Theorem mdegleb 23824

Description: Property of being of limited degree. (Contributed by Stefan O'Rear, 19-Mar-2015.)

Hypotheses

Ref	Expression
mdegval.d	|- D = ( I mDeg R )
mdegval.p	|- P = ( I mPoly R )
mdegval.b	|- B = ( Base ` P )
mdegval.z	|- .0. = ( 0g ` R )
mdegval.a	|- A = { m e. ( NN0 ^m I ) | ( `' m " NN ) e. Fin }
mdegval.h	|- H = ( h e. A |-> (ℂfld gsumg h ) )

Assertion

Ref	Expression
mdegleb	|- ( ( F e. B /\ G e. RR* ) -> ( ( D ` F ) <_ G <-> A. x e. A ( G < ( H ` x ) -> ( F ` x ) = .0. ) ) )

Distinct variable groups:   A, h   m, I   .0. , h   x, A   x, B   x, F   x, G   x, H   h, I   x, R   x, .0.   h, m

Allowed substitution hints:   A( m)   B( h, m)   D( x, h, m)   P( x, h, m)   R( h, m)   F( h, m)   G( h, m)   H( h, m)   I( x)   .0. ( m)

Proof of Theorem mdegleb

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mdegval.d	. . . . 5 |- D = ( I mDeg R )
2		mdegval.p	. . . . 5 |- P = ( I mPoly R )
3		mdegval.b	. . . . 5 |- B = ( Base ` P )
4		mdegval.z	. . . . 5 |- .0. = ( 0g ` R )
5		mdegval.a	. . . . 5 |- A = { m e. ( NN0 ^m I ) | ( `' m " NN ) e. Fin }
6		mdegval.h	. . . . 5 |- H = ( h e. A |-> (ℂfld gsumg h ) )
7	1, 2, 3, 4, 5, 6	mdegval 23823	. . . 4 |- ( F e. B -> ( D ` F ) = sup ( ( H " ( F supp .0. ) ) , RR* , < ) )
8	7	adantr 481	. . 3 |- ( ( F e. B /\ G e. RR* ) -> ( D ` F ) = sup ( ( H " ( F supp .0. ) ) , RR* , < ) )
9	8	breq1d 4663	. 2 |- ( ( F e. B /\ G e. RR* ) -> ( ( D ` F ) <_ G <-> sup ( ( H " ( F supp .0. ) ) , RR* , < ) <_ G ) )
10		imassrn 5477	. . . 4 |- ( H " ( F supp .0. ) ) C_ ran H
11	2, 3	mplrcl 19490	. . . . . . . 8 |- ( F e. B -> I e. _V )
12	11	adantr 481	. . . . . . 7 |- ( ( F e. B /\ G e. RR* ) -> I e. _V )
13	5, 6	tdeglem1 23818	. . . . . . 7 |- ( I e. _V -> H : A --> NN0 )
14	12, 13	syl 17	. . . . . 6 |- ( ( F e. B /\ G e. RR* ) -> H : A --> NN0 )
15		frn 6053	. . . . . 6 |- ( H : A --> NN0 -> ran H C_ NN0 )
16	14, 15	syl 17	. . . . 5 |- ( ( F e. B /\ G e. RR* ) -> ran H C_ NN0 )
17		nn0ssre 11296	. . . . . 6 |- NN0 C_ RR
18		ressxr 10083	. . . . . 6 |- RR C_ RR*
19	17, 18	sstri 3612	. . . . 5 |- NN0 C_ RR*
20	16, 19	syl6ss 3615	. . . 4 |- ( ( F e. B /\ G e. RR* ) -> ran H C_ RR* )
21	10, 20	syl5ss 3614	. . 3 |- ( ( F e. B /\ G e. RR* ) -> ( H " ( F supp .0. ) ) C_ RR* )
22		supxrleub 12156	. . 3 |- ( ( ( H " ( F supp .0. ) ) C_ RR* /\ G e. RR* ) -> ( sup ( ( H " ( F supp .0. ) ) , RR* , < ) <_ G <-> A. y e. ( H " ( F supp .0. ) ) y <_ G ) )
23	21, 22	sylancom 701	. 2 |- ( ( F e. B /\ G e. RR* ) -> ( sup ( ( H " ( F supp .0. ) ) , RR* , < ) <_ G <-> A. y e. ( H " ( F supp .0. ) ) y <_ G ) )
24		ffn 6045	. . . . 5 |- ( H : A --> NN0 -> H Fn A )
25	14, 24	syl 17	. . . 4 |- ( ( F e. B /\ G e. RR* ) -> H Fn A )
26		suppssdm 7308	. . . . 5 |- ( F supp .0. ) C_ dom F
27		eqid 2622	. . . . . . 7 |- ( Base ` R ) = ( Base ` R )
28		simpl 473	. . . . . . 7 |- ( ( F e. B /\ G e. RR* ) -> F e. B )
29	2, 27, 3, 5, 28	mplelf 19433	. . . . . 6 |- ( ( F e. B /\ G e. RR* ) -> F : A --> ( Base ` R ) )
30		fdm 6051	. . . . . 6 |- ( F : A --> ( Base ` R ) -> dom F = A )
31	29, 30	syl 17	. . . . 5 |- ( ( F e. B /\ G e. RR* ) -> dom F = A )
32	26, 31	syl5sseq 3653	. . . 4 |- ( ( F e. B /\ G e. RR* ) -> ( F supp .0. ) C_ A )
33		breq1 4656	. . . . 5 |- ( y = ( H ` x ) -> ( y <_ G <-> ( H ` x ) <_ G ) )
34	33	ralima 6498	. . . 4 |- ( ( H Fn A /\ ( F supp .0. ) C_ A ) -> ( A. y e. ( H " ( F supp .0. ) ) y <_ G <-> A. x e. ( F supp .0. ) ( H ` x ) <_ G ) )
35	25, 32, 34	syl2anc 693	. . 3 |- ( ( F e. B /\ G e. RR* ) -> ( A. y e. ( H " ( F supp .0. ) ) y <_ G <-> A. x e. ( F supp .0. ) ( H ` x ) <_ G ) )
36		ffn 6045	. . . . . . . 8 |- ( F : A --> ( Base ` R ) -> F Fn A )
37	29, 36	syl 17	. . . . . . 7 |- ( ( F e. B /\ G e. RR* ) -> F Fn A )
38		ovex 6678	. . . . . . . . . 10 |- ( NN0 ^m I ) e. _V
39	38	rabex 4813	. . . . . . . . 9 |- { m e. ( NN0 ^m I ) | ( `' m " NN ) e. Fin } e. _V
40	39	a1i 11	. . . . . . . 8 |- ( ( F e. B /\ G e. RR* ) -> { m e. ( NN0 ^m I ) | ( `' m " NN ) e. Fin } e. _V )
41	5, 40	syl5eqel 2705	. . . . . . 7 |- ( ( F e. B /\ G e. RR* ) -> A e. _V )
42		fvex 6201	. . . . . . . . 9 |- ( 0g ` R ) e. _V
43	4, 42	eqeltri 2697	. . . . . . . 8 |- .0. e. _V
44	43	a1i 11	. . . . . . 7 |- ( ( F e. B /\ G e. RR* ) -> .0. e. _V )
45		elsuppfn 7303	. . . . . . . 8 |- ( ( F Fn A /\ A e. _V /\ .0. e. _V ) -> ( x e. ( F supp .0. ) <-> ( x e. A /\ ( F ` x ) =/= .0. ) ) )
46		fvex 6201	. . . . . . . . . . . 12 |- ( F ` x ) e. _V
47	46	biantrur 527	. . . . . . . . . . 11 |- ( ( F ` x ) =/= .0. <-> ( ( F ` x ) e. _V /\ ( F ` x ) =/= .0. ) )
48		eldifsn 4317	. . . . . . . . . . 11 |- ( ( F ` x ) e. ( _V \ { .0. } ) <-> ( ( F ` x ) e. _V /\ ( F ` x ) =/= .0. ) )
49	47, 48	bitr4i 267	. . . . . . . . . 10 |- ( ( F ` x ) =/= .0. <-> ( F ` x ) e. ( _V \ { .0. } ) )
50	49	a1i 11	. . . . . . . . 9 |- ( ( F Fn A /\ A e. _V /\ .0. e. _V ) -> ( ( F ` x ) =/= .0. <-> ( F ` x ) e. ( _V \ { .0. } ) ) )
51	50	anbi2d 740	. . . . . . . 8 |- ( ( F Fn A /\ A e. _V /\ .0. e. _V ) -> ( ( x e. A /\ ( F ` x ) =/= .0. ) <-> ( x e. A /\ ( F ` x ) e. ( _V \ { .0. } ) ) ) )
52	45, 51	bitrd 268	. . . . . . 7 |- ( ( F Fn A /\ A e. _V /\ .0. e. _V ) -> ( x e. ( F supp .0. ) <-> ( x e. A /\ ( F ` x ) e. ( _V \ { .0. } ) ) ) )
53	37, 41, 44, 52	syl3anc 1326	. . . . . 6 |- ( ( F e. B /\ G e. RR* ) -> ( x e. ( F supp .0. ) <-> ( x e. A /\ ( F ` x ) e. ( _V \ { .0. } ) ) ) )
54	53	imbi1d 331	. . . . 5 |- ( ( F e. B /\ G e. RR* ) -> ( ( x e. ( F supp .0. ) -> ( H ` x ) <_ G ) <-> ( ( x e. A /\ ( F ` x ) e. ( _V \ { .0. } ) ) -> ( H ` x ) <_ G ) ) )
55		impexp 462	. . . . . 6 |- ( ( ( x e. A /\ ( F ` x ) e. ( _V \ { .0. } ) ) -> ( H ` x ) <_ G ) <-> ( x e. A -> ( ( F ` x ) e. ( _V \ { .0. } ) -> ( H ` x ) <_ G ) ) )
56		con34b 306	. . . . . . . 8 |- ( ( ( F ` x ) e. ( _V \ { .0. } ) -> ( H ` x ) <_ G ) <-> ( -. ( H ` x ) <_ G -> -. ( F ` x ) e. ( _V \ { .0. } ) ) )
57		simplr 792	. . . . . . . . . . 11 |- ( ( ( F e. B /\ G e. RR* ) /\ x e. A ) -> G e. RR* )
58	14	ffvelrnda 6359	. . . . . . . . . . . 12 |- ( ( ( F e. B /\ G e. RR* ) /\ x e. A ) -> ( H ` x ) e. NN0 )
59	19, 58	sseldi 3601	. . . . . . . . . . 11 |- ( ( ( F e. B /\ G e. RR* ) /\ x e. A ) -> ( H ` x ) e. RR* )
60		xrltnle 10105	. . . . . . . . . . 11 |- ( ( G e. RR* /\ ( H ` x ) e. RR* ) -> ( G < ( H ` x ) <-> -. ( H ` x ) <_ G ) )
61	57, 59, 60	syl2anc 693	. . . . . . . . . 10 |- ( ( ( F e. B /\ G e. RR* ) /\ x e. A ) -> ( G < ( H ` x ) <-> -. ( H ` x ) <_ G ) )
62	61	bicomd 213	. . . . . . . . 9 |- ( ( ( F e. B /\ G e. RR* ) /\ x e. A ) -> ( -. ( H ` x ) <_ G <-> G < ( H ` x ) ) )
63		ianor 509	. . . . . . . . . . 11 |- ( -. ( ( F ` x ) e. _V /\ ( F ` x ) =/= .0. ) <-> ( -. ( F ` x ) e. _V \/ -. ( F ` x ) =/= .0. ) )
64	63, 48	xchnxbir 323	. . . . . . . . . 10 |- ( -. ( F ` x ) e. ( _V \ { .0. } ) <-> ( -. ( F ` x ) e. _V \/ -. ( F ` x ) =/= .0. ) )
65		orcom 402	. . . . . . . . . . . 12 |- ( ( -. ( F ` x ) e. _V \/ -. ( F ` x ) =/= .0. ) <-> ( -. ( F ` x ) =/= .0. \/ -. ( F ` x ) e. _V ) )
66	46	notnoti 137	. . . . . . . . . . . . 13 |- -. -. ( F ` x ) e. _V
67	66	biorfi 422	. . . . . . . . . . . 12 |- ( -. ( F ` x ) =/= .0. <-> ( -. ( F ` x ) =/= .0. \/ -. ( F ` x ) e. _V ) )
68		nne 2798	. . . . . . . . . . . 12 |- ( -. ( F ` x ) =/= .0. <-> ( F ` x ) = .0. )
69	65, 67, 68	3bitr2i 288	. . . . . . . . . . 11 |- ( ( -. ( F ` x ) e. _V \/ -. ( F ` x ) =/= .0. ) <-> ( F ` x ) = .0. )
70	69	a1i 11	. . . . . . . . . 10 |- ( ( ( F e. B /\ G e. RR* ) /\ x e. A ) -> ( ( -. ( F ` x ) e. _V \/ -. ( F ` x ) =/= .0. ) <-> ( F ` x ) = .0. ) )
71	64, 70	syl5bb 272	. . . . . . . . 9 |- ( ( ( F e. B /\ G e. RR* ) /\ x e. A ) -> ( -. ( F ` x ) e. ( _V \ { .0. } ) <-> ( F ` x ) = .0. ) )
72	62, 71	imbi12d 334	. . . . . . . 8 |- ( ( ( F e. B /\ G e. RR* ) /\ x e. A ) -> ( ( -. ( H ` x ) <_ G -> -. ( F ` x ) e. ( _V \ { .0. } ) ) <-> ( G < ( H ` x ) -> ( F ` x ) = .0. ) ) )
73	56, 72	syl5bb 272	. . . . . . 7 |- ( ( ( F e. B /\ G e. RR* ) /\ x e. A ) -> ( ( ( F ` x ) e. ( _V \ { .0. } ) -> ( H ` x ) <_ G ) <-> ( G < ( H ` x ) -> ( F ` x ) = .0. ) ) )
74	73	pm5.74da 723	. . . . . 6 |- ( ( F e. B /\ G e. RR* ) -> ( ( x e. A -> ( ( F ` x ) e. ( _V \ { .0. } ) -> ( H ` x ) <_ G ) ) <-> ( x e. A -> ( G < ( H ` x ) -> ( F ` x ) = .0. ) ) ) )
75	55, 74	syl5bb 272	. . . . 5 |- ( ( F e. B /\ G e. RR* ) -> ( ( ( x e. A /\ ( F ` x ) e. ( _V \ { .0. } ) ) -> ( H ` x ) <_ G ) <-> ( x e. A -> ( G < ( H ` x ) -> ( F ` x ) = .0. ) ) ) )
76	54, 75	bitrd 268	. . . 4 |- ( ( F e. B /\ G e. RR* ) -> ( ( x e. ( F supp .0. ) -> ( H ` x ) <_ G ) <-> ( x e. A -> ( G < ( H ` x ) -> ( F ` x ) = .0. ) ) ) )
77	76	ralbidv2 2984	. . 3 |- ( ( F e. B /\ G e. RR* ) -> ( A. x e. ( F supp .0. ) ( H ` x ) <_ G <-> A. x e. A ( G < ( H ` x ) -> ( F ` x ) = .0. ) ) )
78	35, 77	bitrd 268	. 2 |- ( ( F e. B /\ G e. RR* ) -> ( A. y e. ( H " ( F supp .0. ) ) y <_ G <-> A. x e. A ( G < ( H ` x ) -> ( F ` x ) = .0. ) ) )
79	9, 23, 78	3bitrd 294	1 |- ( ( F e. B /\ G e. RR* ) -> ( ( D ` F ) <_ G <-> A. x e. A ( G < ( H ` x ) -> ( F ` x ) = .0. ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   {crab 2916   _Vcvv 3200   \ cdif 3571   C_ wss 3574   {csn 4177   class class class wbr 4653   |-> cmpt 4729   `'ccnv 5113   dom cdm 5114   ran crn 5115   "cima 5117   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   supp csupp 7295   ^m cmap 7857   Fincfn 7955   supcsup 8346   RRcr 9935   RR*cxr 10073   < clt 10074   <_ cle 10075   NNcn 11020   NN0cn0 11292   Basecbs 15857   0gc0g 16100   gsum_g cgsu 16101   mPoly cmpl 19353  ℂ_fldccnfld 19746   mDeg cmdg 23813

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-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-addf 10015  ax-mulf 10016

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-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  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-int 4476  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-se 5074  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-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-sup 8348  df-oi 8415  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-fz 12327  df-fzo 12466  df-seq 12802  df-hash 13118  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-sca 15957  df-vsca 15958  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-0g 16102  df-gsum 16103  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-grp 17425  df-minusg 17426  df-cntz 17750  df-cmn 18195  df-abl 18196  df-mgp 18490  df-ur 18502  df-ring 18549  df-cring 18550  df-psr 19356  df-mpl 19358  df-cnfld 19747  df-mdeg 23815

This theorem is referenced by:  mdeglt  23825  mdegaddle  23834  mdegvscale  23835  mdegle0  23837  mdegmullem  23838  deg1leb  23855