MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  deg1ldg Structured version   Unicode version

Theorem deg1ldg 20007
Description: A nonzero univariate polynomial always has a nonzero leading coefficient. (Contributed by Stefan O'Rear, 23-Mar-2015.)
Hypotheses
Ref Expression
deg1z.d  |-  D  =  ( deg1  `  R )
deg1z.p  |-  P  =  (Poly1 `  R )
deg1z.z  |-  .0.  =  ( 0g `  P )
deg1nn0cl.b  |-  B  =  ( Base `  P
)
deg1ldg.y  |-  Y  =  ( 0g `  R
)
deg1ldg.a  |-  A  =  (coe1 `  F )
Assertion
Ref Expression
deg1ldg  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
.0.  )  ->  ( A `  ( D `  F ) )  =/= 
Y )

Proof of Theorem deg1ldg
Dummy variables  b 
d  a  c are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 deg1z.d . . . 4  |-  D  =  ( deg1  `  R )
21deg1fval 19995 . . 3  |-  D  =  ( 1o mDeg  R )
3 eqid 2435 . . 3  |-  ( 1o mPoly  R )  =  ( 1o mPoly  R )
4 deg1z.p . . . 4  |-  P  =  (Poly1 `  R )
5 eqid 2435 . . . 4  |-  (PwSer1 `  R
)  =  (PwSer1 `  R
)
6 deg1nn0cl.b . . . 4  |-  B  =  ( Base `  P
)
74, 5, 6ply1bas 16585 . . 3  |-  B  =  ( Base `  ( 1o mPoly  R ) )
8 deg1ldg.y . . 3  |-  Y  =  ( 0g `  R
)
9 psr1baslem 16575 . . 3  |-  ( NN0 
^m  1o )  =  { c  e.  ( NN0  ^m  1o )  |  ( `' c
" NN )  e. 
Fin }
10 tdeglem2 19976 . . 3  |-  ( a  e.  ( NN0  ^m  1o )  |->  ( a `
 (/) ) )  =  ( a  e.  ( NN0  ^m  1o ) 
|->  (fld 
gsumg  a ) )
11 deg1z.z . . . 4  |-  .0.  =  ( 0g `  P )
123, 4, 11ply1mpl0 16641 . . 3  |-  .0.  =  ( 0g `  ( 1o mPoly  R ) )
132, 3, 7, 8, 9, 10, 12mdegldg 19981 . 2  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
.0.  )  ->  E. b  e.  ( NN0  ^m  1o ) ( ( F `
 b )  =/= 
Y  /\  ( (
a  e.  ( NN0 
^m  1o )  |->  ( a `  (/) ) ) `
 b )  =  ( D `  F
) ) )
14 deg1ldg.a . . . . . . . . . . 11  |-  A  =  (coe1 `  F )
1514fvcoe1 16597 . . . . . . . . . 10  |-  ( ( F  e.  B  /\  b  e.  ( NN0  ^m  1o ) )  -> 
( F `  b
)  =  ( A `
 ( b `  (/) ) ) )
16153ad2antl2 1120 . . . . . . . . 9  |-  ( ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/=  .0.  )  /\  b  e.  ( NN0  ^m  1o ) )  -> 
( F `  b
)  =  ( A `
 ( b `  (/) ) ) )
17 fveq1 5719 . . . . . . . . . . . 12  |-  ( a  =  b  ->  (
a `  (/) )  =  ( b `  (/) ) )
18 eqid 2435 . . . . . . . . . . . 12  |-  ( a  e.  ( NN0  ^m  1o )  |->  ( a `
 (/) ) )  =  ( a  e.  ( NN0  ^m  1o ) 
|->  ( a `  (/) ) )
19 fvex 5734 . . . . . . . . . . . 12  |-  ( b `
 (/) )  e.  _V
2017, 18, 19fvmpt 5798 . . . . . . . . . . 11  |-  ( b  e.  ( NN0  ^m  1o )  ->  ( ( a  e.  ( NN0 
^m  1o )  |->  ( a `  (/) ) ) `
 b )  =  ( b `  (/) ) )
2120fveq2d 5724 . . . . . . . . . 10  |-  ( b  e.  ( NN0  ^m  1o )  ->  ( A `
 ( ( a  e.  ( NN0  ^m  1o )  |->  ( a `
 (/) ) ) `  b ) )  =  ( A `  (
b `  (/) ) ) )
2221adantl 453 . . . . . . . . 9  |-  ( ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/=  .0.  )  /\  b  e.  ( NN0  ^m  1o ) )  -> 
( A `  (
( a  e.  ( NN0  ^m  1o ) 
|->  ( a `  (/) ) ) `
 b ) )  =  ( A `  ( b `  (/) ) ) )
2316, 22eqtr4d 2470 . . . . . . . 8  |-  ( ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/=  .0.  )  /\  b  e.  ( NN0  ^m  1o ) )  -> 
( F `  b
)  =  ( A `
 ( ( a  e.  ( NN0  ^m  1o )  |->  ( a `
 (/) ) ) `  b ) ) )
2423neeq1d 2611 . . . . . . 7  |-  ( ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/=  .0.  )  /\  b  e.  ( NN0  ^m  1o ) )  -> 
( ( F `  b )  =/=  Y  <->  ( A `  ( ( a  e.  ( NN0 
^m  1o )  |->  ( a `  (/) ) ) `
 b ) )  =/=  Y ) )
2524anbi1d 686 . . . . . 6  |-  ( ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/=  .0.  )  /\  b  e.  ( NN0  ^m  1o ) )  -> 
( ( ( F `
 b )  =/= 
Y  /\  ( (
a  e.  ( NN0 
^m  1o )  |->  ( a `  (/) ) ) `
 b )  =  ( D `  F
) )  <->  ( ( A `  ( (
a  e.  ( NN0 
^m  1o )  |->  ( a `  (/) ) ) `
 b ) )  =/=  Y  /\  (
( a  e.  ( NN0  ^m  1o ) 
|->  ( a `  (/) ) ) `
 b )  =  ( D `  F
) ) ) )
26 ancom 438 . . . . . 6  |-  ( ( ( A `  (
( a  e.  ( NN0  ^m  1o ) 
|->  ( a `  (/) ) ) `
 b ) )  =/=  Y  /\  (
( a  e.  ( NN0  ^m  1o ) 
|->  ( a `  (/) ) ) `
 b )  =  ( D `  F
) )  <->  ( (
( a  e.  ( NN0  ^m  1o ) 
|->  ( a `  (/) ) ) `
 b )  =  ( D `  F
)  /\  ( A `  ( ( a  e.  ( NN0  ^m  1o )  |->  ( a `  (/) ) ) `  b
) )  =/=  Y
) )
2725, 26syl6bb 253 . . . . 5  |-  ( ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/=  .0.  )  /\  b  e.  ( NN0  ^m  1o ) )  -> 
( ( ( F `
 b )  =/= 
Y  /\  ( (
a  e.  ( NN0 
^m  1o )  |->  ( a `  (/) ) ) `
 b )  =  ( D `  F
) )  <->  ( (
( a  e.  ( NN0  ^m  1o ) 
|->  ( a `  (/) ) ) `
 b )  =  ( D `  F
)  /\  ( A `  ( ( a  e.  ( NN0  ^m  1o )  |->  ( a `  (/) ) ) `  b
) )  =/=  Y
) ) )
2827rexbidva 2714 . . . 4  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
.0.  )  ->  ( E. b  e.  ( NN0  ^m  1o ) ( ( F `  b
)  =/=  Y  /\  ( ( a  e.  ( NN0  ^m  1o )  |->  ( a `  (/) ) ) `  b
)  =  ( D `
 F ) )  <->  E. b  e.  ( NN0  ^m  1o ) ( ( ( a  e.  ( NN0  ^m  1o )  |->  ( a `  (/) ) ) `  b
)  =  ( D `
 F )  /\  ( A `  ( ( a  e.  ( NN0 
^m  1o )  |->  ( a `  (/) ) ) `
 b ) )  =/=  Y ) ) )
29 df1o2 6728 . . . . . 6  |-  1o  =  { (/) }
30 nn0ex 10219 . . . . . 6  |-  NN0  e.  _V
31 0ex 4331 . . . . . 6  |-  (/)  e.  _V
3229, 30, 31, 18mapsnf1o2 7053 . . . . 5  |-  ( a  e.  ( NN0  ^m  1o )  |->  ( a `
 (/) ) ) : ( NN0  ^m  1o )
-1-1-onto-> NN0
33 f1ofo 5673 . . . . 5  |-  ( ( a  e.  ( NN0 
^m  1o )  |->  ( a `  (/) ) ) : ( NN0  ^m  1o ) -1-1-onto-> NN0  ->  ( a  e.  ( NN0  ^m  1o )  |->  ( a `  (/) ) ) : ( NN0  ^m  1o )
-onto->
NN0 )
34 eqeq1 2441 . . . . . . 7  |-  ( ( ( a  e.  ( NN0  ^m  1o ) 
|->  ( a `  (/) ) ) `
 b )  =  d  ->  ( (
( a  e.  ( NN0  ^m  1o ) 
|->  ( a `  (/) ) ) `
 b )  =  ( D `  F
)  <->  d  =  ( D `  F ) ) )
35 fveq2 5720 . . . . . . . 8  |-  ( ( ( a  e.  ( NN0  ^m  1o ) 
|->  ( a `  (/) ) ) `
 b )  =  d  ->  ( A `  ( ( a  e.  ( NN0  ^m  1o )  |->  ( a `  (/) ) ) `  b
) )  =  ( A `  d ) )
3635neeq1d 2611 . . . . . . 7  |-  ( ( ( a  e.  ( NN0  ^m  1o ) 
|->  ( a `  (/) ) ) `
 b )  =  d  ->  ( ( A `  ( (
a  e.  ( NN0 
^m  1o )  |->  ( a `  (/) ) ) `
 b ) )  =/=  Y  <->  ( A `  d )  =/=  Y
) )
3734, 36anbi12d 692 . . . . . 6  |-  ( ( ( a  e.  ( NN0  ^m  1o ) 
|->  ( a `  (/) ) ) `
 b )  =  d  ->  ( (
( ( a  e.  ( NN0  ^m  1o )  |->  ( a `  (/) ) ) `  b
)  =  ( D `
 F )  /\  ( A `  ( ( a  e.  ( NN0 
^m  1o )  |->  ( a `  (/) ) ) `
 b ) )  =/=  Y )  <->  ( d  =  ( D `  F )  /\  ( A `  d )  =/=  Y ) ) )
3837cbvexfo 6015 . . . . 5  |-  ( ( a  e.  ( NN0 
^m  1o )  |->  ( a `  (/) ) ) : ( NN0  ^m  1o ) -onto-> NN0  ->  ( E. b  e.  ( NN0  ^m  1o ) ( ( ( a  e.  ( NN0  ^m  1o ) 
|->  ( a `  (/) ) ) `
 b )  =  ( D `  F
)  /\  ( A `  ( ( a  e.  ( NN0  ^m  1o )  |->  ( a `  (/) ) ) `  b
) )  =/=  Y
)  <->  E. d  e.  NN0  ( d  =  ( D `  F )  /\  ( A `  d )  =/=  Y
) ) )
3932, 33, 38mp2b 10 . . . 4  |-  ( E. b  e.  ( NN0 
^m  1o ) ( ( ( a  e.  ( NN0  ^m  1o )  |->  ( a `  (/) ) ) `  b
)  =  ( D `
 F )  /\  ( A `  ( ( a  e.  ( NN0 
^m  1o )  |->  ( a `  (/) ) ) `
 b ) )  =/=  Y )  <->  E. d  e.  NN0  ( d  =  ( D `  F
)  /\  ( A `  d )  =/=  Y
) )
4028, 39syl6bb 253 . . 3  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
.0.  )  ->  ( E. b  e.  ( NN0  ^m  1o ) ( ( F `  b
)  =/=  Y  /\  ( ( a  e.  ( NN0  ^m  1o )  |->  ( a `  (/) ) ) `  b
)  =  ( D `
 F ) )  <->  E. d  e.  NN0  ( d  =  ( D `  F )  /\  ( A `  d )  =/=  Y
) ) )
411, 4, 11, 6deg1nn0cl 20003 . . . 4  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
.0.  )  ->  ( D `  F )  e.  NN0 )
42 fveq2 5720 . . . . . 6  |-  ( d  =  ( D `  F )  ->  ( A `  d )  =  ( A `  ( D `  F ) ) )
4342neeq1d 2611 . . . . 5  |-  ( d  =  ( D `  F )  ->  (
( A `  d
)  =/=  Y  <->  ( A `  ( D `  F
) )  =/=  Y
) )
4443ceqsrexv 3061 . . . 4  |-  ( ( D `  F )  e.  NN0  ->  ( E. d  e.  NN0  (
d  =  ( D `
 F )  /\  ( A `  d )  =/=  Y )  <->  ( A `  ( D `  F
) )  =/=  Y
) )
4541, 44syl 16 . . 3  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
.0.  )  ->  ( E. d  e.  NN0  ( d  =  ( D `  F )  /\  ( A `  d )  =/=  Y
)  <->  ( A `  ( D `  F ) )  =/=  Y ) )
4640, 45bitrd 245 . 2  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
.0.  )  ->  ( E. b  e.  ( NN0  ^m  1o ) ( ( F `  b
)  =/=  Y  /\  ( ( a  e.  ( NN0  ^m  1o )  |->  ( a `  (/) ) ) `  b
)  =  ( D `
 F ) )  <-> 
( A `  ( D `  F )
)  =/=  Y ) )
4713, 46mpbid 202 1  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
.0.  )  ->  ( A `  ( D `  F ) )  =/= 
Y )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2598   E.wrex 2698   (/)c0 3620    e. cmpt 4258   -onto->wfo 5444   -1-1-onto->wf1o 5445   ` cfv 5446  (class class class)co 6073   1oc1o 6709    ^m cmap 7010   NN0cn0 10213   Basecbs 13461   0gc0g 13715   Ringcrg 15652   mPoly cmpl 16400  PwSer1cps1 16561  Poly1cpl1 16563  coe1cco1 16566   deg1 cdg1 19969
This theorem is referenced by:  deg1ldgn  20008  deg1ldgdomn  20009  deg1add  20018  deg1mul2  20029  drnguc1p  20085
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7588  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-addf 9061  ax-mulf 9062
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-of 6297  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-map 7012  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-sup 7438  df-oi 7471  df-card 7818  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-2 10050  df-3 10051  df-4 10052  df-5 10053  df-6 10054  df-7 10055  df-8 10056  df-9 10057  df-10 10058  df-n0 10214  df-z 10275  df-dec 10375  df-uz 10481  df-fz 11036  df-fzo 11128  df-seq 11316  df-hash 11611  df-struct 13463  df-ndx 13464  df-slot 13465  df-base 13466  df-sets 13467  df-ress 13468  df-plusg 13534  df-mulr 13535  df-starv 13536  df-sca 13537  df-vsca 13538  df-tset 13540  df-ple 13541  df-ds 13543  df-unif 13544  df-0g 13719  df-gsum 13720  df-mnd 14682  df-submnd 14731  df-grp 14804  df-minusg 14805  df-mulg 14807  df-subg 14933  df-cntz 15108  df-cmn 15406  df-abl 15407  df-mgp 15641  df-rng 15655  df-cring 15656  df-ur 15657  df-psr 16409  df-mpl 16411  df-opsr 16417  df-psr1 16568  df-ply1 16570  df-coe1 16573  df-cnfld 16696  df-mdeg 19970  df-deg1 19971
  Copyright terms: Public domain W3C validator