Theorem dgrub 23990

Description: If the M-th coefficient of F is nonzero, then the degree of F is at least M. (Contributed by Mario Carneiro, 22-Jul-2014.)

Hypotheses

Ref	Expression
dgrub.1	|- A = (coeff ` F )
dgrub.2	|- N = (deg ` F )

Assertion

Ref	Expression
dgrub	|- ( ( F e. (Poly ` S ) /\ M e. NN0 /\ ( A ` M ) =/= 0 ) -> M <_ N )

Proof of Theorem dgrub

Dummy variables n x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp1 1061	. . . 4 |- ( ( F e. (Poly ` S ) /\ M e. NN0 /\ ( A ` M ) =/= 0 ) -> F e. (Poly ` S ) )
2		simp2 1062	. . . . 5 |- ( ( F e. (Poly ` S ) /\ M e. NN0 /\ ( A ` M ) =/= 0 ) -> M e. NN0 )
3		dgrub.1	. . . . . . . . 9 |- A = (coeff ` F )
4	3	coef3 23988	. . . . . . . 8 |- ( F e. (Poly ` S ) -> A : NN0 --> CC )
5	1, 4	syl 17	. . . . . . 7 |- ( ( F e. (Poly ` S ) /\ M e. NN0 /\ ( A ` M ) =/= 0 ) -> A : NN0 --> CC )
6	5, 2	ffvelrnd 6360	. . . . . 6 |- ( ( F e. (Poly ` S ) /\ M e. NN0 /\ ( A ` M ) =/= 0 ) -> ( A ` M ) e. CC )
7		simp3 1063	. . . . . 6 |- ( ( F e. (Poly ` S ) /\ M e. NN0 /\ ( A ` M ) =/= 0 ) -> ( A ` M ) =/= 0 )
8		eldifsn 4317	. . . . . 6 |- ( ( A ` M ) e. ( CC \ { 0 } ) <-> ( ( A ` M ) e. CC /\ ( A ` M ) =/= 0 ) )
9	6, 7, 8	sylanbrc 698	. . . . 5 |- ( ( F e. (Poly ` S ) /\ M e. NN0 /\ ( A ` M ) =/= 0 ) -> ( A ` M ) e. ( CC \ { 0 } ) )
10	3	coef 23986	. . . . . 6 |- ( F e. (Poly ` S ) -> A : NN0 --> ( S u. { 0 } ) )
11		ffn 6045	. . . . . 6 |- ( A : NN0 --> ( S u. { 0 } ) -> A Fn NN0 )
12		elpreima 6337	. . . . . 6 |- ( A Fn NN0 -> ( M e. ( `' A " ( CC \ { 0 } ) ) <-> ( M e. NN0 /\ ( A ` M ) e. ( CC \ { 0 } ) ) ) )
13	1, 10, 11, 12	4syl 19	. . . . 5 |- ( ( F e. (Poly ` S ) /\ M e. NN0 /\ ( A ` M ) =/= 0 ) -> ( M e. ( `' A " ( CC \ { 0 } ) ) <-> ( M e. NN0 /\ ( A ` M ) e. ( CC \ { 0 } ) ) ) )
14	2, 9, 13	mpbir2and 957	. . . 4 |- ( ( F e. (Poly ` S ) /\ M e. NN0 /\ ( A ` M ) =/= 0 ) -> M e. ( `' A " ( CC \ { 0 } ) ) )
15		nn0ssre 11296	. . . . . . 7 |- NN0 C_ RR
16		ltso 10118	. . . . . . 7 |- < Or RR
17		soss 5053	. . . . . . 7 |- ( NN0 C_ RR -> ( < Or RR -> < Or NN0 ) )
18	15, 16, 17	mp2 9	. . . . . 6 |- < Or NN0
19	18	a1i 11	. . . . 5 |- ( F e. (Poly ` S ) -> < Or NN0 )
20		0zd 11389	. . . . . 6 |- ( F e. (Poly ` S ) -> 0 e. ZZ )
21		cnvimass 5485	. . . . . . 7 |- ( `' A " ( CC \ { 0 } ) ) C_ dom A
22		fdm 6051	. . . . . . . 8 |- ( A : NN0 --> ( S u. { 0 } ) -> dom A = NN0 )
23	10, 22	syl 17	. . . . . . 7 |- ( F e. (Poly ` S ) -> dom A = NN0 )
24	21, 23	syl5sseq 3653	. . . . . 6 |- ( F e. (Poly ` S ) -> ( `' A " ( CC \ { 0 } ) ) C_ NN0 )
25	3	dgrlem 23985	. . . . . . 7 |- ( F e. (Poly ` S ) -> ( A : NN0 --> ( S u. { 0 } ) /\ E. n e. ZZ A. x e. ( `' A " ( CC \ { 0 } ) ) x <_ n ) )
26	25	simprd 479	. . . . . 6 |- ( F e. (Poly ` S ) -> E. n e. ZZ A. x e. ( `' A " ( CC \ { 0 } ) ) x <_ n )
27		nn0uz 11722	. . . . . . 7 |- NN0 = ( ZZ>= ` 0 )
28	27	uzsupss 11780	. . . . . 6 |- ( ( 0 e. ZZ /\ ( `' A " ( CC \ { 0 } ) ) C_ NN0 /\ E. n e. ZZ A. x e. ( `' A " ( CC \ { 0 } ) ) x <_ n ) -> E. n e. NN0 ( A. x e. ( `' A " ( CC \ { 0 } ) ) -. n < x /\ A. x e. NN0 ( x < n -> E. y e. ( `' A " ( CC \ { 0 } ) ) x < y ) ) )
29	20, 24, 26, 28	syl3anc 1326	. . . . 5 |- ( F e. (Poly ` S ) -> E. n e. NN0 ( A. x e. ( `' A " ( CC \ { 0 } ) ) -. n < x /\ A. x e. NN0 ( x < n -> E. y e. ( `' A " ( CC \ { 0 } ) ) x < y ) ) )
30	19, 29	supub 8365	. . . 4 |- ( F e. (Poly ` S ) -> ( M e. ( `' A " ( CC \ { 0 } ) ) -> -. sup ( ( `' A " ( CC \ { 0 } ) ) , NN0 , < ) < M ) )
31	1, 14, 30	sylc 65	. . 3 |- ( ( F e. (Poly ` S ) /\ M e. NN0 /\ ( A ` M ) =/= 0 ) -> -. sup ( ( `' A " ( CC \ { 0 } ) ) , NN0 , < ) < M )
32		dgrub.2	. . . . . 6 |- N = (deg ` F )
33	3	dgrval 23984	. . . . . 6 |- ( F e. (Poly ` S ) -> (deg ` F ) = sup ( ( `' A " ( CC \ { 0 } ) ) , NN0 , < ) )
34	32, 33	syl5eq 2668	. . . . 5 |- ( F e. (Poly ` S ) -> N = sup ( ( `' A " ( CC \ { 0 } ) ) , NN0 , < ) )
35	1, 34	syl 17	. . . 4 |- ( ( F e. (Poly ` S ) /\ M e. NN0 /\ ( A ` M ) =/= 0 ) -> N = sup ( ( `' A " ( CC \ { 0 } ) ) , NN0 , < ) )
36	35	breq1d 4663	. . 3 |- ( ( F e. (Poly ` S ) /\ M e. NN0 /\ ( A ` M ) =/= 0 ) -> ( N < M <-> sup ( ( `' A " ( CC \ { 0 } ) ) , NN0 , < ) < M ) )
37	31, 36	mtbird 315	. 2 |- ( ( F e. (Poly ` S ) /\ M e. NN0 /\ ( A ` M ) =/= 0 ) -> -. N < M )
38	2	nn0red 11352	. . 3 |- ( ( F e. (Poly ` S ) /\ M e. NN0 /\ ( A ` M ) =/= 0 ) -> M e. RR )
39		dgrcl 23989	. . . . . 6 |- ( F e. (Poly ` S ) -> (deg ` F ) e. NN0 )
40	32, 39	syl5eqel 2705	. . . . 5 |- ( F e. (Poly ` S ) -> N e. NN0 )
41	1, 40	syl 17	. . . 4 |- ( ( F e. (Poly ` S ) /\ M e. NN0 /\ ( A ` M ) =/= 0 ) -> N e. NN0 )
42	41	nn0red 11352	. . 3 |- ( ( F e. (Poly ` S ) /\ M e. NN0 /\ ( A ` M ) =/= 0 ) -> N e. RR )
43	38, 42	lenltd 10183	. 2 |- ( ( F e. (Poly ` S ) /\ M e. NN0 /\ ( A ` M ) =/= 0 ) -> ( M <_ N <-> -. N < M ) )
44	37, 43	mpbird 247	1 |- ( ( F e. (Poly ` S ) /\ M e. NN0 /\ ( A ` M ) =/= 0 ) -> M <_ N )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   \ cdif 3571   u. cun 3572   C_ wss 3574   {csn 4177   class class class wbr 4653   Or wor 5034   `'ccnv 5113   dom cdm 5114   "cima 5117   Fn wfn 5883   -->wf 5884   ` cfv 5888   supcsup 8346   CCcc 9934   RRcr 9935   0cc0 9936   < clt 10074   <_ cle 10075   NN0cn0 11292   ZZcz 11377  Polycply 23940  coeffccoe 23942  degcdgr 23943

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-inf2 8538  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

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  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-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  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-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-fzo 12466  df-fl 12593  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-rlim 14220  df-sum 14417  df-0p 23437  df-ply 23944  df-coe 23946  df-dgr 23947

This theorem is referenced by:  dgrub2  23991  coeidlem  23993  coeid3  23996  dgreq  24000  coemullem  24006  coemulhi  24010  coemulc  24011  dgreq0  24021  dgrlt  24022  dgradd2  24024  dgrmul  24026  vieta1lem2  24066  aannenlem2  24084