Theorem uc1pval 23169

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 5879	. . . . . . . 8 |- ( r = R -> (Poly1 ` r ) = (Poly1 ` R ) )
3		uc1pval.p	. . . . . . . 8 |- P = (Poly1 ` R )
4	2, 3	syl6eqr 2523	. . . . . . 7 |- ( r = R -> (Poly1 ` r ) = P )
5	4	fveq2d 5883	. . . . . 6 |- ( r = R -> ( Base ` (Poly1 ` r ) ) = ( Base ` P ) )
6		uc1pval.b	. . . . . 6 |- B = ( Base ` P )
7	5, 6	syl6eqr 2523	. . . . 5 |- ( r = R -> ( Base ` (Poly1 ` r ) ) = B )
8	4	fveq2d 5883	. . . . . . . 8 |- ( r = R -> ( 0g ` (Poly1 ` r ) ) = ( 0g ` P ) )
9		uc1pval.z	. . . . . . . 8 |- .0. = ( 0g ` P )
10	8, 9	syl6eqr 2523	. . . . . . 7 |- ( r = R -> ( 0g ` (Poly1 ` r ) ) = .0. )
11	10	neeq2d 2703	. . . . . 6 |- ( r = R -> ( f =/= ( 0g ` (Poly1 ` r ) ) <-> f =/= .0. ) )
12		fveq2 5879	. . . . . . . . . 10 |- ( r = R -> ( deg1 ` r ) = ( deg1 ` R ) )
13		uc1pval.d	. . . . . . . . . 10 |- D = ( deg1 ` R )
14	12, 13	syl6eqr 2523	. . . . . . . . 9 |- ( r = R -> ( deg1 ` r ) = D )
15	14	fveq1d 5881	. . . . . . . 8 |- ( r = R -> ( ( deg1 ` r ) ` f ) = ( D ` f ) )
16	15	fveq2d 5883	. . . . . . 7 |- ( r = R -> ( (coe1 ` f ) ` ( ( deg1 ` r ) ` f ) ) = ( (coe1 ` f ) ` ( D ` f ) ) )
17		fveq2 5879	. . . . . . . 8 |- ( r = R -> (Unit ` r ) = (Unit ` R ) )
18		uc1pval.u	. . . . . . . 8 |- U = (Unit ` R )
19	17, 18	syl6eqr 2523	. . . . . . 7 |- ( r = R -> (Unit ` r ) = U )
20	16, 19	eleq12d 2543	. . . . . 6 |- ( r = R -> ( ( (coe1 ` f ) ` ( ( deg1 ` r ) ` f ) ) e. (Unit ` r ) <-> ( (coe1 ` f ) ` ( D ` f ) ) e. U ) )
21	11, 20	anbi12d 725	. . . . 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 3026	. . . 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 23160	. . . 4 |- Unic1p = ( r e. _V |-> { f e. ( Base ` (Poly1 ` r ) ) | ( f =/= ( 0g ` (Poly1 ` r ) ) /\ ( (coe1 ` f ) ` ( ( deg1 ` r ) ` f ) ) e. (Unit ` r ) ) } )
24		fvex 5889	. . . . . 6 |- ( Base ` P ) e. _V
25	6, 24	eqeltri 2545	. . . . 5 |- B e. _V
26	25	rabex 4550	. . . 4 |- { f e. B | ( f =/= .0. /\ ( (coe1 ` f ) ` ( D ` f ) ) e. U ) } e. _V
27	22, 23, 26	fvmpt 5963	. . 3 |- ( R e. _V -> (Unic1p ` R ) = { f e. B | ( f =/= .0. /\ ( (coe1 ` f ) ` ( D ` f ) ) e. U ) } )
28		fvprc 5873	. . . 4 |- ( -. R e. _V -> (Unic1p ` R ) = (/) )
29		ssrab2 3500	. . . . . 6 |- { f e. B | ( f =/= .0. /\ ( (coe1 ` f ) ` ( D ` f ) ) e. U ) } C_ B
30		fvprc 5873	. . . . . . . . . 10 |- ( -. R e. _V -> (Poly1 ` R ) = (/) )
31	3, 30	syl5eq 2517	. . . . . . . . 9 |- ( -. R e. _V -> P = (/) )
32	31	fveq2d 5883	. . . . . . . 8 |- ( -. R e. _V -> ( Base ` P ) = ( Base ` (/) ) )
33		base0 15240	. . . . . . . 8 |- (/) = ( Base ` (/) )
34	32, 33	syl6eqr 2523	. . . . . . 7 |- ( -. R e. _V -> ( Base ` P ) = (/) )
35	6, 34	syl5eq 2517	. . . . . 6 |- ( -. R e. _V -> B = (/) )
36	29, 35	syl5sseq 3466	. . . . 5 |- ( -. R e. _V -> { f e. B | ( f =/= .0. /\ ( (coe1 ` f ) ` ( D ` f ) ) e. U ) } C_ (/) )
37		ss0 3768	. . . . 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 17	. . . 4 |- ( -. R e. _V -> { f e. B | ( f =/= .0. /\ ( (coe1 ` f ) ` ( D ` f ) ) e. U ) } = (/) )
39	28, 38	eqtr4d 2508	. . 3 |- ( -. R e. _V -> (Unic1p ` R ) = { f e. B | ( f =/= .0. /\ ( (coe1 ` f ) ` ( D ` f ) ) e. U ) } )
40	27, 39	pm2.61i 169	. 2 |- (Unic1p ` R ) = { f e. B | ( f =/= .0. /\ ( (coe1 ` f ) ` ( D ` f ) ) e. U ) }
41	1, 40	eqtri 2493	1 |- C = { f e. B | ( f =/= .0. /\ ( (coe1 ` f ) ` ( D ` f ) ) e. U ) }

Syntax hints:   -. wn 3   /\ wa 376   = wceq 1452   e. wcel 1904   =/= wne 2641   {crab 2760   _Vcvv 3031   C_ wss 3390   (/)c0 3722   ` cfv 5589   Basecbs 15199   0gc0g 15416  Unitcui 17945  Poly₁cpl1 18847  coe₁cco1 18848   deg₁ cdg1 23082  Unic_1pcuc1p 23154

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-iota 5553  df-fun 5591  df-fv 5597  df-slot 15203  df-base 15204  df-uc1p 23160

This theorem is referenced by:  isuc1p  23170