rngoisoco - Mathbox for Jeff Madsen

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Theorem rngoisoco 38653

Description: Obsolete theorem, use rimco 20595 instead. The composition of two ring isomorphisms is a ring isomorphism. (Contributed by Jeff Madsen, 16-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

**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 38651	. . . . . 6 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsIso 𝑆)) → 𝐹 ∈ (𝑅 RingOpsHom 𝑆))
2	1	3expa 1136	. . . . 5 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹 ∈ (𝑅 RingOpsIso 𝑆)) → 𝐹 ∈ (𝑅 RingOpsHom 𝑆))
3	2	3adantl3 1187	. . . 4 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ 𝐹 ∈ (𝑅 RingOpsIso 𝑆)) → 𝐹 ∈ (𝑅 RingOpsHom 𝑆))
4		rngoisohom 38651	. . . . . 6 ⊢ ((𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps ∧ 𝐺 ∈ (𝑆 RingOpsIso 𝑇)) → 𝐺 ∈ (𝑆 RingOpsHom 𝑇))
5	4	3expa 1136	. . . . 5 ⊢ (((𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ 𝐺 ∈ (𝑆 RingOpsIso 𝑇)) → 𝐺 ∈ (𝑆 RingOpsHom 𝑇))
6	5	3adantl1 1185	. . . 4 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ 𝐺 ∈ (𝑆 RingOpsIso 𝑇)) → 𝐺 ∈ (𝑆 RingOpsHom 𝑇))
7	3, 6	anim12dan 630	. . 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 38645	. . 3 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RingOpsHom 𝑆) ∧ 𝐺 ∈ (𝑆 RingOpsHom 𝑇))) → (𝐺 ∘ 𝐹) ∈ (𝑅 RingOpsHom 𝑇))
9	7, 8	syldan 602	. 2 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RingOpsIso 𝑆) ∧ 𝐺 ∈ (𝑆 RingOpsIso 𝑇))) → (𝐺 ∘ 𝐹) ∈ (𝑅 RingOpsHom 𝑇))
10		eqid 2763	. . . . . . 7 ⊢ (1^st ‘𝑆) = (1^st ‘𝑆)
11		eqid 2763	. . . . . . 7 ⊢ ran (1^st ‘𝑆) = ran (1^st ‘𝑆)
12		eqid 2763	. . . . . . 7 ⊢ (1^st ‘𝑇) = (1^st ‘𝑇)
13		eqid 2763	. . . . . . 7 ⊢ ran (1^st ‘𝑇) = ran (1^st ‘𝑇)
14	10, 11, 12, 13	rngoiso1o 38650	. . . . . 6 ⊢ ((𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps ∧ 𝐺 ∈ (𝑆 RingOpsIso 𝑇)) → 𝐺:ran (1^st ‘𝑆)–1-1-onto→ran (1^st ‘𝑇))
15	14	3expa 1136	. . . . 5 ⊢ (((𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ 𝐺 ∈ (𝑆 RingOpsIso 𝑇)) → 𝐺:ran (1^st ‘𝑆)–1-1-onto→ran (1^st ‘𝑇))
16	15	3adantl1 1185	. . . 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 728	. . 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 2763	. . . . . . 7 ⊢ (1^st ‘𝑅) = (1^st ‘𝑅)
19		eqid 2763	. . . . . . 7 ⊢ ran (1^st ‘𝑅) = ran (1^st ‘𝑅)
20	18, 19, 10, 11	rngoiso1o 38650	. . . . . 6 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsIso 𝑆)) → 𝐹:ran (1^st ‘𝑅)–1-1-onto→ran (1^st ‘𝑆))
21	20	3expa 1136	. . . . 5 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹 ∈ (𝑅 RingOpsIso 𝑆)) → 𝐹:ran (1^st ‘𝑅)–1-1-onto→ran (1^st ‘𝑆))
22	21	3adantl3 1187	. . . 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 729	. . 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 6844	. . 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 595	. 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 38649	. . . 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 1149	. . 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 485	. 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 725	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 209 ∧ wa 400 ∧ w3a 1103 ∈ wcel 2143 ran crn 5662 ∘ ccom 5665 –1-1-onto→wf1o 6535 ‘cfv 6536 (class class class)co 7410 1^st c1st 7980 RingOpscrngo 38565 RingOpsHom crngohom 38631 RingOpsIso crngoiso 38632

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-sep 5257 ax-nul 5269 ax-pow 5336 ax-pr 5404 ax-un 7732

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-ral 3080 df-rex 3090 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-nul 4287 df-if 4488 df-pw 4564 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4873 df-iun 4958 df-br 5110 df-opab 5174 df-mpt 5193 df-id 5556 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-1st 7982 df-2nd 7983 df-map 8822 df-grpo 30845 df-gid 30846 df-ablo 30897 df-ass 38514 df-exid 38516 df-mgmOLD 38520 df-sgrOLD 38532 df-mndo 38538 df-rngo 38566 df-rngohom 38634 df-rngoiso 38647

This theorem is referenced by: riscer 38659

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