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Theorem rngoisoco 28926
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 28924 . . . . . 6  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngIso  S ) )  ->  F  e.  ( R  RngHom  S ) )
213expa 1188 . . . . 5  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps )  /\  F  e.  ( R  RngIso  S ) )  ->  F  e.  ( R  RngHom  S ) )
323adantl3 1146 . . . 4  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  T  e.  RingOps )  /\  F  e.  ( R  RngIso  S ) )  ->  F  e.  ( R  RngHom  S ) )
4 rngoisohom 28924 . . . . . 6  |-  ( ( S  e.  RingOps  /\  T  e.  RingOps  /\  G  e.  ( S  RngIso  T ) )  ->  G  e.  ( S  RngHom  T ) )
543expa 1188 . . . . 5  |-  ( ( ( S  e.  RingOps  /\  T  e.  RingOps )  /\  G  e.  ( S  RngIso  T ) )  ->  G  e.  ( S  RngHom  T ) )
653adantl1 1144 . . . 4  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  T  e.  RingOps )  /\  G  e.  ( S  RngIso  T ) )  ->  G  e.  ( S  RngHom  T ) )
73, 6anim12da 28742 . . 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 28918 . . 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 470 . 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 2451 . . . . . . 7  |-  ( 1st `  S )  =  ( 1st `  S )
11 eqid 2451 . . . . . . 7  |-  ran  ( 1st `  S )  =  ran  ( 1st `  S
)
12 eqid 2451 . . . . . . 7  |-  ( 1st `  T )  =  ( 1st `  T )
13 eqid 2451 . . . . . . 7  |-  ran  ( 1st `  T )  =  ran  ( 1st `  T
)
1410, 11, 12, 13rngoiso1o 28923 . . . . . 6  |-  ( ( S  e.  RingOps  /\  T  e.  RingOps  /\  G  e.  ( S  RngIso  T ) )  ->  G : ran  ( 1st `  S
)
-1-1-onto-> ran  ( 1st `  T
) )
15143expa 1188 . . . . 5  |-  ( ( ( S  e.  RingOps  /\  T  e.  RingOps )  /\  G  e.  ( S  RngIso  T ) )  ->  G : ran  ( 1st `  S ) -1-1-onto-> ran  ( 1st `  T
) )
16153adantl1 1144 . . . 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 715 . . 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 2451 . . . . . . 7  |-  ( 1st `  R )  =  ( 1st `  R )
19 eqid 2451 . . . . . . 7  |-  ran  ( 1st `  R )  =  ran  ( 1st `  R
)
2018, 19, 10, 11rngoiso1o 28923 . . . . . 6  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngIso  S ) )  ->  F : ran  ( 1st `  R
)
-1-1-onto-> ran  ( 1st `  S
) )
21203expa 1188 . . . . 5  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps )  /\  F  e.  ( R  RngIso  S ) )  ->  F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
) )
22213adantl3 1146 . . . 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 716 . . 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 5761 . . 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 661 . 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 28922 . . . 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 1007 . . 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 465 . 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 913 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 184    /\ wa 369    /\ w3a 965    e. wcel 1758   ran crn 4939    o. ccom 4942   -1-1-onto->wf1o 5515   ` cfv 5516  (class class class)co 6190   1stc1st 6675   RingOpscrngo 23997    RngHom crnghom 28904    RngIso crngiso 28905
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-op 3982  df-uni 4190  df-iun 4271  df-br 4391  df-opab 4449  df-mpt 4450  df-id 4734  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-riota 6151  df-ov 6193  df-oprab 6194  df-mpt2 6195  df-1st 6677  df-2nd 6678  df-map 7316  df-grpo 23813  df-gid 23814  df-ablo 23904  df-ass 23935  df-exid 23937  df-mgm 23941  df-sgr 23953  df-mndo 23960  df-rngo 23998  df-rngohom 28907  df-rngoiso 28920
This theorem is referenced by:  riscer  28932
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