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Theorem ressply1vsca 18400
Description: A restricted power series algebra has the same scalar multiplication operation. (Contributed by Mario Carneiro, 3-Jul-2015.)
Hypotheses
Ref Expression
ressply1.s  |-  S  =  (Poly1 `  R )
ressply1.h  |-  H  =  ( Rs  T )
ressply1.u  |-  U  =  (Poly1 `  H )
ressply1.b  |-  B  =  ( Base `  U
)
ressply1.2  |-  ( ph  ->  T  e.  (SubRing `  R
) )
ressply1.p  |-  P  =  ( Ss  B )
Assertion
Ref Expression
ressply1vsca  |-  ( (
ph  /\  ( X  e.  T  /\  Y  e.  B ) )  -> 
( X ( .s
`  U ) Y )  =  ( X ( .s `  P
) Y ) )

Proof of Theorem ressply1vsca
StepHypRef Expression
1 eqid 2457 . . 3  |-  ( 1o mPoly  R )  =  ( 1o mPoly  R )
2 ressply1.h . . 3  |-  H  =  ( Rs  T )
3 eqid 2457 . . 3  |-  ( 1o mPoly  H )  =  ( 1o mPoly  H )
4 ressply1.u . . . 4  |-  U  =  (Poly1 `  H )
5 eqid 2457 . . . 4  |-  (PwSer1 `  H
)  =  (PwSer1 `  H
)
6 ressply1.b . . . 4  |-  B  =  ( Base `  U
)
74, 5, 6ply1bas 18361 . . 3  |-  B  =  ( Base `  ( 1o mPoly  H ) )
8 1on 7155 . . . 4  |-  1o  e.  On
98a1i 11 . . 3  |-  ( ph  ->  1o  e.  On )
10 ressply1.2 . . 3  |-  ( ph  ->  T  e.  (SubRing `  R
) )
11 eqid 2457 . . 3  |-  ( ( 1o mPoly  R )s  B )  =  ( ( 1o mPoly  R )s  B )
121, 2, 3, 7, 9, 10, 11ressmplvsca 18248 . 2  |-  ( (
ph  /\  ( X  e.  T  /\  Y  e.  B ) )  -> 
( X ( .s
`  ( 1o mPoly  H
) ) Y )  =  ( X ( .s `  ( ( 1o mPoly  R )s  B ) ) Y ) )
13 eqid 2457 . . . 4  |-  ( .s
`  U )  =  ( .s `  U
)
144, 3, 13ply1vsca 18394 . . 3  |-  ( .s
`  U )  =  ( .s `  ( 1o mPoly  H ) )
1514oveqi 6309 . 2  |-  ( X ( .s `  U
) Y )  =  ( X ( .s
`  ( 1o mPoly  H
) ) Y )
16 ressply1.s . . . . 5  |-  S  =  (Poly1 `  R )
17 eqid 2457 . . . . 5  |-  ( .s
`  S )  =  ( .s `  S
)
1816, 1, 17ply1vsca 18394 . . . 4  |-  ( .s
`  S )  =  ( .s `  ( 1o mPoly  R ) )
19 fvex 5882 . . . . . 6  |-  ( Base `  U )  e.  _V
206, 19eqeltri 2541 . . . . 5  |-  B  e. 
_V
21 ressply1.p . . . . . 6  |-  P  =  ( Ss  B )
2221, 17ressvsca 14795 . . . . 5  |-  ( B  e.  _V  ->  ( .s `  S )  =  ( .s `  P
) )
2320, 22ax-mp 5 . . . 4  |-  ( .s
`  S )  =  ( .s `  P
)
24 eqid 2457 . . . . . 6  |-  ( .s
`  ( 1o mPoly  R
) )  =  ( .s `  ( 1o mPoly  R ) )
2511, 24ressvsca 14795 . . . . 5  |-  ( B  e.  _V  ->  ( .s `  ( 1o mPoly  R
) )  =  ( .s `  ( ( 1o mPoly  R )s  B ) ) )
2620, 25ax-mp 5 . . . 4  |-  ( .s
`  ( 1o mPoly  R
) )  =  ( .s `  ( ( 1o mPoly  R )s  B ) )
2718, 23, 263eqtr3i 2494 . . 3  |-  ( .s
`  P )  =  ( .s `  (
( 1o mPoly  R )s  B
) )
2827oveqi 6309 . 2  |-  ( X ( .s `  P
) Y )  =  ( X ( .s
`  ( ( 1o mPoly  R )s  B ) ) Y )
2912, 15, 283eqtr4g 2523 1  |-  ( (
ph  /\  ( X  e.  T  /\  Y  e.  B ) )  -> 
( X ( .s
`  U ) Y )  =  ( X ( .s `  P
) Y ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1395    e. wcel 1819   _Vcvv 3109   Oncon0 4887   ` cfv 5594  (class class class)co 6296   1oc1o 7141   Basecbs 14644   ↾s cress 14645   .scvsca 14716  SubRingcsubrg 17552   mPoly cmpl 18129  PwSer1cps1 18341  Poly1cpl1 18343
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-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-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-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-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-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-of 6539  df-om 6700  df-1st 6799  df-2nd 6800  df-supp 6918  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-map 7440  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-fsupp 7848  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-uz 11107  df-fz 11698  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-tset 14731  df-ple 14732  df-subg 16325  df-ring 17327  df-subrg 17554  df-psr 18132  df-mpl 18134  df-opsr 18136  df-psr1 18346  df-ply1 18348
This theorem is referenced by: (None)
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