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Theorem evl1gsumadd 18264
Description: Univariate polynomial evaluation maps (additive) group sums to group sums. Remark: the proof would be shorter if the theorem is proved directly instead of using evls1gsumadd 18231. (Contributed by AV, 15-Sep-2019.)
Hypotheses
Ref Expression
evl1gsumadd.q  |-  Q  =  (eval1 `  R )
evl1gsumadd.k  |-  K  =  ( Base `  R
)
evl1gsumadd.w  |-  W  =  (Poly1 `  R )
evl1gsumadd.p  |-  P  =  ( R  ^s  K )
evl1gsumadd.b  |-  B  =  ( Base `  W
)
evl1gsumadd.r  |-  ( ph  ->  R  e.  CRing )
evl1gsumadd.y  |-  ( (
ph  /\  x  e.  N )  ->  Y  e.  B )
evl1gsumadd.n  |-  ( ph  ->  N  C_  NN0 )
evl1gsumadd.0  |-  .0.  =  ( 0g `  W )
evl1gsumadd.f  |-  ( ph  ->  ( x  e.  N  |->  Y ) finSupp  .0.  )
Assertion
Ref Expression
evl1gsumadd  |-  ( ph  ->  ( Q `  ( W  gsumg  ( x  e.  N  |->  Y ) ) )  =  ( P  gsumg  ( x  e.  N  |->  ( Q `
 Y ) ) ) )
Distinct variable groups:    x, B    x, K    x, N    x, Q    x, R    ph, x
Allowed substitution hints:    P( x)    W( x)    Y( x)    .0. ( x)

Proof of Theorem evl1gsumadd
StepHypRef Expression
1 evl1gsumadd.q . . . . 5  |-  Q  =  (eval1 `  R )
2 evl1gsumadd.k . . . . 5  |-  K  =  ( Base `  R
)
31, 2evl1fval1 18237 . . . 4  |-  Q  =  ( R evalSub1  K )
43a1i 11 . . 3  |-  ( ph  ->  Q  =  ( R evalSub1  K ) )
54fveq1d 5874 . 2  |-  ( ph  ->  ( Q `  ( W  gsumg  ( x  e.  N  |->  Y ) ) )  =  ( ( R evalSub1  K ) `  ( W 
gsumg  ( x  e.  N  |->  Y ) ) ) )
6 evl1gsumadd.w . . . . . 6  |-  W  =  (Poly1 `  R )
7 evl1gsumadd.r . . . . . . . . 9  |-  ( ph  ->  R  e.  CRing )
82ressid 14567 . . . . . . . . 9  |-  ( R  e.  CRing  ->  ( Rs  K
)  =  R )
97, 8syl 16 . . . . . . . 8  |-  ( ph  ->  ( Rs  K )  =  R )
109eqcomd 2475 . . . . . . 7  |-  ( ph  ->  R  =  ( Rs  K ) )
1110fveq2d 5876 . . . . . 6  |-  ( ph  ->  (Poly1 `  R )  =  (Poly1 `  ( Rs  K ) ) )
126, 11syl5eq 2520 . . . . 5  |-  ( ph  ->  W  =  (Poly1 `  ( Rs  K ) ) )
1312oveq1d 6310 . . . 4  |-  ( ph  ->  ( W  gsumg  ( x  e.  N  |->  Y ) )  =  ( (Poly1 `  ( Rs  K ) )  gsumg  ( x  e.  N  |->  Y ) ) )
1413fveq2d 5876 . . 3  |-  ( ph  ->  ( ( R evalSub1  K ) `
 ( W  gsumg  ( x  e.  N  |->  Y ) ) )  =  ( ( R evalSub1  K ) `  (
(Poly1 `
 ( Rs  K ) )  gsumg  ( x  e.  N  |->  Y ) ) ) )
15 eqid 2467 . . . 4  |-  ( R evalSub1  K )  =  ( R evalSub1  K )
16 eqid 2467 . . . 4  |-  (Poly1 `  ( Rs  K ) )  =  (Poly1 `  ( Rs  K ) )
17 eqid 2467 . . . 4  |-  ( 0g
`  (Poly1 `  ( Rs  K ) ) )  =  ( 0g `  (Poly1 `  ( Rs  K ) ) )
18 eqid 2467 . . . 4  |-  ( Rs  K )  =  ( Rs  K )
19 evl1gsumadd.p . . . 4  |-  P  =  ( R  ^s  K )
20 eqid 2467 . . . 4  |-  ( Base `  (Poly1 `  ( Rs  K ) ) )  =  (
Base `  (Poly1 `  ( Rs  K ) ) )
21 crngring 17081 . . . . 5  |-  ( R  e.  CRing  ->  R  e.  Ring )
222subrgid 17302 . . . . 5  |-  ( R  e.  Ring  ->  K  e.  (SubRing `  R )
)
237, 21, 223syl 20 . . . 4  |-  ( ph  ->  K  e.  (SubRing `  R
) )
24 evl1gsumadd.y . . . . 5  |-  ( (
ph  /\  x  e.  N )  ->  Y  e.  B )
25 evl1gsumadd.b . . . . . 6  |-  B  =  ( Base `  W
)
2612adantr 465 . . . . . . 7  |-  ( (
ph  /\  x  e.  N )  ->  W  =  (Poly1 `  ( Rs  K ) ) )
2726fveq2d 5876 . . . . . 6  |-  ( (
ph  /\  x  e.  N )  ->  ( Base `  W )  =  ( Base `  (Poly1 `  ( Rs  K ) ) ) )
2825, 27syl5eq 2520 . . . . 5  |-  ( (
ph  /\  x  e.  N )  ->  B  =  ( Base `  (Poly1 `  ( Rs  K ) ) ) )
2924, 28eleqtrd 2557 . . . 4  |-  ( (
ph  /\  x  e.  N )  ->  Y  e.  ( Base `  (Poly1 `  ( Rs  K ) ) ) )
30 evl1gsumadd.n . . . 4  |-  ( ph  ->  N  C_  NN0 )
31 evl1gsumadd.f . . . . 5  |-  ( ph  ->  ( x  e.  N  |->  Y ) finSupp  .0.  )
3212eqcomd 2475 . . . . . . 7  |-  ( ph  ->  (Poly1 `  ( Rs  K ) )  =  W )
3332fveq2d 5876 . . . . . 6  |-  ( ph  ->  ( 0g `  (Poly1 `  ( Rs  K ) ) )  =  ( 0g `  W ) )
34 evl1gsumadd.0 . . . . . 6  |-  .0.  =  ( 0g `  W )
3533, 34syl6eqr 2526 . . . . 5  |-  ( ph  ->  ( 0g `  (Poly1 `  ( Rs  K ) ) )  =  .0.  )
3631, 35breqtrrd 4479 . . . 4  |-  ( ph  ->  ( x  e.  N  |->  Y ) finSupp  ( 0g
`  (Poly1 `  ( Rs  K ) ) ) )
3715, 2, 16, 17, 18, 19, 20, 7, 23, 29, 30, 36evls1gsumadd 18231 . . 3  |-  ( ph  ->  ( ( R evalSub1  K ) `
 ( (Poly1 `  ( Rs  K ) )  gsumg  ( x  e.  N  |->  Y ) ) )  =  ( P  gsumg  ( x  e.  N  |->  ( ( R evalSub1  K ) `
 Y ) ) ) )
3814, 37eqtrd 2508 . 2  |-  ( ph  ->  ( ( R evalSub1  K ) `
 ( W  gsumg  ( x  e.  N  |->  Y ) ) )  =  ( P  gsumg  ( x  e.  N  |->  ( ( R evalSub1  K ) `
 Y ) ) ) )
394fveq1d 5874 . . . . 5  |-  ( ph  ->  ( Q `  Y
)  =  ( ( R evalSub1  K ) `  Y
) )
4039eqcomd 2475 . . . 4  |-  ( ph  ->  ( ( R evalSub1  K ) `
 Y )  =  ( Q `  Y
) )
4140mpteq2dv 4540 . . 3  |-  ( ph  ->  ( x  e.  N  |->  ( ( R evalSub1  K ) `
 Y ) )  =  ( x  e.  N  |->  ( Q `  Y ) ) )
4241oveq2d 6311 . 2  |-  ( ph  ->  ( P  gsumg  ( x  e.  N  |->  ( ( R evalSub1  K ) `
 Y ) ) )  =  ( P 
gsumg  ( x  e.  N  |->  ( Q `  Y
) ) ) )
435, 38, 423eqtrd 2512 1  |-  ( ph  ->  ( Q `  ( W  gsumg  ( x  e.  N  |->  Y ) ) )  =  ( P  gsumg  ( x  e.  N  |->  ( Q `
 Y ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767    C_ wss 3481   class class class wbr 4453    |-> cmpt 4511   ` cfv 5594  (class class class)co 6295   finSupp cfsupp 7841   NN0cn0 10807   Basecbs 14507   ↾s cress 14508   0gc0g 14712    gsumg cgsu 14713    ^s cpws 14719   Ringcrg 17070   CRingccrg 17071  SubRingcsubrg 17296  Poly1cpl1 18086   evalSub1 ces1 18220  eval1ce1 18221
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-inf2 8070  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-iin 4334  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-se 4845  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  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 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-of 6535  df-ofr 6536  df-om 6696  df-1st 6795  df-2nd 6796  df-supp 6914  df-recs 7054  df-rdg 7088  df-1o 7142  df-2o 7143  df-oadd 7146  df-er 7323  df-map 7434  df-pm 7435  df-ixp 7482  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-fsupp 7842  df-sup 7913  df-oi 7947  df-card 8332  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-2 10606  df-3 10607  df-4 10608  df-5 10609  df-6 10610  df-7 10611  df-8 10612  df-9 10613  df-10 10614  df-n0 10808  df-z 10877  df-dec 10989  df-uz 11095  df-fz 11685  df-fzo 11805  df-seq 12088  df-hash 12386  df-struct 14509  df-ndx 14510  df-slot 14511  df-base 14512  df-sets 14513  df-ress 14514  df-plusg 14585  df-mulr 14586  df-sca 14588  df-vsca 14589  df-ip 14590  df-tset 14591  df-ple 14592  df-ds 14594  df-hom 14596  df-cco 14597  df-0g 14714  df-gsum 14715  df-prds 14720  df-pws 14722  df-mre 14858  df-mrc 14859  df-acs 14861  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-mhm 15839  df-submnd 15840  df-grp 15929  df-minusg 15930  df-sbg 15931  df-mulg 15932  df-subg 16070  df-ghm 16137  df-cntz 16227  df-cmn 16673  df-abl 16674  df-mgp 17014  df-ur 17026  df-srg 17030  df-ring 17072  df-cring 17073  df-rnghom 17236  df-subrg 17298  df-lmod 17385  df-lss 17450  df-lsp 17489  df-assa 17831  df-asp 17832  df-ascl 17833  df-psr 17875  df-mvr 17876  df-mpl 17877  df-opsr 17879  df-evls 18041  df-evl 18042  df-psr1 18089  df-ply1 18091  df-evls1 18222  df-evl1 18223
This theorem is referenced by: (None)
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