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Theorem rngoisoco 36444
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 36442 . . . . . 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 36442 . . . . . 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 36436 . . 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 2737 . . . . . . 7 (1st β€˜π‘†) = (1st β€˜π‘†)
11 eqid 2737 . . . . . . 7 ran (1st β€˜π‘†) = ran (1st β€˜π‘†)
12 eqid 2737 . . . . . . 7 (1st β€˜π‘‡) = (1st β€˜π‘‡)
13 eqid 2737 . . . . . . 7 ran (1st β€˜π‘‡) = ran (1st β€˜π‘‡)
1410, 11, 12, 13rngoiso1o 36441 . . . . . 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 2737 . . . . . . 7 (1st β€˜π‘…) = (1st β€˜π‘…)
19 eqid 2737 . . . . . . 7 ran (1st β€˜π‘…) = ran (1st β€˜π‘…)
2018, 19, 10, 11rngoiso1o 36441 . . . . . 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 6808 . . 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 36440 . . . 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 5635   ∘ ccom 5638  β€“1-1-ontoβ†’wf1o 6496  β€˜cfv 6497  (class class class)co 7358  1st c1st 7920  RingOpscrngo 36356   RngHom crnghom 36422   RngIso crngiso 36423
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 2708  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
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 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rmo 3354  df-reu 3355  df-rab 3409  df-v 3448  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1st 7922  df-2nd 7923  df-map 8768  df-grpo 29438  df-gid 29439  df-ablo 29490  df-ass 36305  df-exid 36307  df-mgmOLD 36311  df-sgrOLD 36323  df-mndo 36329  df-rngo 36357  df-rngohom 36425  df-rngoiso 36438
This theorem is referenced by:  riscer  36450
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