Theorem dgreq0 24021

Description: The leading coefficient of a polynomial is nonzero, unless the entire polynomial is zero. (Contributed by Mario Carneiro, 22-Jul-2014.) (Proof shortened by Fan Zheng, 21-Jun-2016.)

Hypotheses

Ref	Expression
dgreq0.1	|- N = (deg ` F )
dgreq0.2	|- A = (coeff ` F )

Assertion

Ref	Expression
dgreq0	|- ( F e. (Poly ` S ) -> ( F = 0p <-> ( A ` N ) = 0 ) )

Proof of Theorem dgreq0

Dummy variable k is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dgreq0.2	. . . . . 6 |- A = (coeff ` F )
2		fveq2 6191	. . . . . 6 |- ( F = 0p -> (coeff ` F ) = (coeff ` 0p ) )
3	1, 2	syl5eq 2668	. . . . 5 |- ( F = 0p -> A = (coeff ` 0p ) )
4		coe0 24012	. . . . 5 |- (coeff ` 0p ) = ( NN0 X. { 0 } )
5	3, 4	syl6eq 2672	. . . 4 |- ( F = 0p -> A = ( NN0 X. { 0 } ) )
6		dgreq0.1	. . . . . 6 |- N = (deg ` F )
7		fveq2 6191	. . . . . 6 |- ( F = 0p -> (deg ` F ) = (deg ` 0p ) )
8	6, 7	syl5eq 2668	. . . . 5 |- ( F = 0p -> N = (deg ` 0p ) )
9		dgr0 24018	. . . . 5 |- (deg ` 0p ) = 0
10	8, 9	syl6eq 2672	. . . 4 |- ( F = 0p -> N = 0 )
11	5, 10	fveq12d 6197	. . 3 |- ( F = 0p -> ( A ` N ) = ( ( NN0 X. { 0 } ) ` 0 ) )
12		0nn0 11307	. . . 4 |- 0 e. NN0
13		fvconst2g 6467	. . . 4 |- ( ( 0 e. NN0 /\ 0 e. NN0 ) -> ( ( NN0 X. { 0 } ) ` 0 ) = 0 )
14	12, 12, 13	mp2an 708	. . 3 |- ( ( NN0 X. { 0 } ) ` 0 ) = 0
15	11, 14	syl6eq 2672	. 2 |- ( F = 0p -> ( A ` N ) = 0 )
16	1	coefv0 24004	. . . . . . . 8 |- ( F e. (Poly ` S ) -> ( F ` 0 ) = ( A ` 0 ) )
17	16	adantr 481	. . . . . . 7 |- ( ( F e. (Poly ` S ) /\ ( A ` N ) = 0 ) -> ( F ` 0 ) = ( A ` 0 ) )
18		simpr 477	. . . . . . . . . . . 12 |- ( ( ( F e. (Poly ` S ) /\ ( A ` N ) = 0 ) /\ N e. NN ) -> N e. NN )
19	18	nnred 11035	. . . . . . . . . . 11 |- ( ( ( F e. (Poly ` S ) /\ ( A ` N ) = 0 ) /\ N e. NN ) -> N e. RR )
20	19	ltm1d 10956	. . . . . . . . . 10 |- ( ( ( F e. (Poly ` S ) /\ ( A ` N ) = 0 ) /\ N e. NN ) -> ( N - 1 ) < N )
21		simpll 790	. . . . . . . . . . . 12 |- ( ( ( F e. (Poly ` S ) /\ ( A ` N ) = 0 ) /\ N e. NN ) -> F e. (Poly ` S ) )
22		nnm1nn0 11334	. . . . . . . . . . . . 13 |- ( N e. NN -> ( N - 1 ) e. NN0 )
23	22	adantl 482	. . . . . . . . . . . 12 |- ( ( ( F e. (Poly ` S ) /\ ( A ` N ) = 0 ) /\ N e. NN ) -> ( N - 1 ) e. NN0 )
24	1, 6	dgrub 23990	. . . . . . . . . . . . . . . . . . 19 |- ( ( F e. (Poly ` S ) /\ k e. NN0 /\ ( A ` k ) =/= 0 ) -> k <_ N )
25	24	3expia 1267	. . . . . . . . . . . . . . . . . 18 |- ( ( F e. (Poly ` S ) /\ k e. NN0 ) -> ( ( A ` k ) =/= 0 -> k <_ N ) )
26	25	ad2ant2rl 785	. . . . . . . . . . . . . . . . 17 |- ( ( ( F e. (Poly ` S ) /\ ( A ` N ) = 0 ) /\ ( N e. NN /\ k e. NN0 ) ) -> ( ( A ` k ) =/= 0 -> k <_ N ) )
27		simplr 792	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( F e. (Poly ` S ) /\ ( A ` N ) = 0 ) /\ ( N e. NN /\ k e. NN0 ) ) -> ( A ` N ) = 0 )
28		fveq2 6191	. . . . . . . . . . . . . . . . . . . 20 |- ( N = k -> ( A ` N ) = ( A ` k ) )
29	28	eqeq1d 2624	. . . . . . . . . . . . . . . . . . 19 |- ( N = k -> ( ( A ` N ) = 0 <-> ( A ` k ) = 0 ) )
30	27, 29	syl5ibcom 235	. . . . . . . . . . . . . . . . . 18 |- ( ( ( F e. (Poly ` S ) /\ ( A ` N ) = 0 ) /\ ( N e. NN /\ k e. NN0 ) ) -> ( N = k -> ( A ` k ) = 0 ) )
31	30	necon3d 2815	. . . . . . . . . . . . . . . . 17 |- ( ( ( F e. (Poly ` S ) /\ ( A ` N ) = 0 ) /\ ( N e. NN /\ k e. NN0 ) ) -> ( ( A ` k ) =/= 0 -> N =/= k ) )
32	26, 31	jcad 555	. . . . . . . . . . . . . . . 16 |- ( ( ( F e. (Poly ` S ) /\ ( A ` N ) = 0 ) /\ ( N e. NN /\ k e. NN0 ) ) -> ( ( A ` k ) =/= 0 -> ( k <_ N /\ N =/= k ) ) )
33		nn0re 11301	. . . . . . . . . . . . . . . . . . 19 |- ( k e. NN0 -> k e. RR )
34	33	ad2antll 765	. . . . . . . . . . . . . . . . . 18 |- ( ( ( F e. (Poly ` S ) /\ ( A ` N ) = 0 ) /\ ( N e. NN /\ k e. NN0 ) ) -> k e. RR )
35		nnre 11027	. . . . . . . . . . . . . . . . . . 19 |- ( N e. NN -> N e. RR )
36	35	ad2antrl 764	. . . . . . . . . . . . . . . . . 18 |- ( ( ( F e. (Poly ` S ) /\ ( A ` N ) = 0 ) /\ ( N e. NN /\ k e. NN0 ) ) -> N e. RR )
37	34, 36	ltlend 10182	. . . . . . . . . . . . . . . . 17 |- ( ( ( F e. (Poly ` S ) /\ ( A ` N ) = 0 ) /\ ( N e. NN /\ k e. NN0 ) ) -> ( k < N <-> ( k <_ N /\ N =/= k ) ) )
38		nn0z 11400	. . . . . . . . . . . . . . . . . . 19 |- ( k e. NN0 -> k e. ZZ )
39	38	ad2antll 765	. . . . . . . . . . . . . . . . . 18 |- ( ( ( F e. (Poly ` S ) /\ ( A ` N ) = 0 ) /\ ( N e. NN /\ k e. NN0 ) ) -> k e. ZZ )
40		nnz 11399	. . . . . . . . . . . . . . . . . . 19 |- ( N e. NN -> N e. ZZ )
41	40	ad2antrl 764	. . . . . . . . . . . . . . . . . 18 |- ( ( ( F e. (Poly ` S ) /\ ( A ` N ) = 0 ) /\ ( N e. NN /\ k e. NN0 ) ) -> N e. ZZ )
42		zltlem1 11430	. . . . . . . . . . . . . . . . . 18 |- ( ( k e. ZZ /\ N e. ZZ ) -> ( k < N <-> k <_ ( N - 1 ) ) )
43	39, 41, 42	syl2anc 693	. . . . . . . . . . . . . . . . 17 |- ( ( ( F e. (Poly ` S ) /\ ( A ` N ) = 0 ) /\ ( N e. NN /\ k e. NN0 ) ) -> ( k < N <-> k <_ ( N - 1 ) ) )
44	37, 43	bitr3d 270	. . . . . . . . . . . . . . . 16 |- ( ( ( F e. (Poly ` S ) /\ ( A ` N ) = 0 ) /\ ( N e. NN /\ k e. NN0 ) ) -> ( ( k <_ N /\ N =/= k ) <-> k <_ ( N - 1 ) ) )
45	32, 44	sylibd 229	. . . . . . . . . . . . . . 15 |- ( ( ( F e. (Poly ` S ) /\ ( A ` N ) = 0 ) /\ ( N e. NN /\ k e. NN0 ) ) -> ( ( A ` k ) =/= 0 -> k <_ ( N - 1 ) ) )
46	45	expr 643	. . . . . . . . . . . . . 14 |- ( ( ( F e. (Poly ` S ) /\ ( A ` N ) = 0 ) /\ N e. NN ) -> ( k e. NN0 -> ( ( A ` k ) =/= 0 -> k <_ ( N - 1 ) ) ) )
47	46	ralrimiv 2965	. . . . . . . . . . . . 13 |- ( ( ( F e. (Poly ` S ) /\ ( A ` N ) = 0 ) /\ N e. NN ) -> A. k e. NN0 ( ( A ` k ) =/= 0 -> k <_ ( N - 1 ) ) )
48	1	coef3 23988	. . . . . . . . . . . . . . 15 |- ( F e. (Poly ` S ) -> A : NN0 --> CC )
49	48	ad2antrr 762	. . . . . . . . . . . . . 14 |- ( ( ( F e. (Poly ` S ) /\ ( A ` N ) = 0 ) /\ N e. NN ) -> A : NN0 --> CC )
50		plyco0 23948	. . . . . . . . . . . . . 14 |- ( ( ( N - 1 ) e. NN0 /\ A : NN0 --> CC ) -> ( ( A " ( ZZ>= ` ( ( N - 1 ) + 1 ) ) ) = { 0 } <-> A. k e. NN0 ( ( A ` k ) =/= 0 -> k <_ ( N - 1 ) ) ) )
51	23, 49, 50	syl2anc 693	. . . . . . . . . . . . 13 |- ( ( ( F e. (Poly ` S ) /\ ( A ` N ) = 0 ) /\ N e. NN ) -> ( ( A " ( ZZ>= ` ( ( N - 1 ) + 1 ) ) ) = { 0 } <-> A. k e. NN0 ( ( A ` k ) =/= 0 -> k <_ ( N - 1 ) ) ) )
52	47, 51	mpbird 247	. . . . . . . . . . . 12 |- ( ( ( F e. (Poly ` S ) /\ ( A ` N ) = 0 ) /\ N e. NN ) -> ( A " ( ZZ>= ` ( ( N - 1 ) + 1 ) ) ) = { 0 } )
53	1, 6	dgrlb 23992	. . . . . . . . . . . 12 |- ( ( F e. (Poly ` S ) /\ ( N - 1 ) e. NN0 /\ ( A " ( ZZ>= ` ( ( N - 1 ) + 1 ) ) ) = { 0 } ) -> N <_ ( N - 1 ) )
54	21, 23, 52, 53	syl3anc 1326	. . . . . . . . . . 11 |- ( ( ( F e. (Poly ` S ) /\ ( A ` N ) = 0 ) /\ N e. NN ) -> N <_ ( N - 1 ) )
55	35	adantl 482	. . . . . . . . . . . 12 |- ( ( ( F e. (Poly ` S ) /\ ( A ` N ) = 0 ) /\ N e. NN ) -> N e. RR )
56		peano2rem 10348	. . . . . . . . . . . . 13 |- ( N e. RR -> ( N - 1 ) e. RR )
57	55, 56	syl 17	. . . . . . . . . . . 12 |- ( ( ( F e. (Poly ` S ) /\ ( A ` N ) = 0 ) /\ N e. NN ) -> ( N - 1 ) e. RR )
58	55, 57	lenltd 10183	. . . . . . . . . . 11 |- ( ( ( F e. (Poly ` S ) /\ ( A ` N ) = 0 ) /\ N e. NN ) -> ( N <_ ( N - 1 ) <-> -. ( N - 1 ) < N ) )
59	54, 58	mpbid 222	. . . . . . . . . 10 |- ( ( ( F e. (Poly ` S ) /\ ( A ` N ) = 0 ) /\ N e. NN ) -> -. ( N - 1 ) < N )
60	20, 59	pm2.65da 600	. . . . . . . . 9 |- ( ( F e. (Poly ` S ) /\ ( A ` N ) = 0 ) -> -. N e. NN )
61		dgrcl 23989	. . . . . . . . . . . . 13 |- ( F e. (Poly ` S ) -> (deg ` F ) e. NN0 )
62	6, 61	syl5eqel 2705	. . . . . . . . . . . 12 |- ( F e. (Poly ` S ) -> N e. NN0 )
63	62	adantr 481	. . . . . . . . . . 11 |- ( ( F e. (Poly ` S ) /\ ( A ` N ) = 0 ) -> N e. NN0 )
64		elnn0 11294	. . . . . . . . . . 11 |- ( N e. NN0 <-> ( N e. NN \/ N = 0 ) )
65	63, 64	sylib 208	. . . . . . . . . 10 |- ( ( F e. (Poly ` S ) /\ ( A ` N ) = 0 ) -> ( N e. NN \/ N = 0 ) )
66	65	ord 392	. . . . . . . . 9 |- ( ( F e. (Poly ` S ) /\ ( A ` N ) = 0 ) -> ( -. N e. NN -> N = 0 ) )
67	60, 66	mpd 15	. . . . . . . 8 |- ( ( F e. (Poly ` S ) /\ ( A ` N ) = 0 ) -> N = 0 )
68	67	fveq2d 6195	. . . . . . 7 |- ( ( F e. (Poly ` S ) /\ ( A ` N ) = 0 ) -> ( A ` N ) = ( A ` 0 ) )
69		simpr 477	. . . . . . 7 |- ( ( F e. (Poly ` S ) /\ ( A ` N ) = 0 ) -> ( A ` N ) = 0 )
70	17, 68, 69	3eqtr2d 2662	. . . . . 6 |- ( ( F e. (Poly ` S ) /\ ( A ` N ) = 0 ) -> ( F ` 0 ) = 0 )
71	70	sneqd 4189	. . . . 5 |- ( ( F e. (Poly ` S ) /\ ( A ` N ) = 0 ) -> { ( F ` 0 ) } = { 0 } )
72	71	xpeq2d 5139	. . . 4 |- ( ( F e. (Poly ` S ) /\ ( A ` N ) = 0 ) -> ( CC X. { ( F ` 0 ) } ) = ( CC X. { 0 } ) )
73	6, 67	syl5eqr 2670	. . . . 5 |- ( ( F e. (Poly ` S ) /\ ( A ` N ) = 0 ) -> (deg ` F ) = 0 )
74		0dgrb 24002	. . . . . 6 |- ( F e. (Poly ` S ) -> ( (deg ` F ) = 0 <-> F = ( CC X. { ( F ` 0 ) } ) ) )
75	74	adantr 481	. . . . 5 |- ( ( F e. (Poly ` S ) /\ ( A ` N ) = 0 ) -> ( (deg ` F ) = 0 <-> F = ( CC X. { ( F ` 0 ) } ) ) )
76	73, 75	mpbid 222	. . . 4 |- ( ( F e. (Poly ` S ) /\ ( A ` N ) = 0 ) -> F = ( CC X. { ( F ` 0 ) } ) )
77		df-0p 23437	. . . . 5 |- 0p = ( CC X. { 0 } )
78	77	a1i 11	. . . 4 |- ( ( F e. (Poly ` S ) /\ ( A ` N ) = 0 ) -> 0p = ( CC X. { 0 } ) )
79	72, 76, 78	3eqtr4d 2666	. . 3 |- ( ( F e. (Poly ` S ) /\ ( A ` N ) = 0 ) -> F = 0p )
80	79	ex 450	. 2 |- ( F e. (Poly ` S ) -> ( ( A ` N ) = 0 -> F = 0p ) )
81	15, 80	impbid2 216	1 |- ( F e. (Poly ` S ) -> ( F = 0p <-> ( A ` N ) = 0 ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   {csn 4177   class class class wbr 4653   X. cxp 5112   "cima 5117   -->wf 5884   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   < clt 10074   <_ cle 10075   - cmin 10266   NNcn 11020   NN0cn0 11292   ZZcz 11377   ZZ>=cuz 11687   0pc0p 23436  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:  dgrlt  24022  dgradd2  24024  dgrmul  24026  dgrcolem2  24030  plymul0or  24036  plydivlem4  24051  plydiveu  24053  vieta1lem2  24066  vieta1  24067  aareccl  24081  ftalem2  24800  ftalem4  24802  ftalem5  24803  signsply0  30628  mpaaeu  37720  elaa2lem  40450