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Theorem isrhm2d 16802
Description: Demonstration of ring homomorphism. (Contributed by Mario Carneiro, 13-Jun-2015.)
Hypotheses
Ref Expression
isrhmd.b  |-  B  =  ( Base `  R
)
isrhmd.o  |-  .1.  =  ( 1r `  R )
isrhmd.n  |-  N  =  ( 1r `  S
)
isrhmd.t  |-  .x.  =  ( .r `  R )
isrhmd.u  |-  .X.  =  ( .r `  S )
isrhmd.r  |-  ( ph  ->  R  e.  Ring )
isrhmd.s  |-  ( ph  ->  S  e.  Ring )
isrhmd.ho  |-  ( ph  ->  ( F `  .1.  )  =  N )
isrhmd.ht  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( F `  (
x  .x.  y )
)  =  ( ( F `  x ) 
.X.  ( F `  y ) ) )
isrhm2d.f  |-  ( ph  ->  F  e.  ( R 
GrpHom  S ) )
Assertion
Ref Expression
isrhm2d  |-  ( ph  ->  F  e.  ( R RingHom  S ) )
Distinct variable groups:    ph, x, y   
x, B, y    x, F, y    x, R, y   
x, S, y
Allowed substitution hints:    .x. ( x, y)    .X. ( x, y)    .1. ( x, y)    N( x, y)

Proof of Theorem isrhm2d
StepHypRef Expression
1 isrhmd.r . . 3  |-  ( ph  ->  R  e.  Ring )
2 isrhmd.s . . 3  |-  ( ph  ->  S  e.  Ring )
31, 2jca 532 . 2  |-  ( ph  ->  ( R  e.  Ring  /\  S  e.  Ring )
)
4 isrhm2d.f . . 3  |-  ( ph  ->  F  e.  ( R 
GrpHom  S ) )
5 eqid 2437 . . . . . . 7  |-  (mulGrp `  R )  =  (mulGrp `  R )
65rngmgp 16637 . . . . . 6  |-  ( R  e.  Ring  ->  (mulGrp `  R )  e.  Mnd )
71, 6syl 16 . . . . 5  |-  ( ph  ->  (mulGrp `  R )  e.  Mnd )
8 eqid 2437 . . . . . . 7  |-  (mulGrp `  S )  =  (mulGrp `  S )
98rngmgp 16637 . . . . . 6  |-  ( S  e.  Ring  ->  (mulGrp `  S )  e.  Mnd )
102, 9syl 16 . . . . 5  |-  ( ph  ->  (mulGrp `  S )  e.  Mnd )
117, 10jca 532 . . . 4  |-  ( ph  ->  ( (mulGrp `  R
)  e.  Mnd  /\  (mulGrp `  S )  e. 
Mnd ) )
12 isrhmd.b . . . . . . 7  |-  B  =  ( Base `  R
)
13 eqid 2437 . . . . . . 7  |-  ( Base `  S )  =  (
Base `  S )
1412, 13ghmf 15740 . . . . . 6  |-  ( F  e.  ( R  GrpHom  S )  ->  F : B
--> ( Base `  S
) )
154, 14syl 16 . . . . 5  |-  ( ph  ->  F : B --> ( Base `  S ) )
16 isrhmd.ht . . . . . 6  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( F `  (
x  .x.  y )
)  =  ( ( F `  x ) 
.X.  ( F `  y ) ) )
1716ralrimivva 2802 . . . . 5  |-  ( ph  ->  A. x  e.  B  A. y  e.  B  ( F `  ( x 
.x.  y ) )  =  ( ( F `
 x )  .X.  ( F `  y ) ) )
18 isrhmd.ho . . . . . 6  |-  ( ph  ->  ( F `  .1.  )  =  N )
19 isrhmd.o . . . . . . . 8  |-  .1.  =  ( 1r `  R )
205, 19rngidval 16591 . . . . . . 7  |-  .1.  =  ( 0g `  (mulGrp `  R ) )
2120fveq2i 5687 . . . . . 6  |-  ( F `
 .1.  )  =  ( F `  ( 0g `  (mulGrp `  R
) ) )
22 isrhmd.n . . . . . . 7  |-  N  =  ( 1r `  S
)
238, 22rngidval 16591 . . . . . 6  |-  N  =  ( 0g `  (mulGrp `  S ) )
2418, 21, 233eqtr3g 2492 . . . . 5  |-  ( ph  ->  ( F `  ( 0g `  (mulGrp `  R
) ) )  =  ( 0g `  (mulGrp `  S ) ) )
2515, 17, 243jca 1168 . . . 4  |-  ( ph  ->  ( F : B --> ( Base `  S )  /\  A. x  e.  B  A. y  e.  B  ( F `  ( x 
.x.  y ) )  =  ( ( F `
 x )  .X.  ( F `  y ) )  /\  ( F `
 ( 0g `  (mulGrp `  R ) ) )  =  ( 0g
`  (mulGrp `  S )
) ) )
265, 12mgpbas 16583 . . . . 5  |-  B  =  ( Base `  (mulGrp `  R ) )
278, 13mgpbas 16583 . . . . 5  |-  ( Base `  S )  =  (
Base `  (mulGrp `  S
) )
28 isrhmd.t . . . . . 6  |-  .x.  =  ( .r `  R )
295, 28mgpplusg 16581 . . . . 5  |-  .x.  =  ( +g  `  (mulGrp `  R ) )
30 isrhmd.u . . . . . 6  |-  .X.  =  ( .r `  S )
318, 30mgpplusg 16581 . . . . 5  |-  .X.  =  ( +g  `  (mulGrp `  S ) )
32 eqid 2437 . . . . 5  |-  ( 0g
`  (mulGrp `  R )
)  =  ( 0g
`  (mulGrp `  R )
)
33 eqid 2437 . . . . 5  |-  ( 0g
`  (mulGrp `  S )
)  =  ( 0g
`  (mulGrp `  S )
)
3426, 27, 29, 31, 32, 33ismhm 15458 . . . 4  |-  ( F  e.  ( (mulGrp `  R ) MndHom  (mulGrp `  S
) )  <->  ( (
(mulGrp `  R )  e.  Mnd  /\  (mulGrp `  S )  e.  Mnd )  /\  ( F : B
--> ( Base `  S
)  /\  A. x  e.  B  A. y  e.  B  ( F `  ( x  .x.  y
) )  =  ( ( F `  x
)  .X.  ( F `  y ) )  /\  ( F `  ( 0g
`  (mulGrp `  R )
) )  =  ( 0g `  (mulGrp `  S ) ) ) ) )
3511, 25, 34sylanbrc 664 . . 3  |-  ( ph  ->  F  e.  ( (mulGrp `  R ) MndHom  (mulGrp `  S ) ) )
364, 35jca 532 . 2  |-  ( ph  ->  ( F  e.  ( R  GrpHom  S )  /\  F  e.  ( (mulGrp `  R ) MndHom  (mulGrp `  S ) ) ) )
375, 8isrhm 16797 . 2  |-  ( F  e.  ( R RingHom  S
)  <->  ( ( R  e.  Ring  /\  S  e. 
Ring )  /\  ( F  e.  ( R  GrpHom  S )  /\  F  e.  ( (mulGrp `  R
) MndHom  (mulGrp `  S )
) ) ) )
383, 36, 37sylanbrc 664 1  |-  ( ph  ->  F  e.  ( R RingHom  S ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   A.wral 2709   -->wf 5407   ` cfv 5411  (class class class)co 6086   Basecbs 14166   .rcmulr 14231   0gc0g 14370   Mndcmnd 15401   MndHom cmhm 15454    GrpHom cghm 15733  mulGrpcmgp 16577   1rcur 16589   Ringcrg 16631   RingHom crh 16790
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2418  ax-rep 4396  ax-sep 4406  ax-nul 4414  ax-pow 4463  ax-pr 4524  ax-un 6367  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-nel 2603  df-ral 2714  df-rex 2715  df-reu 2716  df-rab 2718  df-v 2968  df-sbc 3180  df-csb 3282  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-pss 3337  df-nul 3631  df-if 3785  df-pw 3855  df-sn 3871  df-pr 3873  df-tp 3875  df-op 3877  df-uni 4085  df-iun 4166  df-br 4286  df-opab 4344  df-mpt 4345  df-tr 4379  df-eprel 4624  df-id 4628  df-po 4633  df-so 4634  df-fr 4671  df-we 4673  df-ord 4714  df-on 4715  df-lim 4716  df-suc 4717  df-xp 4838  df-rel 4839  df-cnv 4840  df-co 4841  df-dm 4842  df-rn 4843  df-res 4844  df-ima 4845  df-iota 5374  df-fun 5413  df-fn 5414  df-f 5415  df-f1 5416  df-fo 5417  df-f1o 5418  df-fv 5419  df-riota 6045  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-recs 6824  df-rdg 6858  df-er 7093  df-map 7208  df-en 7303  df-dom 7304  df-sdom 7305  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-nn 10315  df-2 10372  df-ndx 14169  df-slot 14170  df-base 14171  df-sets 14172  df-plusg 14243  df-0g 14372  df-mhm 15456  df-ghm 15734  df-mgp 16578  df-ur 16590  df-rng 16633  df-rnghom 16792
This theorem is referenced by:  isrhmd  16803  divsrhm  17293  asclrhm  17386  mulgrhm  17895  mulgrhmOLD  17898  rhmopp  26234
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