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Theorem rngoisoco 26613
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 26611 . . . . . 6  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngIso  S ) )  ->  F  e.  ( R  RngHom  S ) )
213expa 1151 . . . . 5  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps )  /\  F  e.  ( R  RngIso  S ) )  ->  F  e.  ( R  RngHom  S ) )
323adantl3 1113 . . . 4  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  T  e.  RingOps )  /\  F  e.  ( R  RngIso  S ) )  ->  F  e.  ( R  RngHom  S ) )
4 rngoisohom 26611 . . . . . 6  |-  ( ( S  e.  RingOps  /\  T  e.  RingOps  /\  G  e.  ( S  RngIso  T ) )  ->  G  e.  ( S  RngHom  T ) )
543expa 1151 . . . . 5  |-  ( ( ( S  e.  RingOps  /\  T  e.  RingOps )  /\  G  e.  ( S  RngIso  T ) )  ->  G  e.  ( S  RngHom  T ) )
653adantl1 1111 . . . 4  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  T  e.  RingOps )  /\  G  e.  ( S  RngIso  T ) )  ->  G  e.  ( S  RngHom  T ) )
73, 6anim12da 26332 . . 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 26605 . . 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 456 . 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 2283 . . . . . . 7  |-  ( 1st `  S )  =  ( 1st `  S )
11 eqid 2283 . . . . . . 7  |-  ran  ( 1st `  S )  =  ran  ( 1st `  S
)
12 eqid 2283 . . . . . . 7  |-  ( 1st `  T )  =  ( 1st `  T )
13 eqid 2283 . . . . . . 7  |-  ran  ( 1st `  T )  =  ran  ( 1st `  T
)
1410, 11, 12, 13rngoiso1o 26610 . . . . . 6  |-  ( ( S  e.  RingOps  /\  T  e.  RingOps  /\  G  e.  ( S  RngIso  T ) )  ->  G : ran  ( 1st `  S
)
-1-1-onto-> ran  ( 1st `  T
) )
15143expa 1151 . . . . 5  |-  ( ( ( S  e.  RingOps  /\  T  e.  RingOps )  /\  G  e.  ( S  RngIso  T ) )  ->  G : ran  ( 1st `  S ) -1-1-onto-> ran  ( 1st `  T
) )
16153adantl1 1111 . . . 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 696 . . 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 2283 . . . . . . 7  |-  ( 1st `  R )  =  ( 1st `  R )
19 eqid 2283 . . . . . . 7  |-  ran  ( 1st `  R )  =  ran  ( 1st `  R
)
2018, 19, 10, 11rngoiso1o 26610 . . . . . 6  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngIso  S ) )  ->  F : ran  ( 1st `  R
)
-1-1-onto-> ran  ( 1st `  S
) )
21203expa 1151 . . . . 5  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps )  /\  F  e.  ( R  RngIso  S ) )  ->  F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
) )
22213adantl3 1113 . . . 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 697 . . 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 5496 . . 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 642 . 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 26609 . . . 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 974 . . 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 451 . 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 888 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 176    /\ wa 358    /\ w3a 934    e. wcel 1684   ran crn 4690    o. ccom 4693   -1-1-onto->wf1o 5254   ` cfv 5255  (class class class)co 5858   1stc1st 6120   RingOpscrngo 21042    RngHom crnghom 26591    RngIso crngiso 26592
This theorem is referenced by:  riscer  26619
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-map 6774  df-grpo 20858  df-gid 20859  df-ablo 20949  df-ass 20980  df-exid 20982  df-mgm 20986  df-sgr 20998  df-mndo 21005  df-rngo 21043  df-rngohom 26594  df-rngoiso 26607
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