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Theorem subrgply1 16521
Description: A subring of the base ring induces a subring of polynomials. (Contributed by Mario Carneiro, 3-Jul-2015.)
Hypotheses
Ref Expression
subrgply1.s  |-  S  =  (Poly1 `  R )
subrgply1.h  |-  H  =  ( Rs  T )
subrgply1.u  |-  U  =  (Poly1 `  H )
subrgply1.b  |-  B  =  ( Base `  U
)
Assertion
Ref Expression
subrgply1  |-  ( T  e.  (SubRing `  R
)  ->  B  e.  (SubRing `  S ) )

Proof of Theorem subrgply1
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 1on 6628 . . 3  |-  1o  e.  On
2 eqid 2366 . . . 4  |-  ( 1o mPoly  R )  =  ( 1o mPoly  R )
3 subrgply1.h . . . 4  |-  H  =  ( Rs  T )
4 eqid 2366 . . . 4  |-  ( 1o mPoly  H )  =  ( 1o mPoly  H )
5 subrgply1.u . . . . 5  |-  U  =  (Poly1 `  H )
6 eqid 2366 . . . . 5  |-  (PwSer1 `  H
)  =  (PwSer1 `  H
)
7 subrgply1.b . . . . 5  |-  B  =  ( Base `  U
)
85, 6, 7ply1bas 16484 . . . 4  |-  B  =  ( Base `  ( 1o mPoly  H ) )
92, 3, 4, 8subrgmpl 16414 . . 3  |-  ( ( 1o  e.  On  /\  T  e.  (SubRing `  R
) )  ->  B  e.  (SubRing `  ( 1o mPoly  R ) ) )
101, 9mpan 651 . 2  |-  ( T  e.  (SubRing `  R
)  ->  B  e.  (SubRing `  ( 1o mPoly  R
) ) )
11 eqidd 2367 . . 3  |-  ( T  e.  (SubRing `  R
)  ->  ( Base `  S )  =  (
Base `  S )
)
12 subrgply1.s . . . . 5  |-  S  =  (Poly1 `  R )
13 eqid 2366 . . . . 5  |-  (PwSer1 `  R
)  =  (PwSer1 `  R
)
14 eqid 2366 . . . . 5  |-  ( Base `  S )  =  (
Base `  S )
1512, 13, 14ply1bas 16484 . . . 4  |-  ( Base `  S )  =  (
Base `  ( 1o mPoly  R ) )
1615a1i 10 . . 3  |-  ( T  e.  (SubRing `  R
)  ->  ( Base `  S )  =  (
Base `  ( 1o mPoly  R ) ) )
17 eqid 2366 . . . . . 6  |-  ( +g  `  S )  =  ( +g  `  S )
1812, 2, 17ply1plusg 16513 . . . . 5  |-  ( +g  `  S )  =  ( +g  `  ( 1o mPoly  R ) )
1918a1i 10 . . . 4  |-  ( T  e.  (SubRing `  R
)  ->  ( +g  `  S )  =  ( +g  `  ( 1o mPoly  R ) ) )
2019proplem3 13803 . . 3  |-  ( ( T  e.  (SubRing `  R
)  /\  ( x  e.  ( Base `  S
)  /\  y  e.  ( Base `  S )
) )  ->  (
x ( +g  `  S
) y )  =  ( x ( +g  `  ( 1o mPoly  R )
) y ) )
21 eqid 2366 . . . . . 6  |-  ( .r
`  S )  =  ( .r `  S
)
2212, 2, 21ply1mulr 16515 . . . . 5  |-  ( .r
`  S )  =  ( .r `  ( 1o mPoly  R ) )
2322a1i 10 . . . 4  |-  ( T  e.  (SubRing `  R
)  ->  ( .r `  S )  =  ( .r `  ( 1o mPoly  R ) ) )
2423proplem3 13803 . . 3  |-  ( ( T  e.  (SubRing `  R
)  /\  ( x  e.  ( Base `  S
)  /\  y  e.  ( Base `  S )
) )  ->  (
x ( .r `  S ) y )  =  ( x ( .r `  ( 1o mPoly  R ) ) y ) )
2511, 16, 20, 24subrgpropd 15789 . 2  |-  ( T  e.  (SubRing `  R
)  ->  (SubRing `  S
)  =  (SubRing `  ( 1o mPoly  R ) ) )
2610, 25eleqtrrd 2443 1  |-  ( T  e.  (SubRing `  R
)  ->  B  e.  (SubRing `  S ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1647    e. wcel 1715   Oncon0 4495   ` cfv 5358  (class class class)co 5981   1oc1o 6614   Basecbs 13356   ↾s cress 13357   +g cplusg 13416   .rcmulr 13417  SubRingcsubrg 15751   mPoly cmpl 16299  PwSer1cps1 16460  Poly1cpl1 16462
This theorem is referenced by:  plypf1  19809
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615  ax-inf2 7489  ax-cnex 8940  ax-resscn 8941  ax-1cn 8942  ax-icn 8943  ax-addcl 8944  ax-addrcl 8945  ax-mulcl 8946  ax-mulrcl 8947  ax-mulcom 8948  ax-addass 8949  ax-mulass 8950  ax-distr 8951  ax-i2m1 8952  ax-1ne0 8953  ax-1rid 8954  ax-rnegex 8955  ax-rrecex 8956  ax-cnre 8957  ax-pre-lttri 8958  ax-pre-lttrn 8959  ax-pre-ltadd 8960  ax-pre-mulgt0 8961
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 936  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-nel 2532  df-ral 2633  df-rex 2634  df-reu 2635  df-rmo 2636  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-pss 3254  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-tp 3737  df-op 3738  df-uni 3930  df-int 3965  df-iun 4009  df-iin 4010  df-br 4126  df-opab 4180  df-mpt 4181  df-tr 4216  df-eprel 4408  df-id 4412  df-po 4417  df-so 4418  df-fr 4455  df-se 4456  df-we 4457  df-ord 4498  df-on 4499  df-lim 4500  df-suc 4501  df-om 4760  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-isom 5367  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-of 6205  df-ofr 6206  df-1st 6249  df-2nd 6250  df-riota 6446  df-recs 6530  df-rdg 6565  df-1o 6621  df-2o 6622  df-oadd 6625  df-er 6802  df-map 6917  df-pm 6918  df-ixp 6961  df-en 7007  df-dom 7008  df-sdom 7009  df-fin 7010  df-oi 7372  df-card 7719  df-pnf 9016  df-mnf 9017  df-xr 9018  df-ltxr 9019  df-le 9020  df-sub 9186  df-neg 9187  df-nn 9894  df-2 9951  df-3 9952  df-4 9953  df-5 9954  df-6 9955  df-7 9956  df-8 9957  df-9 9958  df-10 9959  df-n0 10115  df-z 10176  df-uz 10382  df-fz 10936  df-fzo 11026  df-seq 11211  df-hash 11506  df-struct 13358  df-ndx 13359  df-slot 13360  df-base 13361  df-sets 13362  df-ress 13363  df-plusg 13429  df-mulr 13430  df-sca 13432  df-vsca 13433  df-tset 13435  df-ple 13436  df-0g 13614  df-gsum 13615  df-mre 13698  df-mrc 13699  df-acs 13701  df-mnd 14577  df-mhm 14625  df-submnd 14626  df-grp 14699  df-minusg 14700  df-mulg 14702  df-subg 14828  df-ghm 14891  df-cntz 15003  df-cmn 15301  df-abl 15302  df-mgp 15536  df-rng 15550  df-ur 15552  df-subrg 15753  df-psr 16308  df-mpl 16310  df-opsr 16316  df-psr1 16467  df-ply1 16469
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