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Theorem itgoss 27037
Description: An integral element is integral over a subset. (Contributed by Stefan O'Rear, 27-Nov-2014.)
Assertion
Ref Expression
itgoss  |-  ( ( S  C_  T  /\  T  C_  CC )  -> 
(IntgOver `  S )  C_  (IntgOver `  T ) )

Proof of Theorem itgoss
Dummy variables  a 
b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 plyss 19985 . . . . 5  |-  ( ( S  C_  T  /\  T  C_  CC )  -> 
(Poly `  S )  C_  (Poly `  T )
)
2 ssrexv 3351 . . . . 5  |-  ( (Poly `  S )  C_  (Poly `  T )  ->  ( E. b  e.  (Poly `  S ) ( ( b `  a )  =  0  /\  (
(coeff `  b ) `  (deg `  b )
)  =  1 )  ->  E. b  e.  (Poly `  T ) ( ( b `  a )  =  0  /\  (
(coeff `  b ) `  (deg `  b )
)  =  1 ) ) )
31, 2syl 16 . . . 4  |-  ( ( S  C_  T  /\  T  C_  CC )  -> 
( E. b  e.  (Poly `  S )
( ( b `  a )  =  0  /\  ( (coeff `  b ) `  (deg `  b ) )  =  1 )  ->  E. b  e.  (Poly `  T )
( ( b `  a )  =  0  /\  ( (coeff `  b ) `  (deg `  b ) )  =  1 ) ) )
43ralrimivw 2733 . . 3  |-  ( ( S  C_  T  /\  T  C_  CC )  ->  A. a  e.  CC  ( E. b  e.  (Poly `  S ) ( ( b `  a )  =  0  /\  (
(coeff `  b ) `  (deg `  b )
)  =  1 )  ->  E. b  e.  (Poly `  T ) ( ( b `  a )  =  0  /\  (
(coeff `  b ) `  (deg `  b )
)  =  1 ) ) )
5 ss2rab 3362 . . 3  |-  ( { a  e.  CC  |  E. b  e.  (Poly `  S ) ( ( b `  a )  =  0  /\  (
(coeff `  b ) `  (deg `  b )
)  =  1 ) }  C_  { a  e.  CC  |  E. b  e.  (Poly `  T )
( ( b `  a )  =  0  /\  ( (coeff `  b ) `  (deg `  b ) )  =  1 ) }  <->  A. a  e.  CC  ( E. b  e.  (Poly `  S )
( ( b `  a )  =  0  /\  ( (coeff `  b ) `  (deg `  b ) )  =  1 )  ->  E. b  e.  (Poly `  T )
( ( b `  a )  =  0  /\  ( (coeff `  b ) `  (deg `  b ) )  =  1 ) ) )
64, 5sylibr 204 . 2  |-  ( ( S  C_  T  /\  T  C_  CC )  ->  { a  e.  CC  |  E. b  e.  (Poly `  S ) ( ( b `  a )  =  0  /\  (
(coeff `  b ) `  (deg `  b )
)  =  1 ) }  C_  { a  e.  CC  |  E. b  e.  (Poly `  T )
( ( b `  a )  =  0  /\  ( (coeff `  b ) `  (deg `  b ) )  =  1 ) } )
7 sstr 3299 . . 3  |-  ( ( S  C_  T  /\  T  C_  CC )  ->  S  C_  CC )
8 itgoval 27035 . . 3  |-  ( S 
C_  CC  ->  (IntgOver `  S
)  =  { a  e.  CC  |  E. b  e.  (Poly `  S
) ( ( b `
 a )  =  0  /\  ( (coeff `  b ) `  (deg `  b ) )  =  1 ) } )
97, 8syl 16 . 2  |-  ( ( S  C_  T  /\  T  C_  CC )  -> 
(IntgOver `  S )  =  { a  e.  CC  |  E. b  e.  (Poly `  S ) ( ( b `  a )  =  0  /\  (
(coeff `  b ) `  (deg `  b )
)  =  1 ) } )
10 itgoval 27035 . . 3  |-  ( T 
C_  CC  ->  (IntgOver `  T
)  =  { a  e.  CC  |  E. b  e.  (Poly `  T
) ( ( b `
 a )  =  0  /\  ( (coeff `  b ) `  (deg `  b ) )  =  1 ) } )
1110adantl 453 . 2  |-  ( ( S  C_  T  /\  T  C_  CC )  -> 
(IntgOver `  T )  =  { a  e.  CC  |  E. b  e.  (Poly `  T ) ( ( b `  a )  =  0  /\  (
(coeff `  b ) `  (deg `  b )
)  =  1 ) } )
126, 9, 113sstr4d 3334 1  |-  ( ( S  C_  T  /\  T  C_  CC )  -> 
(IntgOver `  S )  C_  (IntgOver `  T ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649   A.wral 2649   E.wrex 2650   {crab 2653    C_ wss 3263   ` cfv 5394   CCcc 8921   0cc0 8923   1c1 8924  Polycply 19970  coeffccoe 19972  degcdgr 19973  IntgOvercitgo 27031
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-cnex 8979  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-i2m1 8991  ax-1ne0 8992  ax-rrecex 8995  ax-cnre 8996
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-reu 2656  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-recs 6569  df-rdg 6604  df-map 6956  df-nn 9933  df-n0 10154  df-ply 19974  df-itgo 27033
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