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Theorem itgoss 27336
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 20110 . . . . 5  |-  ( ( S  C_  T  /\  T  C_  CC )  -> 
(Poly `  S )  C_  (Poly `  T )
)
2 ssrexv 3400 . . . . 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 2782 . . 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 3411 . . 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 3348 . . 3  |-  ( ( S  C_  T  /\  T  C_  CC )  ->  S  C_  CC )
8 itgoval 27334 . . 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 27334 . . 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 3383 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 1652   A.wral 2697   E.wrex 2698   {crab 2701    C_ wss 3312   ` cfv 5446   CCcc 8980   0cc0 8982   1c1 8983  Polycply 20095  coeffccoe 20097  degcdgr 20098  IntgOvercitgo 27330
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-i2m1 9050  ax-1ne0 9051  ax-rrecex 9054  ax-cnre 9055
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-recs 6625  df-rdg 6660  df-map 7012  df-nn 9993  df-n0 10214  df-ply 20099  df-itgo 27332
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