Theorem itgoss 37733

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.

Step	Hyp	Ref	Expression
1		plyss 23955	. . . . 5 |- ( ( S C_ T /\ T C_ CC ) -> (Poly ` S ) C_ (Poly ` T ) )
2		ssrexv 3667	. . . . 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 ) ) )
3	1, 2	syl 17	. . . 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 ) ) )
4	3	ralrimivw 2967	. . 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 3678	. . 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 ) ) )
6	4, 5	sylibr 224	. 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 3611	. . 3 |- ( ( S C_ T /\ T C_ CC ) -> S C_ CC )
8		itgoval 37731	. . 3 |- ( S C_ CC -> (IntgOver ` S ) = { a e. CC | E. b e. (Poly ` S ) ( ( b ` a ) = 0 /\ ( (coeff ` b ) ` (deg ` b ) ) = 1 ) } )
9	7, 8	syl 17	. 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 37731	. . 3 |- ( T C_ CC -> (IntgOver ` T ) = { a e. CC | E. b e. (Poly ` T ) ( ( b ` a ) = 0 /\ ( (coeff ` b ) ` (deg ` b ) ) = 1 ) } )
11	10	adantl 482	. 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 ) } )
12	6, 9, 11	3sstr4d 3648	1 |- ( ( S C_ T /\ T C_ CC ) -> (IntgOver ` S ) C_ (IntgOver ` T ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   A.wral 2912   E.wrex 2913   {crab 2916   C_ wss 3574   ` cfv 5888   CCcc 9934   0cc0 9936   1c1 9937  Polycply 23940  coeffccoe 23942  degcdgr 23943  IntgOvercitgo 37727

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-i2m1 10004  ax-1ne0 10005  ax-rrecex 10008  ax-cnre 10009

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-map 7859  df-nn 11021  df-n0 11293  df-ply 23944  df-itgo 37729

This theorem is referenced by: (None)