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Theorem gsumply1subr 17781
Description: Evaluate a group sum in a polynomial ring over a subring. (Contributed by AV, 22-Sep-2019.)
Hypotheses
Ref Expression
subrgply1.s  |-  S  =  (Poly1 `  R )
subrgply1.h  |-  H  =  ( Rs  T )
subrgply1.u  |-  U  =  (Poly1 `  H )
subrgply1.b  |-  B  =  ( Base `  U
)
gsumply1subr.s  |-  ( ph  ->  T  e.  (SubRing `  R
) )
gsumply1subr.a  |-  ( ph  ->  A  e.  V )
gsumply1subr.f  |-  ( ph  ->  F : A --> B )
Assertion
Ref Expression
gsumply1subr  |-  ( ph  ->  ( S  gsumg  F )  =  ( U  gsumg  F ) )

Proof of Theorem gsumply1subr
Dummy variables  s 
t are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 gsumply1subr.a . . 3  |-  ( ph  ->  A  e.  V )
2 gsumply1subr.s . . . 4  |-  ( ph  ->  T  e.  (SubRing `  R
) )
3 subrgply1.s . . . . 5  |-  S  =  (Poly1 `  R )
4 subrgply1.h . . . . 5  |-  H  =  ( Rs  T )
5 subrgply1.u . . . . 5  |-  U  =  (Poly1 `  H )
6 subrgply1.b . . . . 5  |-  B  =  ( Base `  U
)
73, 4, 5, 6subrgply1 17780 . . . 4  |-  ( T  e.  (SubRing `  R
)  ->  B  e.  (SubRing `  S ) )
8 subrgsubg 16963 . . . . 5  |-  ( B  e.  (SubRing `  S
)  ->  B  e.  (SubGrp `  S ) )
9 subgsubm 15791 . . . . 5  |-  ( B  e.  (SubGrp `  S
)  ->  B  e.  (SubMnd `  S ) )
108, 9syl 16 . . . 4  |-  ( B  e.  (SubRing `  S
)  ->  B  e.  (SubMnd `  S ) )
112, 7, 103syl 20 . . 3  |-  ( ph  ->  B  e.  (SubMnd `  S ) )
12 gsumply1subr.f . . 3  |-  ( ph  ->  F : A --> B )
13 eqid 2450 . . 3  |-  ( Ss  B )  =  ( Ss  B )
141, 11, 12, 13gsumsubm 15596 . 2  |-  ( ph  ->  ( S  gsumg  F )  =  ( ( Ss  B )  gsumg  F ) )
15 fex 6035 . . . 4  |-  ( ( F : A --> B  /\  A  e.  V )  ->  F  e.  _V )
1612, 1, 15syl2anc 661 . . 3  |-  ( ph  ->  F  e.  _V )
17 ovex 6201 . . . 4  |-  ( Ss  B )  e.  _V
1817a1i 11 . . 3  |-  ( ph  ->  ( Ss  B )  e.  _V )
19 fvex 5785 . . . . 5  |-  (Poly1 `  H
)  e.  _V
205, 19eqeltri 2532 . . . 4  |-  U  e. 
_V
2120a1i 11 . . 3  |-  ( ph  ->  U  e.  _V )
22 eqid 2450 . . . . 5  |-  ( Base `  U )  =  (
Base `  U )
236oveq2i 6187 . . . . 5  |-  ( Ss  B )  =  ( Ss  (
Base `  U )
)
243, 4, 5, 22, 2, 23ressply1bas 17776 . . . 4  |-  ( ph  ->  ( Base `  U
)  =  ( Base `  ( Ss  B ) ) )
2524eqcomd 2457 . . 3  |-  ( ph  ->  ( Base `  ( Ss  B ) )  =  ( Base `  U
) )
267, 8, 93syl 20 . . . . . 6  |-  ( T  e.  (SubRing `  R
)  ->  B  e.  (SubMnd `  S ) )
2713submmnd 15570 . . . . . 6  |-  ( B  e.  (SubMnd `  S
)  ->  ( Ss  B
)  e.  Mnd )
282, 26, 273syl 20 . . . . 5  |-  ( ph  ->  ( Ss  B )  e.  Mnd )
2928adantr 465 . . . 4  |-  ( (
ph  /\  ( s  e.  ( Base `  ( Ss  B ) )  /\  t  e.  ( Base `  ( Ss  B ) ) ) )  ->  ( Ss  B
)  e.  Mnd )
30 simpl 457 . . . . 5  |-  ( ( s  e.  ( Base `  ( Ss  B ) )  /\  t  e.  ( Base `  ( Ss  B ) ) )  ->  s  e.  (
Base `  ( Ss  B
) ) )
3130adantl 466 . . . 4  |-  ( (
ph  /\  ( s  e.  ( Base `  ( Ss  B ) )  /\  t  e.  ( Base `  ( Ss  B ) ) ) )  ->  s  e.  ( Base `  ( Ss  B
) ) )
32 simprr 756 . . . 4  |-  ( (
ph  /\  ( s  e.  ( Base `  ( Ss  B ) )  /\  t  e.  ( Base `  ( Ss  B ) ) ) )  ->  t  e.  ( Base `  ( Ss  B
) ) )
33 eqid 2450 . . . . 5  |-  ( Base `  ( Ss  B ) )  =  ( Base `  ( Ss  B ) )
34 eqid 2450 . . . . 5  |-  ( +g  `  ( Ss  B ) )  =  ( +g  `  ( Ss  B ) )
3533, 34mndcl 15508 . . . 4  |-  ( ( ( Ss  B )  e.  Mnd  /\  s  e.  ( Base `  ( Ss  B ) )  /\  t  e.  ( Base `  ( Ss  B ) ) )  ->  ( s ( +g  `  ( Ss  B ) ) t )  e.  ( Base `  ( Ss  B ) ) )
3629, 31, 32, 35syl3anc 1219 . . 3  |-  ( (
ph  /\  ( s  e.  ( Base `  ( Ss  B ) )  /\  t  e.  ( Base `  ( Ss  B ) ) ) )  ->  ( s
( +g  `  ( Ss  B ) ) t )  e.  ( Base `  ( Ss  B ) ) )
37 simpl 457 . . . . 5  |-  ( (
ph  /\  ( s  e.  ( Base `  ( Ss  B ) )  /\  t  e.  ( Base `  ( Ss  B ) ) ) )  ->  ph )
383, 4, 5, 6, 2, 13ressply1bas 17776 . . . . . . . . . 10  |-  ( ph  ->  B  =  ( Base `  ( Ss  B ) ) )
3938eqcomd 2457 . . . . . . . . 9  |-  ( ph  ->  ( Base `  ( Ss  B ) )  =  B )
4039eleq2d 2519 . . . . . . . 8  |-  ( ph  ->  ( s  e.  (
Base `  ( Ss  B
) )  <->  s  e.  B ) )
4140biimpcd 224 . . . . . . 7  |-  ( s  e.  ( Base `  ( Ss  B ) )  -> 
( ph  ->  s  e.  B ) )
4241adantr 465 . . . . . 6  |-  ( ( s  e.  ( Base `  ( Ss  B ) )  /\  t  e.  ( Base `  ( Ss  B ) ) )  ->  ( ph  ->  s  e.  B ) )
4342impcom 430 . . . . 5  |-  ( (
ph  /\  ( s  e.  ( Base `  ( Ss  B ) )  /\  t  e.  ( Base `  ( Ss  B ) ) ) )  ->  s  e.  B )
4439eleq2d 2519 . . . . . . . 8  |-  ( ph  ->  ( t  e.  (
Base `  ( Ss  B
) )  <->  t  e.  B ) )
4544biimpcd 224 . . . . . . 7  |-  ( t  e.  ( Base `  ( Ss  B ) )  -> 
( ph  ->  t  e.  B ) )
4645adantl 466 . . . . . 6  |-  ( ( s  e.  ( Base `  ( Ss  B ) )  /\  t  e.  ( Base `  ( Ss  B ) ) )  ->  ( ph  ->  t  e.  B ) )
4746impcom 430 . . . . 5  |-  ( (
ph  /\  ( s  e.  ( Base `  ( Ss  B ) )  /\  t  e.  ( Base `  ( Ss  B ) ) ) )  ->  t  e.  B )
483, 4, 5, 6, 2, 13ressply1add 17777 . . . . 5  |-  ( (
ph  /\  ( s  e.  B  /\  t  e.  B ) )  -> 
( s ( +g  `  U ) t )  =  ( s ( +g  `  ( Ss  B ) ) t ) )
4937, 43, 47, 48syl12anc 1217 . . . 4  |-  ( (
ph  /\  ( s  e.  ( Base `  ( Ss  B ) )  /\  t  e.  ( Base `  ( Ss  B ) ) ) )  ->  ( s
( +g  `  U ) t )  =  ( s ( +g  `  ( Ss  B ) ) t ) )
5049eqcomd 2457 . . 3  |-  ( (
ph  /\  ( s  e.  ( Base `  ( Ss  B ) )  /\  t  e.  ( Base `  ( Ss  B ) ) ) )  ->  ( s
( +g  `  ( Ss  B ) ) t )  =  ( s ( +g  `  U ) t ) )
51 ffun 5645 . . . 4  |-  ( F : A --> B  ->  Fun  F )
5212, 51syl 16 . . 3  |-  ( ph  ->  Fun  F )
53 frn 5649 . . . . 5  |-  ( F : A --> B  ->  ran  F  C_  B )
5412, 53syl 16 . . . 4  |-  ( ph  ->  ran  F  C_  B
)
5554, 38sseqtrd 3476 . . 3  |-  ( ph  ->  ran  F  C_  ( Base `  ( Ss  B ) ) )
5616, 18, 21, 25, 36, 50, 52, 55gsumpropd2 15594 . 2  |-  ( ph  ->  ( ( Ss  B ) 
gsumg  F )  =  ( U  gsumg  F ) )
5714, 56eqtrd 2490 1  |-  ( ph  ->  ( S  gsumg  F )  =  ( U  gsumg  F ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1757   _Vcvv 3054    C_ wss 3412   ran crn 4925   Fun wfun 5496   -->wf 5498   ` cfv 5502  (class class class)co 6176   Basecbs 14262   ↾s cress 14263   +g cplusg 14326    gsumg cgsu 14467   Mndcmnd 15497  SubMndcsubmnd 15551  SubGrpcsubg 15763  SubRingcsubrg 16953  Poly1cpl1 17726
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1709  ax-7 1729  ax-8 1759  ax-9 1761  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2429  ax-rep 4487  ax-sep 4497  ax-nul 4505  ax-pow 4554  ax-pr 4615  ax-un 6458  ax-inf2 7934  ax-cnex 9425  ax-resscn 9426  ax-1cn 9427  ax-icn 9428  ax-addcl 9429  ax-addrcl 9430  ax-mulcl 9431  ax-mulrcl 9432  ax-mulcom 9433  ax-addass 9434  ax-mulass 9435  ax-distr 9436  ax-i2m1 9437  ax-1ne0 9438  ax-1rid 9439  ax-rnegex 9440  ax-rrecex 9441  ax-cnre 9442  ax-pre-lttri 9443  ax-pre-lttrn 9444  ax-pre-ltadd 9445  ax-pre-mulgt0 9446
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1702  df-eu 2263  df-mo 2264  df-clab 2436  df-cleq 2442  df-clel 2445  df-nfc 2598  df-ne 2643  df-nel 2644  df-ral 2797  df-rex 2798  df-reu 2799  df-rmo 2800  df-rab 2801  df-v 3056  df-sbc 3271  df-csb 3373  df-dif 3415  df-un 3417  df-in 3419  df-ss 3426  df-pss 3428  df-nul 3722  df-if 3876  df-pw 3946  df-sn 3962  df-pr 3964  df-tp 3966  df-op 3968  df-uni 4176  df-int 4213  df-iun 4257  df-iin 4258  df-br 4377  df-opab 4435  df-mpt 4436  df-tr 4470  df-eprel 4716  df-id 4720  df-po 4725  df-so 4726  df-fr 4763  df-se 4764  df-we 4765  df-ord 4806  df-on 4807  df-lim 4808  df-suc 4809  df-xp 4930  df-rel 4931  df-cnv 4932  df-co 4933  df-dm 4934  df-rn 4935  df-res 4936  df-ima 4937  df-iota 5465  df-fun 5504  df-fn 5505  df-f 5506  df-f1 5507  df-fo 5508  df-f1o 5509  df-fv 5510  df-isom 5511  df-riota 6137  df-ov 6179  df-oprab 6180  df-mpt2 6181  df-of 6406  df-ofr 6407  df-om 6563  df-1st 6663  df-2nd 6664  df-supp 6777  df-recs 6918  df-rdg 6952  df-1o 7006  df-2o 7007  df-oadd 7010  df-er 7187  df-map 7302  df-pm 7303  df-ixp 7350  df-en 7397  df-dom 7398  df-sdom 7399  df-fin 7400  df-fsupp 7708  df-oi 7811  df-card 8196  df-pnf 9507  df-mnf 9508  df-xr 9509  df-ltxr 9510  df-le 9511  df-sub 9684  df-neg 9685  df-nn 10410  df-2 10467  df-3 10468  df-4 10469  df-5 10470  df-6 10471  df-7 10472  df-8 10473  df-9 10474  df-10 10475  df-n0 10667  df-z 10734  df-uz 10949  df-fz 11525  df-fzo 11636  df-seq 11894  df-hash 12191  df-struct 14264  df-ndx 14265  df-slot 14266  df-base 14267  df-sets 14268  df-ress 14269  df-plusg 14339  df-mulr 14340  df-sca 14342  df-vsca 14343  df-tset 14345  df-ple 14346  df-0g 14468  df-gsum 14469  df-mre 14612  df-mrc 14613  df-acs 14615  df-mnd 15503  df-mhm 15552  df-submnd 15553  df-grp 15633  df-minusg 15634  df-mulg 15636  df-subg 15766  df-ghm 15833  df-cntz 15923  df-cmn 16369  df-abl 16370  df-mgp 16683  df-ur 16695  df-rng 16739  df-subrg 16955  df-psr 17515  df-mpl 17517  df-opsr 17519  df-psr1 17729  df-ply1 17731
This theorem is referenced by: (None)
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