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Theorem evls1rhm 17868
Description: Polynomial evaluation is a homomorphism (into the product ring). (Contributed by AV, 11-Sep-2019.)
Hypotheses
Ref Expression
evls1rhm.q  |-  Q  =  ( S evalSub1  R )
evls1rhm.b  |-  B  =  ( Base `  S
)
evls1rhm.t  |-  T  =  ( S  ^s  B )
evls1rhm.u  |-  U  =  ( Ss  R )
evls1rhm.w  |-  W  =  (Poly1 `  U )
Assertion
Ref Expression
evls1rhm  |-  ( ( S  e.  CRing  /\  R  e.  (SubRing `  S )
)  ->  Q  e.  ( W RingHom  T ) )

Proof of Theorem evls1rhm
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 evls1rhm.b . . . . . 6  |-  B  =  ( Base `  S
)
21subrgss 16974 . . . . 5  |-  ( R  e.  (SubRing `  S
)  ->  R  C_  B
)
32adantl 466 . . . 4  |-  ( ( S  e.  CRing  /\  R  e.  (SubRing `  S )
)  ->  R  C_  B
)
4 elpwg 3968 . . . . 5  |-  ( R  e.  (SubRing `  S
)  ->  ( R  e.  ~P B  <->  R  C_  B
) )
54adantl 466 . . . 4  |-  ( ( S  e.  CRing  /\  R  e.  (SubRing `  S )
)  ->  ( R  e.  ~P B  <->  R  C_  B
) )
63, 5mpbird 232 . . 3  |-  ( ( S  e.  CRing  /\  R  e.  (SubRing `  S )
)  ->  R  e.  ~P B )
7 evls1rhm.q . . . 4  |-  Q  =  ( S evalSub1  R )
8 eqid 2451 . . . 4  |-  ( 1o evalSub  S )  =  ( 1o evalSub  S )
97, 8, 1evls1fval 17865 . . 3  |-  ( ( S  e.  CRing  /\  R  e.  ~P B )  ->  Q  =  ( (
x  e.  ( B  ^m  ( B  ^m  1o ) )  |->  ( x  o.  ( y  e.  B  |->  ( 1o  X.  { y } ) ) ) )  o.  ( ( 1o evalSub  S ) `
 R ) ) )
106, 9syldan 470 . 2  |-  ( ( S  e.  CRing  /\  R  e.  (SubRing `  S )
)  ->  Q  =  ( ( x  e.  ( B  ^m  ( B  ^m  1o ) ) 
|->  ( x  o.  (
y  e.  B  |->  ( 1o  X.  { y } ) ) ) )  o.  ( ( 1o evalSub  S ) `  R
) ) )
11 evls1rhm.t . . . . 5  |-  T  =  ( S  ^s  B )
12 eqid 2451 . . . . 5  |-  ( x  e.  ( B  ^m  ( B  ^m  1o ) )  |->  ( x  o.  ( y  e.  B  |->  ( 1o  X.  {
y } ) ) ) )  =  ( x  e.  ( B  ^m  ( B  ^m  1o ) )  |->  ( x  o.  ( y  e.  B  |->  ( 1o  X.  { y } ) ) ) )
131, 11, 12evls1rhmlem 17867 . . . 4  |-  ( S  e.  CRing  ->  ( x  e.  ( B  ^m  ( B  ^m  1o ) ) 
|->  ( x  o.  (
y  e.  B  |->  ( 1o  X.  { y } ) ) ) )  e.  ( ( S  ^s  ( B  ^m  1o ) ) RingHom  T ) )
1413adantr 465 . . 3  |-  ( ( S  e.  CRing  /\  R  e.  (SubRing `  S )
)  ->  ( x  e.  ( B  ^m  ( B  ^m  1o ) ) 
|->  ( x  o.  (
y  e.  B  |->  ( 1o  X.  { y } ) ) ) )  e.  ( ( S  ^s  ( B  ^m  1o ) ) RingHom  T ) )
15 1on 7029 . . . . 5  |-  1o  e.  On
16 eqid 2451 . . . . . 6  |-  ( ( 1o evalSub  S ) `  R
)  =  ( ( 1o evalSub  S ) `  R
)
17 eqid 2451 . . . . . 6  |-  ( 1o mPoly  U )  =  ( 1o mPoly  U )
18 evls1rhm.u . . . . . 6  |-  U  =  ( Ss  R )
19 eqid 2451 . . . . . 6  |-  ( S  ^s  ( B  ^m  1o ) )  =  ( S  ^s  ( B  ^m  1o ) )
2016, 17, 18, 19, 1evlsrhm 17716 . . . . 5  |-  ( ( 1o  e.  On  /\  S  e.  CRing  /\  R  e.  (SubRing `  S )
)  ->  ( ( 1o evalSub  S ) `  R
)  e.  ( ( 1o mPoly  U ) RingHom  ( S  ^s  ( B  ^m  1o ) ) ) )
2115, 20mp3an1 1302 . . . 4  |-  ( ( S  e.  CRing  /\  R  e.  (SubRing `  S )
)  ->  ( ( 1o evalSub  S ) `  R
)  e.  ( ( 1o mPoly  U ) RingHom  ( S  ^s  ( B  ^m  1o ) ) ) )
22 eqidd 2452 . . . . 5  |-  ( ( S  e.  CRing  /\  R  e.  (SubRing `  S )
)  ->  ( Base `  W )  =  (
Base `  W )
)
23 eqidd 2452 . . . . 5  |-  ( ( S  e.  CRing  /\  R  e.  (SubRing `  S )
)  ->  ( Base `  ( S  ^s  ( B  ^m  1o ) ) )  =  ( Base `  ( S  ^s  ( B  ^m  1o ) ) ) )
24 evls1rhm.w . . . . . . 7  |-  W  =  (Poly1 `  U )
25 eqid 2451 . . . . . . 7  |-  (PwSer1 `  U
)  =  (PwSer1 `  U
)
26 eqid 2451 . . . . . . 7  |-  ( Base `  W )  =  (
Base `  W )
2724, 25, 26ply1bas 17760 . . . . . 6  |-  ( Base `  W )  =  (
Base `  ( 1o mPoly  U ) )
2827a1i 11 . . . . 5  |-  ( ( S  e.  CRing  /\  R  e.  (SubRing `  S )
)  ->  ( Base `  W )  =  (
Base `  ( 1o mPoly  U ) ) )
29 eqid 2451 . . . . . . . 8  |-  ( +g  `  W )  =  ( +g  `  W )
3024, 17, 29ply1plusg 17788 . . . . . . 7  |-  ( +g  `  W )  =  ( +g  `  ( 1o mPoly  U ) )
3130a1i 11 . . . . . 6  |-  ( ( S  e.  CRing  /\  R  e.  (SubRing `  S )
)  ->  ( +g  `  W )  =  ( +g  `  ( 1o mPoly  U ) ) )
3231proplem3 14733 . . . . 5  |-  ( ( ( S  e.  CRing  /\  R  e.  (SubRing `  S
) )  /\  (
x  e.  ( Base `  W )  /\  y  e.  ( Base `  W
) ) )  -> 
( x ( +g  `  W ) y )  =  ( x ( +g  `  ( 1o mPoly  U ) ) y ) )
33 eqidd 2452 . . . . 5  |-  ( ( ( S  e.  CRing  /\  R  e.  (SubRing `  S
) )  /\  (
x  e.  ( Base `  ( S  ^s  ( B  ^m  1o ) ) )  /\  y  e.  ( Base `  ( S  ^s  ( B  ^m  1o ) ) ) ) )  ->  ( x
( +g  `  ( S  ^s  ( B  ^m  1o ) ) ) y )  =  ( x ( +g  `  ( S  ^s  ( B  ^m  1o ) ) ) y ) )
34 eqid 2451 . . . . . . . 8  |-  ( .r
`  W )  =  ( .r `  W
)
3524, 17, 34ply1mulr 17790 . . . . . . 7  |-  ( .r
`  W )  =  ( .r `  ( 1o mPoly  U ) )
3635a1i 11 . . . . . 6  |-  ( ( S  e.  CRing  /\  R  e.  (SubRing `  S )
)  ->  ( .r `  W )  =  ( .r `  ( 1o mPoly  U ) ) )
3736proplem3 14733 . . . . 5  |-  ( ( ( S  e.  CRing  /\  R  e.  (SubRing `  S
) )  /\  (
x  e.  ( Base `  W )  /\  y  e.  ( Base `  W
) ) )  -> 
( x ( .r
`  W ) y )  =  ( x ( .r `  ( 1o mPoly  U ) ) y ) )
38 eqidd 2452 . . . . 5  |-  ( ( ( S  e.  CRing  /\  R  e.  (SubRing `  S
) )  /\  (
x  e.  ( Base `  ( S  ^s  ( B  ^m  1o ) ) )  /\  y  e.  ( Base `  ( S  ^s  ( B  ^m  1o ) ) ) ) )  ->  ( x
( .r `  ( S  ^s  ( B  ^m  1o ) ) ) y )  =  ( x ( .r `  ( S  ^s  ( B  ^m  1o ) ) ) y ) )
3922, 23, 28, 23, 32, 33, 37, 38rhmpropd 17008 . . . 4  |-  ( ( S  e.  CRing  /\  R  e.  (SubRing `  S )
)  ->  ( W RingHom  ( S  ^s  ( B  ^m  1o ) ) )  =  ( ( 1o mPoly  U
) RingHom  ( S  ^s  ( B  ^m  1o ) ) ) )
4021, 39eleqtrrd 2542 . . 3  |-  ( ( S  e.  CRing  /\  R  e.  (SubRing `  S )
)  ->  ( ( 1o evalSub  S ) `  R
)  e.  ( W RingHom 
( S  ^s  ( B  ^m  1o ) ) ) )
41 rhmco 16933 . . 3  |-  ( ( ( x  e.  ( B  ^m  ( B  ^m  1o ) ) 
|->  ( x  o.  (
y  e.  B  |->  ( 1o  X.  { y } ) ) ) )  e.  ( ( S  ^s  ( B  ^m  1o ) ) RingHom  T )  /\  ( ( 1o evalSub  S ) `
 R )  e.  ( W RingHom  ( S  ^s  ( B  ^m  1o ) ) ) )  -> 
( ( x  e.  ( B  ^m  ( B  ^m  1o ) ) 
|->  ( x  o.  (
y  e.  B  |->  ( 1o  X.  { y } ) ) ) )  o.  ( ( 1o evalSub  S ) `  R
) )  e.  ( W RingHom  T ) )
4214, 40, 41syl2anc 661 . 2  |-  ( ( S  e.  CRing  /\  R  e.  (SubRing `  S )
)  ->  ( (
x  e.  ( B  ^m  ( B  ^m  1o ) )  |->  ( x  o.  ( y  e.  B  |->  ( 1o  X.  { y } ) ) ) )  o.  ( ( 1o evalSub  S ) `
 R ) )  e.  ( W RingHom  T
) )
4310, 42eqeltrd 2539 1  |-  ( ( S  e.  CRing  /\  R  e.  (SubRing `  S )
)  ->  Q  e.  ( W RingHom  T ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758    C_ wss 3428   ~Pcpw 3960   {csn 3977    |-> cmpt 4450   Oncon0 4819    X. cxp 4938    o. ccom 4944   ` cfv 5518  (class class class)co 6192   1oc1o 7015    ^m cmap 7316   Basecbs 14278   ↾s cress 14279   +g cplusg 14342   .rcmulr 14343    ^s cpws 14489   CRingccrg 16754   RingHom crh 16912  SubRingcsubrg 16969   mPoly cmpl 17528   evalSub ces 17695  PwSer1cps1 17740  Poly1cpl1 17742   evalSub1 ces1 17859
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4503  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474  ax-inf2 7950  ax-cnex 9441  ax-resscn 9442  ax-1cn 9443  ax-icn 9444  ax-addcl 9445  ax-addrcl 9446  ax-mulcl 9447  ax-mulrcl 9448  ax-mulcom 9449  ax-addass 9450  ax-mulass 9451  ax-distr 9452  ax-i2m1 9453  ax-1ne0 9454  ax-1rid 9455  ax-rnegex 9456  ax-rrecex 9457  ax-cnre 9458  ax-pre-lttri 9459  ax-pre-lttrn 9460  ax-pre-ltadd 9461  ax-pre-mulgt0 9462
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-pss 3444  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-tp 3982  df-op 3984  df-uni 4192  df-int 4229  df-iun 4273  df-iin 4274  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4486  df-eprel 4732  df-id 4736  df-po 4741  df-so 4742  df-fr 4779  df-se 4780  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-isom 5527  df-riota 6153  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-of 6422  df-ofr 6423  df-om 6579  df-1st 6679  df-2nd 6680  df-supp 6793  df-recs 6934  df-rdg 6968  df-1o 7022  df-2o 7023  df-oadd 7026  df-er 7203  df-map 7318  df-pm 7319  df-ixp 7366  df-en 7413  df-dom 7414  df-sdom 7415  df-fin 7416  df-fsupp 7724  df-sup 7794  df-oi 7827  df-card 8212  df-pnf 9523  df-mnf 9524  df-xr 9525  df-ltxr 9526  df-le 9527  df-sub 9700  df-neg 9701  df-nn 10426  df-2 10483  df-3 10484  df-4 10485  df-5 10486  df-6 10487  df-7 10488  df-8 10489  df-9 10490  df-10 10491  df-n0 10683  df-z 10750  df-dec 10859  df-uz 10965  df-fz 11541  df-fzo 11652  df-seq 11910  df-hash 12207  df-struct 14280  df-ndx 14281  df-slot 14282  df-base 14283  df-sets 14284  df-ress 14285  df-plusg 14355  df-mulr 14356  df-sca 14358  df-vsca 14359  df-ip 14360  df-tset 14361  df-ple 14362  df-ds 14364  df-hom 14366  df-cco 14367  df-0g 14484  df-gsum 14485  df-prds 14490  df-pws 14492  df-mre 14628  df-mrc 14629  df-acs 14631  df-mnd 15519  df-mhm 15568  df-submnd 15569  df-grp 15649  df-minusg 15650  df-sbg 15651  df-mulg 15652  df-subg 15782  df-ghm 15849  df-cntz 15939  df-cmn 16385  df-abl 16386  df-mgp 16699  df-ur 16711  df-srg 16715  df-rng 16755  df-cring 16756  df-rnghom 16914  df-subrg 16971  df-lmod 17058  df-lss 17122  df-lsp 17161  df-assa 17492  df-asp 17493  df-ascl 17494  df-psr 17531  df-mvr 17532  df-mpl 17533  df-opsr 17535  df-evls 17697  df-psr1 17745  df-ply1 17747  df-evls1 17861
This theorem is referenced by:  evls1gsumadd  17870  evls1gsummul  17871  evls1varpw  17872
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