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Theorem asclrhm 17334
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 2433 . 2  |-  ( Base `  F )  =  (
Base `  F )
2 eqid 2433 . 2  |-  ( 1r
`  F )  =  ( 1r `  F
)
3 eqid 2433 . 2  |-  ( 1r
`  W )  =  ( 1r `  W
)
4 eqid 2433 . 2  |-  ( .r
`  F )  =  ( .r `  F
)
5 eqid 2433 . 2  |-  ( .r
`  W )  =  ( .r `  W
)
6 asclrhm.f . . . 4  |-  F  =  (Scalar `  W )
76assasca 17315 . . 3  |-  ( W  e. AssAlg  ->  F  e.  CRing )
8 crngrng 16591 . . 3  |-  ( F  e.  CRing  ->  F  e.  Ring )
97, 8syl 16 . 2  |-  ( W  e. AssAlg  ->  F  e.  Ring )
10 assarng 17314 . 2  |-  ( W  e. AssAlg  ->  W  e.  Ring )
111, 2rngidcl 16601 . . . 4  |-  ( F  e.  Ring  ->  ( 1r
`  F )  e.  ( Base `  F
) )
12 asclrhm.a . . . . 5  |-  A  =  (algSc `  W )
13 eqid 2433 . . . . 5  |-  ( .s
`  W )  =  ( .s `  W
)
1412, 6, 1, 13, 3asclval 17328 . . . 4  |-  ( ( 1r `  F )  e.  ( Base `  F
)  ->  ( A `  ( 1r `  F
) )  =  ( ( 1r `  F
) ( .s `  W ) ( 1r
`  W ) ) )
159, 11, 143syl 20 . . 3  |-  ( W  e. AssAlg  ->  ( A `  ( 1r `  F ) )  =  ( ( 1r `  F ) ( .s `  W
) ( 1r `  W ) ) )
16 assalmod 17313 . . . 4  |-  ( W  e. AssAlg  ->  W  e.  LMod )
17 eqid 2433 . . . . . 6  |-  ( Base `  W )  =  (
Base `  W )
1817, 3rngidcl 16601 . . . . 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 16900 . . . 4  |-  ( ( W  e.  LMod  /\  ( 1r `  W )  e.  ( Base `  W
) )  ->  (
( 1r `  F
) ( .s `  W ) ( 1r
`  W ) )  =  ( 1r `  W ) )
2116, 19, 20syl2anc 654 . . 3  |-  ( W  e. AssAlg  ->  ( ( 1r
`  F ) ( .s `  W ) ( 1r `  W
) )  =  ( 1r `  W ) )
2215, 21eqtrd 2465 . 2  |-  ( W  e. AssAlg  ->  ( A `  ( 1r `  F ) )  =  ( 1r
`  W ) )
2317, 5, 3rnglidm 16604 . . . . . . . 8  |-  ( ( W  e.  Ring  /\  ( 1r `  W )  e.  ( Base `  W
) )  ->  (
( 1r `  W
) ( .r `  W ) ( 1r
`  W ) )  =  ( 1r `  W ) )
2410, 19, 23syl2anc 654 . . . . . . 7  |-  ( W  e. AssAlg  ->  ( ( 1r
`  W ) ( .r `  W ) ( 1r `  W
) )  =  ( 1r `  W ) )
2524adantr 462 . . . . . 6  |-  ( ( W  e. AssAlg  /\  (
x  e.  ( Base `  F )  /\  y  e.  ( Base `  F
) ) )  -> 
( ( 1r `  W ) ( .r
`  W ) ( 1r `  W ) )  =  ( 1r
`  W ) )
2625oveq2d 6096 . . . . 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 6096 . . . 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 454 . . . . . 6  |-  ( ( W  e. AssAlg  /\  (
x  e.  ( Base `  F )  /\  y  e.  ( Base `  F
) ) )  ->  W  e. AssAlg )
29 simprl 748 . . . . . 6  |-  ( ( W  e. AssAlg  /\  (
x  e.  ( Base `  F )  /\  y  e.  ( Base `  F
) ) )  ->  x  e.  ( Base `  F ) )
3019adantr 462 . . . . . 6  |-  ( ( W  e. AssAlg  /\  (
x  e.  ( Base `  F )  /\  y  e.  ( Base `  F
) ) )  -> 
( 1r `  W
)  e.  ( Base `  W ) )
3116adantr 462 . . . . . . 7  |-  ( ( W  e. AssAlg  /\  (
x  e.  ( Base `  F )  /\  y  e.  ( Base `  F
) ) )  ->  W  e.  LMod )
32 simprr 749 . . . . . . 7  |-  ( ( W  e. AssAlg  /\  (
x  e.  ( Base `  F )  /\  y  e.  ( Base `  F
) ) )  -> 
y  e.  ( Base `  F ) )
3317, 6, 13, 1lmodvscl 16889 . . . . . . 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 1211 . . . . . 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 17311 . . . . . 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 1213 . . . . 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 17312 . . . . . . 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 1213 . . . . . 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 6096 . . . . 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 2465 . . . 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 16897 . . . . 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 1213 . . . 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 2476 . . 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 16594 . . . . . 6  |-  ( ( F  e.  Ring  /\  x  e.  ( Base `  F
)  /\  y  e.  ( Base `  F )
)  ->  ( x
( .r `  F
) y )  e.  ( Base `  F
) )
45443expb 1181 . . . . 5  |-  ( ( F  e.  Ring  /\  (
x  e.  ( Base `  F )  /\  y  e.  ( Base `  F
) ) )  -> 
( x ( .r
`  F ) y )  e.  ( Base `  F ) )
469, 45sylan 468 . . . 4  |-  ( ( W  e. AssAlg  /\  (
x  e.  ( Base `  F )  /\  y  e.  ( Base `  F
) ) )  -> 
( x ( .r
`  F ) y )  e.  ( Base `  F ) )
4712, 6, 1, 13, 3asclval 17328 . . . 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 17328 . . . . 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 17328 . . . . 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 6098 . . 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 2475 . 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 17331 . 2  |-  ( W  e. AssAlg  ->  A  e.  ( F  GrpHom  W ) )
561, 2, 3, 4, 5, 9, 10, 22, 54, 55isrhm2d 16750 1  |-  ( W  e. AssAlg  ->  A  e.  ( F RingHom  W ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1362    e. wcel 1755   ` cfv 5406  (class class class)co 6080   Basecbs 14157   .rcmulr 14222  Scalarcsca 14224   .scvsca 14225   Ringcrg 16577   CRingccrg 16578   1rcur 16579   RingHom crh 16738   LModclmod 16872  AssAlgcasa 17303  algSccascl 17305
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-cnex 9326  ax-resscn 9327  ax-1cn 9328  ax-icn 9329  ax-addcl 9330  ax-addrcl 9331  ax-mulcl 9332  ax-mulrcl 9333  ax-mulcom 9334  ax-addass 9335  ax-mulass 9336  ax-distr 9337  ax-i2m1 9338  ax-1ne0 9339  ax-1rid 9340  ax-rnegex 9341  ax-rrecex 9342  ax-cnre 9343  ax-pre-lttri 9344  ax-pre-lttrn 9345  ax-pre-ltadd 9346  ax-pre-mulgt0 9347
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-uni 4080  df-iun 4161  df-br 4281  df-opab 4339  df-mpt 4340  df-tr 4374  df-eprel 4619  df-id 4623  df-po 4628  df-so 4629  df-fr 4666  df-we 4668  df-ord 4709  df-on 4710  df-lim 4711  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-om 6466  df-recs 6818  df-rdg 6852  df-er 7089  df-map 7204  df-en 7299  df-dom 7300  df-sdom 7301  df-pnf 9408  df-mnf 9409  df-xr 9410  df-ltxr 9411  df-le 9412  df-sub 9585  df-neg 9586  df-nn 10311  df-2 10368  df-ndx 14160  df-slot 14161  df-base 14162  df-sets 14163  df-plusg 14234  df-0g 14363  df-mnd 15398  df-mhm 15447  df-grp 15525  df-ghm 15725  df-mgp 16566  df-rng 16580  df-cring 16581  df-ur 16582  df-rnghom 16740  df-lmod 16874  df-assa 17306  df-ascl 17308
This theorem is referenced by:  mplind  17516  evlslem1  21367  mpfind  21396  pf1ind  21406
  Copyright terms: Public domain W3C validator