MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  evl1gsumadd Structured version   Unicode version

Theorem evl1gsumadd 18262
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 18229. (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 18235 . . . 4  |-  Q  =  ( R evalSub1  K )
43a1i 11 . . 3  |-  ( ph  ->  Q  =  ( R evalSub1  K ) )
54fveq1d 5854 . 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 14564 . . . . . . . . 9  |-  ( R  e.  CRing  ->  ( Rs  K
)  =  R )
97, 8syl 16 . . . . . . . 8  |-  ( ph  ->  ( Rs  K )  =  R )
109eqcomd 2449 . . . . . . 7  |-  ( ph  ->  R  =  ( Rs  K ) )
1110fveq2d 5856 . . . . . 6  |-  ( ph  ->  (Poly1 `  R )  =  (Poly1 `  ( Rs  K ) ) )
126, 11syl5eq 2494 . . . . 5  |-  ( ph  ->  W  =  (Poly1 `  ( Rs  K ) ) )
1312oveq1d 6292 . . . 4  |-  ( ph  ->  ( W  gsumg  ( x  e.  N  |->  Y ) )  =  ( (Poly1 `  ( Rs  K ) )  gsumg  ( x  e.  N  |->  Y ) ) )
1413fveq2d 5856 . . 3  |-  ( ph  ->  ( ( R evalSub1  K ) `
 ( W  gsumg  ( x  e.  N  |->  Y ) ) )  =  ( ( R evalSub1  K ) `  (
(Poly1 `
 ( Rs  K ) )  gsumg  ( x  e.  N  |->  Y ) ) ) )
15 eqid 2441 . . . 4  |-  ( R evalSub1  K )  =  ( R evalSub1  K )
16 eqid 2441 . . . 4  |-  (Poly1 `  ( Rs  K ) )  =  (Poly1 `  ( Rs  K ) )
17 eqid 2441 . . . 4  |-  ( 0g
`  (Poly1 `  ( Rs  K ) ) )  =  ( 0g `  (Poly1 `  ( Rs  K ) ) )
18 eqid 2441 . . . 4  |-  ( Rs  K )  =  ( Rs  K )
19 evl1gsumadd.p . . . 4  |-  P  =  ( R  ^s  K )
20 eqid 2441 . . . 4  |-  ( Base `  (Poly1 `  ( Rs  K ) ) )  =  (
Base `  (Poly1 `  ( Rs  K ) ) )
21 crngring 17077 . . . . 5  |-  ( R  e.  CRing  ->  R  e.  Ring )
222subrgid 17299 . . . . 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 5856 . . . . . 6  |-  ( (
ph  /\  x  e.  N )  ->  ( Base `  W )  =  ( Base `  (Poly1 `  ( Rs  K ) ) ) )
2825, 27syl5eq 2494 . . . . 5  |-  ( (
ph  /\  x  e.  N )  ->  B  =  ( Base `  (Poly1 `  ( Rs  K ) ) ) )
2924, 28eleqtrd 2531 . . . 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 2449 . . . . . . 7  |-  ( ph  ->  (Poly1 `  ( Rs  K ) )  =  W )
3332fveq2d 5856 . . . . . 6  |-  ( ph  ->  ( 0g `  (Poly1 `  ( Rs  K ) ) )  =  ( 0g `  W ) )
34 evl1gsumadd.0 . . . . . 6  |-  .0.  =  ( 0g `  W )
3533, 34syl6eqr 2500 . . . . 5  |-  ( ph  ->  ( 0g `  (Poly1 `  ( Rs  K ) ) )  =  .0.  )
3631, 35breqtrrd 4459 . . . 4  |-  ( ph  ->  ( x  e.  N  |->  Y ) finSupp  ( 0g
`  (Poly1 `  ( Rs  K ) ) ) )
3715, 2, 16, 17, 18, 19, 20, 7, 23, 29, 30, 36evls1gsumadd 18229 . . 3  |-  ( ph  ->  ( ( R evalSub1  K ) `
 ( (Poly1 `  ( Rs  K ) )  gsumg  ( x  e.  N  |->  Y ) ) )  =  ( P  gsumg  ( x  e.  N  |->  ( ( R evalSub1  K ) `
 Y ) ) ) )
3814, 37eqtrd 2482 . 2  |-  ( ph  ->  ( ( R evalSub1  K ) `
 ( W  gsumg  ( x  e.  N  |->  Y ) ) )  =  ( P  gsumg  ( x  e.  N  |->  ( ( R evalSub1  K ) `
 Y ) ) ) )
394fveq1d 5854 . . . . 5  |-  ( ph  ->  ( Q `  Y
)  =  ( ( R evalSub1  K ) `  Y
) )
4039eqcomd 2449 . . . 4  |-  ( ph  ->  ( ( R evalSub1  K ) `
 Y )  =  ( Q `  Y
) )
4140mpteq2dv 4520 . . 3  |-  ( ph  ->  ( x  e.  N  |->  ( ( R evalSub1  K ) `
 Y ) )  =  ( x  e.  N  |->  ( Q `  Y ) ) )
4241oveq2d 6293 . 2  |-  ( ph  ->  ( P  gsumg  ( x  e.  N  |->  ( ( R evalSub1  K ) `
 Y ) ) )  =  ( P 
gsumg  ( x  e.  N  |->  ( Q `  Y
) ) ) )
435, 38, 423eqtrd 2486 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 1381    e. wcel 1802    C_ wss 3458   class class class wbr 4433    |-> cmpt 4491   ` cfv 5574  (class class class)co 6277   finSupp cfsupp 7827   NN0cn0 10796   Basecbs 14504   ↾s cress 14505   0gc0g 14709    gsumg cgsu 14710    ^s cpws 14716   Ringcrg 17066   CRingccrg 17067  SubRingcsubrg 17293  Poly1cpl1 18084   evalSub1 ces1 18218  eval1ce1 18219
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573  ax-inf2 8056  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-pss 3474  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-tp 4015  df-op 4017  df-uni 4231  df-int 4268  df-iun 4313  df-iin 4314  df-br 4434  df-opab 4492  df-mpt 4493  df-tr 4527  df-eprel 4777  df-id 4781  df-po 4786  df-so 4787  df-fr 4824  df-se 4825  df-we 4826  df-ord 4867  df-on 4868  df-lim 4869  df-suc 4870  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-isom 5583  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-of 6521  df-ofr 6522  df-om 6682  df-1st 6781  df-2nd 6782  df-supp 6900  df-recs 7040  df-rdg 7074  df-1o 7128  df-2o 7129  df-oadd 7132  df-er 7309  df-map 7420  df-pm 7421  df-ixp 7468  df-en 7515  df-dom 7516  df-sdom 7517  df-fin 7518  df-fsupp 7828  df-sup 7899  df-oi 7933  df-card 8318  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9807  df-neg 9808  df-nn 10538  df-2 10595  df-3 10596  df-4 10597  df-5 10598  df-6 10599  df-7 10600  df-8 10601  df-9 10602  df-10 10603  df-n0 10797  df-z 10866  df-dec 10980  df-uz 11086  df-fz 11677  df-fzo 11799  df-seq 12082  df-hash 12380  df-struct 14506  df-ndx 14507  df-slot 14508  df-base 14509  df-sets 14510  df-ress 14511  df-plusg 14582  df-mulr 14583  df-sca 14585  df-vsca 14586  df-ip 14587  df-tset 14588  df-ple 14589  df-ds 14591  df-hom 14593  df-cco 14594  df-0g 14711  df-gsum 14712  df-prds 14717  df-pws 14719  df-mre 14855  df-mrc 14856  df-acs 14858  df-mgm 15741  df-sgrp 15780  df-mnd 15790  df-mhm 15835  df-submnd 15836  df-grp 15926  df-minusg 15927  df-sbg 15928  df-mulg 15929  df-subg 16067  df-ghm 16134  df-cntz 16224  df-cmn 16669  df-abl 16670  df-mgp 17010  df-ur 17022  df-srg 17026  df-ring 17068  df-cring 17069  df-rnghom 17232  df-subrg 17295  df-lmod 17382  df-lss 17447  df-lsp 17486  df-assa 17829  df-asp 17830  df-ascl 17831  df-psr 17873  df-mvr 17874  df-mpl 17875  df-opsr 17877  df-evls 18039  df-evl 18040  df-psr1 18087  df-ply1 18089  df-evls1 18220  df-evl1 18221
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator