Theorem itgocn 29692

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 29687	. . . . 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 5445	. . . 4 |- dom IntgOver C_ ~P CC
3	2	sseli 3463	. . 3 |- ( S e. dom IntgOver -> S e. ~P CC )
4		cnex 9478	. . . . 5 |- CC e. _V
5	4	elpw2 4567	. . . 4 |- ( S e. ~P CC <-> S C_ CC )
6		itgoval 29689	. . . . 5 |- ( S C_ CC -> (IntgOver ` S ) = { a e. CC | E. b e. (Poly ` S ) ( ( b ` a ) = 0 /\ ( (coeff ` b ) ` (deg ` b ) ) = 1 ) } )
7		ssrab2 3548	. . . . 5 |- { a e. CC | E. b e. (Poly ` S ) ( ( b ` a ) = 0 /\ ( (coeff ` b ) ` (deg ` b ) ) = 1 ) } C_ CC
8	6, 7	syl6eqss 3517	. . . 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 5826	. . 3 |- ( -. S e. dom IntgOver -> (IntgOver ` S ) = (/) )
12		0ss 3777	. . 3 |- (/) C_ CC
13	11, 12	syl6eqss 3517	. 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 1370   e. wcel 1758   E.wrex 2800   {crab 2803   C_ wss 3439   (/)c0 3748   ~Pcpw 3971   dom cdm 4951   ` cfv 5529   CCcc 9395   0cc0 9397   1c1 9398  Polycply 21795  coeffccoe 21797  degcdgr 21798  IntgOvercitgo 29685

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-cnex 9453

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fv 5537  df-itgo 29687

This theorem is referenced by: (None)