Theorem mdegldg 23826

Description: A nonzero polynomial has some coefficient which witnesses its degree. (Contributed by Stefan O'Rear, 23-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 ) )
mdegldg.y	|- Y = ( 0g ` P )

Assertion

Ref	Expression
mdegldg	|- ( ( R e. Ring /\ F e. B /\ F =/= Y ) -> E. x e. A ( ( F ` x ) =/= .0. /\ ( H ` x ) = ( D ` F ) ) )

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

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

Proof of Theorem mdegldg

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	3ad2ant2 1083	. . 3 |- ( ( R e. Ring /\ F e. B /\ F =/= Y ) -> ( D ` F ) = sup ( ( H " ( F supp .0. ) ) , RR* , < ) )
9	2, 3	mplrcl 19490	. . . . . . . 8 |- ( F e. B -> I e. _V )
10	9	3ad2ant2 1083	. . . . . . 7 |- ( ( R e. Ring /\ F e. B /\ F =/= Y ) -> I e. _V )
11	5, 6	tdeglem1 23818	. . . . . . 7 |- ( I e. _V -> H : A --> NN0 )
12	10, 11	syl 17	. . . . . 6 |- ( ( R e. Ring /\ F e. B /\ F =/= Y ) -> H : A --> NN0 )
13	12	ffund 6049	. . . . 5 |- ( ( R e. Ring /\ F e. B /\ F =/= Y ) -> Fun H )
14		simp2 1062	. . . . . . 7 |- ( ( R e. Ring /\ F e. B /\ F =/= Y ) -> F e. B )
15		simp1 1061	. . . . . . 7 |- ( ( R e. Ring /\ F e. B /\ F =/= Y ) -> R e. Ring )
16	2, 3, 4, 14, 15	mplelsfi 19491	. . . . . 6 |- ( ( R e. Ring /\ F e. B /\ F =/= Y ) -> F finSupp .0. )
17	16	fsuppimpd 8282	. . . . 5 |- ( ( R e. Ring /\ F e. B /\ F =/= Y ) -> ( F supp .0. ) e. Fin )
18		imafi 8259	. . . . 5 |- ( ( Fun H /\ ( F supp .0. ) e. Fin ) -> ( H " ( F supp .0. ) ) e. Fin )
19	13, 17, 18	syl2anc 693	. . . 4 |- ( ( R e. Ring /\ F e. B /\ F =/= Y ) -> ( H " ( F supp .0. ) ) e. Fin )
20		simp3 1063	. . . . . . 7 |- ( ( R e. Ring /\ F e. B /\ F =/= Y ) -> F =/= Y )
21		mdegldg.y	. . . . . . . 8 |- Y = ( 0g ` P )
22		ringgrp 18552	. . . . . . . . 9 |- ( R e. Ring -> R e. Grp )
23	22	3ad2ant1 1082	. . . . . . . 8 |- ( ( R e. Ring /\ F e. B /\ F =/= Y ) -> R e. Grp )
24	2, 5, 4, 21, 10, 23	mpl0 19441	. . . . . . 7 |- ( ( R e. Ring /\ F e. B /\ F =/= Y ) -> Y = ( A X. { .0. } ) )
25	20, 24	neeqtrd 2863	. . . . . 6 |- ( ( R e. Ring /\ F e. B /\ F =/= Y ) -> F =/= ( A X. { .0. } ) )
26		eqid 2622	. . . . . . . . . 10 |- ( Base ` R ) = ( Base ` R )
27	2, 26, 3, 5, 14	mplelf 19433	. . . . . . . . 9 |- ( ( R e. Ring /\ F e. B /\ F =/= Y ) -> F : A --> ( Base ` R ) )
28	27	ffnd 6046	. . . . . . . 8 |- ( ( R e. Ring /\ F e. B /\ F =/= Y ) -> F Fn A )
29		fvex 6201	. . . . . . . . 9 |- ( 0g ` R ) e. _V
30	4, 29	eqeltri 2697	. . . . . . . 8 |- .0. e. _V
31		ovex 6678	. . . . . . . . . 10 |- ( NN0 ^m I ) e. _V
32	5, 31	rabex2 4815	. . . . . . . . 9 |- A e. _V
33		fnsuppeq0 7323	. . . . . . . . 9 |- ( ( F Fn A /\ A e. _V /\ .0. e. _V ) -> ( ( F supp .0. ) = (/) <-> F = ( A X. { .0. } ) ) )
34	32, 33	mp3an2 1412	. . . . . . . 8 |- ( ( F Fn A /\ .0. e. _V ) -> ( ( F supp .0. ) = (/) <-> F = ( A X. { .0. } ) ) )
35	28, 30, 34	sylancl 694	. . . . . . 7 |- ( ( R e. Ring /\ F e. B /\ F =/= Y ) -> ( ( F supp .0. ) = (/) <-> F = ( A X. { .0. } ) ) )
36	35	necon3bid 2838	. . . . . 6 |- ( ( R e. Ring /\ F e. B /\ F =/= Y ) -> ( ( F supp .0. ) =/= (/) <-> F =/= ( A X. { .0. } ) ) )
37	25, 36	mpbird 247	. . . . 5 |- ( ( R e. Ring /\ F e. B /\ F =/= Y ) -> ( F supp .0. ) =/= (/) )
38	12	ffnd 6046	. . . . . . 7 |- ( ( R e. Ring /\ F e. B /\ F =/= Y ) -> H Fn A )
39		suppssdm 7308	. . . . . . . 8 |- ( F supp .0. ) C_ dom F
40		fdm 6051	. . . . . . . . 9 |- ( F : A --> ( Base ` R ) -> dom F = A )
41	27, 40	syl 17	. . . . . . . 8 |- ( ( R e. Ring /\ F e. B /\ F =/= Y ) -> dom F = A )
42	39, 41	syl5sseq 3653	. . . . . . 7 |- ( ( R e. Ring /\ F e. B /\ F =/= Y ) -> ( F supp .0. ) C_ A )
43		fnimaeq0 6013	. . . . . . 7 |- ( ( H Fn A /\ ( F supp .0. ) C_ A ) -> ( ( H " ( F supp .0. ) ) = (/) <-> ( F supp .0. ) = (/) ) )
44	38, 42, 43	syl2anc 693	. . . . . 6 |- ( ( R e. Ring /\ F e. B /\ F =/= Y ) -> ( ( H " ( F supp .0. ) ) = (/) <-> ( F supp .0. ) = (/) ) )
45	44	necon3bid 2838	. . . . 5 |- ( ( R e. Ring /\ F e. B /\ F =/= Y ) -> ( ( H " ( F supp .0. ) ) =/= (/) <-> ( F supp .0. ) =/= (/) ) )
46	37, 45	mpbird 247	. . . 4 |- ( ( R e. Ring /\ F e. B /\ F =/= Y ) -> ( H " ( F supp .0. ) ) =/= (/) )
47		imassrn 5477	. . . . . 6 |- ( H " ( F supp .0. ) ) C_ ran H
48		frn 6053	. . . . . . 7 |- ( H : A --> NN0 -> ran H C_ NN0 )
49	12, 48	syl 17	. . . . . 6 |- ( ( R e. Ring /\ F e. B /\ F =/= Y ) -> ran H C_ NN0 )
50	47, 49	syl5ss 3614	. . . . 5 |- ( ( R e. Ring /\ F e. B /\ F =/= Y ) -> ( H " ( F supp .0. ) ) C_ NN0 )
51		nn0ssre 11296	. . . . . 6 |- NN0 C_ RR
52		ressxr 10083	. . . . . 6 |- RR C_ RR*
53	51, 52	sstri 3612	. . . . 5 |- NN0 C_ RR*
54	50, 53	syl6ss 3615	. . . 4 |- ( ( R e. Ring /\ F e. B /\ F =/= Y ) -> ( H " ( F supp .0. ) ) C_ RR* )
55		xrltso 11974	. . . . 5 |- < Or RR*
56		fisupcl 8375	. . . . 5 |- ( ( < Or RR* /\ ( ( H " ( F supp .0. ) ) e. Fin /\ ( H " ( F supp .0. ) ) =/= (/) /\ ( H " ( F supp .0. ) ) C_ RR* ) ) -> sup ( ( H " ( F supp .0. ) ) , RR* , < ) e. ( H " ( F supp .0. ) ) )
57	55, 56	mpan 706	. . . 4 |- ( ( ( H " ( F supp .0. ) ) e. Fin /\ ( H " ( F supp .0. ) ) =/= (/) /\ ( H " ( F supp .0. ) ) C_ RR* ) -> sup ( ( H " ( F supp .0. ) ) , RR* , < ) e. ( H " ( F supp .0. ) ) )
58	19, 46, 54, 57	syl3anc 1326	. . 3 |- ( ( R e. Ring /\ F e. B /\ F =/= Y ) -> sup ( ( H " ( F supp .0. ) ) , RR* , < ) e. ( H " ( F supp .0. ) ) )
59	8, 58	eqeltrd 2701	. 2 |- ( ( R e. Ring /\ F e. B /\ F =/= Y ) -> ( D ` F ) e. ( H " ( F supp .0. ) ) )
60		fvelimab 6253	. . . 4 |- ( ( H Fn A /\ ( F supp .0. ) C_ A ) -> ( ( D ` F ) e. ( H " ( F supp .0. ) ) <-> E. x e. ( F supp .0. ) ( H ` x ) = ( D ` F ) ) )
61	38, 42, 60	syl2anc 693	. . 3 |- ( ( R e. Ring /\ F e. B /\ F =/= Y ) -> ( ( D ` F ) e. ( H " ( F supp .0. ) ) <-> E. x e. ( F supp .0. ) ( H ` x ) = ( D ` F ) ) )
62		rexsupp 7313	. . . . 5 |- ( ( F Fn A /\ A e. _V /\ .0. e. _V ) -> ( E. x e. ( F supp .0. ) ( H ` x ) = ( D ` F ) <-> E. x e. A ( ( F ` x ) =/= .0. /\ ( H ` x ) = ( D ` F ) ) ) )
63	32, 30, 62	mp3an23 1416	. . . 4 |- ( F Fn A -> ( E. x e. ( F supp .0. ) ( H ` x ) = ( D ` F ) <-> E. x e. A ( ( F ` x ) =/= .0. /\ ( H ` x ) = ( D ` F ) ) ) )
64	28, 63	syl 17	. . 3 |- ( ( R e. Ring /\ F e. B /\ F =/= Y ) -> ( E. x e. ( F supp .0. ) ( H ` x ) = ( D ` F ) <-> E. x e. A ( ( F ` x ) =/= .0. /\ ( H ` x ) = ( D ` F ) ) ) )
65	61, 64	bitrd 268	. 2 |- ( ( R e. Ring /\ F e. B /\ F =/= Y ) -> ( ( D ` F ) e. ( H " ( F supp .0. ) ) <-> E. x e. A ( ( F ` x ) =/= .0. /\ ( H ` x ) = ( D ` F ) ) ) )
66	59, 65	mpbid 222	1 |- ( ( R e. Ring /\ F e. B /\ F =/= Y ) -> E. x e. A ( ( F ` x ) =/= .0. /\ ( H ` x ) = ( D ` F ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   {crab 2916   _Vcvv 3200   C_ wss 3574   (/)c0 3915   {csn 4177   |-> cmpt 4729   Or wor 5034   X. cxp 5112   `'ccnv 5113   dom cdm 5114   ran crn 5115   "cima 5117   Fun wfun 5882   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   NNcn 11020   NN0cn0 11292   Basecbs 15857   0gc0g 16100   gsum_g cgsu 16101   Grpcgrp 17422   Ringcrg 18547   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-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-subg 17591  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:  mdegnn0cl  23831  deg1ldg  23852