Theorem itgocn 31042

Description: All integral elements are complex numbers. (Contributed by Stefan O'Rear, 27-Nov-2014.)

Assertion

Ref	Expression
itgocn	|- (IntgOver ` S ) C_ CC

Proof of Theorem itgocn

Dummy variables a b c are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-itgo 31037	. . . . 5 |- IntgOver = ( a e. ~P CC |-> { b e. CC | E. c e. (Poly ` a ) ( ( c ` b ) = 0 /\ ( (coeff ` c ) ` (deg ` c ) ) = 1 ) } )
2	1	dmmptss 5509	. . . 4 |- dom IntgOver C_ ~P CC
3	2	sseli 3505	. . 3 |- ( S e. dom IntgOver -> S e. ~P CC )
4		cnex 9585	. . . . 5 |- CC e. _V
5	4	elpw2 4617	. . . 4 |- ( S e. ~P CC <-> S C_ CC )
6		itgoval 31039	. . . . 5 |- ( S C_ CC -> (IntgOver ` S ) = { a e. CC | E. b e. (Poly ` S ) ( ( b ` a ) = 0 /\ ( (coeff ` b ) ` (deg ` b ) ) = 1 ) } )
7		ssrab2 3590	. . . . 5 |- { a e. CC | E. b e. (Poly ` S ) ( ( b ` a ) = 0 /\ ( (coeff ` b ) ` (deg ` b ) ) = 1 ) } C_ CC
8	6, 7	syl6eqss 3559	. . . 4 |- ( S C_ CC -> (IntgOver ` S ) C_ CC )
9	5, 8	sylbi 195	. . 3 |- ( S e. ~P CC -> (IntgOver ` S ) C_ CC )
10	3, 9	syl 16	. 2 |- ( S e. dom IntgOver -> (IntgOver ` S ) C_ CC )
11		ndmfv 5896	. . 3 |- ( -. S e. dom IntgOver -> (IntgOver ` S ) = (/) )
12		0ss 3819	. . 3 |- (/) C_ CC
13	11, 12	syl6eqss 3559	. 2 |- ( -. S e. dom IntgOver -> (IntgOver ` S ) C_ CC )
14	10, 13	pm2.61i 164	1 |- (IntgOver ` S ) C_ CC

Syntax hints:   -. wn 3   /\ wa 369   = wceq 1379   e. wcel 1767   E.wrex 2818   {crab 2821   C_ wss 3481   (/)c0 3790   ~Pcpw 4016   dom cdm 5005   ` cfv 5594   CCcc 9502   0cc0 9504   1c1 9505  Polycply 22449  coeffccoe 22451  degcdgr 22452  IntgOvercitgo 31035

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-cnex 9560

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fv 5602  df-itgo 31037

This theorem is referenced by: (None)