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Theorem subrgply1 17806
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 7032 . . 3  |-  1o  e.  On
2 eqid 2452 . . . 4  |-  ( 1o mPoly  R )  =  ( 1o mPoly  R )
3 subrgply1.h . . . 4  |-  H  =  ( Rs  T )
4 eqid 2452 . . . 4  |-  ( 1o mPoly  H )  =  ( 1o mPoly  H )
5 subrgply1.u . . . . 5  |-  U  =  (Poly1 `  H )
6 eqid 2452 . . . . 5  |-  (PwSer1 `  H
)  =  (PwSer1 `  H
)
7 subrgply1.b . . . . 5  |-  B  =  ( Base `  U
)
85, 6, 7ply1bas 17770 . . . 4  |-  B  =  ( Base `  ( 1o mPoly  H ) )
92, 3, 4, 8subrgmpl 17658 . . 3  |-  ( ( 1o  e.  On  /\  T  e.  (SubRing `  R
) )  ->  B  e.  (SubRing `  ( 1o mPoly  R ) ) )
101, 9mpan 670 . 2  |-  ( T  e.  (SubRing `  R
)  ->  B  e.  (SubRing `  ( 1o mPoly  R
) ) )
11 eqidd 2453 . . 3  |-  ( T  e.  (SubRing `  R
)  ->  ( Base `  S )  =  (
Base `  S )
)
12 subrgply1.s . . . . 5  |-  S  =  (Poly1 `  R )
13 eqid 2452 . . . . 5  |-  (PwSer1 `  R
)  =  (PwSer1 `  R
)
14 eqid 2452 . . . . 5  |-  ( Base `  S )  =  (
Base `  S )
1512, 13, 14ply1bas 17770 . . . 4  |-  ( Base `  S )  =  (
Base `  ( 1o mPoly  R ) )
1615a1i 11 . . 3  |-  ( T  e.  (SubRing `  R
)  ->  ( Base `  S )  =  (
Base `  ( 1o mPoly  R ) ) )
17 eqid 2452 . . . . . 6  |-  ( +g  `  S )  =  ( +g  `  S )
1812, 2, 17ply1plusg 17798 . . . . 5  |-  ( +g  `  S )  =  ( +g  `  ( 1o mPoly  R ) )
1918a1i 11 . . . 4  |-  ( T  e.  (SubRing `  R
)  ->  ( +g  `  S )  =  ( +g  `  ( 1o mPoly  R ) ) )
2019proplem3 14743 . . 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 2452 . . . . . 6  |-  ( .r
`  S )  =  ( .r `  S
)
2212, 2, 21ply1mulr 17800 . . . . 5  |-  ( .r
`  S )  =  ( .r `  ( 1o mPoly  R ) )
2322a1i 11 . . . 4  |-  ( T  e.  (SubRing `  R
)  ->  ( .r `  S )  =  ( .r `  ( 1o mPoly  R ) ) )
2423proplem3 14743 . . 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 17017 . 2  |-  ( T  e.  (SubRing `  R
)  ->  (SubRing `  S
)  =  (SubRing `  ( 1o mPoly  R ) ) )
2610, 25eleqtrrd 2543 1  |-  ( T  e.  (SubRing `  R
)  ->  B  e.  (SubRing `  S ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   Oncon0 4822   ` cfv 5521  (class class class)co 6195   1oc1o 7018   Basecbs 14287   ↾s cress 14288   +g cplusg 14352   .rcmulr 14353  SubRingcsubrg 16979   mPoly cmpl 17538  PwSer1cps1 17750  Poly1cpl1 17752
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 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477  ax-inf2 7953  ax-cnex 9444  ax-resscn 9445  ax-1cn 9446  ax-icn 9447  ax-addcl 9448  ax-addrcl 9449  ax-mulcl 9450  ax-mulrcl 9451  ax-mulcom 9452  ax-addass 9453  ax-mulass 9454  ax-distr 9455  ax-i2m1 9456  ax-1ne0 9457  ax-1rid 9458  ax-rnegex 9459  ax-rrecex 9460  ax-cnre 9461  ax-pre-lttri 9462  ax-pre-lttrn 9463  ax-pre-ltadd 9464  ax-pre-mulgt0 9465
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 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-nel 2648  df-ral 2801  df-rex 2802  df-reu 2803  df-rmo 2804  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-pss 3447  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4195  df-int 4232  df-iun 4276  df-iin 4277  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4489  df-eprel 4735  df-id 4739  df-po 4744  df-so 4745  df-fr 4782  df-se 4783  df-we 4784  df-ord 4825  df-on 4826  df-lim 4827  df-suc 4828  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-isom 5530  df-riota 6156  df-ov 6198  df-oprab 6199  df-mpt2 6200  df-of 6425  df-ofr 6426  df-om 6582  df-1st 6682  df-2nd 6683  df-supp 6796  df-recs 6937  df-rdg 6971  df-1o 7025  df-2o 7026  df-oadd 7029  df-er 7206  df-map 7321  df-pm 7322  df-ixp 7369  df-en 7416  df-dom 7417  df-sdom 7418  df-fin 7419  df-fsupp 7727  df-oi 7830  df-card 8215  df-pnf 9526  df-mnf 9527  df-xr 9528  df-ltxr 9529  df-le 9530  df-sub 9703  df-neg 9704  df-nn 10429  df-2 10486  df-3 10487  df-4 10488  df-5 10489  df-6 10490  df-7 10491  df-8 10492  df-9 10493  df-10 10494  df-n0 10686  df-z 10753  df-uz 10968  df-fz 11550  df-fzo 11661  df-seq 11919  df-hash 12216  df-struct 14289  df-ndx 14290  df-slot 14291  df-base 14292  df-sets 14293  df-ress 14294  df-plusg 14365  df-mulr 14366  df-sca 14368  df-vsca 14369  df-tset 14371  df-ple 14372  df-0g 14494  df-gsum 14495  df-mre 14638  df-mrc 14639  df-acs 14641  df-mnd 15529  df-mhm 15578  df-submnd 15579  df-grp 15659  df-minusg 15660  df-mulg 15662  df-subg 15792  df-ghm 15859  df-cntz 15949  df-cmn 16395  df-abl 16396  df-mgp 16709  df-ur 16721  df-rng 16765  df-subrg 16981  df-psr 17541  df-mpl 17543  df-opsr 17545  df-psr1 17755  df-ply1 17757
This theorem is referenced by:  gsumply1subr  17807  plypf1  21808
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