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Theorem asclrhm 16097
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 2296 . 2  |-  ( Base `  F )  =  (
Base `  F )
2 eqid 2296 . 2  |-  ( 1r
`  F )  =  ( 1r `  F
)
3 eqid 2296 . 2  |-  ( 1r
`  W )  =  ( 1r `  W
)
4 eqid 2296 . 2  |-  ( .r
`  F )  =  ( .r `  F
)
5 eqid 2296 . 2  |-  ( .r
`  W )  =  ( .r `  W
)
6 asclrhm.f . . . 4  |-  F  =  (Scalar `  W )
76assasca 16078 . . 3  |-  ( W  e. AssAlg  ->  F  e.  CRing )
8 crngrng 15367 . . 3  |-  ( F  e.  CRing  ->  F  e.  Ring )
97, 8syl 15 . 2  |-  ( W  e. AssAlg  ->  F  e.  Ring )
10 assarng 16077 . 2  |-  ( W  e. AssAlg  ->  W  e.  Ring )
111, 2rngidcl 15377 . . . 4  |-  ( F  e.  Ring  ->  ( 1r
`  F )  e.  ( Base `  F
) )
12 asclrhm.a . . . . 5  |-  A  =  (algSc `  W )
13 eqid 2296 . . . . 5  |-  ( .s
`  W )  =  ( .s `  W
)
1412, 6, 1, 13, 3asclval 16091 . . . 4  |-  ( ( 1r `  F )  e.  ( Base `  F
)  ->  ( A `  ( 1r `  F
) )  =  ( ( 1r `  F
) ( .s `  W ) ( 1r
`  W ) ) )
159, 11, 143syl 18 . . 3  |-  ( W  e. AssAlg  ->  ( A `  ( 1r `  F ) )  =  ( ( 1r `  F ) ( .s `  W
) ( 1r `  W ) ) )
16 assalmod 16076 . . . 4  |-  ( W  e. AssAlg  ->  W  e.  LMod )
17 eqid 2296 . . . . . 6  |-  ( Base `  W )  =  (
Base `  W )
1817, 3rngidcl 15377 . . . . 5  |-  ( W  e.  Ring  ->  ( 1r
`  W )  e.  ( Base `  W
) )
1910, 18syl 15 . . . 4  |-  ( W  e. AssAlg  ->  ( 1r `  W )  e.  (
Base `  W )
)
2017, 6, 13, 2lmodvs1 15674 . . . 4  |-  ( ( W  e.  LMod  /\  ( 1r `  W )  e.  ( Base `  W
) )  ->  (
( 1r `  F
) ( .s `  W ) ( 1r
`  W ) )  =  ( 1r `  W ) )
2116, 19, 20syl2anc 642 . . 3  |-  ( W  e. AssAlg  ->  ( ( 1r
`  F ) ( .s `  W ) ( 1r `  W
) )  =  ( 1r `  W ) )
2215, 21eqtrd 2328 . 2  |-  ( W  e. AssAlg  ->  ( A `  ( 1r `  F ) )  =  ( 1r
`  W ) )
2317, 5, 3rnglidm 15380 . . . . . . . 8  |-  ( ( W  e.  Ring  /\  ( 1r `  W )  e.  ( Base `  W
) )  ->  (
( 1r `  W
) ( .r `  W ) ( 1r
`  W ) )  =  ( 1r `  W ) )
2410, 19, 23syl2anc 642 . . . . . . 7  |-  ( W  e. AssAlg  ->  ( ( 1r
`  W ) ( .r `  W ) ( 1r `  W
) )  =  ( 1r `  W ) )
2524adantr 451 . . . . . 6  |-  ( ( W  e. AssAlg  /\  (
x  e.  ( Base `  F )  /\  y  e.  ( Base `  F
) ) )  -> 
( ( 1r `  W ) ( .r
`  W ) ( 1r `  W ) )  =  ( 1r
`  W ) )
2625oveq2d 5890 . . . . 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 5890 . . . 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 443 . . . . . 6  |-  ( ( W  e. AssAlg  /\  (
x  e.  ( Base `  F )  /\  y  e.  ( Base `  F
) ) )  ->  W  e. AssAlg )
29 simprl 732 . . . . . 6  |-  ( ( W  e. AssAlg  /\  (
x  e.  ( Base `  F )  /\  y  e.  ( Base `  F
) ) )  ->  x  e.  ( Base `  F ) )
3019adantr 451 . . . . . 6  |-  ( ( W  e. AssAlg  /\  (
x  e.  ( Base `  F )  /\  y  e.  ( Base `  F
) ) )  -> 
( 1r `  W
)  e.  ( Base `  W ) )
3116adantr 451 . . . . . . 7  |-  ( ( W  e. AssAlg  /\  (
x  e.  ( Base `  F )  /\  y  e.  ( Base `  F
) ) )  ->  W  e.  LMod )
32 simprr 733 . . . . . . 7  |-  ( ( W  e. AssAlg  /\  (
x  e.  ( Base `  F )  /\  y  e.  ( Base `  F
) ) )  -> 
y  e.  ( Base `  F ) )
3317, 6, 13, 1lmodvscl 15660 . . . . . . 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 1182 . . . . . 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 16074 . . . . . 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 1184 . . . . 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 16075 . . . . . . 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 1184 . . . . . 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 5890 . . . . 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 2328 . . . 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 15670 . . . . 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 1184 . . . 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 2339 . . 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 15370 . . . . . 6  |-  ( ( F  e.  Ring  /\  x  e.  ( Base `  F
)  /\  y  e.  ( Base `  F )
)  ->  ( x
( .r `  F
) y )  e.  ( Base `  F
) )
45443expb 1152 . . . . 5  |-  ( ( F  e.  Ring  /\  (
x  e.  ( Base `  F )  /\  y  e.  ( Base `  F
) ) )  -> 
( x ( .r
`  F ) y )  e.  ( Base `  F ) )
469, 45sylan 457 . . . 4  |-  ( ( W  e. AssAlg  /\  (
x  e.  ( Base `  F )  /\  y  e.  ( Base `  F
) ) )  -> 
( x ( .r
`  F ) y )  e.  ( Base `  F ) )
4712, 6, 1, 13, 3asclval 16091 . . . 4  |-  ( ( x ( .r `  F ) y )  e.  ( Base `  F
)  ->  ( A `  ( x ( .r
`  F ) y ) )  =  ( ( x ( .r
`  F ) y ) ( .s `  W ) ( 1r
`  W ) ) )
4846, 47syl 15 . . 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 16091 . . . . 5  |-  ( x  e.  ( Base `  F
)  ->  ( A `  x )  =  ( x ( .s `  W ) ( 1r
`  W ) ) )
5029, 49syl 15 . . . 4  |-  ( ( W  e. AssAlg  /\  (
x  e.  ( Base `  F )  /\  y  e.  ( Base `  F
) ) )  -> 
( A `  x
)  =  ( x ( .s `  W
) ( 1r `  W ) ) )
5112, 6, 1, 13, 3asclval 16091 . . . . 5  |-  ( y  e.  ( Base `  F
)  ->  ( A `  y )  =  ( y ( .s `  W ) ( 1r
`  W ) ) )
5232, 51syl 15 . . . 4  |-  ( ( W  e. AssAlg  /\  (
x  e.  ( Base `  F )  /\  y  e.  ( Base `  F
) ) )  -> 
( A `  y
)  =  ( y ( .s `  W
) ( 1r `  W ) ) )
5350, 52oveq12d 5892 . . 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 2338 . 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 16094 . 2  |-  ( W  e. AssAlg  ->  A  e.  ( F  GrpHom  W ) )
561, 2, 3, 4, 5, 9, 10, 22, 54, 55isrhm2d 15522 1  |-  ( W  e. AssAlg  ->  A  e.  ( F RingHom  W ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   ` cfv 5271  (class class class)co 5874   Basecbs 13164   .rcmulr 13225  Scalarcsca 13227   .scvsca 13228   Ringcrg 15353   CRingccrg 15354   1rcur 15355   RingHom crh 15510   LModclmod 15643  AssAlgcasa 16066  algSccascl 16068
This theorem is referenced by:  mplind  16259  evlslem1  19415  mpfind  19444  pf1ind  19454
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-plusg 13237  df-0g 13420  df-mnd 14383  df-mhm 14431  df-grp 14505  df-ghm 14697  df-mgp 15342  df-rng 15356  df-cring 15357  df-ur 15358  df-rnghom 15512  df-lmod 15645  df-assa 16069  df-ascl 16071
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