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Theorem evls1rhm 18486
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 17557 . . . . 5  |-  ( R  e.  (SubRing `  S
)  ->  R  C_  B
)
32adantl 466 . . . 4  |-  ( ( S  e.  CRing  /\  R  e.  (SubRing `  S )
)  ->  R  C_  B
)
4 elpwg 4023 . . . . 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 2457 . . . 4  |-  ( 1o evalSub  S )  =  ( 1o evalSub  S )
97, 8, 1evls1fval 18483 . . 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 2457 . . . . 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 18485 . . . 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 7155 . . . . 5  |-  1o  e.  On
16 eqid 2457 . . . . . 6  |-  ( ( 1o evalSub  S ) `  R
)  =  ( ( 1o evalSub  S ) `  R
)
17 eqid 2457 . . . . . 6  |-  ( 1o mPoly  U )  =  ( 1o mPoly  U )
18 evls1rhm.u . . . . . 6  |-  U  =  ( Ss  R )
19 eqid 2457 . . . . . 6  |-  ( S  ^s  ( B  ^m  1o ) )  =  ( S  ^s  ( B  ^m  1o ) )
2016, 17, 18, 19, 1evlsrhm 18317 . . . . 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 1311 . . . 4  |-  ( ( S  e.  CRing  /\  R  e.  (SubRing `  S )
)  ->  ( ( 1o evalSub  S ) `  R
)  e.  ( ( 1o mPoly  U ) RingHom  ( S  ^s  ( B  ^m  1o ) ) ) )
22 eqidd 2458 . . . . 5  |-  ( ( S  e.  CRing  /\  R  e.  (SubRing `  S )
)  ->  ( Base `  W )  =  (
Base `  W )
)
23 eqidd 2458 . . . . 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 2457 . . . . . . 7  |-  (PwSer1 `  U
)  =  (PwSer1 `  U
)
26 eqid 2457 . . . . . . 7  |-  ( Base `  W )  =  (
Base `  W )
2724, 25, 26ply1bas 18361 . . . . . 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 2457 . . . . . . . 8  |-  ( +g  `  W )  =  ( +g  `  W )
3024, 17, 29ply1plusg 18393 . . . . . . 7  |-  ( +g  `  W )  =  ( +g  `  ( 1o mPoly  U ) )
3130a1i 11 . . . . . 6  |-  ( ( S  e.  CRing  /\  R  e.  (SubRing `  S )
)  ->  ( +g  `  W )  =  ( +g  `  ( 1o mPoly  U ) ) )
3231oveqdr 6320 . . . . 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 2458 . . . . 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 2457 . . . . . . . 8  |-  ( .r
`  W )  =  ( .r `  W
)
3524, 17, 34ply1mulr 18395 . . . . . . 7  |-  ( .r
`  W )  =  ( .r `  ( 1o mPoly  U ) )
3635a1i 11 . . . . . 6  |-  ( ( S  e.  CRing  /\  R  e.  (SubRing `  S )
)  ->  ( .r `  W )  =  ( .r `  ( 1o mPoly  U ) ) )
3736oveqdr 6320 . . . . 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 2458 . . . . 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 17591 . . . 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 2548 . . 3  |-  ( ( S  e.  CRing  /\  R  e.  (SubRing `  S )
)  ->  ( ( 1o evalSub  S ) `  R
)  e.  ( W RingHom 
( S  ^s  ( B  ^m  1o ) ) ) )
41 rhmco 17513 . . 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 2545 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 1395    e. wcel 1819    C_ wss 3471   ~Pcpw 4015   {csn 4032    |-> cmpt 4515   Oncon0 4887    X. cxp 5006    o. ccom 5012   ` cfv 5594  (class class class)co 6296   1oc1o 7141    ^m cmap 7438   Basecbs 14644   ↾s cress 14645   +g cplusg 14712   .rcmulr 14713    ^s cpws 14864   CRingccrg 17326   RingHom crh 17488  SubRingcsubrg 17552   mPoly cmpl 18129   evalSub ces 18296  PwSer1cps1 18341  Poly1cpl1 18343   evalSub1 ces1 18477
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-inf2 8075  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-iin 4335  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-se 4848  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-of 6539  df-ofr 6540  df-om 6700  df-1st 6799  df-2nd 6800  df-supp 6918  df-recs 7060  df-rdg 7094  df-1o 7148  df-2o 7149  df-oadd 7152  df-er 7329  df-map 7440  df-pm 7441  df-ixp 7489  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-fsupp 7848  df-sup 7919  df-oi 7953  df-card 8337  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-3 10616  df-4 10617  df-5 10618  df-6 10619  df-7 10620  df-8 10621  df-9 10622  df-10 10623  df-n0 10817  df-z 10886  df-dec 11001  df-uz 11107  df-fz 11698  df-fzo 11822  df-seq 12111  df-hash 12409  df-struct 14646  df-ndx 14647  df-slot 14648  df-base 14649  df-sets 14650  df-ress 14651  df-plusg 14725  df-mulr 14726  df-sca 14728  df-vsca 14729  df-ip 14730  df-tset 14731  df-ple 14732  df-ds 14734  df-hom 14736  df-cco 14737  df-0g 14859  df-gsum 14860  df-prds 14865  df-pws 14867  df-mre 15003  df-mrc 15004  df-acs 15006  df-mgm 15999  df-sgrp 16038  df-mnd 16048  df-mhm 16093  df-submnd 16094  df-grp 16184  df-minusg 16185  df-sbg 16186  df-mulg 16187  df-subg 16325  df-ghm 16392  df-cntz 16482  df-cmn 16927  df-abl 16928  df-mgp 17269  df-ur 17281  df-srg 17285  df-ring 17327  df-cring 17328  df-rnghom 17491  df-subrg 17554  df-lmod 17641  df-lss 17706  df-lsp 17745  df-assa 18088  df-asp 18089  df-ascl 18090  df-psr 18132  df-mvr 18133  df-mpl 18134  df-opsr 18136  df-evls 18298  df-psr1 18346  df-ply1 18348  df-evls1 18479
This theorem is referenced by:  evls1gsumadd  18488  evls1gsummul  18489  evls1varpw  18490
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