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Theorem isrhm2d 15506
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 518 . 2  |-  ( ph  ->  ( R  e.  Ring  /\  S  e.  Ring )
)
4 isrhm2d.f . . 3  |-  ( ph  ->  F  e.  ( R 
GrpHom  S ) )
5 eqid 2283 . . . . . . 7  |-  (mulGrp `  R )  =  (mulGrp `  R )
65rngmgp 15347 . . . . . 6  |-  ( R  e.  Ring  ->  (mulGrp `  R )  e.  Mnd )
71, 6syl 15 . . . . 5  |-  ( ph  ->  (mulGrp `  R )  e.  Mnd )
8 eqid 2283 . . . . . . 7  |-  (mulGrp `  S )  =  (mulGrp `  S )
98rngmgp 15347 . . . . . 6  |-  ( S  e.  Ring  ->  (mulGrp `  S )  e.  Mnd )
102, 9syl 15 . . . . 5  |-  ( ph  ->  (mulGrp `  S )  e.  Mnd )
117, 10jca 518 . . . 4  |-  ( ph  ->  ( (mulGrp `  R
)  e.  Mnd  /\  (mulGrp `  S )  e. 
Mnd ) )
12 isrhmd.b . . . . . . 7  |-  B  =  ( Base `  R
)
13 eqid 2283 . . . . . . 7  |-  ( Base `  S )  =  (
Base `  S )
1412, 13ghmf 14687 . . . . . 6  |-  ( F  e.  ( R  GrpHom  S )  ->  F : B
--> ( Base `  S
) )
154, 14syl 15 . . . . 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 2635 . . . . 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 15343 . . . . . . 7  |-  .1.  =  ( 0g `  (mulGrp `  R ) )
2120fveq2i 5528 . . . . . 6  |-  ( F `
 .1.  )  =  ( F `  ( 0g `  (mulGrp `  R
) ) )
22 isrhmd.n . . . . . . 7  |-  N  =  ( 1r `  S
)
238, 22rngidval 15343 . . . . . 6  |-  N  =  ( 0g `  (mulGrp `  S ) )
2418, 21, 233eqtr3g 2338 . . . . 5  |-  ( ph  ->  ( F `  ( 0g `  (mulGrp `  R
) ) )  =  ( 0g `  (mulGrp `  S ) ) )
2515, 17, 243jca 1132 . . . 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 15331 . . . . 5  |-  B  =  ( Base `  (mulGrp `  R ) )
278, 13mgpbas 15331 . . . . 5  |-  ( Base `  S )  =  (
Base `  (mulGrp `  S
) )
28 isrhmd.t . . . . . 6  |-  .x.  =  ( .r `  R )
295, 28mgpplusg 15329 . . . . 5  |-  .x.  =  ( +g  `  (mulGrp `  R ) )
30 isrhmd.u . . . . . 6  |-  .X.  =  ( .r `  S )
318, 30mgpplusg 15329 . . . . 5  |-  .X.  =  ( +g  `  (mulGrp `  S ) )
32 eqid 2283 . . . . 5  |-  ( 0g
`  (mulGrp `  R )
)  =  ( 0g
`  (mulGrp `  R )
)
33 eqid 2283 . . . . 5  |-  ( 0g
`  (mulGrp `  S )
)  =  ( 0g
`  (mulGrp `  S )
)
3426, 27, 29, 31, 32, 33ismhm 14417 . . . 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 645 . . 3  |-  ( ph  ->  F  e.  ( (mulGrp `  R ) MndHom  (mulGrp `  S ) ) )
364, 35jca 518 . 2  |-  ( ph  ->  ( F  e.  ( R  GrpHom  S )  /\  F  e.  ( (mulGrp `  R ) MndHom  (mulGrp `  S ) ) ) )
375, 8isrhm 15501 . 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 645 1  |-  ( ph  ->  F  e.  ( R RingHom  S ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543   -->wf 5251   ` cfv 5255  (class class class)co 5858   Basecbs 13148   .rcmulr 13209   0gc0g 13400   Mndcmnd 14361   MndHom cmhm 14413    GrpHom cghm 14680  mulGrpcmgp 15325   Ringcrg 15337   1rcur 15339   RingHom crh 15494
This theorem is referenced by:  isrhmd  15507  divsrhm  15989  asclrhm  16081  mulgrhm  16460
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-plusg 13221  df-0g 13404  df-mhm 14415  df-ghm 14681  df-mgp 15326  df-rng 15340  df-ur 15342  df-rnghom 15496
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