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Theorem rngoisoco 36850
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))) β†’ (𝐺 ∘ 𝐹) ∈ (𝑅 RngIso 𝑇))

Proof of Theorem rngoisoco
StepHypRef Expression
1 rngoisohom 36848 . . . . . 6 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) β†’ 𝐹 ∈ (𝑅 RngHom 𝑆))
213expa 1119 . . . . 5 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) β†’ 𝐹 ∈ (𝑅 RngHom 𝑆))
323adantl3 1169 . . . 4 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) β†’ 𝐹 ∈ (𝑅 RngHom 𝑆))
4 rngoisohom 36848 . . . . . 6 ((𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps ∧ 𝐺 ∈ (𝑆 RngIso 𝑇)) β†’ 𝐺 ∈ (𝑆 RngHom 𝑇))
543expa 1119 . . . . 5 (((𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ 𝐺 ∈ (𝑆 RngIso 𝑇)) β†’ 𝐺 ∈ (𝑆 RngHom 𝑇))
653adantl1 1167 . . . 4 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ 𝐺 ∈ (𝑆 RngIso 𝑇)) β†’ 𝐺 ∈ (𝑆 RngHom 𝑇))
73, 6anim12dan 620 . . 3 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngIso 𝑆) ∧ 𝐺 ∈ (𝑆 RngIso 𝑇))) β†’ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐺 ∈ (𝑆 RngHom 𝑇)))
8 rngohomco 36842 . . 3 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐺 ∈ (𝑆 RngHom 𝑇))) β†’ (𝐺 ∘ 𝐹) ∈ (𝑅 RngHom 𝑇))
97, 8syldan 592 . 2 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngIso 𝑆) ∧ 𝐺 ∈ (𝑆 RngIso 𝑇))) β†’ (𝐺 ∘ 𝐹) ∈ (𝑅 RngHom 𝑇))
10 eqid 2733 . . . . . . 7 (1st β€˜π‘†) = (1st β€˜π‘†)
11 eqid 2733 . . . . . . 7 ran (1st β€˜π‘†) = ran (1st β€˜π‘†)
12 eqid 2733 . . . . . . 7 (1st β€˜π‘‡) = (1st β€˜π‘‡)
13 eqid 2733 . . . . . . 7 ran (1st β€˜π‘‡) = ran (1st β€˜π‘‡)
1410, 11, 12, 13rngoiso1o 36847 . . . . . 6 ((𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps ∧ 𝐺 ∈ (𝑆 RngIso 𝑇)) β†’ 𝐺:ran (1st β€˜π‘†)–1-1-ontoβ†’ran (1st β€˜π‘‡))
15143expa 1119 . . . . 5 (((𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ 𝐺 ∈ (𝑆 RngIso 𝑇)) β†’ 𝐺:ran (1st β€˜π‘†)–1-1-ontoβ†’ran (1st β€˜π‘‡))
16153adantl1 1167 . . . 4 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ 𝐺 ∈ (𝑆 RngIso 𝑇)) β†’ 𝐺:ran (1st β€˜π‘†)–1-1-ontoβ†’ran (1st β€˜π‘‡))
1716adantrl 715 . . 3 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngIso 𝑆) ∧ 𝐺 ∈ (𝑆 RngIso 𝑇))) β†’ 𝐺:ran (1st β€˜π‘†)–1-1-ontoβ†’ran (1st β€˜π‘‡))
18 eqid 2733 . . . . . . 7 (1st β€˜π‘…) = (1st β€˜π‘…)
19 eqid 2733 . . . . . . 7 ran (1st β€˜π‘…) = ran (1st β€˜π‘…)
2018, 19, 10, 11rngoiso1o 36847 . . . . . 6 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) β†’ 𝐹:ran (1st β€˜π‘…)–1-1-ontoβ†’ran (1st β€˜π‘†))
21203expa 1119 . . . . 5 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) β†’ 𝐹:ran (1st β€˜π‘…)–1-1-ontoβ†’ran (1st β€˜π‘†))
22213adantl3 1169 . . . 4 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) β†’ 𝐹:ran (1st β€˜π‘…)–1-1-ontoβ†’ran (1st β€˜π‘†))
2322adantrr 716 . . 3 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngIso 𝑆) ∧ 𝐺 ∈ (𝑆 RngIso 𝑇))) β†’ 𝐹:ran (1st β€˜π‘…)–1-1-ontoβ†’ran (1st β€˜π‘†))
24 f1oco 6857 . . 3 ((𝐺:ran (1st β€˜π‘†)–1-1-ontoβ†’ran (1st β€˜π‘‡) ∧ 𝐹:ran (1st β€˜π‘…)–1-1-ontoβ†’ran (1st β€˜π‘†)) β†’ (𝐺 ∘ 𝐹):ran (1st β€˜π‘…)–1-1-ontoβ†’ran (1st β€˜π‘‡))
2517, 23, 24syl2anc 585 . 2 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngIso 𝑆) ∧ 𝐺 ∈ (𝑆 RngIso 𝑇))) β†’ (𝐺 ∘ 𝐹):ran (1st β€˜π‘…)–1-1-ontoβ†’ran (1st β€˜π‘‡))
2618, 19, 12, 13isrngoiso 36846 . . . 4 ((𝑅 ∈ RingOps ∧ 𝑇 ∈ RingOps) β†’ ((𝐺 ∘ 𝐹) ∈ (𝑅 RngIso 𝑇) ↔ ((𝐺 ∘ 𝐹) ∈ (𝑅 RngHom 𝑇) ∧ (𝐺 ∘ 𝐹):ran (1st β€˜π‘…)–1-1-ontoβ†’ran (1st β€˜π‘‡))))
27263adant2 1132 . . 3 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) β†’ ((𝐺 ∘ 𝐹) ∈ (𝑅 RngIso 𝑇) ↔ ((𝐺 ∘ 𝐹) ∈ (𝑅 RngHom 𝑇) ∧ (𝐺 ∘ 𝐹):ran (1st β€˜π‘…)–1-1-ontoβ†’ran (1st β€˜π‘‡))))
2827adantr 482 . 2 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngIso 𝑆) ∧ 𝐺 ∈ (𝑆 RngIso 𝑇))) β†’ ((𝐺 ∘ 𝐹) ∈ (𝑅 RngIso 𝑇) ↔ ((𝐺 ∘ 𝐹) ∈ (𝑅 RngHom 𝑇) ∧ (𝐺 ∘ 𝐹):ran (1st β€˜π‘…)–1-1-ontoβ†’ran (1st β€˜π‘‡))))
299, 25, 28mpbir2and 712 1 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngIso 𝑆) ∧ 𝐺 ∈ (𝑆 RngIso 𝑇))) β†’ (𝐺 ∘ 𝐹) ∈ (𝑅 RngIso 𝑇))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   ∈ wcel 2107  ran crn 5678   ∘ ccom 5681  β€“1-1-ontoβ†’wf1o 6543  β€˜cfv 6544  (class class class)co 7409  1st c1st 7973  RingOpscrngo 36762   RngHom crnghom 36828   RngIso crngiso 36829
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-1st 7975  df-2nd 7976  df-map 8822  df-grpo 29746  df-gid 29747  df-ablo 29798  df-ass 36711  df-exid 36713  df-mgmOLD 36717  df-sgrOLD 36729  df-mndo 36735  df-rngo 36763  df-rngohom 36831  df-rngoiso 36844
This theorem is referenced by:  riscer  36856
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