Theorem hbtlem1 37693

Description: Value of the leading coefficient sequence function. (Contributed by Stefan O'Rear, 31-Mar-2015.)

Hypotheses

Ref	Expression
hbtlem.p	|- P = (Poly1 ` R )
hbtlem.u	|- U = (LIdeal ` P )
hbtlem.s	|- S = (ldgIdlSeq ` R )
hbtlem.d	|- D = ( deg1 ` R )

Assertion

Ref	Expression
hbtlem1	|- ( ( R e. V /\ I e. U /\ X e. NN0 ) -> ( ( S ` I ) ` X ) = { j | E. k e. I ( ( D ` k ) <_ X /\ j = ( (coe1 ` k ) ` X ) ) } )

Distinct variable groups:   j, I, k   R, j, k   j, X, k

Allowed substitution hints:   D( j, k)   P( j, k)   S( j, k)   U( j, k)   V( j, k)

Proof of Theorem hbtlem1

Dummy variables i r x are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		hbtlem.s	. . . . . 6 |- S = (ldgIdlSeq ` R )
2		elex 3212	. . . . . . 7 |- ( R e. V -> R e. _V )
3		fveq2 6191	. . . . . . . . . . . 12 |- ( r = R -> (Poly1 ` r ) = (Poly1 ` R ) )
4		hbtlem.p	. . . . . . . . . . . 12 |- P = (Poly1 ` R )
5	3, 4	syl6eqr 2674	. . . . . . . . . . 11 |- ( r = R -> (Poly1 ` r ) = P )
6	5	fveq2d 6195	. . . . . . . . . 10 |- ( r = R -> (LIdeal ` (Poly1 ` r ) ) = (LIdeal ` P ) )
7		hbtlem.u	. . . . . . . . . 10 |- U = (LIdeal ` P )
8	6, 7	syl6eqr 2674	. . . . . . . . 9 |- ( r = R -> (LIdeal ` (Poly1 ` r ) ) = U )
9		fveq2 6191	. . . . . . . . . . . . . . . 16 |- ( r = R -> ( deg1 ` r ) = ( deg1 ` R ) )
10		hbtlem.d	. . . . . . . . . . . . . . . 16 |- D = ( deg1 ` R )
11	9, 10	syl6eqr 2674	. . . . . . . . . . . . . . 15 |- ( r = R -> ( deg1 ` r ) = D )
12	11	fveq1d 6193	. . . . . . . . . . . . . 14 |- ( r = R -> ( ( deg1 ` r ) ` k ) = ( D ` k ) )
13	12	breq1d 4663	. . . . . . . . . . . . 13 |- ( r = R -> ( ( ( deg1 ` r ) ` k ) <_ x <-> ( D ` k ) <_ x ) )
14	13	anbi1d 741	. . . . . . . . . . . 12 |- ( r = R -> ( ( ( ( deg1 ` r ) ` k ) <_ x /\ j = ( (coe1 ` k ) ` x ) ) <-> ( ( D ` k ) <_ x /\ j = ( (coe1 ` k ) ` x ) ) ) )
15	14	rexbidv 3052	. . . . . . . . . . 11 |- ( r = R -> ( E. k e. i ( ( ( deg1 ` r ) ` k ) <_ x /\ j = ( (coe1 ` k ) ` x ) ) <-> E. k e. i ( ( D ` k ) <_ x /\ j = ( (coe1 ` k ) ` x ) ) ) )
16	15	abbidv 2741	. . . . . . . . . 10 |- ( r = R -> { j | E. k e. i ( ( ( deg1 ` r ) ` k ) <_ x /\ j = ( (coe1 ` k ) ` x ) ) } = { j | E. k e. i ( ( D ` k ) <_ x /\ j = ( (coe1 ` k ) ` x ) ) } )
17	16	mpteq2dv 4745	. . . . . . . . 9 |- ( r = R -> ( x e. NN0 |-> { j | E. k e. i ( ( ( deg1 ` r ) ` k ) <_ x /\ j = ( (coe1 ` k ) ` x ) ) } ) = ( x e. NN0 |-> { j | E. k e. i ( ( D ` k ) <_ x /\ j = ( (coe1 ` k ) ` x ) ) } ) )
18	8, 17	mpteq12dv 4733	. . . . . . . 8 |- ( r = R -> ( i e. (LIdeal ` (Poly1 ` r ) ) |-> ( x e. NN0 |-> { j | E. k e. i ( ( ( deg1 ` r ) ` k ) <_ x /\ j = ( (coe1 ` k ) ` x ) ) } ) ) = ( i e. U |-> ( x e. NN0 |-> { j | E. k e. i ( ( D ` k ) <_ x /\ j = ( (coe1 ` k ) ` x ) ) } ) ) )
19		df-ldgis 37692	. . . . . . . 8 |- ldgIdlSeq = ( r e. _V |-> ( i e. (LIdeal ` (Poly1 ` r ) ) |-> ( x e. NN0 |-> { j | E. k e. i ( ( ( deg1 ` r ) ` k ) <_ x /\ j = ( (coe1 ` k ) ` x ) ) } ) ) )
20		fvex 6201	. . . . . . . . . 10 |- (LIdeal ` P ) e. _V
21	7, 20	eqeltri 2697	. . . . . . . . 9 |- U e. _V
22	21	mptex 6486	. . . . . . . 8 |- ( i e. U |-> ( x e. NN0 |-> { j | E. k e. i ( ( D ` k ) <_ x /\ j = ( (coe1 ` k ) ` x ) ) } ) ) e. _V
23	18, 19, 22	fvmpt 6282	. . . . . . 7 |- ( R e. _V -> (ldgIdlSeq ` R ) = ( i e. U |-> ( x e. NN0 |-> { j | E. k e. i ( ( D ` k ) <_ x /\ j = ( (coe1 ` k ) ` x ) ) } ) ) )
24	2, 23	syl 17	. . . . . 6 |- ( R e. V -> (ldgIdlSeq ` R ) = ( i e. U |-> ( x e. NN0 |-> { j | E. k e. i ( ( D ` k ) <_ x /\ j = ( (coe1 ` k ) ` x ) ) } ) ) )
25	1, 24	syl5eq 2668	. . . . 5 |- ( R e. V -> S = ( i e. U |-> ( x e. NN0 |-> { j | E. k e. i ( ( D ` k ) <_ x /\ j = ( (coe1 ` k ) ` x ) ) } ) ) )
26	25	fveq1d 6193	. . . 4 |- ( R e. V -> ( S ` I ) = ( ( i e. U |-> ( x e. NN0 |-> { j | E. k e. i ( ( D ` k ) <_ x /\ j = ( (coe1 ` k ) ` x ) ) } ) ) ` I ) )
27	26	fveq1d 6193	. . 3 |- ( R e. V -> ( ( S ` I ) ` X ) = ( ( ( i e. U |-> ( x e. NN0 |-> { j | E. k e. i ( ( D ` k ) <_ x /\ j = ( (coe1 ` k ) ` x ) ) } ) ) ` I ) ` X ) )
28	27	3ad2ant1 1082	. 2 |- ( ( R e. V /\ I e. U /\ X e. NN0 ) -> ( ( S ` I ) ` X ) = ( ( ( i e. U |-> ( x e. NN0 |-> { j | E. k e. i ( ( D ` k ) <_ x /\ j = ( (coe1 ` k ) ` x ) ) } ) ) ` I ) ` X ) )
29		rexeq 3139	. . . . . . 7 |- ( i = I -> ( E. k e. i ( ( D ` k ) <_ x /\ j = ( (coe1 ` k ) ` x ) ) <-> E. k e. I ( ( D ` k ) <_ x /\ j = ( (coe1 ` k ) ` x ) ) ) )
30	29	abbidv 2741	. . . . . 6 |- ( i = I -> { j | E. k e. i ( ( D ` k ) <_ x /\ j = ( (coe1 ` k ) ` x ) ) } = { j | E. k e. I ( ( D ` k ) <_ x /\ j = ( (coe1 ` k ) ` x ) ) } )
31	30	mpteq2dv 4745	. . . . 5 |- ( i = I -> ( x e. NN0 |-> { j | E. k e. i ( ( D ` k ) <_ x /\ j = ( (coe1 ` k ) ` x ) ) } ) = ( x e. NN0 |-> { j | E. k e. I ( ( D ` k ) <_ x /\ j = ( (coe1 ` k ) ` x ) ) } ) )
32		eqid 2622	. . . . 5 |- ( i e. U |-> ( x e. NN0 |-> { j | E. k e. i ( ( D ` k ) <_ x /\ j = ( (coe1 ` k ) ` x ) ) } ) ) = ( i e. U |-> ( x e. NN0 |-> { j | E. k e. i ( ( D ` k ) <_ x /\ j = ( (coe1 ` k ) ` x ) ) } ) )
33		nn0ex 11298	. . . . . 6 |- NN0 e. _V
34	33	mptex 6486	. . . . 5 |- ( x e. NN0 |-> { j | E. k e. I ( ( D ` k ) <_ x /\ j = ( (coe1 ` k ) ` x ) ) } ) e. _V
35	31, 32, 34	fvmpt 6282	. . . 4 |- ( I e. U -> ( ( i e. U |-> ( x e. NN0 |-> { j | E. k e. i ( ( D ` k ) <_ x /\ j = ( (coe1 ` k ) ` x ) ) } ) ) ` I ) = ( x e. NN0 |-> { j | E. k e. I ( ( D ` k ) <_ x /\ j = ( (coe1 ` k ) ` x ) ) } ) )
36	35	fveq1d 6193	. . 3 |- ( I e. U -> ( ( ( i e. U |-> ( x e. NN0 |-> { j | E. k e. i ( ( D ` k ) <_ x /\ j = ( (coe1 ` k ) ` x ) ) } ) ) ` I ) ` X ) = ( ( x e. NN0 |-> { j | E. k e. I ( ( D ` k ) <_ x /\ j = ( (coe1 ` k ) ` x ) ) } ) ` X ) )
37	36	3ad2ant2 1083	. 2 |- ( ( R e. V /\ I e. U /\ X e. NN0 ) -> ( ( ( i e. U |-> ( x e. NN0 |-> { j | E. k e. i ( ( D ` k ) <_ x /\ j = ( (coe1 ` k ) ` x ) ) } ) ) ` I ) ` X ) = ( ( x e. NN0 |-> { j | E. k e. I ( ( D ` k ) <_ x /\ j = ( (coe1 ` k ) ` x ) ) } ) ` X ) )
38		simp3 1063	. . 3 |- ( ( R e. V /\ I e. U /\ X e. NN0 ) -> X e. NN0 )
39		simpr 477	. . . . . 6 |- ( ( ( D ` k ) <_ X /\ j = ( (coe1 ` k ) ` X ) ) -> j = ( (coe1 ` k ) ` X ) )
40	39	reximi 3011	. . . . 5 |- ( E. k e. I ( ( D ` k ) <_ X /\ j = ( (coe1 ` k ) ` X ) ) -> E. k e. I j = ( (coe1 ` k ) ` X ) )
41	40	ss2abi 3674	. . . 4 |- { j | E. k e. I ( ( D ` k ) <_ X /\ j = ( (coe1 ` k ) ` X ) ) } C_ { j | E. k e. I j = ( (coe1 ` k ) ` X ) }
42		abrexexg 7140	. . . . 5 |- ( I e. U -> { j | E. k e. I j = ( (coe1 ` k ) ` X ) } e. _V )
43	42	3ad2ant2 1083	. . . 4 |- ( ( R e. V /\ I e. U /\ X e. NN0 ) -> { j | E. k e. I j = ( (coe1 ` k ) ` X ) } e. _V )
44		ssexg 4804	. . . 4 |- ( ( { j | E. k e. I ( ( D ` k ) <_ X /\ j = ( (coe1 ` k ) ` X ) ) } C_ { j | E. k e. I j = ( (coe1 ` k ) ` X ) } /\ { j | E. k e. I j = ( (coe1 ` k ) ` X ) } e. _V ) -> { j | E. k e. I ( ( D ` k ) <_ X /\ j = ( (coe1 ` k ) ` X ) ) } e. _V )
45	41, 43, 44	sylancr 695	. . 3 |- ( ( R e. V /\ I e. U /\ X e. NN0 ) -> { j | E. k e. I ( ( D ` k ) <_ X /\ j = ( (coe1 ` k ) ` X ) ) } e. _V )
46		breq2 4657	. . . . . . 7 |- ( x = X -> ( ( D ` k ) <_ x <-> ( D ` k ) <_ X ) )
47		fveq2 6191	. . . . . . . 8 |- ( x = X -> ( (coe1 ` k ) ` x ) = ( (coe1 ` k ) ` X ) )
48	47	eqeq2d 2632	. . . . . . 7 |- ( x = X -> ( j = ( (coe1 ` k ) ` x ) <-> j = ( (coe1 ` k ) ` X ) ) )
49	46, 48	anbi12d 747	. . . . . 6 |- ( x = X -> ( ( ( D ` k ) <_ x /\ j = ( (coe1 ` k ) ` x ) ) <-> ( ( D ` k ) <_ X /\ j = ( (coe1 ` k ) ` X ) ) ) )
50	49	rexbidv 3052	. . . . 5 |- ( x = X -> ( E. k e. I ( ( D ` k ) <_ x /\ j = ( (coe1 ` k ) ` x ) ) <-> E. k e. I ( ( D ` k ) <_ X /\ j = ( (coe1 ` k ) ` X ) ) ) )
51	50	abbidv 2741	. . . 4 |- ( x = X -> { j | E. k e. I ( ( D ` k ) <_ x /\ j = ( (coe1 ` k ) ` x ) ) } = { j | E. k e. I ( ( D ` k ) <_ X /\ j = ( (coe1 ` k ) ` X ) ) } )
52		eqid 2622	. . . 4 |- ( x e. NN0 |-> { j | E. k e. I ( ( D ` k ) <_ x /\ j = ( (coe1 ` k ) ` x ) ) } ) = ( x e. NN0 |-> { j | E. k e. I ( ( D ` k ) <_ x /\ j = ( (coe1 ` k ) ` x ) ) } )
53	51, 52	fvmptg 6280	. . 3 |- ( ( X e. NN0 /\ { j | E. k e. I ( ( D ` k ) <_ X /\ j = ( (coe1 ` k ) ` X ) ) } e. _V ) -> ( ( x e. NN0 |-> { j | E. k e. I ( ( D ` k ) <_ x /\ j = ( (coe1 ` k ) ` x ) ) } ) ` X ) = { j | E. k e. I ( ( D ` k ) <_ X /\ j = ( (coe1 ` k ) ` X ) ) } )
54	38, 45, 53	syl2anc 693	. 2 |- ( ( R e. V /\ I e. U /\ X e. NN0 ) -> ( ( x e. NN0 |-> { j | E. k e. I ( ( D ` k ) <_ x /\ j = ( (coe1 ` k ) ` x ) ) } ) ` X ) = { j | E. k e. I ( ( D ` k ) <_ X /\ j = ( (coe1 ` k ) ` X ) ) } )
55	28, 37, 54	3eqtrd 2660	1 |- ( ( R e. V /\ I e. U /\ X e. NN0 ) -> ( ( S ` I ) ` X ) = { j | E. k e. I ( ( D ` k ) <_ X /\ j = ( (coe1 ` k ) ` X ) ) } )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   {cab 2608   E.wrex 2913   _Vcvv 3200   C_ wss 3574   class class class wbr 4653   |-> cmpt 4729   ` cfv 5888   <_ cle 10075   NN0cn0 11292  LIdealclidl 19170  Poly₁cpl1 19547  coe₁cco1 19548   deg₁ cdg1 23814  ldgIdlSeqcldgis 37691

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-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-i2m1 10004  ax-1ne0 10005  ax-rrecex 10008  ax-cnre 10009

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  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-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-nn 11021  df-n0 11293  df-ldgis 37692

This theorem is referenced by:  hbtlem2  37694  hbtlem4  37696  hbtlem3  37697  hbtlem5  37698  hbtlem6  37699