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Theorem rhmco 15759
Description: The composition of ring homomorphisms is a homomorphism. (Contributed by Mario Carneiro, 12-Jun-2015.)
Assertion
Ref Expression
rhmco  |-  ( ( F  e.  ( T RingHom  U )  /\  G  e.  ( S RingHom  T )
)  ->  ( F  o.  G )  e.  ( S RingHom  U ) )

Proof of Theorem rhmco
StepHypRef Expression
1 rhmrcl2 15750 . . 3  |-  ( F  e.  ( T RingHom  U
)  ->  U  e.  Ring )
2 rhmrcl1 15749 . . 3  |-  ( G  e.  ( S RingHom  T
)  ->  S  e.  Ring )
31, 2anim12ci 551 . 2  |-  ( ( F  e.  ( T RingHom  U )  /\  G  e.  ( S RingHom  T )
)  ->  ( S  e.  Ring  /\  U  e.  Ring ) )
4 rhmghm 15753 . . . 4  |-  ( F  e.  ( T RingHom  U
)  ->  F  e.  ( T  GrpHom  U ) )
5 rhmghm 15753 . . . 4  |-  ( G  e.  ( S RingHom  T
)  ->  G  e.  ( S  GrpHom  T ) )
6 ghmco 14952 . . . 4  |-  ( ( F  e.  ( T 
GrpHom  U )  /\  G  e.  ( S  GrpHom  T ) )  ->  ( F  o.  G )  e.  ( S  GrpHom  U ) )
74, 5, 6syl2an 464 . . 3  |-  ( ( F  e.  ( T RingHom  U )  /\  G  e.  ( S RingHom  T )
)  ->  ( F  o.  G )  e.  ( S  GrpHom  U ) )
8 eqid 2387 . . . . 5  |-  (mulGrp `  T )  =  (mulGrp `  T )
9 eqid 2387 . . . . 5  |-  (mulGrp `  U )  =  (mulGrp `  U )
108, 9rhmmhm 15752 . . . 4  |-  ( F  e.  ( T RingHom  U
)  ->  F  e.  ( (mulGrp `  T ) MndHom  (mulGrp `  U ) ) )
11 eqid 2387 . . . . 5  |-  (mulGrp `  S )  =  (mulGrp `  S )
1211, 8rhmmhm 15752 . . . 4  |-  ( G  e.  ( S RingHom  T
)  ->  G  e.  ( (mulGrp `  S ) MndHom  (mulGrp `  T ) ) )
13 mhmco 14689 . . . 4  |-  ( ( F  e.  ( (mulGrp `  T ) MndHom  (mulGrp `  U ) )  /\  G  e.  ( (mulGrp `  S ) MndHom  (mulGrp `  T ) ) )  ->  ( F  o.  G )  e.  ( (mulGrp `  S ) MndHom  (mulGrp `  U ) ) )
1410, 12, 13syl2an 464 . . 3  |-  ( ( F  e.  ( T RingHom  U )  /\  G  e.  ( S RingHom  T )
)  ->  ( F  o.  G )  e.  ( (mulGrp `  S ) MndHom  (mulGrp `  U ) ) )
157, 14jca 519 . 2  |-  ( ( F  e.  ( T RingHom  U )  /\  G  e.  ( S RingHom  T )
)  ->  ( ( F  o.  G )  e.  ( S  GrpHom  U )  /\  ( F  o.  G )  e.  ( (mulGrp `  S ) MndHom  (mulGrp `  U ) ) ) )
1611, 9isrhm 15751 . 2  |-  ( ( F  o.  G )  e.  ( S RingHom  U
)  <->  ( ( S  e.  Ring  /\  U  e. 
Ring )  /\  (
( F  o.  G
)  e.  ( S 
GrpHom  U )  /\  ( F  o.  G )  e.  ( (mulGrp `  S
) MndHom  (mulGrp `  U )
) ) ) )
173, 15, 16sylanbrc 646 1  |-  ( ( F  e.  ( T RingHom  U )  /\  G  e.  ( S RingHom  T )
)  ->  ( F  o.  G )  e.  ( S RingHom  U ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    e. wcel 1717    o. ccom 4822   ` cfv 5394  (class class class)co 6020   MndHom cmhm 14663    GrpHom cghm 14930  mulGrpcmgp 15575   Ringcrg 15587   RingHom crh 15744
This theorem is referenced by:  chrrhm  16735  evl1rhm  19816
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-cnex 8979  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-pre-mulgt0 9000
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-riota 6485  df-recs 6569  df-rdg 6604  df-er 6841  df-map 6956  df-en 7046  df-dom 7047  df-sdom 7048  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-sub 9225  df-neg 9226  df-nn 9933  df-2 9990  df-ndx 13399  df-slot 13400  df-base 13401  df-sets 13402  df-plusg 13469  df-0g 13654  df-mnd 14617  df-mhm 14665  df-grp 14739  df-ghm 14931  df-mgp 15576  df-rng 15590  df-ur 15592  df-rnghom 15746
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