Theorem uc1pval 22666

Description: Value of the set of unitic 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 )
uc1pval.c	|- C = (Unic1p ` R )
uc1pval.u	|- U = (Unit ` R )

Assertion

Ref	Expression
uc1pval	|- C = { f e. B | ( f =/= .0. /\ ( (coe1 ` f ) ` ( D ` f ) ) e. U ) }

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

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

Proof of Theorem uc1pval

Dummy variable r is distinct from all other variables.

Step	Hyp	Ref	Expression
1		uc1pval.c	. 2 |- C = (Unic1p ` R )
2		fveq2 5872	. . . . . . . 8 |- ( r = R -> (Poly1 ` r ) = (Poly1 ` R ) )
3		uc1pval.p	. . . . . . . 8 |- P = (Poly1 ` R )
4	2, 3	syl6eqr 2516	. . . . . . 7 |- ( r = R -> (Poly1 ` r ) = P )
5	4	fveq2d 5876	. . . . . 6 |- ( r = R -> ( Base ` (Poly1 ` r ) ) = ( Base ` P ) )
6		uc1pval.b	. . . . . 6 |- B = ( Base ` P )
7	5, 6	syl6eqr 2516	. . . . 5 |- ( r = R -> ( Base ` (Poly1 ` r ) ) = B )
8	4	fveq2d 5876	. . . . . . . 8 |- ( r = R -> ( 0g ` (Poly1 ` r ) ) = ( 0g ` P ) )
9		uc1pval.z	. . . . . . . 8 |- .0. = ( 0g ` P )
10	8, 9	syl6eqr 2516	. . . . . . 7 |- ( r = R -> ( 0g ` (Poly1 ` r ) ) = .0. )
11	10	neeq2d 2735	. . . . . 6 |- ( r = R -> ( f =/= ( 0g ` (Poly1 ` r ) ) <-> f =/= .0. ) )
12		fveq2 5872	. . . . . . . . . 10 |- ( r = R -> ( deg1 ` r ) = ( deg1 ` R ) )
13		uc1pval.d	. . . . . . . . . 10 |- D = ( deg1 ` R )
14	12, 13	syl6eqr 2516	. . . . . . . . 9 |- ( r = R -> ( deg1 ` r ) = D )
15	14	fveq1d 5874	. . . . . . . 8 |- ( r = R -> ( ( deg1 ` r ) ` f ) = ( D ` f ) )
16	15	fveq2d 5876	. . . . . . 7 |- ( r = R -> ( (coe1 ` f ) ` ( ( deg1 ` r ) ` f ) ) = ( (coe1 ` f ) ` ( D ` f ) ) )
17		fveq2 5872	. . . . . . . 8 |- ( r = R -> (Unit ` r ) = (Unit ` R ) )
18		uc1pval.u	. . . . . . . 8 |- U = (Unit ` R )
19	17, 18	syl6eqr 2516	. . . . . . 7 |- ( r = R -> (Unit ` r ) = U )
20	16, 19	eleq12d 2539	. . . . . 6 |- ( r = R -> ( ( (coe1 ` f ) ` ( ( deg1 ` r ) ` f ) ) e. (Unit ` r ) <-> ( (coe1 ` f ) ` ( D ` f ) ) e. U ) )
21	11, 20	anbi12d 710	. . . . 5 |- ( r = R -> ( ( f =/= ( 0g ` (Poly1 ` r ) ) /\ ( (coe1 ` f ) ` ( ( deg1 ` r ) ` f ) ) e. (Unit ` r ) ) <-> ( f =/= .0. /\ ( (coe1 ` f ) ` ( D ` f ) ) e. U ) ) )
22	7, 21	rabeqbidv 3104	. . . 4 |- ( r = R -> { f e. ( Base ` (Poly1 ` r ) ) | ( f =/= ( 0g ` (Poly1 ` r ) ) /\ ( (coe1 ` f ) ` ( ( deg1 ` r ) ` f ) ) e. (Unit ` r ) ) } = { f e. B | ( f =/= .0. /\ ( (coe1 ` f ) ` ( D ` f ) ) e. U ) } )
23		df-uc1p 22658	. . . 4 |- Unic1p = ( r e. _V |-> { f e. ( Base ` (Poly1 ` r ) ) | ( f =/= ( 0g ` (Poly1 ` r ) ) /\ ( (coe1 ` f ) ` ( ( deg1 ` r ) ` f ) ) e. (Unit ` r ) ) } )
24		fvex 5882	. . . . . 6 |- ( Base ` P ) e. _V
25	6, 24	eqeltri 2541	. . . . 5 |- B e. _V
26	25	rabex 4607	. . . 4 |- { f e. B | ( f =/= .0. /\ ( (coe1 ` f ) ` ( D ` f ) ) e. U ) } e. _V
27	22, 23, 26	fvmpt 5956	. . 3 |- ( R e. _V -> (Unic1p ` R ) = { f e. B | ( f =/= .0. /\ ( (coe1 ` f ) ` ( D ` f ) ) e. U ) } )
28		fvprc 5866	. . . 4 |- ( -. R e. _V -> (Unic1p ` R ) = (/) )
29		ssrab2 3581	. . . . . 6 |- { f e. B | ( f =/= .0. /\ ( (coe1 ` f ) ` ( D ` f ) ) e. U ) } C_ B
30		fvprc 5866	. . . . . . . . . 10 |- ( -. R e. _V -> (Poly1 ` R ) = (/) )
31	3, 30	syl5eq 2510	. . . . . . . . 9 |- ( -. R e. _V -> P = (/) )
32	31	fveq2d 5876	. . . . . . . 8 |- ( -. R e. _V -> ( Base ` P ) = ( Base ` (/) ) )
33		base0 14685	. . . . . . . 8 |- (/) = ( Base ` (/) )
34	32, 33	syl6eqr 2516	. . . . . . 7 |- ( -. R e. _V -> ( Base ` P ) = (/) )
35	6, 34	syl5eq 2510	. . . . . 6 |- ( -. R e. _V -> B = (/) )
36	29, 35	syl5sseq 3547	. . . . 5 |- ( -. R e. _V -> { f e. B | ( f =/= .0. /\ ( (coe1 ` f ) ` ( D ` f ) ) e. U ) } C_ (/) )
37		ss0 3825	. . . . 5 |- ( { f e. B | ( f =/= .0. /\ ( (coe1 ` f ) ` ( D ` f ) ) e. U ) } C_ (/) -> { f e. B | ( f =/= .0. /\ ( (coe1 ` f ) ` ( D ` f ) ) e. U ) } = (/) )
38	36, 37	syl 16	. . . 4 |- ( -. R e. _V -> { f e. B | ( f =/= .0. /\ ( (coe1 ` f ) ` ( D ` f ) ) e. U ) } = (/) )
39	28, 38	eqtr4d 2501	. . 3 |- ( -. R e. _V -> (Unic1p ` R ) = { f e. B | ( f =/= .0. /\ ( (coe1 ` f ) ` ( D ` f ) ) e. U ) } )
40	27, 39	pm2.61i 164	. 2 |- (Unic1p ` R ) = { f e. B | ( f =/= .0. /\ ( (coe1 ` f ) ` ( D ` f ) ) e. U ) }
41	1, 40	eqtri 2486	1 |- C = { f e. B | ( f =/= .0. /\ ( (coe1 ` f ) ` ( D ` f ) ) e. U ) }

Syntax hints:   -. wn 3   /\ wa 369   = wceq 1395   e. wcel 1819   =/= wne 2652   {crab 2811   _Vcvv 3109   C_ wss 3471   (/)c0 3793   ` cfv 5594   Basecbs 14644   0gc0g 14857  Unitcui 17415  Poly₁cpl1 18343  coe₁cco1 18344   deg₁ cdg1 22578  Unic_1pcuc1p 22653

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-iota 5557  df-fun 5596  df-fv 5602  df-slot 14648  df-base 14649  df-uc1p 22658

This theorem is referenced by:  isuc1p  22667