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Theorem evl1gsumadd 17791
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 17758. (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 17764 . . . 4  |-  Q  =  ( R evalSub1  K )
43a1i 11 . . 3  |-  ( ph  ->  Q  =  ( R evalSub1  K ) )
54fveq1d 5692 . 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 14232 . . . . . . . . 9  |-  ( R  e.  CRing  ->  ( Rs  K
)  =  R )
97, 8syl 16 . . . . . . . 8  |-  ( ph  ->  ( Rs  K )  =  R )
109eqcomd 2447 . . . . . . 7  |-  ( ph  ->  R  =  ( Rs  K ) )
1110fveq2d 5694 . . . . . 6  |-  ( ph  ->  (Poly1 `  R )  =  (Poly1 `  ( Rs  K ) ) )
126, 11syl5eq 2486 . . . . 5  |-  ( ph  ->  W  =  (Poly1 `  ( Rs  K ) ) )
1312oveq1d 6105 . . . 4  |-  ( ph  ->  ( W  gsumg  ( x  e.  N  |->  Y ) )  =  ( (Poly1 `  ( Rs  K ) )  gsumg  ( x  e.  N  |->  Y ) ) )
1413fveq2d 5694 . . 3  |-  ( ph  ->  ( ( R evalSub1  K ) `
 ( W  gsumg  ( x  e.  N  |->  Y ) ) )  =  ( ( R evalSub1  K ) `  (
(Poly1 `
 ( Rs  K ) )  gsumg  ( x  e.  N  |->  Y ) ) ) )
15 eqid 2442 . . . 4  |-  ( R evalSub1  K )  =  ( R evalSub1  K )
16 eqid 2442 . . . 4  |-  (Poly1 `  ( Rs  K ) )  =  (Poly1 `  ( Rs  K ) )
17 eqid 2442 . . . 4  |-  ( 0g
`  (Poly1 `  ( Rs  K ) ) )  =  ( 0g `  (Poly1 `  ( Rs  K ) ) )
18 eqid 2442 . . . 4  |-  ( Rs  K )  =  ( Rs  K )
19 evl1gsumadd.p . . . 4  |-  P  =  ( R  ^s  K )
20 eqid 2442 . . . 4  |-  ( Base `  (Poly1 `  ( Rs  K ) ) )  =  (
Base `  (Poly1 `  ( Rs  K ) ) )
21 crngrng 16654 . . . . 5  |-  ( R  e.  CRing  ->  R  e.  Ring )
222subrgid 16866 . . . . 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 5694 . . . . . 6  |-  ( (
ph  /\  x  e.  N )  ->  ( Base `  W )  =  ( Base `  (Poly1 `  ( Rs  K ) ) ) )
2825, 27syl5eq 2486 . . . . 5  |-  ( (
ph  /\  x  e.  N )  ->  B  =  ( Base `  (Poly1 `  ( Rs  K ) ) ) )
2924, 28eleqtrd 2518 . . . 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 2447 . . . . . . 7  |-  ( ph  ->  (Poly1 `  ( Rs  K ) )  =  W )
3332fveq2d 5694 . . . . . 6  |-  ( ph  ->  ( 0g `  (Poly1 `  ( Rs  K ) ) )  =  ( 0g `  W ) )
34 evl1gsumadd.0 . . . . . 6  |-  .0.  =  ( 0g `  W )
3533, 34syl6eqr 2492 . . . . 5  |-  ( ph  ->  ( 0g `  (Poly1 `  ( Rs  K ) ) )  =  .0.  )
3631, 35breqtrrd 4317 . . . 4  |-  ( ph  ->  ( x  e.  N  |->  Y ) finSupp  ( 0g
`  (Poly1 `  ( Rs  K ) ) ) )
3715, 2, 16, 17, 18, 19, 20, 7, 23, 29, 30, 36evls1gsumadd 17758 . . 3  |-  ( ph  ->  ( ( R evalSub1  K ) `
 ( (Poly1 `  ( Rs  K ) )  gsumg  ( x  e.  N  |->  Y ) ) )  =  ( P  gsumg  ( x  e.  N  |->  ( ( R evalSub1  K ) `
 Y ) ) ) )
3814, 37eqtrd 2474 . 2  |-  ( ph  ->  ( ( R evalSub1  K ) `
 ( W  gsumg  ( x  e.  N  |->  Y ) ) )  =  ( P  gsumg  ( x  e.  N  |->  ( ( R evalSub1  K ) `
 Y ) ) ) )
394fveq1d 5692 . . . . 5  |-  ( ph  ->  ( Q `  Y
)  =  ( ( R evalSub1  K ) `  Y
) )
4039eqcomd 2447 . . . 4  |-  ( ph  ->  ( ( R evalSub1  K ) `
 Y )  =  ( Q `  Y
) )
4140mpteq2dv 4378 . . 3  |-  ( ph  ->  ( x  e.  N  |->  ( ( R evalSub1  K ) `
 Y ) )  =  ( x  e.  N  |->  ( Q `  Y ) ) )
4241oveq2d 6106 . 2  |-  ( ph  ->  ( P  gsumg  ( x  e.  N  |->  ( ( R evalSub1  K ) `
 Y ) ) )  =  ( P 
gsumg  ( x  e.  N  |->  ( Q `  Y
) ) ) )
435, 38, 423eqtrd 2478 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 1369    e. wcel 1756    C_ wss 3327   class class class wbr 4291    e. cmpt 4349   ` cfv 5417  (class class class)co 6090   finSupp cfsupp 7619   NN0cn0 10578   Basecbs 14173   ↾s cress 14174   0gc0g 14377    gsumg cgsu 14378    ^s cpws 14384   Ringcrg 16644   CRingccrg 16645  SubRingcsubrg 16860  Poly1cpl1 17632   evalSub1 ces1 17747  eval1ce1 17748
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4402  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371  ax-inf2 7846  ax-cnex 9337  ax-resscn 9338  ax-1cn 9339  ax-icn 9340  ax-addcl 9341  ax-addrcl 9342  ax-mulcl 9343  ax-mulrcl 9344  ax-mulcom 9345  ax-addass 9346  ax-mulass 9347  ax-distr 9348  ax-i2m1 9349  ax-1ne0 9350  ax-1rid 9351  ax-rnegex 9352  ax-rrecex 9353  ax-cnre 9354  ax-pre-lttri 9355  ax-pre-lttrn 9356  ax-pre-ltadd 9357  ax-pre-mulgt0 9358
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2719  df-rex 2720  df-reu 2721  df-rmo 2722  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-pss 3343  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-tp 3881  df-op 3883  df-uni 4091  df-int 4128  df-iun 4172  df-iin 4173  df-br 4292  df-opab 4350  df-mpt 4351  df-tr 4385  df-eprel 4631  df-id 4635  df-po 4640  df-so 4641  df-fr 4678  df-se 4679  df-we 4680  df-ord 4721  df-on 4722  df-lim 4723  df-suc 4724  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-isom 5426  df-riota 6051  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-of 6319  df-ofr 6320  df-om 6476  df-1st 6576  df-2nd 6577  df-supp 6690  df-recs 6831  df-rdg 6865  df-1o 6919  df-2o 6920  df-oadd 6923  df-er 7100  df-map 7215  df-pm 7216  df-ixp 7263  df-en 7310  df-dom 7311  df-sdom 7312  df-fin 7313  df-fsupp 7620  df-sup 7690  df-oi 7723  df-card 8108  df-pnf 9419  df-mnf 9420  df-xr 9421  df-ltxr 9422  df-le 9423  df-sub 9596  df-neg 9597  df-nn 10322  df-2 10379  df-3 10380  df-4 10381  df-5 10382  df-6 10383  df-7 10384  df-8 10385  df-9 10386  df-10 10387  df-n0 10579  df-z 10646  df-dec 10755  df-uz 10861  df-fz 11437  df-fzo 11548  df-seq 11806  df-hash 12103  df-struct 14175  df-ndx 14176  df-slot 14177  df-base 14178  df-sets 14179  df-ress 14180  df-plusg 14250  df-mulr 14251  df-sca 14253  df-vsca 14254  df-ip 14255  df-tset 14256  df-ple 14257  df-ds 14259  df-hom 14261  df-cco 14262  df-0g 14379  df-gsum 14380  df-prds 14385  df-pws 14387  df-mre 14523  df-mrc 14524  df-acs 14526  df-mnd 15414  df-mhm 15463  df-submnd 15464  df-grp 15544  df-minusg 15545  df-sbg 15546  df-mulg 15547  df-subg 15677  df-ghm 15744  df-cntz 15834  df-cmn 16278  df-abl 16279  df-mgp 16591  df-ur 16603  df-srg 16607  df-rng 16646  df-cring 16647  df-rnghom 16805  df-subrg 16862  df-lmod 16949  df-lss 17013  df-lsp 17052  df-assa 17383  df-asp 17384  df-ascl 17385  df-psr 17422  df-mvr 17423  df-mpl 17424  df-opsr 17426  df-evls 17587  df-evl 17588  df-psr1 17635  df-ply1 17637  df-evls1 17749  df-evl1 17750
This theorem is referenced by: (None)
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