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Theorem asclrhm 17392
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 2438 . 2  |-  ( Base `  F )  =  (
Base `  F )
2 eqid 2438 . 2  |-  ( 1r
`  F )  =  ( 1r `  F
)
3 eqid 2438 . 2  |-  ( 1r
`  W )  =  ( 1r `  W
)
4 eqid 2438 . 2  |-  ( .r
`  F )  =  ( .r `  F
)
5 eqid 2438 . 2  |-  ( .r
`  W )  =  ( .r `  W
)
6 asclrhm.f . . . 4  |-  F  =  (Scalar `  W )
76assasca 17373 . . 3  |-  ( W  e. AssAlg  ->  F  e.  CRing )
8 crngrng 16645 . . 3  |-  ( F  e.  CRing  ->  F  e.  Ring )
97, 8syl 16 . 2  |-  ( W  e. AssAlg  ->  F  e.  Ring )
10 assarng 17372 . 2  |-  ( W  e. AssAlg  ->  W  e.  Ring )
111, 2rngidcl 16655 . . . 4  |-  ( F  e.  Ring  ->  ( 1r
`  F )  e.  ( Base `  F
) )
12 asclrhm.a . . . . 5  |-  A  =  (algSc `  W )
13 eqid 2438 . . . . 5  |-  ( .s
`  W )  =  ( .s `  W
)
1412, 6, 1, 13, 3asclval 17386 . . . 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 17371 . . . 4  |-  ( W  e. AssAlg  ->  W  e.  LMod )
17 eqid 2438 . . . . . 6  |-  ( Base `  W )  =  (
Base `  W )
1817, 3rngidcl 16655 . . . . 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 16956 . . . 4  |-  ( ( W  e.  LMod  /\  ( 1r `  W )  e.  ( Base `  W
) )  ->  (
( 1r `  F
) ( .s `  W ) ( 1r
`  W ) )  =  ( 1r `  W ) )
2116, 19, 20syl2anc 661 . . 3  |-  ( W  e. AssAlg  ->  ( ( 1r
`  F ) ( .s `  W ) ( 1r `  W
) )  =  ( 1r `  W ) )
2215, 21eqtrd 2470 . 2  |-  ( W  e. AssAlg  ->  ( A `  ( 1r `  F ) )  =  ( 1r
`  W ) )
2317, 5, 3rnglidm 16658 . . . . . . . 8  |-  ( ( W  e.  Ring  /\  ( 1r `  W )  e.  ( Base `  W
) )  ->  (
( 1r `  W
) ( .r `  W ) ( 1r
`  W ) )  =  ( 1r `  W ) )
2410, 19, 23syl2anc 661 . . . . . . 7  |-  ( W  e. AssAlg  ->  ( ( 1r
`  W ) ( .r `  W ) ( 1r `  W
) )  =  ( 1r `  W ) )
2524adantr 465 . . . . . 6  |-  ( ( W  e. AssAlg  /\  (
x  e.  ( Base `  F )  /\  y  e.  ( Base `  F
) ) )  -> 
( ( 1r `  W ) ( .r
`  W ) ( 1r `  W ) )  =  ( 1r
`  W ) )
2625oveq2d 6102 . . . . 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 6102 . . . 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 457 . . . . . 6  |-  ( ( W  e. AssAlg  /\  (
x  e.  ( Base `  F )  /\  y  e.  ( Base `  F
) ) )  ->  W  e. AssAlg )
29 simprl 755 . . . . . 6  |-  ( ( W  e. AssAlg  /\  (
x  e.  ( Base `  F )  /\  y  e.  ( Base `  F
) ) )  ->  x  e.  ( Base `  F ) )
3019adantr 465 . . . . . 6  |-  ( ( W  e. AssAlg  /\  (
x  e.  ( Base `  F )  /\  y  e.  ( Base `  F
) ) )  -> 
( 1r `  W
)  e.  ( Base `  W ) )
3116adantr 465 . . . . . . 7  |-  ( ( W  e. AssAlg  /\  (
x  e.  ( Base `  F )  /\  y  e.  ( Base `  F
) ) )  ->  W  e.  LMod )
32 simprr 756 . . . . . . 7  |-  ( ( W  e. AssAlg  /\  (
x  e.  ( Base `  F )  /\  y  e.  ( Base `  F
) ) )  -> 
y  e.  ( Base `  F ) )
3317, 6, 13, 1lmodvscl 16945 . . . . . . 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 1218 . . . . . 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 17369 . . . . . 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 1220 . . . . 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 17370 . . . . . . 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 1220 . . . . . 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 6102 . . . . 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 2470 . . . 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 16953 . . . . 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 1220 . . . 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 2481 . . 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 16648 . . . . . 6  |-  ( ( F  e.  Ring  /\  x  e.  ( Base `  F
)  /\  y  e.  ( Base `  F )
)  ->  ( x
( .r `  F
) y )  e.  ( Base `  F
) )
45443expb 1188 . . . . 5  |-  ( ( F  e.  Ring  /\  (
x  e.  ( Base `  F )  /\  y  e.  ( Base `  F
) ) )  -> 
( x ( .r
`  F ) y )  e.  ( Base `  F ) )
469, 45sylan 471 . . . 4  |-  ( ( W  e. AssAlg  /\  (
x  e.  ( Base `  F )  /\  y  e.  ( Base `  F
) ) )  -> 
( x ( .r
`  F ) y )  e.  ( Base `  F ) )
4712, 6, 1, 13, 3asclval 17386 . . . 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 17386 . . . . 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 17386 . . . . 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 6104 . . 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 2480 . 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 17389 . 2  |-  ( W  e. AssAlg  ->  A  e.  ( F  GrpHom  W ) )
561, 2, 3, 4, 5, 9, 10, 22, 54, 55isrhm2d 16806 1  |-  ( W  e. AssAlg  ->  A  e.  ( F RingHom  W ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   ` cfv 5413  (class class class)co 6086   Basecbs 14166   .rcmulr 14231  Scalarcsca 14233   .scvsca 14234   1rcur 16593   Ringcrg 16635   CRingccrg 16636   RingHom crh 16794   LModclmod 16928  AssAlgcasa 17361  algSccascl 17363
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 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  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 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  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-mnd 15407  df-mhm 15456  df-grp 15536  df-ghm 15736  df-mgp 16582  df-ur 16594  df-rng 16637  df-cring 16638  df-rnghom 16796  df-lmod 16930  df-assa 17364  df-ascl 17366
This theorem is referenced by:  mplind  17564  evlslem1  17581  mpfind  17602  pf1ind  17769
  Copyright terms: Public domain W3C validator