

This theorem is referenced by: riscer 38191

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