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Theorem dgreq0 21732
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.
StepHypRef Expression
1 dgreq0.2 . . . . . 6  |-  A  =  (coeff `  F )
2 fveq2 5691 . . . . . 6  |-  ( F  =  0p  -> 
(coeff `  F )  =  (coeff `  0p
) )
31, 2syl5eq 2487 . . . . 5  |-  ( F  =  0p  ->  A  =  (coeff `  0p ) )
4 coe0 21723 . . . . 5  |-  (coeff ` 
0p )  =  ( NN0  X.  {
0 } )
53, 4syl6eq 2491 . . . 4  |-  ( F  =  0p  ->  A  =  ( NN0  X. 
{ 0 } ) )
6 dgreq0.1 . . . . . 6  |-  N  =  (deg `  F )
7 fveq2 5691 . . . . . 6  |-  ( F  =  0p  -> 
(deg `  F )  =  (deg `  0p
) )
86, 7syl5eq 2487 . . . . 5  |-  ( F  =  0p  ->  N  =  (deg `  0p ) )
9 dgr0 21729 . . . . 5  |-  (deg ` 
0p )  =  0
108, 9syl6eq 2491 . . . 4  |-  ( F  =  0p  ->  N  =  0 )
115, 10fveq12d 5697 . . 3  |-  ( F  =  0p  -> 
( A `  N
)  =  ( ( NN0  X.  { 0 } ) `  0
) )
12 0nn0 10594 . . . 4  |-  0  e.  NN0
13 fvconst2g 5931 . . . 4  |-  ( ( 0  e.  NN0  /\  0  e.  NN0 )  -> 
( ( NN0  X.  { 0 } ) `
 0 )  =  0 )
1412, 12, 13mp2an 672 . . 3  |-  ( ( NN0  X.  { 0 } ) `  0
)  =  0
1511, 14syl6eq 2491 . 2  |-  ( F  =  0p  -> 
( A `  N
)  =  0 )
161coefv0 21715 . . . . . . . 8  |-  ( F  e.  (Poly `  S
)  ->  ( F `  0 )  =  ( A `  0
) )
1716adantr 465 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  ( A `  N )  =  0 )  -> 
( F `  0
)  =  ( A `
 0 ) )
18 simpr 461 . . . . . . . . . . . 12  |-  ( ( ( F  e.  (Poly `  S )  /\  ( A `  N )  =  0 )  /\  N  e.  NN )  ->  N  e.  NN )
1918nnred 10337 . . . . . . . . . . 11  |-  ( ( ( F  e.  (Poly `  S )  /\  ( A `  N )  =  0 )  /\  N  e.  NN )  ->  N  e.  RR )
2019ltm1d 10265 . . . . . . . . . 10  |-  ( ( ( F  e.  (Poly `  S )  /\  ( A `  N )  =  0 )  /\  N  e.  NN )  ->  ( N  -  1 )  <  N )
21 simpll 753 . . . . . . . . . . . 12  |-  ( ( ( F  e.  (Poly `  S )  /\  ( A `  N )  =  0 )  /\  N  e.  NN )  ->  F  e.  (Poly `  S ) )
22 nnm1nn0 10621 . . . . . . . . . . . . 13  |-  ( N  e.  NN  ->  ( N  -  1 )  e.  NN0 )
2322adantl 466 . . . . . . . . . . . 12  |-  ( ( ( F  e.  (Poly `  S )  /\  ( A `  N )  =  0 )  /\  N  e.  NN )  ->  ( N  -  1 )  e.  NN0 )
241, 6dgrub 21702 . . . . . . . . . . . . . . . . . . 19  |-  ( ( F  e.  (Poly `  S )  /\  k  e.  NN0  /\  ( A `
 k )  =/=  0 )  ->  k  <_  N )
25243expia 1189 . . . . . . . . . . . . . . . . . 18  |-  ( ( F  e.  (Poly `  S )  /\  k  e.  NN0 )  ->  (
( A `  k
)  =/=  0  -> 
k  <_  N )
)
2625ad2ant2rl 748 . . . . . . . . . . . . . . . . 17  |-  ( ( ( F  e.  (Poly `  S )  /\  ( A `  N )  =  0 )  /\  ( N  e.  NN  /\  k  e.  NN0 )
)  ->  ( ( A `  k )  =/=  0  ->  k  <_  N ) )
27 simplr 754 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( F  e.  (Poly `  S )  /\  ( A `  N )  =  0 )  /\  ( N  e.  NN  /\  k  e.  NN0 )
)  ->  ( A `  N )  =  0 )
28 fveq2 5691 . . . . . . . . . . . . . . . . . . . 20  |-  ( N  =  k  ->  ( A `  N )  =  ( A `  k ) )
2928eqeq1d 2451 . . . . . . . . . . . . . . . . . . 19  |-  ( N  =  k  ->  (
( A `  N
)  =  0  <->  ( A `  k )  =  0 ) )
3027, 29syl5ibcom 220 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( F  e.  (Poly `  S )  /\  ( A `  N )  =  0 )  /\  ( N  e.  NN  /\  k  e.  NN0 )
)  ->  ( N  =  k  ->  ( A `
 k )  =  0 ) )
3130necon3d 2646 . . . . . . . . . . . . . . . . 17  |-  ( ( ( F  e.  (Poly `  S )  /\  ( A `  N )  =  0 )  /\  ( N  e.  NN  /\  k  e.  NN0 )
)  ->  ( ( A `  k )  =/=  0  ->  N  =/=  k ) )
3226, 31jcad 533 . . . . . . . . . . . . . . . 16  |-  ( ( ( F  e.  (Poly `  S )  /\  ( A `  N )  =  0 )  /\  ( N  e.  NN  /\  k  e.  NN0 )
)  ->  ( ( A `  k )  =/=  0  ->  ( k  <_  N  /\  N  =/=  k ) ) )
33 nn0re 10588 . . . . . . . . . . . . . . . . . . 19  |-  ( k  e.  NN0  ->  k  e.  RR )
3433ad2antll 728 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( F  e.  (Poly `  S )  /\  ( A `  N )  =  0 )  /\  ( N  e.  NN  /\  k  e.  NN0 )
)  ->  k  e.  RR )
35 nnre 10329 . . . . . . . . . . . . . . . . . . 19  |-  ( N  e.  NN  ->  N  e.  RR )
3635ad2antrl 727 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( F  e.  (Poly `  S )  /\  ( A `  N )  =  0 )  /\  ( N  e.  NN  /\  k  e.  NN0 )
)  ->  N  e.  RR )
3734, 36ltlend 9519 . . . . . . . . . . . . . . . . 17  |-  ( ( ( F  e.  (Poly `  S )  /\  ( A `  N )  =  0 )  /\  ( N  e.  NN  /\  k  e.  NN0 )
)  ->  ( k  <  N  <->  ( k  <_  N  /\  N  =/=  k
) ) )
38 nn0z 10669 . . . . . . . . . . . . . . . . . . 19  |-  ( k  e.  NN0  ->  k  e.  ZZ )
3938ad2antll 728 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( F  e.  (Poly `  S )  /\  ( A `  N )  =  0 )  /\  ( N  e.  NN  /\  k  e.  NN0 )
)  ->  k  e.  ZZ )
40 nnz 10668 . . . . . . . . . . . . . . . . . . 19  |-  ( N  e.  NN  ->  N  e.  ZZ )
4140ad2antrl 727 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( F  e.  (Poly `  S )  /\  ( A `  N )  =  0 )  /\  ( N  e.  NN  /\  k  e.  NN0 )
)  ->  N  e.  ZZ )
42 zltlem1 10697 . . . . . . . . . . . . . . . . . 18  |-  ( ( k  e.  ZZ  /\  N  e.  ZZ )  ->  ( k  <  N  <->  k  <_  ( N  - 
1 ) ) )
4339, 41, 42syl2anc 661 . . . . . . . . . . . . . . . . 17  |-  ( ( ( F  e.  (Poly `  S )  /\  ( A `  N )  =  0 )  /\  ( N  e.  NN  /\  k  e.  NN0 )
)  ->  ( k  <  N  <->  k  <_  ( N  -  1 ) ) )
4437, 43bitr3d 255 . . . . . . . . . . . . . . . 16  |-  ( ( ( F  e.  (Poly `  S )  /\  ( A `  N )  =  0 )  /\  ( N  e.  NN  /\  k  e.  NN0 )
)  ->  ( (
k  <_  N  /\  N  =/=  k )  <->  k  <_  ( N  -  1 ) ) )
4532, 44sylibd 214 . . . . . . . . . . . . . . 15  |-  ( ( ( F  e.  (Poly `  S )  /\  ( A `  N )  =  0 )  /\  ( N  e.  NN  /\  k  e.  NN0 )
)  ->  ( ( A `  k )  =/=  0  ->  k  <_ 
( N  -  1 ) ) )
4645expr 615 . . . . . . . . . . . . . 14  |-  ( ( ( F  e.  (Poly `  S )  /\  ( A `  N )  =  0 )  /\  N  e.  NN )  ->  ( k  e.  NN0  ->  ( ( A `  k )  =/=  0  ->  k  <_  ( N  -  1 ) ) ) )
4746ralrimiv 2798 . . . . . . . . . . . . 13  |-  ( ( ( F  e.  (Poly `  S )  /\  ( A `  N )  =  0 )  /\  N  e.  NN )  ->  A. k  e.  NN0  ( ( A `  k )  =/=  0  ->  k  <_  ( N  -  1 ) ) )
481coef3 21700 . . . . . . . . . . . . . . 15  |-  ( F  e.  (Poly `  S
)  ->  A : NN0
--> CC )
4948ad2antrr 725 . . . . . . . . . . . . . 14  |-  ( ( ( F  e.  (Poly `  S )  /\  ( A `  N )  =  0 )  /\  N  e.  NN )  ->  A : NN0 --> CC )
50 plyco0 21660 . . . . . . . . . . . . . 14  |-  ( ( ( N  -  1 )  e.  NN0  /\  A : NN0 --> CC )  ->  ( ( A
" ( ZZ>= `  (
( N  -  1 )  +  1 ) ) )  =  {
0 }  <->  A. k  e.  NN0  ( ( A `
 k )  =/=  0  ->  k  <_  ( N  -  1 ) ) ) )
5123, 49, 50syl2anc 661 . . . . . . . . . . . . 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 ) ) ) )
5247, 51mpbird 232 . . . . . . . . . . . 12  |-  ( ( ( F  e.  (Poly `  S )  /\  ( A `  N )  =  0 )  /\  N  e.  NN )  ->  ( A " ( ZZ>=
`  ( ( N  -  1 )  +  1 ) ) )  =  { 0 } )
531, 6dgrlb 21704 . . . . . . . . . . . 12  |-  ( ( F  e.  (Poly `  S )  /\  ( N  -  1 )  e.  NN0  /\  ( A " ( ZZ>= `  (
( N  -  1 )  +  1 ) ) )  =  {
0 } )  ->  N  <_  ( N  - 
1 ) )
5421, 23, 52, 53syl3anc 1218 . . . . . . . . . . 11  |-  ( ( ( F  e.  (Poly `  S )  /\  ( A `  N )  =  0 )  /\  N  e.  NN )  ->  N  <_  ( N  -  1 ) )
5535adantl 466 . . . . . . . . . . . 12  |-  ( ( ( F  e.  (Poly `  S )  /\  ( A `  N )  =  0 )  /\  N  e.  NN )  ->  N  e.  RR )
56 peano2rem 9675 . . . . . . . . . . . . 13  |-  ( N  e.  RR  ->  ( N  -  1 )  e.  RR )
5755, 56syl 16 . . . . . . . . . . . 12  |-  ( ( ( F  e.  (Poly `  S )  /\  ( A `  N )  =  0 )  /\  N  e.  NN )  ->  ( N  -  1 )  e.  RR )
5855, 57lenltd 9520 . . . . . . . . . . 11  |-  ( ( ( F  e.  (Poly `  S )  /\  ( A `  N )  =  0 )  /\  N  e.  NN )  ->  ( N  <_  ( N  -  1 )  <->  -.  ( N  -  1 )  <  N ) )
5954, 58mpbid 210 . . . . . . . . . 10  |-  ( ( ( F  e.  (Poly `  S )  /\  ( A `  N )  =  0 )  /\  N  e.  NN )  ->  -.  ( N  - 
1 )  <  N
)
6020, 59pm2.65da 576 . . . . . . . . 9  |-  ( ( F  e.  (Poly `  S )  /\  ( A `  N )  =  0 )  ->  -.  N  e.  NN )
61 dgrcl 21701 . . . . . . . . . . . . 13  |-  ( F  e.  (Poly `  S
)  ->  (deg `  F
)  e.  NN0 )
626, 61syl5eqel 2527 . . . . . . . . . . . 12  |-  ( F  e.  (Poly `  S
)  ->  N  e.  NN0 )
6362adantr 465 . . . . . . . . . . 11  |-  ( ( F  e.  (Poly `  S )  /\  ( A `  N )  =  0 )  ->  N  e.  NN0 )
64 elnn0 10581 . . . . . . . . . . 11  |-  ( N  e.  NN0  <->  ( N  e.  NN  \/  N  =  0 ) )
6563, 64sylib 196 . . . . . . . . . 10  |-  ( ( F  e.  (Poly `  S )  /\  ( A `  N )  =  0 )  -> 
( N  e.  NN  \/  N  =  0
) )
6665ord 377 . . . . . . . . 9  |-  ( ( F  e.  (Poly `  S )  /\  ( A `  N )  =  0 )  -> 
( -.  N  e.  NN  ->  N  = 
0 ) )
6760, 66mpd 15 . . . . . . . 8  |-  ( ( F  e.  (Poly `  S )  /\  ( A `  N )  =  0 )  ->  N  =  0 )
6867fveq2d 5695 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  ( A `  N )  =  0 )  -> 
( A `  N
)  =  ( A `
 0 ) )
69 simpr 461 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  ( A `  N )  =  0 )  -> 
( A `  N
)  =  0 )
7017, 68, 693eqtr2d 2481 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  ( A `  N )  =  0 )  -> 
( F `  0
)  =  0 )
7170sneqd 3889 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  ( A `  N )  =  0 )  ->  { ( F ` 
0 ) }  =  { 0 } )
7271xpeq2d 4864 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  ( A `  N )  =  0 )  -> 
( CC  X.  {
( F `  0
) } )  =  ( CC  X.  {
0 } ) )
736, 67syl5eqr 2489 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  ( A `  N )  =  0 )  -> 
(deg `  F )  =  0 )
74 0dgrb 21714 . . . . . 6  |-  ( F  e.  (Poly `  S
)  ->  ( (deg `  F )  =  0  <-> 
F  =  ( CC 
X.  { ( F `
 0 ) } ) ) )
7574adantr 465 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  ( A `  N )  =  0 )  -> 
( (deg `  F
)  =  0  <->  F  =  ( CC  X.  { ( F ` 
0 ) } ) ) )
7673, 75mpbid 210 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  ( A `  N )  =  0 )  ->  F  =  ( CC  X.  { ( F ` 
0 ) } ) )
77 df-0p 21148 . . . . 5  |-  0p  =  ( CC  X.  { 0 } )
7877a1i 11 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  ( A `  N )  =  0 )  -> 
0p  =  ( CC  X.  { 0 } ) )
7972, 76, 783eqtr4d 2485 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  ( A `  N )  =  0 )  ->  F  =  0p
)
8079ex 434 . 2  |-  ( F  e.  (Poly `  S
)  ->  ( ( A `  N )  =  0  ->  F  =  0p ) )
8115, 80impbid2 204 1  |-  ( F  e.  (Poly `  S
)  ->  ( F  =  0p  <->  ( A `  N )  =  0 ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1369    e. wcel 1756    =/= wne 2606   A.wral 2715   {csn 3877   class class class wbr 4292    X. cxp 4838   "cima 4843   -->wf 5414   ` cfv 5418  (class class class)co 6091   CCcc 9280   RRcr 9281   0cc0 9282   1c1 9283    + caddc 9285    < clt 9418    <_ cle 9419    - cmin 9595   NNcn 10322   NN0cn0 10579   ZZcz 10646   ZZ>=cuz 10861   0pc0p 21147  Polycply 21652  coeffccoe 21654  degcdgr 21655
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-inf2 7847  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359  ax-pre-sup 9360  ax-addf 9361
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-se 4680  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-isom 5427  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-of 6320  df-om 6477  df-1st 6577  df-2nd 6578  df-recs 6832  df-rdg 6866  df-1o 6920  df-oadd 6924  df-er 7101  df-map 7216  df-pm 7217  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-sup 7691  df-oi 7724  df-card 8109  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-div 9994  df-nn 10323  df-2 10380  df-3 10381  df-n0 10580  df-z 10647  df-uz 10862  df-rp 10992  df-fz 11438  df-fzo 11549  df-fl 11642  df-seq 11807  df-exp 11866  df-hash 12104  df-cj 12588  df-re 12589  df-im 12590  df-sqr 12724  df-abs 12725  df-clim 12966  df-rlim 12967  df-sum 13164  df-0p 21148  df-ply 21656  df-coe 21658  df-dgr 21659
This theorem is referenced by:  dgrlt  21733  dgradd2  21735  dgrmul  21737  dgrcolem2  21741  plymul0or  21747  plydivlem4  21762  plydiveu  21764  vieta1lem2  21777  vieta1  21778  aareccl  21792  ftalem2  22411  ftalem4  22413  ftalem5  22414  signsply0  26952  mpaaeu  29507
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