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Theorem deg1ldg 19478
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 19466 . . 3  |-  D  =  ( 1o mDeg  R )
3 eqid 2283 . . 3  |-  ( 1o mPoly  R )  =  ( 1o mPoly  R )
4 deg1z.p . . . 4  |-  P  =  (Poly1 `  R )
5 eqid 2283 . . . 4  |-  (PwSer1 `  R
)  =  (PwSer1 `  R
)
6 deg1nn0cl.b . . . 4  |-  B  =  ( Base `  P
)
74, 5, 6ply1bas 16274 . . 3  |-  B  =  ( Base `  ( 1o mPoly  R ) )
8 deg1ldg.y . . 3  |-  Y  =  ( 0g `  R
)
9 psr1baslem 16264 . . 3  |-  ( NN0 
^m  1o )  =  { c  e.  ( NN0  ^m  1o )  |  ( `' c
" NN )  e. 
Fin }
10 tdeglem2 19447 . . 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 16333 . . 3  |-  .0.  =  ( 0g `  ( 1o mPoly  R ) )
132, 3, 7, 8, 9, 10, 12mdegldg 19452 . 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 16288 . . . . . . . . . 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 5524 . . . . . . . . . . . 12  |-  ( a  =  b  ->  (
a `  (/) )  =  ( b `  (/) ) )
18 eqid 2283 . . . . . . . . . . . 12  |-  ( a  e.  ( NN0  ^m  1o )  |->  ( a `
 (/) ) )  =  ( a  e.  ( NN0  ^m  1o ) 
|->  ( a `  (/) ) )
19 fvex 5539 . . . . . . . . . . . 12  |-  ( b `
 (/) )  e.  _V
2017, 18, 19fvmpt 5602 . . . . . . . . . . 11  |-  ( b  e.  ( NN0  ^m  1o )  ->  ( ( a  e.  ( NN0 
^m  1o )  |->  ( a `  (/) ) ) `
 b )  =  ( b `  (/) ) )
2120fveq2d 5529 . . . . . . . . . 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 2318 . . . . . . . 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 2459 . . . . . . 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 2560 . . . 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 6491 . . . . . 6  |-  1o  =  { (/) }
30 nn0ex 9971 . . . . . 6  |-  NN0  e.  _V
31 0ex 4150 . . . . . 6  |-  (/)  e.  _V
3229, 30, 31, 18mapsnf1o2 6815 . . . . 5  |-  ( a  e.  ( NN0  ^m  1o )  |->  ( a `
 (/) ) ) : ( NN0  ^m  1o )
-1-1-onto-> NN0
33 f1ofo 5479 . . . . 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 2289 . . . . . . 7  |-  ( ( ( a  e.  ( NN0  ^m  1o ) 
|->  ( a `  (/) ) ) `
 b )  =  d  ->  ( (
( a  e.  ( NN0  ^m  1o ) 
|->  ( a `  (/) ) ) `
 b )  =  ( D `  F
)  <->  d  =  ( D `  F ) ) )
35 fveq2 5525 . . . . . . . 8  |-  ( ( ( a  e.  ( NN0  ^m  1o ) 
|->  ( a `  (/) ) ) `
 b )  =  d  ->  ( A `  ( ( a  e.  ( NN0  ^m  1o )  |->  ( a `  (/) ) ) `  b
) )  =  ( A `  d ) )
3635neeq1d 2459 . . . . . . 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 5800 . . . . 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 19474 . . . 4  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
.0.  )  ->  ( D `  F )  e.  NN0 )
42 fveq2 5525 . . . . . 6  |-  ( d  =  ( D `  F )  ->  ( A `  d )  =  ( A `  ( D `  F ) ) )
4342neeq1d 2459 . . . . 5  |-  ( d  =  ( D `  F )  ->  (
( A `  d
)  =/=  Y  <->  ( A `  ( D `  F
) )  =/=  Y
) )
4443ceqsrexv 2901 . . . 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 1623    e. wcel 1684    =/= wne 2446   E.wrex 2544   (/)c0 3455    e. cmpt 4077   -onto->wfo 5253   -1-1-onto->wf1o 5254   ` cfv 5255  (class class class)co 5858   1oc1o 6472    ^m cmap 6772   NN0cn0 9965   Basecbs 13148   0gc0g 13400   Ringcrg 15337   mPoly cmpl 16089  PwSer1cps1 16250  Poly1cpl1 16252  coe1cco1 16255   deg1 cdg1 19440
This theorem is referenced by:  deg1ldgn  19479  deg1ldgdomn  19480  deg1add  19489  deg1mul2  19500  drnguc1p  19556
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-addf 8816  ax-mulf 8817
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-oi 7225  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-fz 10783  df-fzo 10871  df-seq 11047  df-hash 11338  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-starv 13223  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-0g 13404  df-gsum 13405  df-mnd 14367  df-submnd 14416  df-grp 14489  df-minusg 14490  df-mulg 14492  df-subg 14618  df-cntz 14793  df-cmn 15091  df-abl 15092  df-mgp 15326  df-rng 15340  df-cring 15341  df-ur 15342  df-psr 16098  df-mpl 16100  df-opsr 16106  df-psr1 16257  df-ply1 16259  df-coe1 16262  df-cnfld 16378  df-mdeg 19441  df-deg1 19442
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