Theorem mon1pval 23901

Description: Value of the set of monic polynomials. (Contributed by Stefan O'Rear, 28-Mar-2015.)

Hypotheses

Ref	Expression
uc1pval.p	|- P = (Poly1 ` R )
uc1pval.b	|- B = ( Base ` P )
uc1pval.z	|- .0. = ( 0g ` P )
uc1pval.d	|- D = ( deg1 ` R )
mon1pval.m	|- M = (Monic1p ` R )
mon1pval.o	|- .1. = ( 1r ` R )

Assertion

Ref	Expression
mon1pval	|- M = { f e. B | ( f =/= .0. /\ ( (coe1 ` f ) ` ( D ` f ) ) = .1. ) }

Distinct variable groups:   B, f   D, f   .1. , f   R, f   .0. , f

Allowed substitution hints:   P( f)   M( f)

Proof of Theorem mon1pval

Dummy variable r is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mon1pval.m	. 2 |- M = (Monic1p ` R )
2		fveq2 6191	. . . . . . . 8 |- ( r = R -> (Poly1 ` r ) = (Poly1 ` R ) )
3		uc1pval.p	. . . . . . . 8 |- P = (Poly1 ` R )
4	2, 3	syl6eqr 2674	. . . . . . 7 |- ( r = R -> (Poly1 ` r ) = P )
5	4	fveq2d 6195	. . . . . 6 |- ( r = R -> ( Base ` (Poly1 ` r ) ) = ( Base ` P ) )
6		uc1pval.b	. . . . . 6 |- B = ( Base ` P )
7	5, 6	syl6eqr 2674	. . . . 5 |- ( r = R -> ( Base ` (Poly1 ` r ) ) = B )
8	4	fveq2d 6195	. . . . . . . 8 |- ( r = R -> ( 0g ` (Poly1 ` r ) ) = ( 0g ` P ) )
9		uc1pval.z	. . . . . . . 8 |- .0. = ( 0g ` P )
10	8, 9	syl6eqr 2674	. . . . . . 7 |- ( r = R -> ( 0g ` (Poly1 ` r ) ) = .0. )
11	10	neeq2d 2854	. . . . . 6 |- ( r = R -> ( f =/= ( 0g ` (Poly1 ` r ) ) <-> f =/= .0. ) )
12		fveq2 6191	. . . . . . . . . 10 |- ( r = R -> ( deg1 ` r ) = ( deg1 ` R ) )
13		uc1pval.d	. . . . . . . . . 10 |- D = ( deg1 ` R )
14	12, 13	syl6eqr 2674	. . . . . . . . 9 |- ( r = R -> ( deg1 ` r ) = D )
15	14	fveq1d 6193	. . . . . . . 8 |- ( r = R -> ( ( deg1 ` r ) ` f ) = ( D ` f ) )
16	15	fveq2d 6195	. . . . . . 7 |- ( r = R -> ( (coe1 ` f ) ` ( ( deg1 ` r ) ` f ) ) = ( (coe1 ` f ) ` ( D ` f ) ) )
17		fveq2 6191	. . . . . . . 8 |- ( r = R -> ( 1r ` r ) = ( 1r ` R ) )
18		mon1pval.o	. . . . . . . 8 |- .1. = ( 1r ` R )
19	17, 18	syl6eqr 2674	. . . . . . 7 |- ( r = R -> ( 1r ` r ) = .1. )
20	16, 19	eqeq12d 2637	. . . . . 6 |- ( r = R -> ( ( (coe1 ` f ) ` ( ( deg1 ` r ) ` f ) ) = ( 1r ` r ) <-> ( (coe1 ` f ) ` ( D ` f ) ) = .1. ) )
21	11, 20	anbi12d 747	. . . . 5 |- ( r = R -> ( ( f =/= ( 0g ` (Poly1 ` r ) ) /\ ( (coe1 ` f ) ` ( ( deg1 ` r ) ` f ) ) = ( 1r ` r ) ) <-> ( f =/= .0. /\ ( (coe1 ` f ) ` ( D ` f ) ) = .1. ) ) )
22	7, 21	rabeqbidv 3195	. . . 4 |- ( r = R -> { f e. ( Base ` (Poly1 ` r ) ) | ( f =/= ( 0g ` (Poly1 ` r ) ) /\ ( (coe1 ` f ) ` ( ( deg1 ` r ) ` f ) ) = ( 1r ` r ) ) } = { f e. B | ( f =/= .0. /\ ( (coe1 ` f ) ` ( D ` f ) ) = .1. ) } )
23		df-mon1 23890	. . . 4 |- Monic1p = ( r e. _V |-> { f e. ( Base ` (Poly1 ` r ) ) | ( f =/= ( 0g ` (Poly1 ` r ) ) /\ ( (coe1 ` f ) ` ( ( deg1 ` r ) ` f ) ) = ( 1r ` r ) ) } )
24		fvex 6201	. . . . . 6 |- ( Base ` P ) e. _V
25	6, 24	eqeltri 2697	. . . . 5 |- B e. _V
26	25	rabex 4813	. . . 4 |- { f e. B | ( f =/= .0. /\ ( (coe1 ` f ) ` ( D ` f ) ) = .1. ) } e. _V
27	22, 23, 26	fvmpt 6282	. . 3 |- ( R e. _V -> (Monic1p ` R ) = { f e. B | ( f =/= .0. /\ ( (coe1 ` f ) ` ( D ` f ) ) = .1. ) } )
28		fvprc 6185	. . . 4 |- ( -. R e. _V -> (Monic1p ` R ) = (/) )
29		ssrab2 3687	. . . . . 6 |- { f e. B | ( f =/= .0. /\ ( (coe1 ` f ) ` ( D ` f ) ) = .1. ) } C_ B
30		fvprc 6185	. . . . . . . . . 10 |- ( -. R e. _V -> (Poly1 ` R ) = (/) )
31	3, 30	syl5eq 2668	. . . . . . . . 9 |- ( -. R e. _V -> P = (/) )
32	31	fveq2d 6195	. . . . . . . 8 |- ( -. R e. _V -> ( Base ` P ) = ( Base ` (/) ) )
33	6, 32	syl5eq 2668	. . . . . . 7 |- ( -. R e. _V -> B = ( Base ` (/) ) )
34		base0 15912	. . . . . . 7 |- (/) = ( Base ` (/) )
35	33, 34	syl6eqr 2674	. . . . . 6 |- ( -. R e. _V -> B = (/) )
36	29, 35	syl5sseq 3653	. . . . 5 |- ( -. R e. _V -> { f e. B | ( f =/= .0. /\ ( (coe1 ` f ) ` ( D ` f ) ) = .1. ) } C_ (/) )
37		ss0 3974	. . . . 5 |- ( { f e. B | ( f =/= .0. /\ ( (coe1 ` f ) ` ( D ` f ) ) = .1. ) } C_ (/) -> { f e. B | ( f =/= .0. /\ ( (coe1 ` f ) ` ( D ` f ) ) = .1. ) } = (/) )
38	36, 37	syl 17	. . . 4 |- ( -. R e. _V -> { f e. B | ( f =/= .0. /\ ( (coe1 ` f ) ` ( D ` f ) ) = .1. ) } = (/) )
39	28, 38	eqtr4d 2659	. . 3 |- ( -. R e. _V -> (Monic1p ` R ) = { f e. B | ( f =/= .0. /\ ( (coe1 ` f ) ` ( D ` f ) ) = .1. ) } )
40	27, 39	pm2.61i 176	. 2 |- (Monic1p ` R ) = { f e. B | ( f =/= .0. /\ ( (coe1 ` f ) ` ( D ` f ) ) = .1. ) }
41	1, 40	eqtri 2644	1 |- M = { f e. B | ( f =/= .0. /\ ( (coe1 ` f ) ` ( D ` f ) ) = .1. ) }

Syntax hints:   -. wn 3   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   {crab 2916   _Vcvv 3200   C_ wss 3574   (/)c0 3915   ` cfv 5888   Basecbs 15857   0gc0g 16100   1rcur 18501  Poly₁cpl1 19547  coe₁cco1 19548   deg₁ cdg1 23814  Monic_1pcmn1 23885

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

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-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-iota 5851  df-fun 5890  df-fv 5896  df-slot 15861  df-base 15863  df-mon1 23890

This theorem is referenced by:  ismon1p  23902