MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  asclrhm Unicode version

Theorem asclrhm 16355
Description: The scalar injection is a ring homomorphism. (Contributed by Mario Carneiro, 8-Mar-2015.)
Hypotheses
Ref Expression
asclrhm.a  |-  A  =  (algSc `  W )
asclrhm.f  |-  F  =  (Scalar `  W )
Assertion
Ref Expression
asclrhm  |-  ( W  e. AssAlg  ->  A  e.  ( F RingHom  W ) )

Proof of Theorem asclrhm
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2404 . 2  |-  ( Base `  F )  =  (
Base `  F )
2 eqid 2404 . 2  |-  ( 1r
`  F )  =  ( 1r `  F
)
3 eqid 2404 . 2  |-  ( 1r
`  W )  =  ( 1r `  W
)
4 eqid 2404 . 2  |-  ( .r
`  F )  =  ( .r `  F
)
5 eqid 2404 . 2  |-  ( .r
`  W )  =  ( .r `  W
)
6 asclrhm.f . . . 4  |-  F  =  (Scalar `  W )
76assasca 16336 . . 3  |-  ( W  e. AssAlg  ->  F  e.  CRing )
8 crngrng 15629 . . 3  |-  ( F  e.  CRing  ->  F  e.  Ring )
97, 8syl 16 . 2  |-  ( W  e. AssAlg  ->  F  e.  Ring )
10 assarng 16335 . 2  |-  ( W  e. AssAlg  ->  W  e.  Ring )
111, 2rngidcl 15639 . . . 4  |-  ( F  e.  Ring  ->  ( 1r
`  F )  e.  ( Base `  F
) )
12 asclrhm.a . . . . 5  |-  A  =  (algSc `  W )
13 eqid 2404 . . . . 5  |-  ( .s
`  W )  =  ( .s `  W
)
1412, 6, 1, 13, 3asclval 16349 . . . 4  |-  ( ( 1r `  F )  e.  ( Base `  F
)  ->  ( A `  ( 1r `  F
) )  =  ( ( 1r `  F
) ( .s `  W ) ( 1r
`  W ) ) )
159, 11, 143syl 19 . . 3  |-  ( W  e. AssAlg  ->  ( A `  ( 1r `  F ) )  =  ( ( 1r `  F ) ( .s `  W
) ( 1r `  W ) ) )
16 assalmod 16334 . . . 4  |-  ( W  e. AssAlg  ->  W  e.  LMod )
17 eqid 2404 . . . . . 6  |-  ( Base `  W )  =  (
Base `  W )
1817, 3rngidcl 15639 . . . . 5  |-  ( W  e.  Ring  ->  ( 1r
`  W )  e.  ( Base `  W
) )
1910, 18syl 16 . . . 4  |-  ( W  e. AssAlg  ->  ( 1r `  W )  e.  (
Base `  W )
)
2017, 6, 13, 2lmodvs1 15933 . . . 4  |-  ( ( W  e.  LMod  /\  ( 1r `  W )  e.  ( Base `  W
) )  ->  (
( 1r `  F
) ( .s `  W ) ( 1r
`  W ) )  =  ( 1r `  W ) )
2116, 19, 20syl2anc 643 . . 3  |-  ( W  e. AssAlg  ->  ( ( 1r
`  F ) ( .s `  W ) ( 1r `  W
) )  =  ( 1r `  W ) )
2215, 21eqtrd 2436 . 2  |-  ( W  e. AssAlg  ->  ( A `  ( 1r `  F ) )  =  ( 1r
`  W ) )
2317, 5, 3rnglidm 15642 . . . . . . . 8  |-  ( ( W  e.  Ring  /\  ( 1r `  W )  e.  ( Base `  W
) )  ->  (
( 1r `  W
) ( .r `  W ) ( 1r
`  W ) )  =  ( 1r `  W ) )
2410, 19, 23syl2anc 643 . . . . . . 7  |-  ( W  e. AssAlg  ->  ( ( 1r
`  W ) ( .r `  W ) ( 1r `  W
) )  =  ( 1r `  W ) )
2524adantr 452 . . . . . 6  |-  ( ( W  e. AssAlg  /\  (
x  e.  ( Base `  F )  /\  y  e.  ( Base `  F
) ) )  -> 
( ( 1r `  W ) ( .r
`  W ) ( 1r `  W ) )  =  ( 1r
`  W ) )
2625oveq2d 6056 . . . . 5  |-  ( ( W  e. AssAlg  /\  (
x  e.  ( Base `  F )  /\  y  e.  ( Base `  F
) ) )  -> 
( y ( .s
`  W ) ( ( 1r `  W
) ( .r `  W ) ( 1r
`  W ) ) )  =  ( y ( .s `  W
) ( 1r `  W ) ) )
2726oveq2d 6056 . . . 4  |-  ( ( W  e. AssAlg  /\  (
x  e.  ( Base `  F )  /\  y  e.  ( Base `  F
) ) )  -> 
( x ( .s
`  W ) ( y ( .s `  W ) ( ( 1r `  W ) ( .r `  W
) ( 1r `  W ) ) ) )  =  ( x ( .s `  W
) ( y ( .s `  W ) ( 1r `  W
) ) ) )
28 simpl 444 . . . . . 6  |-  ( ( W  e. AssAlg  /\  (
x  e.  ( Base `  F )  /\  y  e.  ( Base `  F
) ) )  ->  W  e. AssAlg )
29 simprl 733 . . . . . 6  |-  ( ( W  e. AssAlg  /\  (
x  e.  ( Base `  F )  /\  y  e.  ( Base `  F
) ) )  ->  x  e.  ( Base `  F ) )
3019adantr 452 . . . . . 6  |-  ( ( W  e. AssAlg  /\  (
x  e.  ( Base `  F )  /\  y  e.  ( Base `  F
) ) )  -> 
( 1r `  W
)  e.  ( Base `  W ) )
3116adantr 452 . . . . . . 7  |-  ( ( W  e. AssAlg  /\  (
x  e.  ( Base `  F )  /\  y  e.  ( Base `  F
) ) )  ->  W  e.  LMod )
32 simprr 734 . . . . . . 7  |-  ( ( W  e. AssAlg  /\  (
x  e.  ( Base `  F )  /\  y  e.  ( Base `  F
) ) )  -> 
y  e.  ( Base `  F ) )
3317, 6, 13, 1lmodvscl 15922 . . . . . . 7  |-  ( ( W  e.  LMod  /\  y  e.  ( Base `  F
)  /\  ( 1r `  W )  e.  (
Base `  W )
)  ->  ( y
( .s `  W
) ( 1r `  W ) )  e.  ( Base `  W
) )
3431, 32, 30, 33syl3anc 1184 . . . . . 6  |-  ( ( W  e. AssAlg  /\  (
x  e.  ( Base `  F )  /\  y  e.  ( Base `  F
) ) )  -> 
( y ( .s
`  W ) ( 1r `  W ) )  e.  ( Base `  W ) )
3517, 6, 1, 13, 5assaass 16332 . . . . . 6  |-  ( ( W  e. AssAlg  /\  (
x  e.  ( Base `  F )  /\  ( 1r `  W )  e.  ( Base `  W
)  /\  ( y
( .s `  W
) ( 1r `  W ) )  e.  ( Base `  W
) ) )  -> 
( ( x ( .s `  W ) ( 1r `  W
) ) ( .r
`  W ) ( y ( .s `  W ) ( 1r
`  W ) ) )  =  ( x ( .s `  W
) ( ( 1r
`  W ) ( .r `  W ) ( y ( .s
`  W ) ( 1r `  W ) ) ) ) )
3628, 29, 30, 34, 35syl13anc 1186 . . . . 5  |-  ( ( W  e. AssAlg  /\  (
x  e.  ( Base `  F )  /\  y  e.  ( Base `  F
) ) )  -> 
( ( x ( .s `  W ) ( 1r `  W
) ) ( .r
`  W ) ( y ( .s `  W ) ( 1r
`  W ) ) )  =  ( x ( .s `  W
) ( ( 1r
`  W ) ( .r `  W ) ( y ( .s
`  W ) ( 1r `  W ) ) ) ) )
3717, 6, 1, 13, 5assaassr 16333 . . . . . . 7  |-  ( ( W  e. AssAlg  /\  (
y  e.  ( Base `  F )  /\  ( 1r `  W )  e.  ( Base `  W
)  /\  ( 1r `  W )  e.  (
Base `  W )
) )  ->  (
( 1r `  W
) ( .r `  W ) ( y ( .s `  W
) ( 1r `  W ) ) )  =  ( y ( .s `  W ) ( ( 1r `  W ) ( .r
`  W ) ( 1r `  W ) ) ) )
3828, 32, 30, 30, 37syl13anc 1186 . . . . . 6  |-  ( ( W  e. AssAlg  /\  (
x  e.  ( Base `  F )  /\  y  e.  ( Base `  F
) ) )  -> 
( ( 1r `  W ) ( .r
`  W ) ( y ( .s `  W ) ( 1r
`  W ) ) )  =  ( y ( .s `  W
) ( ( 1r
`  W ) ( .r `  W ) ( 1r `  W
) ) ) )
3938oveq2d 6056 . . . . 5  |-  ( ( W  e. AssAlg  /\  (
x  e.  ( Base `  F )  /\  y  e.  ( Base `  F
) ) )  -> 
( x ( .s
`  W ) ( ( 1r `  W
) ( .r `  W ) ( y ( .s `  W
) ( 1r `  W ) ) ) )  =  ( x ( .s `  W
) ( y ( .s `  W ) ( ( 1r `  W ) ( .r
`  W ) ( 1r `  W ) ) ) ) )
4036, 39eqtrd 2436 . . . 4  |-  ( ( W  e. AssAlg  /\  (
x  e.  ( Base `  F )  /\  y  e.  ( Base `  F
) ) )  -> 
( ( x ( .s `  W ) ( 1r `  W
) ) ( .r
`  W ) ( y ( .s `  W ) ( 1r
`  W ) ) )  =  ( x ( .s `  W
) ( y ( .s `  W ) ( ( 1r `  W ) ( .r
`  W ) ( 1r `  W ) ) ) ) )
4117, 6, 13, 1, 4lmodvsass 15930 . . . . 5  |-  ( ( W  e.  LMod  /\  (
x  e.  ( Base `  F )  /\  y  e.  ( Base `  F
)  /\  ( 1r `  W )  e.  (
Base `  W )
) )  ->  (
( x ( .r
`  F ) y ) ( .s `  W ) ( 1r
`  W ) )  =  ( x ( .s `  W ) ( y ( .s
`  W ) ( 1r `  W ) ) ) )
4231, 29, 32, 30, 41syl13anc 1186 . . . 4  |-  ( ( W  e. AssAlg  /\  (
x  e.  ( Base `  F )  /\  y  e.  ( Base `  F
) ) )  -> 
( ( x ( .r `  F ) y ) ( .s
`  W ) ( 1r `  W ) )  =  ( x ( .s `  W
) ( y ( .s `  W ) ( 1r `  W
) ) ) )
4327, 40, 423eqtr4rd 2447 . . 3  |-  ( ( W  e. AssAlg  /\  (
x  e.  ( Base `  F )  /\  y  e.  ( Base `  F
) ) )  -> 
( ( x ( .r `  F ) y ) ( .s
`  W ) ( 1r `  W ) )  =  ( ( x ( .s `  W ) ( 1r
`  W ) ) ( .r `  W
) ( y ( .s `  W ) ( 1r `  W
) ) ) )
441, 4rngcl 15632 . . . . . 6  |-  ( ( F  e.  Ring  /\  x  e.  ( Base `  F
)  /\  y  e.  ( Base `  F )
)  ->  ( x
( .r `  F
) y )  e.  ( Base `  F
) )
45443expb 1154 . . . . 5  |-  ( ( F  e.  Ring  /\  (
x  e.  ( Base `  F )  /\  y  e.  ( Base `  F
) ) )  -> 
( x ( .r
`  F ) y )  e.  ( Base `  F ) )
469, 45sylan 458 . . . 4  |-  ( ( W  e. AssAlg  /\  (
x  e.  ( Base `  F )  /\  y  e.  ( Base `  F
) ) )  -> 
( x ( .r
`  F ) y )  e.  ( Base `  F ) )
4712, 6, 1, 13, 3asclval 16349 . . . 4  |-  ( ( x ( .r `  F ) y )  e.  ( Base `  F
)  ->  ( A `  ( x ( .r
`  F ) y ) )  =  ( ( x ( .r
`  F ) y ) ( .s `  W ) ( 1r
`  W ) ) )
4846, 47syl 16 . . 3  |-  ( ( W  e. AssAlg  /\  (
x  e.  ( Base `  F )  /\  y  e.  ( Base `  F
) ) )  -> 
( A `  (
x ( .r `  F ) y ) )  =  ( ( x ( .r `  F ) y ) ( .s `  W
) ( 1r `  W ) ) )
4912, 6, 1, 13, 3asclval 16349 . . . . 5  |-  ( x  e.  ( Base `  F
)  ->  ( A `  x )  =  ( x ( .s `  W ) ( 1r
`  W ) ) )
5029, 49syl 16 . . . 4  |-  ( ( W  e. AssAlg  /\  (
x  e.  ( Base `  F )  /\  y  e.  ( Base `  F
) ) )  -> 
( A `  x
)  =  ( x ( .s `  W
) ( 1r `  W ) ) )
5112, 6, 1, 13, 3asclval 16349 . . . . 5  |-  ( y  e.  ( Base `  F
)  ->  ( A `  y )  =  ( y ( .s `  W ) ( 1r
`  W ) ) )
5232, 51syl 16 . . . 4  |-  ( ( W  e. AssAlg  /\  (
x  e.  ( Base `  F )  /\  y  e.  ( Base `  F
) ) )  -> 
( A `  y
)  =  ( y ( .s `  W
) ( 1r `  W ) ) )
5350, 52oveq12d 6058 . . 3  |-  ( ( W  e. AssAlg  /\  (
x  e.  ( Base `  F )  /\  y  e.  ( Base `  F
) ) )  -> 
( ( A `  x ) ( .r
`  W ) ( A `  y ) )  =  ( ( x ( .s `  W ) ( 1r
`  W ) ) ( .r `  W
) ( y ( .s `  W ) ( 1r `  W
) ) ) )
5443, 48, 533eqtr4d 2446 . 2  |-  ( ( W  e. AssAlg  /\  (
x  e.  ( Base `  F )  /\  y  e.  ( Base `  F
) ) )  -> 
( A `  (
x ( .r `  F ) y ) )  =  ( ( A `  x ) ( .r `  W
) ( A `  y ) ) )
5512, 6, 10, 16asclghm 16352 . 2  |-  ( W  e. AssAlg  ->  A  e.  ( F  GrpHom  W ) )
561, 2, 3, 4, 5, 9, 10, 22, 54, 55isrhm2d 15784 1  |-  ( W  e. AssAlg  ->  A  e.  ( F RingHom  W ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   ` cfv 5413  (class class class)co 6040   Basecbs 13424   .rcmulr 13485  Scalarcsca 13487   .scvsca 13488   Ringcrg 15615   CRingccrg 15616   1rcur 15617   RingHom crh 15772   LModclmod 15905  AssAlgcasa 16324  algSccascl 16326
This theorem is referenced by:  mplind  16517  evlslem1  19889  mpfind  19918  pf1ind  19928
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-riota 6508  df-recs 6592  df-rdg 6627  df-er 6864  df-map 6979  df-en 7069  df-dom 7070  df-sdom 7071  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-2 10014  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-plusg 13497  df-0g 13682  df-mnd 14645  df-mhm 14693  df-grp 14767  df-ghm 14959  df-mgp 15604  df-rng 15618  df-cring 15619  df-ur 15620  df-rnghom 15774  df-lmod 15907  df-assa 16327  df-ascl 16329
  Copyright terms: Public domain W3C validator