Theorem uc1pval 21611

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 5691	. . . . . . . 8 |- ( r = R -> (Poly1 ` r ) = (Poly1 ` R ) )
3		uc1pval.p	. . . . . . . 8 |- P = (Poly1 ` R )
4	2, 3	syl6eqr 2493	. . . . . . 7 |- ( r = R -> (Poly1 ` r ) = P )
5	4	fveq2d 5695	. . . . . 6 |- ( r = R -> ( Base ` (Poly1 ` r ) ) = ( Base ` P ) )
6		uc1pval.b	. . . . . 6 |- B = ( Base ` P )
7	5, 6	syl6eqr 2493	. . . . 5 |- ( r = R -> ( Base ` (Poly1 ` r ) ) = B )
8	4	fveq2d 5695	. . . . . . . 8 |- ( r = R -> ( 0g ` (Poly1 ` r ) ) = ( 0g ` P ) )
9		uc1pval.z	. . . . . . . 8 |- .0. = ( 0g ` P )
10	8, 9	syl6eqr 2493	. . . . . . 7 |- ( r = R -> ( 0g ` (Poly1 ` r ) ) = .0. )
11	10	neeq2d 2622	. . . . . 6 |- ( r = R -> ( f =/= ( 0g ` (Poly1 ` r ) ) <-> f =/= .0. ) )
12		fveq2 5691	. . . . . . . . . 10 |- ( r = R -> ( deg1 ` r ) = ( deg1 ` R ) )
13		uc1pval.d	. . . . . . . . . 10 |- D = ( deg1 ` R )
14	12, 13	syl6eqr 2493	. . . . . . . . 9 |- ( r = R -> ( deg1 ` r ) = D )
15	14	fveq1d 5693	. . . . . . . 8 |- ( r = R -> ( ( deg1 ` r ) ` f ) = ( D ` f ) )
16	15	fveq2d 5695	. . . . . . 7 |- ( r = R -> ( (coe1 ` f ) ` ( ( deg1 ` r ) ` f ) ) = ( (coe1 ` f ) ` ( D ` f ) ) )
17		fveq2 5691	. . . . . . . 8 |- ( r = R -> (Unit ` r ) = (Unit ` R ) )
18		uc1pval.u	. . . . . . . 8 |- U = (Unit ` R )
19	17, 18	syl6eqr 2493	. . . . . . 7 |- ( r = R -> (Unit ` r ) = U )
20	16, 19	eleq12d 2511	. . . . . 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 2967	. . . 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 21603	. . . 4 |- Unic1p = ( r e. _V |-> { f e. ( Base ` (Poly1 ` r ) ) | ( f =/= ( 0g ` (Poly1 ` r ) ) /\ ( (coe1 ` f ) ` ( ( deg1 ` r ) ` f ) ) e. (Unit ` r ) ) } )
24		fvex 5701	. . . . . 6 |- ( Base ` P ) e. _V
25	6, 24	eqeltri 2513	. . . . 5 |- B e. _V
26	25	rabex 4443	. . . 4 |- { f e. B | ( f =/= .0. /\ ( (coe1 ` f ) ` ( D ` f ) ) e. U ) } e. _V
27	22, 23, 26	fvmpt 5774	. . 3 |- ( R e. _V -> (Unic1p ` R ) = { f e. B | ( f =/= .0. /\ ( (coe1 ` f ) ` ( D ` f ) ) e. U ) } )
28		fvprc 5685	. . . 4 |- ( -. R e. _V -> (Unic1p ` R ) = (/) )
29		ssrab2 3437	. . . . . 6 |- { f e. B | ( f =/= .0. /\ ( (coe1 ` f ) ` ( D ` f ) ) e. U ) } C_ B
30		fvprc 5685	. . . . . . . . . 10 |- ( -. R e. _V -> (Poly1 ` R ) = (/) )
31	3, 30	syl5eq 2487	. . . . . . . . 9 |- ( -. R e. _V -> P = (/) )
32	31	fveq2d 5695	. . . . . . . 8 |- ( -. R e. _V -> ( Base ` P ) = ( Base ` (/) ) )
33		base0 14213	. . . . . . . 8 |- (/) = ( Base ` (/) )
34	32, 33	syl6eqr 2493	. . . . . . 7 |- ( -. R e. _V -> ( Base ` P ) = (/) )
35	6, 34	syl5eq 2487	. . . . . 6 |- ( -. R e. _V -> B = (/) )
36	29, 35	syl5sseq 3404	. . . . 5 |- ( -. R e. _V -> { f e. B | ( f =/= .0. /\ ( (coe1 ` f ) ` ( D ` f ) ) e. U ) } C_ (/) )
37		ss0 3668	. . . . 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 2478	. . 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 2463	1 |- C = { f e. B | ( f =/= .0. /\ ( (coe1 ` f ) ` ( D ` f ) ) e. U ) }

Syntax hints:   -. wn 3   /\ wa 369   = wceq 1369   e. wcel 1756   =/= wne 2606   {crab 2719   _Vcvv 2972   C_ wss 3328   (/)c0 3637   ` cfv 5418   Basecbs 14174   0gc0g 14378  Unitcui 16731  Poly₁cpl1 17633  coe₁cco1 17634   deg₁ cdg1 21523  Unic_1pcuc1p 21598

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-sbc 3187  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-iota 5381  df-fun 5420  df-fv 5426  df-slot 14178  df-base 14179  df-uc1p 21603

This theorem is referenced by:  isuc1p  21612