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Theorem rngoisoco 36936
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 36934 . . . . . 6 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) β†’ 𝐹 ∈ (𝑅 RngHom 𝑆))
213expa 1118 . . . . 5 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) β†’ 𝐹 ∈ (𝑅 RngHom 𝑆))
323adantl3 1168 . . . 4 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) β†’ 𝐹 ∈ (𝑅 RngHom 𝑆))
4 rngoisohom 36934 . . . . . 6 ((𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps ∧ 𝐺 ∈ (𝑆 RngIso 𝑇)) β†’ 𝐺 ∈ (𝑆 RngHom 𝑇))
543expa 1118 . . . . 5 (((𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ 𝐺 ∈ (𝑆 RngIso 𝑇)) β†’ 𝐺 ∈ (𝑆 RngHom 𝑇))
653adantl1 1166 . . . 4 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ 𝐺 ∈ (𝑆 RngIso 𝑇)) β†’ 𝐺 ∈ (𝑆 RngHom 𝑇))
73, 6anim12dan 619 . . 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 36928 . . 3 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐺 ∈ (𝑆 RngHom 𝑇))) β†’ (𝐺 ∘ 𝐹) ∈ (𝑅 RngHom 𝑇))
97, 8syldan 591 . 2 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngIso 𝑆) ∧ 𝐺 ∈ (𝑆 RngIso 𝑇))) β†’ (𝐺 ∘ 𝐹) ∈ (𝑅 RngHom 𝑇))
10 eqid 2732 . . . . . . 7 (1st β€˜π‘†) = (1st β€˜π‘†)
11 eqid 2732 . . . . . . 7 ran (1st β€˜π‘†) = ran (1st β€˜π‘†)
12 eqid 2732 . . . . . . 7 (1st β€˜π‘‡) = (1st β€˜π‘‡)
13 eqid 2732 . . . . . . 7 ran (1st β€˜π‘‡) = ran (1st β€˜π‘‡)
1410, 11, 12, 13rngoiso1o 36933 . . . . . 6 ((𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps ∧ 𝐺 ∈ (𝑆 RngIso 𝑇)) β†’ 𝐺:ran (1st β€˜π‘†)–1-1-ontoβ†’ran (1st β€˜π‘‡))
15143expa 1118 . . . . 5 (((𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ 𝐺 ∈ (𝑆 RngIso 𝑇)) β†’ 𝐺:ran (1st β€˜π‘†)–1-1-ontoβ†’ran (1st β€˜π‘‡))
16153adantl1 1166 . . . 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 714 . . 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 2732 . . . . . . 7 (1st β€˜π‘…) = (1st β€˜π‘…)
19 eqid 2732 . . . . . . 7 ran (1st β€˜π‘…) = ran (1st β€˜π‘…)
2018, 19, 10, 11rngoiso1o 36933 . . . . . 6 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) β†’ 𝐹:ran (1st β€˜π‘…)–1-1-ontoβ†’ran (1st β€˜π‘†))
21203expa 1118 . . . . 5 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) β†’ 𝐹:ran (1st β€˜π‘…)–1-1-ontoβ†’ran (1st β€˜π‘†))
22213adantl3 1168 . . . 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 715 . . 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 6856 . . 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 584 . 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 36932 . . . 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 1131 . . 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 481 . 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 711 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 396   ∧ w3a 1087   ∈ wcel 2106  ran crn 5677   ∘ ccom 5680  β€“1-1-ontoβ†’wf1o 6542  β€˜cfv 6543  (class class class)co 7411  1st c1st 7975  RingOpscrngo 36848   RngHom crnghom 36914   RngIso crngiso 36915
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-1st 7977  df-2nd 7978  df-map 8824  df-grpo 29784  df-gid 29785  df-ablo 29836  df-ass 36797  df-exid 36799  df-mgmOLD 36803  df-sgrOLD 36815  df-mndo 36821  df-rngo 36849  df-rngohom 36917  df-rngoiso 36930
This theorem is referenced by:  riscer  36942
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