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Theorem aaitgo 36022
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 2966 . . 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 11272 . . . . 5  |-  QQ  C_  CC
3 itgoval 36021 . . . . 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 2520 . . 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 23263 . . . . 5  |-  ( a  e.  AA  ->  a  e.  CC )
7 mpaacl 36013 . . . . . 6  |-  ( a  e.  AA  ->  (minPolyAA `  a )  e.  (Poly `  QQ ) )
8 mpaaroot 36015 . . . . . 6  |-  ( a  e.  AA  ->  (
(minPolyAA `  a ) `  a )  =  0 )
9 mpaadgr 36014 . . . . . . . 8  |-  ( a  e.  AA  ->  (deg `  (minPolyAA `  a )
)  =  (degAA `  a
) )
109fveq2d 5867 . . . . . . 7  |-  ( a  e.  AA  ->  (
(coeff `  (minPolyAA `  a
) ) `  (deg `  (minPolyAA `  a )
) )  =  ( (coeff `  (minPolyAA `  a
) ) `  (degAA `  a ) ) )
11 mpaamn 36016 . . . . . . 7  |-  ( a  e.  AA  ->  (
(coeff `  (minPolyAA `  a
) ) `  (degAA `  a ) )  =  1 )
1210, 11eqtrd 2484 . . . . . 6  |-  ( a  e.  AA  ->  (
(coeff `  (minPolyAA `  a
) ) `  (deg `  (minPolyAA `  a )
) )  =  1 )
13 fveq1 5862 . . . . . . . . 9  |-  ( b  =  (minPolyAA `  a
)  ->  ( b `  a )  =  ( (minPolyAA `  a ) `  a ) )
1413eqeq1d 2452 . . . . . . . 8  |-  ( b  =  (minPolyAA `  a
)  ->  ( (
b `  a )  =  0  <->  ( (minPolyAA `  a ) `  a
)  =  0 ) )
15 fveq2 5863 . . . . . . . . . 10  |-  ( b  =  (minPolyAA `  a
)  ->  (coeff `  b
)  =  (coeff `  (minPolyAA `  a ) ) )
16 fveq2 5863 . . . . . . . . . 10  |-  ( b  =  (minPolyAA `  a
)  ->  (deg `  b
)  =  (deg `  (minPolyAA `  a ) ) )
1715, 16fveq12d 5869 . . . . . . . . 9  |-  ( b  =  (minPolyAA `  a
)  ->  ( (coeff `  b ) `  (deg `  b ) )  =  ( (coeff `  (minPolyAA `  a ) ) `  (deg `  (minPolyAA `  a
) ) ) )
1817eqeq1d 2452 . . . . . . . 8  |-  ( b  =  (minPolyAA `  a
)  ->  ( (
(coeff `  b ) `  (deg `  b )
)  =  1  <->  (
(coeff `  (minPolyAA `  a
) ) `  (deg `  (minPolyAA `  a )
) )  =  1 ) )
1914, 18anbi12d 716 . . . . . . 7  |-  ( b  =  (minPolyAA `  a
)  ->  ( (
( b `  a
)  =  0  /\  ( (coeff `  b
) `  (deg `  b
) )  =  1 )  <->  ( ( (minPolyAA `  a ) `  a
)  =  0  /\  ( (coeff `  (minPolyAA `  a ) ) `  (deg `  (minPolyAA `  a
) ) )  =  1 ) ) )
2019rspcev 3149 . . . . . 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 1265 . . . . 5  |-  ( a  e.  AA  ->  E. b  e.  (Poly `  QQ )
( ( b `  a )  =  0  /\  ( (coeff `  b ) `  (deg `  b ) )  =  1 ) )
226, 21jca 535 . . . 4  |-  ( a  e.  AA  ->  (
a  e.  CC  /\  E. b  e.  (Poly `  QQ ) ( ( b `
 a )  =  0  /\  ( (coeff `  b ) `  (deg `  b ) )  =  1 ) ) )
23 simpl 459 . . . . . . . . 9  |-  ( ( b  e.  (Poly `  QQ )  /\  (
( b `  a
)  =  0  /\  ( (coeff `  b
) `  (deg `  b
) )  =  1 ) )  ->  b  e.  (Poly `  QQ )
)
24 coe0 23203 . . . . . . . . . . . . . . 15  |-  (coeff ` 
0p )  =  ( NN0  X.  {
0 } )
2524fveq1i 5864 . . . . . . . . . . . . . 14  |-  ( (coeff `  0p ) `
 (deg `  0p ) )  =  ( ( NN0  X.  { 0 } ) `
 (deg `  0p ) )
26 dgr0 23209 . . . . . . . . . . . . . . . 16  |-  (deg ` 
0p )  =  0
27 0nn0 10881 . . . . . . . . . . . . . . . 16  |-  0  e.  NN0
2826, 27eqeltri 2524 . . . . . . . . . . . . . . 15  |-  (deg ` 
0p )  e. 
NN0
29 c0ex 9634 . . . . . . . . . . . . . . . 16  |-  0  e.  _V
3029fvconst2 6118 . . . . . . . . . . . . . . 15  |-  ( (deg
`  0p )  e.  NN0  ->  ( ( NN0  X.  { 0 } ) `  (deg `  0p ) )  =  0 )
3128, 30ax-mp 5 . . . . . . . . . . . . . 14  |-  ( ( NN0  X.  { 0 } ) `  (deg `  0p ) )  =  0
3225, 31eqtri 2472 . . . . . . . . . . . . 13  |-  ( (coeff `  0p ) `
 (deg `  0p ) )  =  0
33 0ne1 10674 . . . . . . . . . . . . 13  |-  0  =/=  1
3432, 33eqnetri 2693 . . . . . . . . . . . 12  |-  ( (coeff `  0p ) `
 (deg `  0p ) )  =/=  1
35 fveq2 5863 . . . . . . . . . . . . . 14  |-  ( b  =  0p  -> 
(coeff `  b )  =  (coeff `  0p
) )
36 fveq2 5863 . . . . . . . . . . . . . 14  |-  ( b  =  0p  -> 
(deg `  b )  =  (deg `  0p
) )
3735, 36fveq12d 5869 . . . . . . . . . . . . 13  |-  ( b  =  0p  -> 
( (coeff `  b
) `  (deg `  b
) )  =  ( (coeff `  0p
) `  (deg `  0p ) ) )
3837neeq1d 2682 . . . . . . . . . . . 12  |-  ( b  =  0p  -> 
( ( (coeff `  b ) `  (deg `  b ) )  =/=  1  <->  ( (coeff ` 
0p ) `  (deg `  0p ) )  =/=  1 ) )
3934, 38mpbiri 237 . . . . . . . . . . 11  |-  ( b  =  0p  -> 
( (coeff `  b
) `  (deg `  b
) )  =/=  1
)
4039necon2i 2657 . . . . . . . . . 10  |-  ( ( (coeff `  b ) `  (deg `  b )
)  =  1  -> 
b  =/=  0p )
4140ad2antll 734 . . . . . . . . 9  |-  ( ( b  e.  (Poly `  QQ )  /\  (
( b `  a
)  =  0  /\  ( (coeff `  b
) `  (deg `  b
) )  =  1 ) )  ->  b  =/=  0p )
42 eldifsn 4096 . . . . . . . . 9  |-  ( b  e.  ( (Poly `  QQ )  \  { 0p } )  <->  ( b  e.  (Poly `  QQ )  /\  b  =/=  0p ) )
4323, 41, 42sylanbrc 669 . . . . . . . 8  |-  ( ( b  e.  (Poly `  QQ )  /\  (
( b `  a
)  =  0  /\  ( (coeff `  b
) `  (deg `  b
) )  =  1 ) )  ->  b  e.  ( (Poly `  QQ )  \  { 0p } ) )
44 simprl 763 . . . . . . . 8  |-  ( ( b  e.  (Poly `  QQ )  /\  (
( b `  a
)  =  0  /\  ( (coeff `  b
) `  (deg `  b
) )  =  1 ) )  ->  (
b `  a )  =  0 )
4543, 44jca 535 . . . . . . 7  |-  ( ( b  e.  (Poly `  QQ )  /\  (
( b `  a
)  =  0  /\  ( (coeff `  b
) `  (deg `  b
) )  =  1 ) )  ->  (
b  e.  ( (Poly `  QQ )  \  {
0p } )  /\  ( b `  a )  =  0 ) )
4645reximi2 2853 . . . . . 6  |-  ( E. b  e.  (Poly `  QQ ) ( ( b `
 a )  =  0  /\  ( (coeff `  b ) `  (deg `  b ) )  =  1 )  ->  E. b  e.  ( (Poly `  QQ )  \  { 0p } ) ( b `
 a )  =  0 )
4746anim2i 572 . . . . 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 23271 . . . . 5  |-  ( a  e.  AA  <->  ( a  e.  CC  /\  E. b  e.  ( (Poly `  QQ )  \  { 0p } ) ( b `
 a )  =  0 ) )
4947, 48sylibr 216 . . . 4  |-  ( ( a  e.  CC  /\  E. b  e.  (Poly `  QQ ) ( ( b `
 a )  =  0  /\  ( (coeff `  b ) `  (deg `  b ) )  =  1 ) )  -> 
a  e.  AA )
5022, 49impbii 191 . . 3  |-  ( a  e.  AA  <->  ( a  e.  CC  /\  E. b  e.  (Poly `  QQ )
( ( b `  a )  =  0  /\  ( (coeff `  b ) `  (deg `  b ) )  =  1 ) ) )
511, 5, 503bitr4ri 282 . 2  |-  ( a  e.  AA  <->  a  e.  (IntgOver `  QQ ) )
5251eqriv 2447 1  |-  AA  =  (IntgOver `  QQ )
Colors of variables: wff setvar class
Syntax hints:    /\ wa 371    = wceq 1443    e. wcel 1886    =/= wne 2621   E.wrex 2737   {crab 2740    \ cdif 3400    C_ wss 3403   {csn 3967    X. cxp 4831   ` cfv 5581   CCcc 9534   0cc0 9536   1c1 9537   NN0cn0 10866   QQcq 11261   0pc0p 22620  Polycply 23131  coeffccoe 23133  degcdgr 23134   AAcaa 23260  degAAcdgraa 35993  minPolyAAcmpaa 35995  IntgOvercitgo 36017
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-rep 4514  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580  ax-inf2 8143  ax-cnex 9592  ax-resscn 9593  ax-1cn 9594  ax-icn 9595  ax-addcl 9596  ax-addrcl 9597  ax-mulcl 9598  ax-mulrcl 9599  ax-mulcom 9600  ax-addass 9601  ax-mulass 9602  ax-distr 9603  ax-i2m1 9604  ax-1ne0 9605  ax-1rid 9606  ax-rnegex 9607  ax-rrecex 9608  ax-cnre 9609  ax-pre-lttri 9610  ax-pre-lttrn 9611  ax-pre-ltadd 9612  ax-pre-mulgt0 9613  ax-pre-sup 9614  ax-addf 9615
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-fal 1449  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-nel 2624  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-pss 3419  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-tp 3972  df-op 3974  df-uni 4198  df-int 4234  df-iun 4279  df-br 4402  df-opab 4461  df-mpt 4462  df-tr 4497  df-eprel 4744  df-id 4748  df-po 4754  df-so 4755  df-fr 4792  df-se 4793  df-we 4794  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-pred 5379  df-ord 5425  df-on 5426  df-lim 5427  df-suc 5428  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-isom 5590  df-riota 6250  df-ov 6291  df-oprab 6292  df-mpt2 6293  df-of 6528  df-om 6690  df-1st 6790  df-2nd 6791  df-wrecs 7025  df-recs 7087  df-rdg 7125  df-1o 7179  df-oadd 7183  df-er 7360  df-map 7471  df-pm 7472  df-en 7567  df-dom 7568  df-sdom 7569  df-fin 7570  df-sup 7953  df-inf 7954  df-oi 8022  df-card 8370  df-pnf 9674  df-mnf 9675  df-xr 9676  df-ltxr 9677  df-le 9678  df-sub 9859  df-neg 9860  df-div 10267  df-nn 10607  df-2 10665  df-3 10666  df-n0 10867  df-z 10935  df-uz 11157  df-q 11262  df-rp 11300  df-fz 11782  df-fzo 11913  df-fl 12025  df-mod 12094  df-seq 12211  df-exp 12270  df-hash 12513  df-cj 13155  df-re 13156  df-im 13157  df-sqrt 13291  df-abs 13292  df-clim 13545  df-rlim 13546  df-sum 13746  df-0p 22621  df-ply 23135  df-coe 23137  df-dgr 23138  df-aa 23261  df-dgraa 35996  df-mpaa 35998  df-itgo 36019
This theorem is referenced by: (None)
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