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

Theorem deg1ldg 22224
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 22212 . . 3  |-  D  =  ( 1o mDeg  R )
3 eqid 2467 . . 3  |-  ( 1o mPoly  R )  =  ( 1o mPoly  R )
4 deg1z.p . . . 4  |-  P  =  (Poly1 `  R )
5 eqid 2467 . . . 4  |-  (PwSer1 `  R
)  =  (PwSer1 `  R
)
6 deg1nn0cl.b . . . 4  |-  B  =  ( Base `  P
)
74, 5, 6ply1bas 18002 . . 3  |-  B  =  ( Base `  ( 1o mPoly  R ) )
8 deg1ldg.y . . 3  |-  Y  =  ( 0g `  R
)
9 psr1baslem 17992 . . 3  |-  ( NN0 
^m  1o )  =  { c  e.  ( NN0  ^m  1o )  |  ( `' c
" NN )  e. 
Fin }
10 tdeglem2 22191 . . 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 18064 . . 3  |-  .0.  =  ( 0g `  ( 1o mPoly  R ) )
132, 3, 7, 8, 9, 10, 12mdegldg 22198 . 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 18014 . . . . . . . . . 10  |-  ( ( F  e.  B  /\  b  e.  ( NN0  ^m  1o ) )  -> 
( F `  b
)  =  ( A `
 ( b `  (/) ) ) )
16153ad2antl2 1159 . . . . . . . . 9  |-  ( ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/=  .0.  )  /\  b  e.  ( NN0  ^m  1o ) )  -> 
( F `  b
)  =  ( A `
 ( b `  (/) ) ) )
17 fveq1 5863 . . . . . . . . . . . 12  |-  ( a  =  b  ->  (
a `  (/) )  =  ( b `  (/) ) )
18 eqid 2467 . . . . . . . . . . . 12  |-  ( a  e.  ( NN0  ^m  1o )  |->  ( a `
 (/) ) )  =  ( a  e.  ( NN0  ^m  1o ) 
|->  ( a `  (/) ) )
19 fvex 5874 . . . . . . . . . . . 12  |-  ( b `
 (/) )  e.  _V
2017, 18, 19fvmpt 5948 . . . . . . . . . . 11  |-  ( b  e.  ( NN0  ^m  1o )  ->  ( ( a  e.  ( NN0 
^m  1o )  |->  ( a `  (/) ) ) `
 b )  =  ( b `  (/) ) )
2120fveq2d 5868 . . . . . . . . . 10  |-  ( b  e.  ( NN0  ^m  1o )  ->  ( A `
 ( ( a  e.  ( NN0  ^m  1o )  |->  ( a `
 (/) ) ) `  b ) )  =  ( A `  (
b `  (/) ) ) )
2221adantl 466 . . . . . . . . 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 2511 . . . . . . . 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 2744 . . . . . . 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 704 . . . . . 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 450 . . . . . 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 261 . . . . 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 2970 . . . 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 7139 . . . . . 6  |-  1o  =  { (/) }
30 nn0ex 10797 . . . . . 6  |-  NN0  e.  _V
31 0ex 4577 . . . . . 6  |-  (/)  e.  _V
3229, 30, 31, 18mapsnf1o2 7463 . . . . 5  |-  ( a  e.  ( NN0  ^m  1o )  |->  ( a `
 (/) ) ) : ( NN0  ^m  1o )
-1-1-onto-> NN0
33 f1ofo 5821 . . . . 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 2471 . . . . . . 7  |-  ( ( ( a  e.  ( NN0  ^m  1o ) 
|->  ( a `  (/) ) ) `
 b )  =  d  ->  ( (
( a  e.  ( NN0  ^m  1o ) 
|->  ( a `  (/) ) ) `
 b )  =  ( D `  F
)  <->  d  =  ( D `  F ) ) )
35 fveq2 5864 . . . . . . . 8  |-  ( ( ( a  e.  ( NN0  ^m  1o ) 
|->  ( a `  (/) ) ) `
 b )  =  d  ->  ( A `  ( ( a  e.  ( NN0  ^m  1o )  |->  ( a `  (/) ) ) `  b
) )  =  ( A `  d ) )
3635neeq1d 2744 . . . . . . 7  |-  ( ( ( a  e.  ( NN0  ^m  1o ) 
|->  ( a `  (/) ) ) `
 b )  =  d  ->  ( ( A `  ( (
a  e.  ( NN0 
^m  1o )  |->  ( a `  (/) ) ) `
 b ) )  =/=  Y  <->  ( A `  d )  =/=  Y
) )
3734, 36anbi12d 710 . . . . . 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 6179 . . . . 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 261 . . 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 22220 . . . 4  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
.0.  )  ->  ( D `  F )  e.  NN0 )
42 fveq2 5864 . . . . . 6  |-  ( d  =  ( D `  F )  ->  ( A `  d )  =  ( A `  ( D `  F ) ) )
4342neeq1d 2744 . . . . 5  |-  ( d  =  ( D `  F )  ->  (
( A `  d
)  =/=  Y  <->  ( A `  ( D `  F
) )  =/=  Y
) )
4443ceqsrexv 3237 . . . 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 253 . 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 210 1  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
.0.  )  ->  ( A `  ( D `  F ) )  =/= 
Y )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   E.wrex 2815   (/)c0 3785    |-> cmpt 4505   -onto->wfo 5584   -1-1-onto->wf1o 5585   ` cfv 5586  (class class class)co 6282   1oc1o 7120    ^m cmap 7417   NN0cn0 10791   Basecbs 14483   0gc0g 14688   Ringcrg 16983   mPoly cmpl 17770  PwSer1cps1 17982  Poly1cpl1 17984  coe1cco1 17985   deg1 cdg1 22184
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-inf2 8054  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565  ax-addf 9567  ax-mulf 9568
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-isom 5595  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-of 6522  df-om 6679  df-1st 6781  df-2nd 6782  df-supp 6899  df-recs 7039  df-rdg 7073  df-1o 7127  df-oadd 7131  df-er 7308  df-map 7419  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-fsupp 7826  df-sup 7897  df-oi 7931  df-card 8316  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-nn 10533  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-n0 10792  df-z 10861  df-dec 10973  df-uz 11079  df-fz 11669  df-fzo 11789  df-seq 12071  df-hash 12368  df-struct 14485  df-ndx 14486  df-slot 14487  df-base 14488  df-sets 14489  df-ress 14490  df-plusg 14561  df-mulr 14562  df-starv 14563  df-sca 14564  df-vsca 14565  df-tset 14567  df-ple 14568  df-ds 14570  df-unif 14571  df-0g 14690  df-gsum 14691  df-mnd 15725  df-submnd 15775  df-grp 15855  df-minusg 15856  df-mulg 15858  df-subg 15990  df-cntz 16147  df-cmn 16593  df-abl 16594  df-mgp 16929  df-ur 16941  df-rng 16985  df-cring 16986  df-psr 17773  df-mpl 17775  df-opsr 17777  df-psr1 17987  df-ply1 17989  df-coe1 17990  df-cnfld 18189  df-mdeg 22185  df-deg1 22186
This theorem is referenced by:  deg1ldgn  22225  deg1ldgdomn  22226  deg1add  22236  deg1mul2  22247  drnguc1p  22303
  Copyright terms: Public domain W3C validator