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Theorem deg1ldg 19494
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 19482 . . 3  |-  D  =  ( 1o mDeg  R )
3 eqid 2296 . . 3  |-  ( 1o mPoly  R )  =  ( 1o mPoly  R )
4 deg1z.p . . . 4  |-  P  =  (Poly1 `  R )
5 eqid 2296 . . . 4  |-  (PwSer1 `  R
)  =  (PwSer1 `  R
)
6 deg1nn0cl.b . . . 4  |-  B  =  ( Base `  P
)
74, 5, 6ply1bas 16290 . . 3  |-  B  =  ( Base `  ( 1o mPoly  R ) )
8 deg1ldg.y . . 3  |-  Y  =  ( 0g `  R
)
9 psr1baslem 16280 . . 3  |-  ( NN0 
^m  1o )  =  { c  e.  ( NN0  ^m  1o )  |  ( `' c
" NN )  e. 
Fin }
10 tdeglem2 19463 . . 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 16349 . . 3  |-  .0.  =  ( 0g `  ( 1o mPoly  R ) )
132, 3, 7, 8, 9, 10, 12mdegldg 19468 . 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 16304 . . . . . . . . . 10  |-  ( ( F  e.  B  /\  b  e.  ( NN0  ^m  1o ) )  -> 
( F `  b
)  =  ( A `
 ( b `  (/) ) ) )
16153ad2antl2 1118 . . . . . . . . 9  |-  ( ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/=  .0.  )  /\  b  e.  ( NN0  ^m  1o ) )  -> 
( F `  b
)  =  ( A `
 ( b `  (/) ) ) )
17 fveq1 5540 . . . . . . . . . . . 12  |-  ( a  =  b  ->  (
a `  (/) )  =  ( b `  (/) ) )
18 eqid 2296 . . . . . . . . . . . 12  |-  ( a  e.  ( NN0  ^m  1o )  |->  ( a `
 (/) ) )  =  ( a  e.  ( NN0  ^m  1o ) 
|->  ( a `  (/) ) )
19 fvex 5555 . . . . . . . . . . . 12  |-  ( b `
 (/) )  e.  _V
2017, 18, 19fvmpt 5618 . . . . . . . . . . 11  |-  ( b  e.  ( NN0  ^m  1o )  ->  ( ( a  e.  ( NN0 
^m  1o )  |->  ( a `  (/) ) ) `
 b )  =  ( b `  (/) ) )
2120fveq2d 5545 . . . . . . . . . 10  |-  ( b  e.  ( NN0  ^m  1o )  ->  ( A `
 ( ( a  e.  ( NN0  ^m  1o )  |->  ( a `
 (/) ) ) `  b ) )  =  ( A `  (
b `  (/) ) ) )
2221adantl 452 . . . . . . . . 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 2331 . . . . . . . 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 2472 . . . . . . 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 685 . . . . . 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 437 . . . . . 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 252 . . . . 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 2573 . . . 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 6507 . . . . . 6  |-  1o  =  { (/) }
30 nn0ex 9987 . . . . . 6  |-  NN0  e.  _V
31 0ex 4166 . . . . . 6  |-  (/)  e.  _V
3229, 30, 31, 18mapsnf1o2 6831 . . . . 5  |-  ( a  e.  ( NN0  ^m  1o )  |->  ( a `
 (/) ) ) : ( NN0  ^m  1o )
-1-1-onto-> NN0
33 f1ofo 5495 . . . . 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 2302 . . . . . . 7  |-  ( ( ( a  e.  ( NN0  ^m  1o ) 
|->  ( a `  (/) ) ) `
 b )  =  d  ->  ( (
( a  e.  ( NN0  ^m  1o ) 
|->  ( a `  (/) ) ) `
 b )  =  ( D `  F
)  <->  d  =  ( D `  F ) ) )
35 fveq2 5541 . . . . . . . 8  |-  ( ( ( a  e.  ( NN0  ^m  1o ) 
|->  ( a `  (/) ) ) `
 b )  =  d  ->  ( A `  ( ( a  e.  ( NN0  ^m  1o )  |->  ( a `  (/) ) ) `  b
) )  =  ( A `  d ) )
3635neeq1d 2472 . . . . . . 7  |-  ( ( ( a  e.  ( NN0  ^m  1o ) 
|->  ( a `  (/) ) ) `
 b )  =  d  ->  ( ( A `  ( (
a  e.  ( NN0 
^m  1o )  |->  ( a `  (/) ) ) `
 b ) )  =/=  Y  <->  ( A `  d )  =/=  Y
) )
3734, 36anbi12d 691 . . . . . 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 5816 . . . . 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 9 . . . 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 252 . . 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 19490 . . . 4  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
.0.  )  ->  ( D `  F )  e.  NN0 )
42 fveq2 5541 . . . . . 6  |-  ( d  =  ( D `  F )  ->  ( A `  d )  =  ( A `  ( D `  F ) ) )
4342neeq1d 2472 . . . . 5  |-  ( d  =  ( D `  F )  ->  (
( A `  d
)  =/=  Y  <->  ( A `  ( D `  F
) )  =/=  Y
) )
4443ceqsrexv 2914 . . . 4  |-  ( ( D `  F )  e.  NN0  ->  ( E. d  e.  NN0  (
d  =  ( D `
 F )  /\  ( A `  d )  =/=  Y )  <->  ( A `  ( D `  F
) )  =/=  Y
) )
4541, 44syl 15 . . 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 244 . 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 201 1  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
.0.  )  ->  ( A `  ( D `  F ) )  =/= 
Y )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   E.wrex 2557   (/)c0 3468    e. cmpt 4093   -onto->wfo 5269   -1-1-onto->wf1o 5270   ` cfv 5271  (class class class)co 5874   1oc1o 6488    ^m cmap 6788   NN0cn0 9981   Basecbs 13164   0gc0g 13416   Ringcrg 15353   mPoly cmpl 16105  PwSer1cps1 16266  Poly1cpl1 16268  coe1cco1 16271   deg1 cdg1 19456
This theorem is referenced by:  deg1ldgn  19495  deg1ldgdomn  19496  deg1add  19505  deg1mul2  19516  drnguc1p  19572
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-addf 8832  ax-mulf 8833
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-oi 7241  df-card 7588  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-fz 10799  df-fzo 10887  df-seq 11063  df-hash 11354  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-starv 13239  df-sca 13240  df-vsca 13241  df-tset 13243  df-ple 13244  df-ds 13246  df-0g 13420  df-gsum 13421  df-mnd 14383  df-submnd 14432  df-grp 14505  df-minusg 14506  df-mulg 14508  df-subg 14634  df-cntz 14809  df-cmn 15107  df-abl 15108  df-mgp 15342  df-rng 15356  df-cring 15357  df-ur 15358  df-psr 16114  df-mpl 16116  df-opsr 16122  df-psr1 16273  df-ply1 16275  df-coe1 16278  df-cnfld 16394  df-mdeg 19457  df-deg1 19458
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