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Theorem rngoisoco 37989
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))) → (𝐺𝐹) ∈ (𝑅 RingOpsIso 𝑇))

Proof of Theorem rngoisoco
StepHypRef Expression
1 rngoisohom 37987 . . . . . 6 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsIso 𝑆)) → 𝐹 ∈ (𝑅 RingOpsHom 𝑆))
213expa 1119 . . . . 5 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹 ∈ (𝑅 RingOpsIso 𝑆)) → 𝐹 ∈ (𝑅 RingOpsHom 𝑆))
323adantl3 1169 . . . 4 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ 𝐹 ∈ (𝑅 RingOpsIso 𝑆)) → 𝐹 ∈ (𝑅 RingOpsHom 𝑆))
4 rngoisohom 37987 . . . . . 6 ((𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps ∧ 𝐺 ∈ (𝑆 RingOpsIso 𝑇)) → 𝐺 ∈ (𝑆 RingOpsHom 𝑇))
543expa 1119 . . . . 5 (((𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ 𝐺 ∈ (𝑆 RingOpsIso 𝑇)) → 𝐺 ∈ (𝑆 RingOpsHom 𝑇))
653adantl1 1167 . . . 4 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ 𝐺 ∈ (𝑆 RingOpsIso 𝑇)) → 𝐺 ∈ (𝑆 RingOpsHom 𝑇))
73, 6anim12dan 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 37981 . . 3 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RingOpsHom 𝑆) ∧ 𝐺 ∈ (𝑆 RingOpsHom 𝑇))) → (𝐺𝐹) ∈ (𝑅 RingOpsHom 𝑇))
97, 8syldan 591 . 2 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RingOpsIso 𝑆) ∧ 𝐺 ∈ (𝑆 RingOpsIso 𝑇))) → (𝐺𝐹) ∈ (𝑅 RingOpsHom 𝑇))
10 eqid 2737 . . . . . . 7 (1st𝑆) = (1st𝑆)
11 eqid 2737 . . . . . . 7 ran (1st𝑆) = ran (1st𝑆)
12 eqid 2737 . . . . . . 7 (1st𝑇) = (1st𝑇)
13 eqid 2737 . . . . . . 7 ran (1st𝑇) = ran (1st𝑇)
1410, 11, 12, 13rngoiso1o 37986 . . . . . 6 ((𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps ∧ 𝐺 ∈ (𝑆 RingOpsIso 𝑇)) → 𝐺:ran (1st𝑆)–1-1-onto→ran (1st𝑇))
15143expa 1119 . . . . 5 (((𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ 𝐺 ∈ (𝑆 RingOpsIso 𝑇)) → 𝐺:ran (1st𝑆)–1-1-onto→ran (1st𝑇))
16153adantl1 1167 . . . 4 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ 𝐺 ∈ (𝑆 RingOpsIso 𝑇)) → 𝐺:ran (1st𝑆)–1-1-onto→ran (1st𝑇))
1716adantrl 716 . . 3 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RingOpsIso 𝑆) ∧ 𝐺 ∈ (𝑆 RingOpsIso 𝑇))) → 𝐺:ran (1st𝑆)–1-1-onto→ran (1st𝑇))
18 eqid 2737 . . . . . . 7 (1st𝑅) = (1st𝑅)
19 eqid 2737 . . . . . . 7 ran (1st𝑅) = ran (1st𝑅)
2018, 19, 10, 11rngoiso1o 37986 . . . . . 6 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsIso 𝑆)) → 𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆))
21203expa 1119 . . . . 5 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹 ∈ (𝑅 RingOpsIso 𝑆)) → 𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆))
22213adantl3 1169 . . . 4 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ 𝐹 ∈ (𝑅 RingOpsIso 𝑆)) → 𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆))
2322adantrr 717 . . 3 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RingOpsIso 𝑆) ∧ 𝐺 ∈ (𝑆 RingOpsIso 𝑇))) → 𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆))
24 f1oco 6871 . . 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 584 . 2 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RingOpsIso 𝑆) ∧ 𝐺 ∈ (𝑆 RingOpsIso 𝑇))) → (𝐺𝐹):ran (1st𝑅)–1-1-onto→ran (1st𝑇))
2618, 19, 12, 13isrngoiso 37985 . . . 4 ((𝑅 ∈ RingOps ∧ 𝑇 ∈ RingOps) → ((𝐺𝐹) ∈ (𝑅 RingOpsIso 𝑇) ↔ ((𝐺𝐹) ∈ (𝑅 RingOpsHom 𝑇) ∧ (𝐺𝐹):ran (1st𝑅)–1-1-onto→ran (1st𝑇))))
27263adant2 1132 . . 3 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) → ((𝐺𝐹) ∈ (𝑅 RingOpsIso 𝑇) ↔ ((𝐺𝐹) ∈ (𝑅 RingOpsHom 𝑇) ∧ (𝐺𝐹):ran (1st𝑅)–1-1-onto→ran (1st𝑇))))
2827adantr 480 . 2 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RingOpsIso 𝑆) ∧ 𝐺 ∈ (𝑆 RingOpsIso 𝑇))) → ((𝐺𝐹) ∈ (𝑅 RingOpsIso 𝑇) ↔ ((𝐺𝐹) ∈ (𝑅 RingOpsHom 𝑇) ∧ (𝐺𝐹):ran (1st𝑅)–1-1-onto→ran (1st𝑇))))
299, 25, 28mpbir2and 713 1 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RingOpsIso 𝑆) ∧ 𝐺 ∈ (𝑆 RingOpsIso 𝑇))) → (𝐺𝐹) ∈ (𝑅 RingOpsIso 𝑇))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1087  wcel 2108  ran crn 5686  ccom 5689  1-1-ontowf1o 6560  cfv 6561  (class class class)co 7431  1st c1st 8012  RingOpscrngo 37901   RingOpsHom crngohom 37967   RingOpsIso crngoiso 37968
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-map 8868  df-grpo 30512  df-gid 30513  df-ablo 30564  df-ass 37850  df-exid 37852  df-mgmOLD 37856  df-sgrOLD 37868  df-mndo 37874  df-rngo 37902  df-rngohom 37970  df-rngoiso 37983
This theorem is referenced by:  riscer  37995
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