Theorem elmnc 37706

Description: Property of a monic polynomial. (Contributed by Stefan O'Rear, 5-Dec-2014.)

Assertion

Ref	Expression
elmnc	|- ( P e. ( Monic ` S ) <-> ( P e. (Poly ` S ) /\ ( (coeff ` P ) ` (deg ` P ) ) = 1 ) )

Proof of Theorem elmnc

Dummy variables s p are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-mnc 37703	. . . . 5 |- Monic = ( s e. ~P CC |-> { p e. (Poly ` s ) | ( (coeff ` p ) ` (deg ` p ) ) = 1 } )
2	1	dmmptss 5631	. . . 4 |- dom Monic C_ ~P CC
3		elfvdm 6220	. . . 4 |- ( P e. ( Monic ` S ) -> S e. dom Monic )
4	2, 3	sseldi 3601	. . 3 |- ( P e. ( Monic ` S ) -> S e. ~P CC )
5	4	elpwid 4170	. 2 |- ( P e. ( Monic ` S ) -> S C_ CC )
6		plybss 23950	. . 3 |- ( P e. (Poly ` S ) -> S C_ CC )
7	6	adantr 481	. 2 |- ( ( P e. (Poly ` S ) /\ ( (coeff ` P ) ` (deg ` P ) ) = 1 ) -> S C_ CC )
8		cnex 10017	. . . . . 6 |- CC e. _V
9	8	elpw2 4828	. . . . 5 |- ( S e. ~P CC <-> S C_ CC )
10		fveq2 6191	. . . . . . 7 |- ( s = S -> (Poly ` s ) = (Poly ` S ) )
11		rabeq 3192	. . . . . . 7 |- ( (Poly ` s ) = (Poly ` S ) -> { p e. (Poly ` s ) | ( (coeff ` p ) ` (deg ` p ) ) = 1 } = { p e. (Poly ` S ) | ( (coeff ` p ) ` (deg ` p ) ) = 1 } )
12	10, 11	syl 17	. . . . . 6 |- ( s = S -> { p e. (Poly ` s ) | ( (coeff ` p ) ` (deg ` p ) ) = 1 } = { p e. (Poly ` S ) | ( (coeff ` p ) ` (deg ` p ) ) = 1 } )
13		fvex 6201	. . . . . . 7 |- (Poly ` S ) e. _V
14	13	rabex 4813	. . . . . 6 |- { p e. (Poly ` S ) | ( (coeff ` p ) ` (deg ` p ) ) = 1 } e. _V
15	12, 1, 14	fvmpt 6282	. . . . 5 |- ( S e. ~P CC -> ( Monic ` S ) = { p e. (Poly ` S ) | ( (coeff ` p ) ` (deg ` p ) ) = 1 } )
16	9, 15	sylbir 225	. . . 4 |- ( S C_ CC -> ( Monic ` S ) = { p e. (Poly ` S ) | ( (coeff ` p ) ` (deg ` p ) ) = 1 } )
17	16	eleq2d 2687	. . 3 |- ( S C_ CC -> ( P e. ( Monic ` S ) <-> P e. { p e. (Poly ` S ) | ( (coeff ` p ) ` (deg ` p ) ) = 1 } ) )
18		fveq2 6191	. . . . . 6 |- ( p = P -> (coeff ` p ) = (coeff ` P ) )
19		fveq2 6191	. . . . . 6 |- ( p = P -> (deg ` p ) = (deg ` P ) )
20	18, 19	fveq12d 6197	. . . . 5 |- ( p = P -> ( (coeff ` p ) ` (deg ` p ) ) = ( (coeff ` P ) ` (deg ` P ) ) )
21	20	eqeq1d 2624	. . . 4 |- ( p = P -> ( ( (coeff ` p ) ` (deg ` p ) ) = 1 <-> ( (coeff ` P ) ` (deg ` P ) ) = 1 ) )
22	21	elrab 3363	. . 3 |- ( P e. { p e. (Poly ` S ) | ( (coeff ` p ) ` (deg ` p ) ) = 1 } <-> ( P e. (Poly ` S ) /\ ( (coeff ` P ) ` (deg ` P ) ) = 1 ) )
23	17, 22	syl6bb 276	. 2 |- ( S C_ CC -> ( P e. ( Monic ` S ) <-> ( P e. (Poly ` S ) /\ ( (coeff ` P ) ` (deg ` P ) ) = 1 ) ) )
24	5, 7, 23	pm5.21nii 368	1 |- ( P e. ( Monic ` S ) <-> ( P e. (Poly ` S ) /\ ( (coeff ` P ) ` (deg ` P ) ) = 1 ) )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   {crab 2916   C_ wss 3574   ~Pcpw 4158   dom cdm 5114   ` cfv 5888   CCcc 9934   1c1 9937  Polycply 23940  coeffccoe 23942  degcdgr 23943   Monic cmnc 37701

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-ne 2795  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-ply 23944  df-mnc 37703

This theorem is referenced by:  mncply  37707  mnccoe  37708