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Theorem rngoisoco 26552
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 26550 . . . . . 6  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngIso  S ) )  ->  F  e.  ( R  RngHom  S ) )
213expa 1153 . . . . 5  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps )  /\  F  e.  ( R  RngIso  S ) )  ->  F  e.  ( R  RngHom  S ) )
323adantl3 1115 . . . 4  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  T  e.  RingOps )  /\  F  e.  ( R  RngIso  S ) )  ->  F  e.  ( R  RngHom  S ) )
4 rngoisohom 26550 . . . . . 6  |-  ( ( S  e.  RingOps  /\  T  e.  RingOps  /\  G  e.  ( S  RngIso  T ) )  ->  G  e.  ( S  RngHom  T ) )
543expa 1153 . . . . 5  |-  ( ( ( S  e.  RingOps  /\  T  e.  RingOps )  /\  G  e.  ( S  RngIso  T ) )  ->  G  e.  ( S  RngHom  T ) )
653adantl1 1113 . . . 4  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  T  e.  RingOps )  /\  G  e.  ( S  RngIso  T ) )  ->  G  e.  ( S  RngHom  T ) )
73, 6anim12da 26366 . . 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 26544 . . 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 457 . 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 2435 . . . . . . 7  |-  ( 1st `  S )  =  ( 1st `  S )
11 eqid 2435 . . . . . . 7  |-  ran  ( 1st `  S )  =  ran  ( 1st `  S
)
12 eqid 2435 . . . . . . 7  |-  ( 1st `  T )  =  ( 1st `  T )
13 eqid 2435 . . . . . . 7  |-  ran  ( 1st `  T )  =  ran  ( 1st `  T
)
1410, 11, 12, 13rngoiso1o 26549 . . . . . 6  |-  ( ( S  e.  RingOps  /\  T  e.  RingOps  /\  G  e.  ( S  RngIso  T ) )  ->  G : ran  ( 1st `  S
)
-1-1-onto-> ran  ( 1st `  T
) )
15143expa 1153 . . . . 5  |-  ( ( ( S  e.  RingOps  /\  T  e.  RingOps )  /\  G  e.  ( S  RngIso  T ) )  ->  G : ran  ( 1st `  S ) -1-1-onto-> ran  ( 1st `  T
) )
16153adantl1 1113 . . . 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 697 . . 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 2435 . . . . . . 7  |-  ( 1st `  R )  =  ( 1st `  R )
19 eqid 2435 . . . . . . 7  |-  ran  ( 1st `  R )  =  ran  ( 1st `  R
)
2018, 19, 10, 11rngoiso1o 26549 . . . . . 6  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngIso  S ) )  ->  F : ran  ( 1st `  R
)
-1-1-onto-> ran  ( 1st `  S
) )
21203expa 1153 . . . . 5  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps )  /\  F  e.  ( R  RngIso  S ) )  ->  F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
) )
22213adantl3 1115 . . . 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 698 . . 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 5690 . . 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 643 . 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 26548 . . . 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 976 . . 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 452 . 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 889 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 177    /\ wa 359    /\ w3a 936    e. wcel 1725   ran crn 4871    o. ccom 4874   -1-1-onto->wf1o 5445   ` cfv 5446  (class class class)co 6073   1stc1st 6339   RingOpscrngo 21953    RngHom crnghom 26530    RngIso crngiso 26531
This theorem is referenced by:  riscer  26558
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-map 7012  df-grpo 21769  df-gid 21770  df-ablo 21860  df-ass 21891  df-exid 21893  df-mgm 21897  df-sgr 21909  df-mndo 21916  df-rngo 21954  df-rngohom 26533  df-rngoiso 26546
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