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Theorem rngohomsub 31960
Description: Ring homomorphisms preserve subtraction. (Contributed by Jeff Madsen, 15-Jun-2011.)
Hypotheses
Ref Expression
rnghomsub.1  |-  G  =  ( 1st `  R
)
rnghomsub.2  |-  X  =  ran  G
rnghomsub.3  |-  H  =  (  /g  `  G
)
rnghomsub.4  |-  J  =  ( 1st `  S
)
rnghomsub.5  |-  K  =  (  /g  `  J
)
Assertion
Ref Expression
rngohomsub  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  /\  ( A  e.  X  /\  B  e.  X ) )  -> 
( F `  ( A H B ) )  =  ( ( F `
 A ) K ( F `  B
) ) )

Proof of Theorem rngohomsub
StepHypRef Expression
1 rnghomsub.1 . . . . 5  |-  G  =  ( 1st `  R
)
21rngogrpo 26004 . . . 4  |-  ( R  e.  RingOps  ->  G  e.  GrpOp )
323ad2ant1 1026 . . 3  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  ->  G  e.  GrpOp
)
4 rnghomsub.4 . . . . 5  |-  J  =  ( 1st `  S
)
54rngogrpo 26004 . . . 4  |-  ( S  e.  RingOps  ->  J  e.  GrpOp )
653ad2ant2 1027 . . 3  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  ->  J  e.  GrpOp
)
71, 4rngogrphom 31958 . . 3  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  ->  F  e.  ( G GrpOpHom  J ) )
83, 6, 73jca 1185 . 2  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  ->  ( G  e.  GrpOp  /\  J  e.  GrpOp  /\  F  e.  ( G GrpOpHom  J ) ) )
9 rnghomsub.2 . . 3  |-  X  =  ran  G
10 rnghomsub.3 . . 3  |-  H  =  (  /g  `  G
)
11 rnghomsub.5 . . 3  |-  K  =  (  /g  `  J
)
129, 10, 11ghomdiv 31930 . 2  |-  ( ( ( G  e.  GrpOp  /\  J  e.  GrpOp  /\  F  e.  ( G GrpOpHom  J )
)  /\  ( A  e.  X  /\  B  e.  X ) )  -> 
( F `  ( A H B ) )  =  ( ( F `
 A ) K ( F `  B
) ) )
138, 12sylan 473 1  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  /\  ( A  e.  X  /\  B  e.  X ) )  -> 
( F `  ( A H B ) )  =  ( ( F `
 A ) K ( F `  B
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1867   ran crn 4846   ` cfv 5592  (class class class)co 6296   1stc1st 6796   GrpOpcgr 25800    /g cgs 25803   GrpOpHom cghomOLD 25971   RingOpscrngo 25989    RngHom crnghom 31947
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 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4529  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588
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 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-reu 2780  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-iun 4295  df-br 4418  df-opab 4476  df-mpt 4477  df-id 4760  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6798  df-2nd 6799  df-map 7473  df-grpo 25805  df-gid 25806  df-ginv 25807  df-gdiv 25808  df-ablo 25896  df-ghomOLD 25972  df-rngo 25990  df-rngohom 31950
This theorem is referenced by: (None)
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