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Theorem deg1ldg 19884
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 19872 . . 3  |-  D  =  ( 1o mDeg  R )
3 eqid 2389 . . 3  |-  ( 1o mPoly  R )  =  ( 1o mPoly  R )
4 deg1z.p . . . 4  |-  P  =  (Poly1 `  R )
5 eqid 2389 . . . 4  |-  (PwSer1 `  R
)  =  (PwSer1 `  R
)
6 deg1nn0cl.b . . . 4  |-  B  =  ( Base `  P
)
74, 5, 6ply1bas 16522 . . 3  |-  B  =  ( Base `  ( 1o mPoly  R ) )
8 deg1ldg.y . . 3  |-  Y  =  ( 0g `  R
)
9 psr1baslem 16512 . . 3  |-  ( NN0 
^m  1o )  =  { c  e.  ( NN0  ^m  1o )  |  ( `' c
" NN )  e. 
Fin }
10 tdeglem2 19853 . . 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 16578 . . 3  |-  .0.  =  ( 0g `  ( 1o mPoly  R ) )
132, 3, 7, 8, 9, 10, 12mdegldg 19858 . 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 16534 . . . . . . . . . 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 5669 . . . . . . . . . . . 12  |-  ( a  =  b  ->  (
a `  (/) )  =  ( b `  (/) ) )
18 eqid 2389 . . . . . . . . . . . 12  |-  ( a  e.  ( NN0  ^m  1o )  |->  ( a `
 (/) ) )  =  ( a  e.  ( NN0  ^m  1o ) 
|->  ( a `  (/) ) )
19 fvex 5684 . . . . . . . . . . . 12  |-  ( b `
 (/) )  e.  _V
2017, 18, 19fvmpt 5747 . . . . . . . . . . 11  |-  ( b  e.  ( NN0  ^m  1o )  ->  ( ( a  e.  ( NN0 
^m  1o )  |->  ( a `  (/) ) ) `
 b )  =  ( b `  (/) ) )
2120fveq2d 5674 . . . . . . . . . 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 2424 . . . . . . . 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 2565 . . . . . . 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 2668 . . . 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 6674 . . . . . 6  |-  1o  =  { (/) }
30 nn0ex 10161 . . . . . 6  |-  NN0  e.  _V
31 0ex 4282 . . . . . 6  |-  (/)  e.  _V
3229, 30, 31, 18mapsnf1o2 6999 . . . . 5  |-  ( a  e.  ( NN0  ^m  1o )  |->  ( a `
 (/) ) ) : ( NN0  ^m  1o )
-1-1-onto-> NN0
33 f1ofo 5623 . . . . 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 2395 . . . . . . 7  |-  ( ( ( a  e.  ( NN0  ^m  1o ) 
|->  ( a `  (/) ) ) `
 b )  =  d  ->  ( (
( a  e.  ( NN0  ^m  1o ) 
|->  ( a `  (/) ) ) `
 b )  =  ( D `  F
)  <->  d  =  ( D `  F ) ) )
35 fveq2 5670 . . . . . . . 8  |-  ( ( ( a  e.  ( NN0  ^m  1o ) 
|->  ( a `  (/) ) ) `
 b )  =  d  ->  ( A `  ( ( a  e.  ( NN0  ^m  1o )  |->  ( a `  (/) ) ) `  b
) )  =  ( A `  d ) )
3635neeq1d 2565 . . . . . . 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 5964 . . . . 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 19880 . . . 4  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
.0.  )  ->  ( D `  F )  e.  NN0 )
42 fveq2 5670 . . . . . 6  |-  ( d  =  ( D `  F )  ->  ( A `  d )  =  ( A `  ( D `  F ) ) )
4342neeq1d 2565 . . . . 5  |-  ( d  =  ( D `  F )  ->  (
( A `  d
)  =/=  Y  <->  ( A `  ( D `  F
) )  =/=  Y
) )
4443ceqsrexv 3014 . . . 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 1649    e. wcel 1717    =/= wne 2552   E.wrex 2652   (/)c0 3573    e. cmpt 4209   -onto->wfo 5394   -1-1-onto->wf1o 5395   ` cfv 5396  (class class class)co 6022   1oc1o 6655    ^m cmap 6956   NN0cn0 10155   Basecbs 13398   0gc0g 13652   Ringcrg 15589   mPoly cmpl 16337  PwSer1cps1 16498  Poly1cpl1 16500  coe1cco1 16503   deg1 cdg1 19846
This theorem is referenced by:  deg1ldgn  19885  deg1ldgdomn  19886  deg1add  19895  deg1mul2  19906  drnguc1p  19962
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643  ax-inf2 7531  ax-cnex 8981  ax-resscn 8982  ax-1cn 8983  ax-icn 8984  ax-addcl 8985  ax-addrcl 8986  ax-mulcl 8987  ax-mulrcl 8988  ax-mulcom 8989  ax-addass 8990  ax-mulass 8991  ax-distr 8992  ax-i2m1 8993  ax-1ne0 8994  ax-1rid 8995  ax-rnegex 8996  ax-rrecex 8997  ax-cnre 8998  ax-pre-lttri 8999  ax-pre-lttrn 9000  ax-pre-ltadd 9001  ax-pre-mulgt0 9002  ax-addf 9004  ax-mulf 9005
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-reu 2658  df-rmo 2659  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-pss 3281  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-tp 3767  df-op 3768  df-uni 3960  df-int 3995  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-tr 4246  df-eprel 4437  df-id 4441  df-po 4446  df-so 4447  df-fr 4484  df-se 4485  df-we 4486  df-ord 4527  df-on 4528  df-lim 4529  df-suc 4530  df-om 4788  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-isom 5405  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-of 6246  df-1st 6290  df-2nd 6291  df-riota 6487  df-recs 6571  df-rdg 6606  df-1o 6662  df-oadd 6666  df-er 6843  df-map 6958  df-en 7048  df-dom 7049  df-sdom 7050  df-fin 7051  df-sup 7383  df-oi 7414  df-card 7761  df-pnf 9057  df-mnf 9058  df-xr 9059  df-ltxr 9060  df-le 9061  df-sub 9227  df-neg 9228  df-nn 9935  df-2 9992  df-3 9993  df-4 9994  df-5 9995  df-6 9996  df-7 9997  df-8 9998  df-9 9999  df-10 10000  df-n0 10156  df-z 10217  df-dec 10317  df-uz 10423  df-fz 10978  df-fzo 11068  df-seq 11253  df-hash 11548  df-struct 13400  df-ndx 13401  df-slot 13402  df-base 13403  df-sets 13404  df-ress 13405  df-plusg 13471  df-mulr 13472  df-starv 13473  df-sca 13474  df-vsca 13475  df-tset 13477  df-ple 13478  df-ds 13480  df-unif 13481  df-0g 13656  df-gsum 13657  df-mnd 14619  df-submnd 14668  df-grp 14741  df-minusg 14742  df-mulg 14744  df-subg 14870  df-cntz 15045  df-cmn 15343  df-abl 15344  df-mgp 15578  df-rng 15592  df-cring 15593  df-ur 15594  df-psr 16346  df-mpl 16348  df-opsr 16354  df-psr1 16505  df-ply1 16507  df-coe1 16510  df-cnfld 16629  df-mdeg 19847  df-deg1 19848
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