Theorem uc1pval 22408

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 2526	. . . . . . 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 2526	. . . . 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 2526	. . . . . . 7 |- ( r = R -> ( 0g ` (Poly1 ` r ) ) = .0. )
11	10	neeq2d 2745	. . . . . 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 2526	. . . . . . . . 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 2526	. . . . . . 7 |- ( r = R -> (Unit ` r ) = U )
20	16, 19	eleq12d 2549	. . . . . 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 3113	. . . 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 22400	. . . 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 2551	. . . . 5 |- B e. _V
26	25	rabex 4604	. . . 4 |- { f e. B | ( f =/= .0. /\ ( (coe1 ` f ) ` ( D ` f ) ) e. U ) } e. _V
27	22, 23, 26	fvmpt 5957	. . 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 3590	. . . . . 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 2520	. . . . . . . . 9 |- ( -. R e. _V -> P = (/) )
32	31	fveq2d 5876	. . . . . . . 8 |- ( -. R e. _V -> ( Base ` P ) = ( Base ` (/) ) )
33		base0 14546	. . . . . . . 8 |- (/) = ( Base ` (/) )
34	32, 33	syl6eqr 2526	. . . . . . 7 |- ( -. R e. _V -> ( Base ` P ) = (/) )
35	6, 34	syl5eq 2520	. . . . . 6 |- ( -. R e. _V -> B = (/) )
36	29, 35	syl5sseq 3557	. . . . 5 |- ( -. R e. _V -> { f e. B | ( f =/= .0. /\ ( (coe1 ` f ) ` ( D ` f ) ) e. U ) } C_ (/) )
37		ss0 3821	. . . . 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 2511	. . 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 2496	1 |- C = { f e. B | ( f =/= .0. /\ ( (coe1 ` f ) ` ( D ` f ) ) e. U ) }

Syntax hints:   -. wn 3   /\ wa 369   = wceq 1379   e. wcel 1767   =/= wne 2662   {crab 2821   _Vcvv 3118   C_ wss 3481   (/)c0 3790   ` cfv 5594   Basecbs 14507   0gc0g 14712  Unitcui 17160  Poly₁cpl1 18086  coe₁cco1 18087   deg₁ cdg1 22320  Unic_1pcuc1p 22395

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-iota 5557  df-fun 5596  df-fv 5602  df-slot 14511  df-base 14512  df-uc1p 22400

This theorem is referenced by:  isuc1p  22409