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Theorem aaitgo 30716
Description: The standard algebraic numbers  AA are generated by IntgOver. (Contributed by Stefan O'Rear, 27-Nov-2014.)
Assertion
Ref Expression
aaitgo  |-  AA  =  (IntgOver `  QQ )

Proof of Theorem aaitgo
Dummy variables  a 
b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rabid 3038 . . 3  |-  ( a  e.  { a  e.  CC  |  E. b  e.  (Poly `  QQ )
( ( b `  a )  =  0  /\  ( (coeff `  b ) `  (deg `  b ) )  =  1 ) }  <->  ( a  e.  CC  /\  E. b  e.  (Poly `  QQ )
( ( b `  a )  =  0  /\  ( (coeff `  b ) `  (deg `  b ) )  =  1 ) ) )
2 qsscn 11189 . . . . 5  |-  QQ  C_  CC
3 itgoval 30715 . . . . 5  |-  ( QQ  C_  CC  ->  (IntgOver `  QQ )  =  { a  e.  CC  |  E. b  e.  (Poly `  QQ )
( ( b `  a )  =  0  /\  ( (coeff `  b ) `  (deg `  b ) )  =  1 ) } )
42, 3ax-mp 5 . . . 4  |-  (IntgOver `  QQ )  =  { a  e.  CC  |  E. b  e.  (Poly `  QQ )
( ( b `  a )  =  0  /\  ( (coeff `  b ) `  (deg `  b ) )  =  1 ) }
54eleq2i 2545 . . 3  |-  ( a  e.  (IntgOver `  QQ ) 
<->  a  e.  { a  e.  CC  |  E. b  e.  (Poly `  QQ ) ( ( b `
 a )  =  0  /\  ( (coeff `  b ) `  (deg `  b ) )  =  1 ) } )
6 aacn 22447 . . . . 5  |-  ( a  e.  AA  ->  a  e.  CC )
7 mpaacl 30707 . . . . . 6  |-  ( a  e.  AA  ->  (minPolyAA `  a )  e.  (Poly `  QQ ) )
8 mpaaroot 30709 . . . . . 6  |-  ( a  e.  AA  ->  (
(minPolyAA `  a ) `  a )  =  0 )
9 mpaadgr 30708 . . . . . . . 8  |-  ( a  e.  AA  ->  (deg `  (minPolyAA `  a )
)  =  (degAA `  a
) )
109fveq2d 5868 . . . . . . 7  |-  ( a  e.  AA  ->  (
(coeff `  (minPolyAA `  a
) ) `  (deg `  (minPolyAA `  a )
) )  =  ( (coeff `  (minPolyAA `  a
) ) `  (degAA `  a ) ) )
11 mpaamn 30710 . . . . . . 7  |-  ( a  e.  AA  ->  (
(coeff `  (minPolyAA `  a
) ) `  (degAA `  a ) )  =  1 )
1210, 11eqtrd 2508 . . . . . 6  |-  ( a  e.  AA  ->  (
(coeff `  (minPolyAA `  a
) ) `  (deg `  (minPolyAA `  a )
) )  =  1 )
13 fveq1 5863 . . . . . . . . 9  |-  ( b  =  (minPolyAA `  a
)  ->  ( b `  a )  =  ( (minPolyAA `  a ) `  a ) )
1413eqeq1d 2469 . . . . . . . 8  |-  ( b  =  (minPolyAA `  a
)  ->  ( (
b `  a )  =  0  <->  ( (minPolyAA `  a ) `  a
)  =  0 ) )
15 fveq2 5864 . . . . . . . . . 10  |-  ( b  =  (minPolyAA `  a
)  ->  (coeff `  b
)  =  (coeff `  (minPolyAA `  a ) ) )
16 fveq2 5864 . . . . . . . . . 10  |-  ( b  =  (minPolyAA `  a
)  ->  (deg `  b
)  =  (deg `  (minPolyAA `  a ) ) )
1715, 16fveq12d 5870 . . . . . . . . 9  |-  ( b  =  (minPolyAA `  a
)  ->  ( (coeff `  b ) `  (deg `  b ) )  =  ( (coeff `  (minPolyAA `  a ) ) `  (deg `  (minPolyAA `  a
) ) ) )
1817eqeq1d 2469 . . . . . . . 8  |-  ( b  =  (minPolyAA `  a
)  ->  ( (
(coeff `  b ) `  (deg `  b )
)  =  1  <->  (
(coeff `  (minPolyAA `  a
) ) `  (deg `  (minPolyAA `  a )
) )  =  1 ) )
1914, 18anbi12d 710 . . . . . . 7  |-  ( b  =  (minPolyAA `  a
)  ->  ( (
( b `  a
)  =  0  /\  ( (coeff `  b
) `  (deg `  b
) )  =  1 )  <->  ( ( (minPolyAA `  a ) `  a
)  =  0  /\  ( (coeff `  (minPolyAA `  a ) ) `  (deg `  (minPolyAA `  a
) ) )  =  1 ) ) )
2019rspcev 3214 . . . . . 6  |-  ( ( (minPolyAA `  a )  e.  (Poly `  QQ )  /\  ( ( (minPolyAA `  a
) `  a )  =  0  /\  (
(coeff `  (minPolyAA `  a
) ) `  (deg `  (minPolyAA `  a )
) )  =  1 ) )  ->  E. b  e.  (Poly `  QQ )
( ( b `  a )  =  0  /\  ( (coeff `  b ) `  (deg `  b ) )  =  1 ) )
217, 8, 12, 20syl12anc 1226 . . . . 5  |-  ( a  e.  AA  ->  E. b  e.  (Poly `  QQ )
( ( b `  a )  =  0  /\  ( (coeff `  b ) `  (deg `  b ) )  =  1 ) )
226, 21jca 532 . . . 4  |-  ( a  e.  AA  ->  (
a  e.  CC  /\  E. b  e.  (Poly `  QQ ) ( ( b `
 a )  =  0  /\  ( (coeff `  b ) `  (deg `  b ) )  =  1 ) ) )
23 simpl 457 . . . . . . . . 9  |-  ( ( b  e.  (Poly `  QQ )  /\  (
( b `  a
)  =  0  /\  ( (coeff `  b
) `  (deg `  b
) )  =  1 ) )  ->  b  e.  (Poly `  QQ )
)
24 coe0 22387 . . . . . . . . . . . . . . 15  |-  (coeff ` 
0p )  =  ( NN0  X.  {
0 } )
2524fveq1i 5865 . . . . . . . . . . . . . 14  |-  ( (coeff `  0p ) `
 (deg `  0p ) )  =  ( ( NN0  X.  { 0 } ) `
 (deg `  0p ) )
26 dgr0 22393 . . . . . . . . . . . . . . . 16  |-  (deg ` 
0p )  =  0
27 0nn0 10806 . . . . . . . . . . . . . . . 16  |-  0  e.  NN0
2826, 27eqeltri 2551 . . . . . . . . . . . . . . 15  |-  (deg ` 
0p )  e. 
NN0
29 c0ex 9586 . . . . . . . . . . . . . . . 16  |-  0  e.  _V
3029fvconst2 6114 . . . . . . . . . . . . . . 15  |-  ( (deg
`  0p )  e.  NN0  ->  ( ( NN0  X.  { 0 } ) `  (deg `  0p ) )  =  0 )
3128, 30ax-mp 5 . . . . . . . . . . . . . 14  |-  ( ( NN0  X.  { 0 } ) `  (deg `  0p ) )  =  0
3225, 31eqtri 2496 . . . . . . . . . . . . 13  |-  ( (coeff `  0p ) `
 (deg `  0p ) )  =  0
33 0ne1 10599 . . . . . . . . . . . . 13  |-  0  =/=  1
3432, 33eqnetri 2763 . . . . . . . . . . . 12  |-  ( (coeff `  0p ) `
 (deg `  0p ) )  =/=  1
35 fveq2 5864 . . . . . . . . . . . . . 14  |-  ( b  =  0p  -> 
(coeff `  b )  =  (coeff `  0p
) )
36 fveq2 5864 . . . . . . . . . . . . . 14  |-  ( b  =  0p  -> 
(deg `  b )  =  (deg `  0p
) )
3735, 36fveq12d 5870 . . . . . . . . . . . . 13  |-  ( b  =  0p  -> 
( (coeff `  b
) `  (deg `  b
) )  =  ( (coeff `  0p
) `  (deg `  0p ) ) )
3837neeq1d 2744 . . . . . . . . . . . 12  |-  ( b  =  0p  -> 
( ( (coeff `  b ) `  (deg `  b ) )  =/=  1  <->  ( (coeff ` 
0p ) `  (deg `  0p ) )  =/=  1 ) )
3934, 38mpbiri 233 . . . . . . . . . . 11  |-  ( b  =  0p  -> 
( (coeff `  b
) `  (deg `  b
) )  =/=  1
)
4039necon2i 2710 . . . . . . . . . 10  |-  ( ( (coeff `  b ) `  (deg `  b )
)  =  1  -> 
b  =/=  0p )
4140ad2antll 728 . . . . . . . . 9  |-  ( ( b  e.  (Poly `  QQ )  /\  (
( b `  a
)  =  0  /\  ( (coeff `  b
) `  (deg `  b
) )  =  1 ) )  ->  b  =/=  0p )
42 eldifsn 4152 . . . . . . . . 9  |-  ( b  e.  ( (Poly `  QQ )  \  { 0p } )  <->  ( b  e.  (Poly `  QQ )  /\  b  =/=  0p ) )
4323, 41, 42sylanbrc 664 . . . . . . . 8  |-  ( ( b  e.  (Poly `  QQ )  /\  (
( b `  a
)  =  0  /\  ( (coeff `  b
) `  (deg `  b
) )  =  1 ) )  ->  b  e.  ( (Poly `  QQ )  \  { 0p } ) )
44 simprl 755 . . . . . . . 8  |-  ( ( b  e.  (Poly `  QQ )  /\  (
( b `  a
)  =  0  /\  ( (coeff `  b
) `  (deg `  b
) )  =  1 ) )  ->  (
b `  a )  =  0 )
4543, 44jca 532 . . . . . . 7  |-  ( ( b  e.  (Poly `  QQ )  /\  (
( b `  a
)  =  0  /\  ( (coeff `  b
) `  (deg `  b
) )  =  1 ) )  ->  (
b  e.  ( (Poly `  QQ )  \  {
0p } )  /\  ( b `  a )  =  0 ) )
4645reximi2 2931 . . . . . 6  |-  ( E. b  e.  (Poly `  QQ ) ( ( b `
 a )  =  0  /\  ( (coeff `  b ) `  (deg `  b ) )  =  1 )  ->  E. b  e.  ( (Poly `  QQ )  \  { 0p } ) ( b `
 a )  =  0 )
4746anim2i 569 . . . . 5  |-  ( ( a  e.  CC  /\  E. b  e.  (Poly `  QQ ) ( ( b `
 a )  =  0  /\  ( (coeff `  b ) `  (deg `  b ) )  =  1 ) )  -> 
( a  e.  CC  /\ 
E. b  e.  ( (Poly `  QQ )  \  { 0p }
) ( b `  a )  =  0 ) )
48 elqaa 22452 . . . . 5  |-  ( a  e.  AA  <->  ( a  e.  CC  /\  E. b  e.  ( (Poly `  QQ )  \  { 0p } ) ( b `
 a )  =  0 ) )
4947, 48sylibr 212 . . . 4  |-  ( ( a  e.  CC  /\  E. b  e.  (Poly `  QQ ) ( ( b `
 a )  =  0  /\  ( (coeff `  b ) `  (deg `  b ) )  =  1 ) )  -> 
a  e.  AA )
5022, 49impbii 188 . . 3  |-  ( a  e.  AA  <->  ( a  e.  CC  /\  E. b  e.  (Poly `  QQ )
( ( b `  a )  =  0  /\  ( (coeff `  b ) `  (deg `  b ) )  =  1 ) ) )
511, 5, 503bitr4ri 278 . 2  |-  ( a  e.  AA  <->  a  e.  (IntgOver `  QQ ) )
5251eqriv 2463 1  |-  AA  =  (IntgOver `  QQ )
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662   E.wrex 2815   {crab 2818    \ cdif 3473    C_ wss 3476   {csn 4027    X. cxp 4997   ` cfv 5586   CCcc 9486   0cc0 9488   1c1 9489   NN0cn0 10791   QQcq 11178   0pc0p 21811  Polycply 22316  coeffccoe 22318  degcdgr 22319   AAcaa 22444  degAAcdgraa 30694  minPolyAAcmpaa 30695  IntgOvercitgo 30711
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-pre-sup 9566  ax-addf 9567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  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-recs 7039  df-rdg 7073  df-1o 7127  df-oadd 7131  df-er 7308  df-map 7419  df-pm 7420  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  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-div 10203  df-nn 10533  df-2 10590  df-3 10591  df-n0 10792  df-z 10861  df-uz 11079  df-q 11179  df-rp 11217  df-fz 11669  df-fzo 11789  df-fl 11893  df-mod 11961  df-seq 12072  df-exp 12131  df-hash 12370  df-cj 12891  df-re 12892  df-im 12893  df-sqrt 13027  df-abs 13028  df-clim 13270  df-rlim 13271  df-sum 13468  df-0p 21812  df-ply 22320  df-coe 22322  df-dgr 22323  df-aa 22445  df-dgraa 30696  df-mpaa 30697  df-itgo 30713
This theorem is referenced by: (None)
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