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Theorem itgoss 27368
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 19581 . . . . 5  |-  ( ( S  C_  T  /\  T  C_  CC )  -> 
(Poly `  S )  C_  (Poly `  T )
)
2 ssrexv 3238 . . . . 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 15 . . . 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 2627 . . 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 3249 . . 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 203 . 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 3187 . . 3  |-  ( ( S  C_  T  /\  T  C_  CC )  ->  S  C_  CC )
8 itgoval 27366 . . 3  |-  ( S 
C_  CC  ->  (IntgOver `  S
)  =  { a  e.  CC  |  E. b  e.  (Poly `  S
) ( ( b `
 a )  =  0  /\  ( (coeff `  b ) `  (deg `  b ) )  =  1 ) } )
97, 8syl 15 . 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 27366 . . 3  |-  ( T 
C_  CC  ->  (IntgOver `  T
)  =  { a  e.  CC  |  E. b  e.  (Poly `  T
) ( ( b `
 a )  =  0  /\  ( (coeff `  b ) `  (deg `  b ) )  =  1 ) } )
1110adantl 452 . 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 3221 1  |-  ( ( S  C_  T  /\  T  C_  CC )  -> 
(IntgOver `  S )  C_  (IntgOver `  T ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623   A.wral 2543   E.wrex 2544   {crab 2547    C_ wss 3152   ` cfv 5255   CCcc 8735   0cc0 8737   1c1 8738  Polycply 19566  coeffccoe 19568  degcdgr 19569  IntgOvercitgo 27362
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-i2m1 8805  ax-1ne0 8806  ax-rrecex 8809  ax-cnre 8810
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-recs 6388  df-rdg 6423  df-map 6774  df-nn 9747  df-n0 9966  df-ply 19570  df-itgo 27364
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