rngoisoco - Mathbox for Jeff Madsen

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Theorem rngoisoco 38237

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 38235	. . . . . 6 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsIso 𝑆)) → 𝐹 ∈ (𝑅 RingOpsHom 𝑆))
2	1	3expa 1119	. . . . 5 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹 ∈ (𝑅 RingOpsIso 𝑆)) → 𝐹 ∈ (𝑅 RingOpsHom 𝑆))
3	2	3adantl3 1170	. . . 4 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ 𝐹 ∈ (𝑅 RingOpsIso 𝑆)) → 𝐹 ∈ (𝑅 RingOpsHom 𝑆))
4		rngoisohom 38235	. . . . . 6 ⊢ ((𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps ∧ 𝐺 ∈ (𝑆 RingOpsIso 𝑇)) → 𝐺 ∈ (𝑆 RingOpsHom 𝑇))
5	4	3expa 1119	. . . . 5 ⊢ (((𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ 𝐺 ∈ (𝑆 RingOpsIso 𝑇)) → 𝐺 ∈ (𝑆 RingOpsHom 𝑇))
6	5	3adantl1 1168	. . . 4 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ 𝐺 ∈ (𝑆 RingOpsIso 𝑇)) → 𝐺 ∈ (𝑆 RingOpsHom 𝑇))
7	3, 6	anim12dan 620	. . 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 38229	. . 3 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RingOpsHom 𝑆) ∧ 𝐺 ∈ (𝑆 RingOpsHom 𝑇))) → (𝐺 ∘ 𝐹) ∈ (𝑅 RingOpsHom 𝑇))
9	7, 8	syldan 592	. 2 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RingOpsIso 𝑆) ∧ 𝐺 ∈ (𝑆 RingOpsIso 𝑇))) → (𝐺 ∘ 𝐹) ∈ (𝑅 RingOpsHom 𝑇))
10		eqid 2737	. . . . . . 7 ⊢ (1^st ‘𝑆) = (1^st ‘𝑆)
11		eqid 2737	. . . . . . 7 ⊢ ran (1^st ‘𝑆) = ran (1^st ‘𝑆)
12		eqid 2737	. . . . . . 7 ⊢ (1^st ‘𝑇) = (1^st ‘𝑇)
13		eqid 2737	. . . . . . 7 ⊢ ran (1^st ‘𝑇) = ran (1^st ‘𝑇)
14	10, 11, 12, 13	rngoiso1o 38234	. . . . . 6 ⊢ ((𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps ∧ 𝐺 ∈ (𝑆 RingOpsIso 𝑇)) → 𝐺:ran (1^st ‘𝑆)–1-1-onto→ran (1^st ‘𝑇))
15	14	3expa 1119	. . . . 5 ⊢ (((𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ 𝐺 ∈ (𝑆 RingOpsIso 𝑇)) → 𝐺:ran (1^st ‘𝑆)–1-1-onto→ran (1^st ‘𝑇))
16	15	3adantl1 1168	. . . 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 717	. . 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 2737	. . . . . . 7 ⊢ (1^st ‘𝑅) = (1^st ‘𝑅)
19		eqid 2737	. . . . . . 7 ⊢ ran (1^st ‘𝑅) = ran (1^st ‘𝑅)
20	18, 19, 10, 11	rngoiso1o 38234	. . . . . 6 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsIso 𝑆)) → 𝐹:ran (1^st ‘𝑅)–1-1-onto→ran (1^st ‘𝑆))
21	20	3expa 1119	. . . . 5 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹 ∈ (𝑅 RingOpsIso 𝑆)) → 𝐹:ran (1^st ‘𝑅)–1-1-onto→ran (1^st ‘𝑆))
22	21	3adantl3 1170	. . . 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 718	. . 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 6805	. . 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 585	. 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 38233	. . . 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 1132	. . 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 714	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 1087 ∈ wcel 2114 ran crn 5633 ∘ ccom 5636 –1-1-onto→wf1o 6499 ‘cfv 6500 (class class class)co 7368 1^st c1st 7941 RingOpscrngo 38149 RingOpsHom crngohom 38215 RingOpsIso crngoiso 38216

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-ral 3053 df-rex 3063 df-rmo 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-iun 4950 df-br 5101 df-opab 5163 df-mpt 5182 df-id 5527 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-1st 7943 df-2nd 7944 df-map 8777 df-grpo 30585 df-gid 30586 df-ablo 30637 df-ass 38098 df-exid 38100 df-mgmOLD 38104 df-sgrOLD 38116 df-mndo 38122 df-rngo 38150 df-rngohom 38218 df-rngoiso 38231

This theorem is referenced by: riscer 38243

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