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Theorem rngoisoco 26612
Description: The composition of two ring isomorphisms is a ring isomorphism. (Contributed by Jeff Madsen, 16-Jun-2011.)
Assertion
Ref Expression
rngoisoco  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  T  e.  RingOps )  /\  ( F  e.  ( R  RngIso  S )  /\  G  e.  ( S  RngIso  T ) ) )  ->  ( G  o.  F )  e.  ( R  RngIso  T ) )

Proof of Theorem rngoisoco
StepHypRef Expression
1 rngoisohom 26610 . . . . . 6  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngIso  S ) )  ->  F  e.  ( R  RngHom  S ) )
213expa 1154 . . . . 5  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps )  /\  F  e.  ( R  RngIso  S ) )  ->  F  e.  ( R  RngHom  S ) )
323adantl3 1116 . . . 4  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  T  e.  RingOps )  /\  F  e.  ( R  RngIso  S ) )  ->  F  e.  ( R  RngHom  S ) )
4 rngoisohom 26610 . . . . . 6  |-  ( ( S  e.  RingOps  /\  T  e.  RingOps  /\  G  e.  ( S  RngIso  T ) )  ->  G  e.  ( S  RngHom  T ) )
543expa 1154 . . . . 5  |-  ( ( ( S  e.  RingOps  /\  T  e.  RingOps )  /\  G  e.  ( S  RngIso  T ) )  ->  G  e.  ( S  RngHom  T ) )
653adantl1 1114 . . . 4  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  T  e.  RingOps )  /\  G  e.  ( S  RngIso  T ) )  ->  G  e.  ( S  RngHom  T ) )
73, 6anim12da 26426 . . 3  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  T  e.  RingOps )  /\  ( F  e.  ( R  RngIso  S )  /\  G  e.  ( S  RngIso  T ) ) )  ->  ( F  e.  ( R  RngHom  S )  /\  G  e.  ( S  RngHom  T ) ) )
8 rngohomco 26604 . . 3  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  T  e.  RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  G  e.  ( S  RngHom  T ) ) )  ->  ( G  o.  F )  e.  ( R  RngHom  T ) )
97, 8syldan 458 . 2  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  T  e.  RingOps )  /\  ( F  e.  ( R  RngIso  S )  /\  G  e.  ( S  RngIso  T ) ) )  ->  ( G  o.  F )  e.  ( R  RngHom  T ) )
10 eqid 2438 . . . . . . 7  |-  ( 1st `  S )  =  ( 1st `  S )
11 eqid 2438 . . . . . . 7  |-  ran  ( 1st `  S )  =  ran  ( 1st `  S
)
12 eqid 2438 . . . . . . 7  |-  ( 1st `  T )  =  ( 1st `  T )
13 eqid 2438 . . . . . . 7  |-  ran  ( 1st `  T )  =  ran  ( 1st `  T
)
1410, 11, 12, 13rngoiso1o 26609 . . . . . 6  |-  ( ( S  e.  RingOps  /\  T  e.  RingOps  /\  G  e.  ( S  RngIso  T ) )  ->  G : ran  ( 1st `  S
)
-1-1-onto-> ran  ( 1st `  T
) )
15143expa 1154 . . . . 5  |-  ( ( ( S  e.  RingOps  /\  T  e.  RingOps )  /\  G  e.  ( S  RngIso  T ) )  ->  G : ran  ( 1st `  S ) -1-1-onto-> ran  ( 1st `  T
) )
16153adantl1 1114 . . . 4  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  T  e.  RingOps )  /\  G  e.  ( S  RngIso  T ) )  ->  G : ran  ( 1st `  S
)
-1-1-onto-> ran  ( 1st `  T
) )
1716adantrl 698 . . 3  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  T  e.  RingOps )  /\  ( F  e.  ( R  RngIso  S )  /\  G  e.  ( S  RngIso  T ) ) )  ->  G : ran  ( 1st `  S
)
-1-1-onto-> ran  ( 1st `  T
) )
18 eqid 2438 . . . . . . 7  |-  ( 1st `  R )  =  ( 1st `  R )
19 eqid 2438 . . . . . . 7  |-  ran  ( 1st `  R )  =  ran  ( 1st `  R
)
2018, 19, 10, 11rngoiso1o 26609 . . . . . 6  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngIso  S ) )  ->  F : ran  ( 1st `  R
)
-1-1-onto-> ran  ( 1st `  S
) )
21203expa 1154 . . . . 5  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps )  /\  F  e.  ( R  RngIso  S ) )  ->  F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
) )
22213adantl3 1116 . . . 4  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  T  e.  RingOps )  /\  F  e.  ( R  RngIso  S ) )  ->  F : ran  ( 1st `  R
)
-1-1-onto-> ran  ( 1st `  S
) )
2322adantrr 699 . . 3  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  T  e.  RingOps )  /\  ( F  e.  ( R  RngIso  S )  /\  G  e.  ( S  RngIso  T ) ) )  ->  F : ran  ( 1st `  R
)
-1-1-onto-> ran  ( 1st `  S
) )
24 f1oco 5701 . . 3  |-  ( ( G : ran  ( 1st `  S ) -1-1-onto-> ran  ( 1st `  T )  /\  F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
) )  ->  ( G  o.  F ) : ran  ( 1st `  R
)
-1-1-onto-> ran  ( 1st `  T
) )
2517, 23, 24syl2anc 644 . 2  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  T  e.  RingOps )  /\  ( F  e.  ( R  RngIso  S )  /\  G  e.  ( S  RngIso  T ) ) )  ->  ( G  o.  F ) : ran  ( 1st `  R
)
-1-1-onto-> ran  ( 1st `  T
) )
2618, 19, 12, 13isrngoiso 26608 . . . 4  |-  ( ( R  e.  RingOps  /\  T  e.  RingOps )  ->  (
( G  o.  F
)  e.  ( R 
RngIso  T )  <->  ( ( G  o.  F )  e.  ( R  RngHom  T )  /\  ( G  o.  F ) : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  T
) ) ) )
27263adant2 977 . . 3  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  T  e.  RingOps )  ->  ( ( G  o.  F )  e.  ( R  RngIso  T )  <-> 
( ( G  o.  F )  e.  ( R  RngHom  T )  /\  ( G  o.  F
) : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  T ) ) ) )
2827adantr 453 . 2  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  T  e.  RingOps )  /\  ( F  e.  ( R  RngIso  S )  /\  G  e.  ( S  RngIso  T ) ) )  ->  (
( G  o.  F
)  e.  ( R 
RngIso  T )  <->  ( ( G  o.  F )  e.  ( R  RngHom  T )  /\  ( G  o.  F ) : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  T
) ) ) )
299, 25, 28mpbir2and 890 1  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  T  e.  RingOps )  /\  ( F  e.  ( R  RngIso  S )  /\  G  e.  ( S  RngIso  T ) ) )  ->  ( G  o.  F )  e.  ( R  RngIso  T ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    e. wcel 1726   ran crn 4882    o. ccom 4885   -1-1-onto->wf1o 5456   ` cfv 5457  (class class class)co 6084   1stc1st 6350   RingOpscrngo 21968    RngHom crnghom 26590    RngIso crngiso 26591
This theorem is referenced by:  riscer  26618
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-riota 6552  df-map 7023  df-grpo 21784  df-gid 21785  df-ablo 21875  df-ass 21906  df-exid 21908  df-mgm 21912  df-sgr 21924  df-mndo 21931  df-rngo 21969  df-rngohom 26593  df-rngoiso 26606
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