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Theorem aaitgo 29522
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 2900 . . 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 10967 . . . . 5  |-  QQ  C_  CC
3 itgoval 29521 . . . . 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 2507 . . 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 21786 . . . . 5  |-  ( a  e.  AA  ->  a  e.  CC )
7 mpaacl 29513 . . . . . 6  |-  ( a  e.  AA  ->  (minPolyAA `  a )  e.  (Poly `  QQ ) )
8 mpaaroot 29515 . . . . . 6  |-  ( a  e.  AA  ->  (
(minPolyAA `  a ) `  a )  =  0 )
9 mpaadgr 29514 . . . . . . . 8  |-  ( a  e.  AA  ->  (deg `  (minPolyAA `  a )
)  =  (degAA `  a
) )
109fveq2d 5698 . . . . . . 7  |-  ( a  e.  AA  ->  (
(coeff `  (minPolyAA `  a
) ) `  (deg `  (minPolyAA `  a )
) )  =  ( (coeff `  (minPolyAA `  a
) ) `  (degAA `  a ) ) )
11 mpaamn 29516 . . . . . . 7  |-  ( a  e.  AA  ->  (
(coeff `  (minPolyAA `  a
) ) `  (degAA `  a ) )  =  1 )
1210, 11eqtrd 2475 . . . . . 6  |-  ( a  e.  AA  ->  (
(coeff `  (minPolyAA `  a
) ) `  (deg `  (minPolyAA `  a )
) )  =  1 )
13 fveq1 5693 . . . . . . . . 9  |-  ( b  =  (minPolyAA `  a
)  ->  ( b `  a )  =  ( (minPolyAA `  a ) `  a ) )
1413eqeq1d 2451 . . . . . . . 8  |-  ( b  =  (minPolyAA `  a
)  ->  ( (
b `  a )  =  0  <->  ( (minPolyAA `  a ) `  a
)  =  0 ) )
15 fveq2 5694 . . . . . . . . . 10  |-  ( b  =  (minPolyAA `  a
)  ->  (coeff `  b
)  =  (coeff `  (minPolyAA `  a ) ) )
16 fveq2 5694 . . . . . . . . . 10  |-  ( b  =  (minPolyAA `  a
)  ->  (deg `  b
)  =  (deg `  (minPolyAA `  a ) ) )
1715, 16fveq12d 5700 . . . . . . . . 9  |-  ( b  =  (minPolyAA `  a
)  ->  ( (coeff `  b ) `  (deg `  b ) )  =  ( (coeff `  (minPolyAA `  a ) ) `  (deg `  (minPolyAA `  a
) ) ) )
1817eqeq1d 2451 . . . . . . . 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 3076 . . . . . 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 1216 . . . . 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 21726 . . . . . . . . . . . . . . 15  |-  (coeff ` 
0p )  =  ( NN0  X.  {
0 } )
2524fveq1i 5695 . . . . . . . . . . . . . 14  |-  ( (coeff `  0p ) `
 (deg `  0p ) )  =  ( ( NN0  X.  { 0 } ) `
 (deg `  0p ) )
26 dgr0 21732 . . . . . . . . . . . . . . . 16  |-  (deg ` 
0p )  =  0
27 0nn0 10597 . . . . . . . . . . . . . . . 16  |-  0  e.  NN0
2826, 27eqeltri 2513 . . . . . . . . . . . . . . 15  |-  (deg ` 
0p )  e. 
NN0
29 c0ex 9383 . . . . . . . . . . . . . . . 16  |-  0  e.  _V
3029fvconst2 5936 . . . . . . . . . . . . . . 15  |-  ( (deg
`  0p )  e.  NN0  ->  ( ( NN0  X.  { 0 } ) `  (deg `  0p ) )  =  0 )
3128, 30ax-mp 5 . . . . . . . . . . . . . 14  |-  ( ( NN0  X.  { 0 } ) `  (deg `  0p ) )  =  0
3225, 31eqtri 2463 . . . . . . . . . . . . 13  |-  ( (coeff `  0p ) `
 (deg `  0p ) )  =  0
33 0ne1 10392 . . . . . . . . . . . . 13  |-  0  =/=  1
3432, 33eqnetri 2628 . . . . . . . . . . . 12  |-  ( (coeff `  0p ) `
 (deg `  0p ) )  =/=  1
35 fveq2 5694 . . . . . . . . . . . . . 14  |-  ( b  =  0p  -> 
(coeff `  b )  =  (coeff `  0p
) )
36 fveq2 5694 . . . . . . . . . . . . . 14  |-  ( b  =  0p  -> 
(deg `  b )  =  (deg `  0p
) )
3735, 36fveq12d 5700 . . . . . . . . . . . . 13  |-  ( b  =  0p  -> 
( (coeff `  b
) `  (deg `  b
) )  =  ( (coeff `  0p
) `  (deg `  0p ) ) )
3837neeq1d 2624 . . . . . . . . . . . 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 2661 . . . . . . . . . 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 4003 . . . . . . . . 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 2825 . . . . . 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 21791 . . . . 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 2440 1  |-  AA  =  (IntgOver `  QQ )
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    = wceq 1369    e. wcel 1756    =/= wne 2609   E.wrex 2719   {crab 2722    \ cdif 3328    C_ wss 3331   {csn 3880    X. cxp 4841   ` cfv 5421   CCcc 9283   0cc0 9285   1c1 9286   NN0cn0 10582   QQcq 10956   0pc0p 21150  Polycply 21655  coeffccoe 21657  degcdgr 21658   AAcaa 21783  degAAcdgraa 29500  minPolyAAcmpaa 29501  IntgOvercitgo 29517
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4406  ax-sep 4416  ax-nul 4424  ax-pow 4473  ax-pr 4534  ax-un 6375  ax-inf2 7850  ax-cnex 9341  ax-resscn 9342  ax-1cn 9343  ax-icn 9344  ax-addcl 9345  ax-addrcl 9346  ax-mulcl 9347  ax-mulrcl 9348  ax-mulcom 9349  ax-addass 9350  ax-mulass 9351  ax-distr 9352  ax-i2m1 9353  ax-1ne0 9354  ax-1rid 9355  ax-rnegex 9356  ax-rrecex 9357  ax-cnre 9358  ax-pre-lttri 9359  ax-pre-lttrn 9360  ax-pre-ltadd 9361  ax-pre-mulgt0 9362  ax-pre-sup 9363  ax-addf 9364
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2571  df-ne 2611  df-nel 2612  df-ral 2723  df-rex 2724  df-reu 2725  df-rmo 2726  df-rab 2727  df-v 2977  df-sbc 3190  df-csb 3292  df-dif 3334  df-un 3336  df-in 3338  df-ss 3345  df-pss 3347  df-nul 3641  df-if 3795  df-pw 3865  df-sn 3881  df-pr 3883  df-tp 3885  df-op 3887  df-uni 4095  df-int 4132  df-iun 4176  df-br 4296  df-opab 4354  df-mpt 4355  df-tr 4389  df-eprel 4635  df-id 4639  df-po 4644  df-so 4645  df-fr 4682  df-se 4683  df-we 4684  df-ord 4725  df-on 4726  df-lim 4727  df-suc 4728  df-xp 4849  df-rel 4850  df-cnv 4851  df-co 4852  df-dm 4853  df-rn 4854  df-res 4855  df-ima 4856  df-iota 5384  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-isom 5430  df-riota 6055  df-ov 6097  df-oprab 6098  df-mpt2 6099  df-of 6323  df-om 6480  df-1st 6580  df-2nd 6581  df-recs 6835  df-rdg 6869  df-1o 6923  df-oadd 6927  df-er 7104  df-map 7219  df-pm 7220  df-en 7314  df-dom 7315  df-sdom 7316  df-fin 7317  df-sup 7694  df-oi 7727  df-card 8112  df-pnf 9423  df-mnf 9424  df-xr 9425  df-ltxr 9426  df-le 9427  df-sub 9600  df-neg 9601  df-div 9997  df-nn 10326  df-2 10383  df-3 10384  df-n0 10583  df-z 10650  df-uz 10865  df-q 10957  df-rp 10995  df-fz 11441  df-fzo 11552  df-fl 11645  df-mod 11712  df-seq 11810  df-exp 11869  df-hash 12107  df-cj 12591  df-re 12592  df-im 12593  df-sqr 12727  df-abs 12728  df-clim 12969  df-rlim 12970  df-sum 13167  df-0p 21151  df-ply 21659  df-coe 21661  df-dgr 21662  df-aa 21784  df-dgraa 29502  df-mpaa 29503  df-itgo 29519
This theorem is referenced by: (None)
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