Theorem itgocn 37734

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 37729	. . . . 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 5631	. . . 4 |- dom IntgOver C_ ~P CC
3	2	sseli 3599	. . 3 |- ( S e. dom IntgOver -> S e. ~P CC )
4		cnex 10017	. . . . 5 |- CC e. _V
5	4	elpw2 4828	. . . 4 |- ( S e. ~P CC <-> S C_ CC )
6		itgoval 37731	. . . . 5 |- ( S C_ CC -> (IntgOver ` S ) = { a e. CC | E. b e. (Poly ` S ) ( ( b ` a ) = 0 /\ ( (coeff ` b ) ` (deg ` b ) ) = 1 ) } )
7		ssrab2 3687	. . . . 5 |- { a e. CC | E. b e. (Poly ` S ) ( ( b ` a ) = 0 /\ ( (coeff ` b ) ` (deg ` b ) ) = 1 ) } C_ CC
8	6, 7	syl6eqss 3655	. . . 4 |- ( S C_ CC -> (IntgOver ` S ) C_ CC )
9	5, 8	sylbi 207	. . 3 |- ( S e. ~P CC -> (IntgOver ` S ) C_ CC )
10	3, 9	syl 17	. 2 |- ( S e. dom IntgOver -> (IntgOver ` S ) C_ CC )
11		ndmfv 6218	. . 3 |- ( -. S e. dom IntgOver -> (IntgOver ` S ) = (/) )
12		0ss 3972	. . 3 |- (/) C_ CC
13	11, 12	syl6eqss 3655	. 2 |- ( -. S e. dom IntgOver -> (IntgOver ` S ) C_ CC )
14	10, 13	pm2.61i 176	1 |- (IntgOver ` S ) C_ CC

Syntax hints:   -. wn 3   /\ wa 384   = wceq 1483   e. wcel 1990   E.wrex 2913   {crab 2916   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   dom cdm 5114   ` 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-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-cnex 9992

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  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-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  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-iota 5851  df-fun 5890  df-fv 5896  df-itgo 37729

This theorem is referenced by: (None)