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Theorem rngoisoco 32134
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 32132 . . . . . 6  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngIso  S ) )  ->  F  e.  ( R  RngHom  S ) )
213expa 1205 . . . . 5  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps )  /\  F  e.  ( R  RngIso  S ) )  ->  F  e.  ( R  RngHom  S ) )
323adantl3 1163 . . . 4  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  T  e.  RingOps )  /\  F  e.  ( R  RngIso  S ) )  ->  F  e.  ( R  RngHom  S ) )
4 rngoisohom 32132 . . . . . 6  |-  ( ( S  e.  RingOps  /\  T  e.  RingOps  /\  G  e.  ( S  RngIso  T ) )  ->  G  e.  ( S  RngHom  T ) )
543expa 1205 . . . . 5  |-  ( ( ( S  e.  RingOps  /\  T  e.  RingOps )  /\  G  e.  ( S  RngIso  T ) )  ->  G  e.  ( S  RngHom  T ) )
653adantl1 1161 . . . 4  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  T  e.  RingOps )  /\  G  e.  ( S  RngIso  T ) )  ->  G  e.  ( S  RngHom  T ) )
73, 6anim12da 31951 . . 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 32126 . . 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 472 . 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 2422 . . . . . . 7  |-  ( 1st `  S )  =  ( 1st `  S )
11 eqid 2422 . . . . . . 7  |-  ran  ( 1st `  S )  =  ran  ( 1st `  S
)
12 eqid 2422 . . . . . . 7  |-  ( 1st `  T )  =  ( 1st `  T )
13 eqid 2422 . . . . . . 7  |-  ran  ( 1st `  T )  =  ran  ( 1st `  T
)
1410, 11, 12, 13rngoiso1o 32131 . . . . . 6  |-  ( ( S  e.  RingOps  /\  T  e.  RingOps  /\  G  e.  ( S  RngIso  T ) )  ->  G : ran  ( 1st `  S
)
-1-1-onto-> ran  ( 1st `  T
) )
15143expa 1205 . . . . 5  |-  ( ( ( S  e.  RingOps  /\  T  e.  RingOps )  /\  G  e.  ( S  RngIso  T ) )  ->  G : ran  ( 1st `  S ) -1-1-onto-> ran  ( 1st `  T
) )
16153adantl1 1161 . . . 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 720 . . 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 2422 . . . . . . 7  |-  ( 1st `  R )  =  ( 1st `  R )
19 eqid 2422 . . . . . . 7  |-  ran  ( 1st `  R )  =  ran  ( 1st `  R
)
2018, 19, 10, 11rngoiso1o 32131 . . . . . 6  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngIso  S ) )  ->  F : ran  ( 1st `  R
)
-1-1-onto-> ran  ( 1st `  S
) )
21203expa 1205 . . . . 5  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps )  /\  F  e.  ( R  RngIso  S ) )  ->  F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
) )
22213adantl3 1163 . . . 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 721 . . 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 5849 . . 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 665 . 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 32130 . . . 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 1024 . . 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 466 . 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 930 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 setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    e. wcel 1868   ran crn 4850    o. ccom 4853   -1-1-onto->wf1o 5596   ` cfv 5597  (class class class)co 6301   1stc1st 6801   RingOpscrngo 26088    RngHom crnghom 32112    RngIso crngiso 32113
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-sep 4543  ax-nul 4551  ax-pow 4598  ax-pr 4656  ax-un 6593
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-ral 2780  df-rex 2781  df-reu 2782  df-rmo 2783  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-op 4003  df-uni 4217  df-iun 4298  df-br 4421  df-opab 4480  df-mpt 4481  df-id 4764  df-xp 4855  df-rel 4856  df-cnv 4857  df-co 4858  df-dm 4859  df-rn 4860  df-res 4861  df-ima 4862  df-iota 5561  df-fun 5599  df-fn 5600  df-f 5601  df-f1 5602  df-fo 5603  df-f1o 5604  df-fv 5605  df-riota 6263  df-ov 6304  df-oprab 6305  df-mpt2 6306  df-1st 6803  df-2nd 6804  df-map 7478  df-grpo 25904  df-gid 25905  df-ablo 25995  df-ass 26026  df-exid 26028  df-mgmOLD 26032  df-sgrOLD 26044  df-mndo 26051  df-rngo 26089  df-rngohom 32115  df-rngoiso 32128
This theorem is referenced by:  riscer  32140
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