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Theorem aaitgo 27470
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 2729 . . 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 10343 . . . . 5  |-  QQ  C_  CC
3 itgoval 27469 . . . . 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 8 . . . 4  |-  (IntgOver `  QQ )  =  { a  e.  CC  |  E. b  e.  (Poly `  QQ )
( ( b `  a )  =  0  /\  ( (coeff `  b ) `  (deg `  b ) )  =  1 ) }
54eleq2i 2360 . . 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 19713 . . . . 5  |-  ( a  e.  AA  ->  a  e.  CC )
7 mpaacl 27461 . . . . . 6  |-  ( a  e.  AA  ->  (minPolyAA `  a )  e.  (Poly `  QQ ) )
8 mpaaroot 27463 . . . . . 6  |-  ( a  e.  AA  ->  (
(minPolyAA `  a ) `  a )  =  0 )
9 mpaadgr 27462 . . . . . . . 8  |-  ( a  e.  AA  ->  (deg `  (minPolyAA `  a )
)  =  (degAA `  a
) )
109fveq2d 5545 . . . . . . 7  |-  ( a  e.  AA  ->  (
(coeff `  (minPolyAA `  a
) ) `  (deg `  (minPolyAA `  a )
) )  =  ( (coeff `  (minPolyAA `  a
) ) `  (degAA `  a ) ) )
11 mpaamn 27464 . . . . . . 7  |-  ( a  e.  AA  ->  (
(coeff `  (minPolyAA `  a
) ) `  (degAA `  a ) )  =  1 )
1210, 11eqtrd 2328 . . . . . 6  |-  ( a  e.  AA  ->  (
(coeff `  (minPolyAA `  a
) ) `  (deg `  (minPolyAA `  a )
) )  =  1 )
13 fveq1 5540 . . . . . . . . 9  |-  ( b  =  (minPolyAA `  a
)  ->  ( b `  a )  =  ( (minPolyAA `  a ) `  a ) )
1413eqeq1d 2304 . . . . . . . 8  |-  ( b  =  (minPolyAA `  a
)  ->  ( (
b `  a )  =  0  <->  ( (minPolyAA `  a ) `  a
)  =  0 ) )
15 fveq2 5541 . . . . . . . . . 10  |-  ( b  =  (minPolyAA `  a
)  ->  (coeff `  b
)  =  (coeff `  (minPolyAA `  a ) ) )
16 fveq2 5541 . . . . . . . . . 10  |-  ( b  =  (minPolyAA `  a
)  ->  (deg `  b
)  =  (deg `  (minPolyAA `  a ) ) )
1715, 16fveq12d 5547 . . . . . . . . 9  |-  ( b  =  (minPolyAA `  a
)  ->  ( (coeff `  b ) `  (deg `  b ) )  =  ( (coeff `  (minPolyAA `  a ) ) `  (deg `  (minPolyAA `  a
) ) ) )
1817eqeq1d 2304 . . . . . . . 8  |-  ( b  =  (minPolyAA `  a
)  ->  ( (
(coeff `  b ) `  (deg `  b )
)  =  1  <->  (
(coeff `  (minPolyAA `  a
) ) `  (deg `  (minPolyAA `  a )
) )  =  1 ) )
1914, 18anbi12d 691 . . . . . . 7  |-  ( b  =  (minPolyAA `  a
)  ->  ( (
( b `  a
)  =  0  /\  ( (coeff `  b
) `  (deg `  b
) )  =  1 )  <->  ( ( (minPolyAA `  a ) `  a
)  =  0  /\  ( (coeff `  (minPolyAA `  a ) ) `  (deg `  (minPolyAA `  a
) ) )  =  1 ) ) )
2019rspcev 2897 . . . . . 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 1180 . . . . 5  |-  ( a  e.  AA  ->  E. b  e.  (Poly `  QQ )
( ( b `  a )  =  0  /\  ( (coeff `  b ) `  (deg `  b ) )  =  1 ) )
226, 21jca 518 . . . 4  |-  ( a  e.  AA  ->  (
a  e.  CC  /\  E. b  e.  (Poly `  QQ ) ( ( b `
 a )  =  0  /\  ( (coeff `  b ) `  (deg `  b ) )  =  1 ) ) )
23 simpl 443 . . . . . . . . 9  |-  ( ( b  e.  (Poly `  QQ )  /\  (
( b `  a
)  =  0  /\  ( (coeff `  b
) `  (deg `  b
) )  =  1 ) )  ->  b  e.  (Poly `  QQ )
)
24 coe0 19653 . . . . . . . . . . . . . . 15  |-  (coeff ` 
0 p )  =  ( NN0  X.  {
0 } )
2524fveq1i 5542 . . . . . . . . . . . . . 14  |-  ( (coeff `  0 p ) `
 (deg `  0 p ) )  =  ( ( NN0  X.  { 0 } ) `
 (deg `  0 p ) )
26 dgr0 19659 . . . . . . . . . . . . . . . 16  |-  (deg ` 
0 p )  =  0
27 0nn0 9996 . . . . . . . . . . . . . . . 16  |-  0  e.  NN0
2826, 27eqeltri 2366 . . . . . . . . . . . . . . 15  |-  (deg ` 
0 p )  e. 
NN0
29 c0ex 8848 . . . . . . . . . . . . . . . 16  |-  0  e.  _V
3029fvconst2 5745 . . . . . . . . . . . . . . 15  |-  ( (deg
`  0 p )  e.  NN0  ->  ( ( NN0  X.  { 0 } ) `  (deg `  0 p ) )  =  0 )
3128, 30ax-mp 8 . . . . . . . . . . . . . 14  |-  ( ( NN0  X.  { 0 } ) `  (deg `  0 p ) )  =  0
3225, 31eqtri 2316 . . . . . . . . . . . . 13  |-  ( (coeff `  0 p ) `
 (deg `  0 p ) )  =  0
33 ax-1ne0 8822 . . . . . . . . . . . . . 14  |-  1  =/=  0
3433necomi 2541 . . . . . . . . . . . . 13  |-  0  =/=  1
3532, 34eqnetri 2476 . . . . . . . . . . . 12  |-  ( (coeff `  0 p ) `
 (deg `  0 p ) )  =/=  1
36 fveq2 5541 . . . . . . . . . . . . . 14  |-  ( b  =  0 p  -> 
(coeff `  b )  =  (coeff `  0 p
) )
37 fveq2 5541 . . . . . . . . . . . . . 14  |-  ( b  =  0 p  -> 
(deg `  b )  =  (deg `  0 p
) )
3836, 37fveq12d 5547 . . . . . . . . . . . . 13  |-  ( b  =  0 p  -> 
( (coeff `  b
) `  (deg `  b
) )  =  ( (coeff `  0 p
) `  (deg `  0 p ) ) )
3938neeq1d 2472 . . . . . . . . . . . 12  |-  ( b  =  0 p  -> 
( ( (coeff `  b ) `  (deg `  b ) )  =/=  1  <->  ( (coeff ` 
0 p ) `  (deg `  0 p ) )  =/=  1 ) )
4035, 39mpbiri 224 . . . . . . . . . . 11  |-  ( b  =  0 p  -> 
( (coeff `  b
) `  (deg `  b
) )  =/=  1
)
4140necon2i 2506 . . . . . . . . . 10  |-  ( ( (coeff `  b ) `  (deg `  b )
)  =  1  -> 
b  =/=  0 p )
4241ad2antll 709 . . . . . . . . 9  |-  ( ( b  e.  (Poly `  QQ )  /\  (
( b `  a
)  =  0  /\  ( (coeff `  b
) `  (deg `  b
) )  =  1 ) )  ->  b  =/=  0 p )
43 eldifsn 3762 . . . . . . . . 9  |-  ( b  e.  ( (Poly `  QQ )  \  { 0 p } )  <->  ( b  e.  (Poly `  QQ )  /\  b  =/=  0 p ) )
4423, 42, 43sylanbrc 645 . . . . . . . 8  |-  ( ( b  e.  (Poly `  QQ )  /\  (
( b `  a
)  =  0  /\  ( (coeff `  b
) `  (deg `  b
) )  =  1 ) )  ->  b  e.  ( (Poly `  QQ )  \  { 0 p } ) )
45 simprl 732 . . . . . . . 8  |-  ( ( b  e.  (Poly `  QQ )  /\  (
( b `  a
)  =  0  /\  ( (coeff `  b
) `  (deg `  b
) )  =  1 ) )  ->  (
b `  a )  =  0 )
4644, 45jca 518 . . . . . . 7  |-  ( ( b  e.  (Poly `  QQ )  /\  (
( b `  a
)  =  0  /\  ( (coeff `  b
) `  (deg `  b
) )  =  1 ) )  ->  (
b  e.  ( (Poly `  QQ )  \  {
0 p } )  /\  ( b `  a )  =  0 ) )
4746reximi2 2662 . . . . . 6  |-  ( E. b  e.  (Poly `  QQ ) ( ( b `
 a )  =  0  /\  ( (coeff `  b ) `  (deg `  b ) )  =  1 )  ->  E. b  e.  ( (Poly `  QQ )  \  { 0 p } ) ( b `
 a )  =  0 )
4847anim2i 552 . . . . 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 )  \  { 0 p }
) ( b `  a )  =  0 ) )
49 elqaa 19718 . . . . 5  |-  ( a  e.  AA  <->  ( a  e.  CC  /\  E. b  e.  ( (Poly `  QQ )  \  { 0 p } ) ( b `
 a )  =  0 ) )
5048, 49sylibr 203 . . . 4  |-  ( ( a  e.  CC  /\  E. b  e.  (Poly `  QQ ) ( ( b `
 a )  =  0  /\  ( (coeff `  b ) `  (deg `  b ) )  =  1 ) )  -> 
a  e.  AA )
5122, 50impbii 180 . . 3  |-  ( a  e.  AA  <->  ( a  e.  CC  /\  E. b  e.  (Poly `  QQ )
( ( b `  a )  =  0  /\  ( (coeff `  b ) `  (deg `  b ) )  =  1 ) ) )
521, 5, 513bitr4ri 269 . 2  |-  ( a  e.  AA  <->  a  e.  (IntgOver `  QQ ) )
5352eqriv 2293 1  |-  AA  =  (IntgOver `  QQ )
Colors of variables: wff set class
Syntax hints:    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459   E.wrex 2557   {crab 2560    \ cdif 3162    C_ wss 3165   {csn 3653    X. cxp 4703   ` cfv 5271   CCcc 8751   0cc0 8753   1c1 8754   NN0cn0 9981   QQcq 10332   0 pc0p 19040  Polycply 19582  coeffccoe 19584  degcdgr 19585   AAcaa 19710  degAAcdgraa 27448  minPolyAAcmpaa 27449  IntgOvercitgo 27465
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831  ax-addf 8832
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-map 6790  df-pm 6791  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-oi 7241  df-card 7588  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-q 10333  df-rp 10371  df-fz 10799  df-fzo 10887  df-fl 10941  df-mod 10990  df-seq 11063  df-exp 11121  df-hash 11354  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-clim 11978  df-rlim 11979  df-sum 12175  df-0p 19041  df-ply 19586  df-coe 19588  df-dgr 19589  df-aa 19711  df-dgraa 27450  df-mpaa 27451  df-itgo 27467
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