rngoisoco - Mathbox for Jeff Madsen

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Theorem rngoisoco 38303

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 38301	. . . . . 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 38301	. . . . . 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 38295	. . 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 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 38300	. . . . . 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 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 38300	. . . . . 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 6803	. . 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 38299	. . . 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 5632 ∘ ccom 5635 –1-1-onto→wf1o 6497 ‘cfv 6498 (class class class)co 7367 1^st c1st 7940 RingOpscrngo 38215 RingOpsHom crngohom 38281 RingOpsIso crngoiso 38282

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 2708 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689

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 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-ral 3052 df-rex 3062 df-rmo 3342 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4851 df-iun 4935 df-br 5086 df-opab 5148 df-mpt 5167 df-id 5526 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-1st 7942 df-2nd 7943 df-map 8775 df-grpo 30564 df-gid 30565 df-ablo 30616 df-ass 38164 df-exid 38166 df-mgmOLD 38170 df-sgrOLD 38182 df-mndo 38188 df-rngo 38216 df-rngohom 38284 df-rngoiso 38297

This theorem is referenced by: riscer 38309

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