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Theorem rngoisoco 33413
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 RngIso 𝑆) ∧ 𝐺 ∈ (𝑆 RngIso 𝑇))) → (𝐺𝐹) ∈ (𝑅 RngIso 𝑇))

Proof of Theorem rngoisoco
StepHypRef Expression
1 rngoisohom 33411 . . . . . 6 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) → 𝐹 ∈ (𝑅 RngHom 𝑆))
213expa 1262 . . . . 5 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) → 𝐹 ∈ (𝑅 RngHom 𝑆))
323adantl3 1217 . . . 4 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) → 𝐹 ∈ (𝑅 RngHom 𝑆))
4 rngoisohom 33411 . . . . . 6 ((𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps ∧ 𝐺 ∈ (𝑆 RngIso 𝑇)) → 𝐺 ∈ (𝑆 RngHom 𝑇))
543expa 1262 . . . . 5 (((𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ 𝐺 ∈ (𝑆 RngIso 𝑇)) → 𝐺 ∈ (𝑆 RngHom 𝑇))
653adantl1 1215 . . . 4 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ 𝐺 ∈ (𝑆 RngIso 𝑇)) → 𝐺 ∈ (𝑆 RngHom 𝑇))
73, 6anim12da 33138 . . 3 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngIso 𝑆) ∧ 𝐺 ∈ (𝑆 RngIso 𝑇))) → (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐺 ∈ (𝑆 RngHom 𝑇)))
8 rngohomco 33405 . . 3 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐺 ∈ (𝑆 RngHom 𝑇))) → (𝐺𝐹) ∈ (𝑅 RngHom 𝑇))
97, 8syldan 487 . 2 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngIso 𝑆) ∧ 𝐺 ∈ (𝑆 RngIso 𝑇))) → (𝐺𝐹) ∈ (𝑅 RngHom 𝑇))
10 eqid 2621 . . . . . . 7 (1st𝑆) = (1st𝑆)
11 eqid 2621 . . . . . . 7 ran (1st𝑆) = ran (1st𝑆)
12 eqid 2621 . . . . . . 7 (1st𝑇) = (1st𝑇)
13 eqid 2621 . . . . . . 7 ran (1st𝑇) = ran (1st𝑇)
1410, 11, 12, 13rngoiso1o 33410 . . . . . 6 ((𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps ∧ 𝐺 ∈ (𝑆 RngIso 𝑇)) → 𝐺:ran (1st𝑆)–1-1-onto→ran (1st𝑇))
15143expa 1262 . . . . 5 (((𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ 𝐺 ∈ (𝑆 RngIso 𝑇)) → 𝐺:ran (1st𝑆)–1-1-onto→ran (1st𝑇))
16153adantl1 1215 . . . 4 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ 𝐺 ∈ (𝑆 RngIso 𝑇)) → 𝐺:ran (1st𝑆)–1-1-onto→ran (1st𝑇))
1716adantrl 751 . . 3 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngIso 𝑆) ∧ 𝐺 ∈ (𝑆 RngIso 𝑇))) → 𝐺:ran (1st𝑆)–1-1-onto→ran (1st𝑇))
18 eqid 2621 . . . . . . 7 (1st𝑅) = (1st𝑅)
19 eqid 2621 . . . . . . 7 ran (1st𝑅) = ran (1st𝑅)
2018, 19, 10, 11rngoiso1o 33410 . . . . . 6 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) → 𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆))
21203expa 1262 . . . . 5 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) → 𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆))
22213adantl3 1217 . . . 4 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) → 𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆))
2322adantrr 752 . . 3 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngIso 𝑆) ∧ 𝐺 ∈ (𝑆 RngIso 𝑇))) → 𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆))
24 f1oco 6116 . . 3 ((𝐺:ran (1st𝑆)–1-1-onto→ran (1st𝑇) ∧ 𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆)) → (𝐺𝐹):ran (1st𝑅)–1-1-onto→ran (1st𝑇))
2517, 23, 24syl2anc 692 . 2 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngIso 𝑆) ∧ 𝐺 ∈ (𝑆 RngIso 𝑇))) → (𝐺𝐹):ran (1st𝑅)–1-1-onto→ran (1st𝑇))
2618, 19, 12, 13isrngoiso 33409 . . . 4 ((𝑅 ∈ RingOps ∧ 𝑇 ∈ RingOps) → ((𝐺𝐹) ∈ (𝑅 RngIso 𝑇) ↔ ((𝐺𝐹) ∈ (𝑅 RngHom 𝑇) ∧ (𝐺𝐹):ran (1st𝑅)–1-1-onto→ran (1st𝑇))))
27263adant2 1078 . . 3 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) → ((𝐺𝐹) ∈ (𝑅 RngIso 𝑇) ↔ ((𝐺𝐹) ∈ (𝑅 RngHom 𝑇) ∧ (𝐺𝐹):ran (1st𝑅)–1-1-onto→ran (1st𝑇))))
2827adantr 481 . 2 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngIso 𝑆) ∧ 𝐺 ∈ (𝑆 RngIso 𝑇))) → ((𝐺𝐹) ∈ (𝑅 RngIso 𝑇) ↔ ((𝐺𝐹) ∈ (𝑅 RngHom 𝑇) ∧ (𝐺𝐹):ran (1st𝑅)–1-1-onto→ran (1st𝑇))))
299, 25, 28mpbir2and 956 1 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngIso 𝑆) ∧ 𝐺 ∈ (𝑆 RngIso 𝑇))) → (𝐺𝐹) ∈ (𝑅 RngIso 𝑇))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036  wcel 1987  ran crn 5075  ccom 5078  1-1-ontowf1o 5846  cfv 5847  (class class class)co 6604  1st c1st 7111  RingOpscrngo 33325   RngHom crnghom 33391   RngIso crngiso 33392
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-1st 7113  df-2nd 7114  df-map 7804  df-grpo 27196  df-gid 27197  df-ablo 27248  df-ass 33274  df-exid 33276  df-mgmOLD 33280  df-sgrOLD 33292  df-mndo 33298  df-rngo 33326  df-rngohom 33394  df-rngoiso 33407
This theorem is referenced by:  riscer  33419
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