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Theorem evls1rhm 18923
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 18021 . . . . 5  |-  ( R  e.  (SubRing `  S
)  ->  R  C_  B
)
32adantl 468 . . . 4  |-  ( ( S  e.  CRing  /\  R  e.  (SubRing `  S )
)  ->  R  C_  B
)
4 elpwg 3961 . . . . 5  |-  ( R  e.  (SubRing `  S
)  ->  ( R  e.  ~P B  <->  R  C_  B
) )
54adantl 468 . . . 4  |-  ( ( S  e.  CRing  /\  R  e.  (SubRing `  S )
)  ->  ( R  e.  ~P B  <->  R  C_  B
) )
63, 5mpbird 236 . . 3  |-  ( ( S  e.  CRing  /\  R  e.  (SubRing `  S )
)  ->  R  e.  ~P B )
7 evls1rhm.q . . . 4  |-  Q  =  ( S evalSub1  R )
8 eqid 2453 . . . 4  |-  ( 1o evalSub  S )  =  ( 1o evalSub  S )
97, 8, 1evls1fval 18920 . . 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 473 . 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 2453 . . . . 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 18922 . . . 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 467 . . 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 7194 . . . . 5  |-  1o  e.  On
16 eqid 2453 . . . . . 6  |-  ( ( 1o evalSub  S ) `  R
)  =  ( ( 1o evalSub  S ) `  R
)
17 eqid 2453 . . . . . 6  |-  ( 1o mPoly  U )  =  ( 1o mPoly  U )
18 evls1rhm.u . . . . . 6  |-  U  =  ( Ss  R )
19 eqid 2453 . . . . . 6  |-  ( S  ^s  ( B  ^m  1o ) )  =  ( S  ^s  ( B  ^m  1o ) )
2016, 17, 18, 19, 1evlsrhm 18756 . . . . 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 1353 . . . 4  |-  ( ( S  e.  CRing  /\  R  e.  (SubRing `  S )
)  ->  ( ( 1o evalSub  S ) `  R
)  e.  ( ( 1o mPoly  U ) RingHom  ( S  ^s  ( B  ^m  1o ) ) ) )
22 eqidd 2454 . . . . 5  |-  ( ( S  e.  CRing  /\  R  e.  (SubRing `  S )
)  ->  ( Base `  W )  =  (
Base `  W )
)
23 eqidd 2454 . . . . 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 2453 . . . . . . 7  |-  (PwSer1 `  U
)  =  (PwSer1 `  U
)
26 eqid 2453 . . . . . . 7  |-  ( Base `  W )  =  (
Base `  W )
2724, 25, 26ply1bas 18800 . . . . . 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 2453 . . . . . . . 8  |-  ( +g  `  W )  =  ( +g  `  W )
3024, 17, 29ply1plusg 18830 . . . . . . 7  |-  ( +g  `  W )  =  ( +g  `  ( 1o mPoly  U ) )
3130a1i 11 . . . . . 6  |-  ( ( S  e.  CRing  /\  R  e.  (SubRing `  S )
)  ->  ( +g  `  W )  =  ( +g  `  ( 1o mPoly  U ) ) )
3231oveqdr 6319 . . . . 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 2454 . . . . 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 2453 . . . . . . . 8  |-  ( .r
`  W )  =  ( .r `  W
)
3524, 17, 34ply1mulr 18832 . . . . . . 7  |-  ( .r
`  W )  =  ( .r `  ( 1o mPoly  U ) )
3635a1i 11 . . . . . 6  |-  ( ( S  e.  CRing  /\  R  e.  (SubRing `  S )
)  ->  ( .r `  W )  =  ( .r `  ( 1o mPoly  U ) ) )
3736oveqdr 6319 . . . . 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 2454 . . . . 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 18055 . . . 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 2534 . . 3  |-  ( ( S  e.  CRing  /\  R  e.  (SubRing `  S )
)  ->  ( ( 1o evalSub  S ) `  R
)  e.  ( W RingHom 
( S  ^s  ( B  ^m  1o ) ) ) )
41 rhmco 17977 . . 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 667 . 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 2531 1  |-  ( ( S  e.  CRing  /\  R  e.  (SubRing `  S )
)  ->  Q  e.  ( W RingHom  T ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    = wceq 1446    e. wcel 1889    C_ wss 3406   ~Pcpw 3953   {csn 3970    |-> cmpt 4464    X. cxp 4835    o. ccom 4841   Oncon0 5426   ` cfv 5585  (class class class)co 6295   1oc1o 7180    ^m cmap 7477   Basecbs 15133   ↾s cress 15134   +g cplusg 15202   .rcmulr 15203    ^s cpws 15357   CRingccrg 17793   RingHom crh 17952  SubRingcsubrg 18016   mPoly cmpl 18589   evalSub ces 18739  PwSer1cps1 18780  Poly1cpl1 18782   evalSub1 ces1 18914
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-rep 4518  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588  ax-inf2 8151  ax-cnex 9600  ax-resscn 9601  ax-1cn 9602  ax-icn 9603  ax-addcl 9604  ax-addrcl 9605  ax-mulcl 9606  ax-mulrcl 9607  ax-mulcom 9608  ax-addass 9609  ax-mulass 9610  ax-distr 9611  ax-i2m1 9612  ax-1ne0 9613  ax-1rid 9614  ax-rnegex 9615  ax-rrecex 9616  ax-cnre 9617  ax-pre-lttri 9618  ax-pre-lttrn 9619  ax-pre-ltadd 9620  ax-pre-mulgt0 9621
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-nel 2627  df-ral 2744  df-rex 2745  df-reu 2746  df-rmo 2747  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-tp 3975  df-op 3977  df-uni 4202  df-int 4238  df-iun 4283  df-iin 4284  df-br 4406  df-opab 4465  df-mpt 4466  df-tr 4501  df-eprel 4748  df-id 4752  df-po 4758  df-so 4759  df-fr 4796  df-se 4797  df-we 4798  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-pred 5383  df-ord 5429  df-on 5430  df-lim 5431  df-suc 5432  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-isom 5594  df-riota 6257  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-of 6536  df-ofr 6537  df-om 6698  df-1st 6798  df-2nd 6799  df-supp 6920  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-1o 7187  df-2o 7188  df-oadd 7191  df-er 7368  df-map 7479  df-pm 7480  df-ixp 7528  df-en 7575  df-dom 7576  df-sdom 7577  df-fin 7578  df-fsupp 7889  df-sup 7961  df-oi 8030  df-card 8378  df-pnf 9682  df-mnf 9683  df-xr 9684  df-ltxr 9685  df-le 9686  df-sub 9867  df-neg 9868  df-nn 10617  df-2 10675  df-3 10676  df-4 10677  df-5 10678  df-6 10679  df-7 10680  df-8 10681  df-9 10682  df-10 10683  df-n0 10877  df-z 10945  df-dec 11059  df-uz 11167  df-fz 11792  df-fzo 11923  df-seq 12221  df-hash 12523  df-struct 15135  df-ndx 15136  df-slot 15137  df-base 15138  df-sets 15139  df-ress 15140  df-plusg 15215  df-mulr 15216  df-sca 15218  df-vsca 15219  df-ip 15220  df-tset 15221  df-ple 15222  df-ds 15224  df-hom 15226  df-cco 15227  df-0g 15352  df-gsum 15353  df-prds 15358  df-pws 15360  df-mre 15504  df-mrc 15505  df-acs 15507  df-mgm 16500  df-sgrp 16539  df-mnd 16549  df-mhm 16594  df-submnd 16595  df-grp 16685  df-minusg 16686  df-sbg 16687  df-mulg 16688  df-subg 16826  df-ghm 16893  df-cntz 16983  df-cmn 17444  df-abl 17445  df-mgp 17736  df-ur 17748  df-srg 17752  df-ring 17794  df-cring 17795  df-rnghom 17955  df-subrg 18018  df-lmod 18105  df-lss 18168  df-lsp 18207  df-assa 18548  df-asp 18549  df-ascl 18550  df-psr 18592  df-mvr 18593  df-mpl 18594  df-opsr 18596  df-evls 18741  df-psr1 18785  df-ply1 18787  df-evls1 18916
This theorem is referenced by:  evls1gsumadd  18925  evls1gsummul  18926  evls1varpw  18927
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