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.

Step	Hyp	Ref	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 ) } )
4	2, 3	ax-mp 5	. . . 4 |- (IntgOver ` QQ ) = { a e. CC | E. b e. (Poly ` QQ ) ( ( b ` a ) = 0 /\ ( (coeff ` b ) ` (deg ` b ) ) = 1 ) }
5	4	eleq2i 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 ) )
10	9	fveq2d 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 )
12	10, 11	eqtrd 2508	. . . . . 6 |- ( a e. AA -> ( (coeff ` (minPolyAA ` a ) ) ` (deg ` (minPolyAA ` a ) ) ) = 1 )
13		fveq1 5863	. . . . . . . . 9 |- ( b = (minPolyAA ` a ) -> ( b ` a ) = ( (minPolyAA ` a ) ` a ) )
14	13	eqeq1d 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 ) ) )
17	15, 16	fveq12d 5870	. . . . . . . . 9 |- ( b = (minPolyAA ` a ) -> ( (coeff ` b ) ` (deg ` b ) ) = ( (coeff ` (minPolyAA ` a ) ) ` (deg ` (minPolyAA ` a ) ) ) )
18	17	eqeq1d 2469	. . . . . . . 8 |- ( b = (minPolyAA ` a ) -> ( ( (coeff ` b ) ` (deg ` b ) ) = 1 <-> ( (coeff ` (minPolyAA ` a ) ) ` (deg ` (minPolyAA ` a ) ) ) = 1 ) )
19	14, 18	anbi12d 710	. . . . . . 7 |- ( b = (minPolyAA ` a ) -> ( ( ( b ` a ) = 0 /\ ( (coeff ` b ) ` (deg ` b ) ) = 1 ) <-> ( ( (minPolyAA ` a ) ` a ) = 0 /\ ( (coeff ` (minPolyAA ` a ) ) ` (deg ` (minPolyAA ` a ) ) ) = 1 ) ) )
20	19	rspcev 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 ) )
21	7, 8, 12, 20	syl12anc 1226	. . . . 5 |- ( a e. AA -> E. b e. (Poly ` QQ ) ( ( b ` a ) = 0 /\ ( (coeff ` b ) ` (deg ` b ) ) = 1 ) )
22	6, 21	jca 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 } )
25	24	fveq1i 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
28	26, 27	eqeltri 2551	. . . . . . . . . . . . . . 15 |- (deg ` 0p ) e. NN0
29		c0ex 9586	. . . . . . . . . . . . . . . 16 |- 0 e. _V
30	29	fvconst2 6114	. . . . . . . . . . . . . . 15 |- ( (deg ` 0p ) e. NN0 -> ( ( NN0 X. { 0 } ) ` (deg ` 0p ) ) = 0 )
31	28, 30	ax-mp 5	. . . . . . . . . . . . . 14 |- ( ( NN0 X. { 0 } ) ` (deg ` 0p ) ) = 0
32	25, 31	eqtri 2496	. . . . . . . . . . . . 13 |- ( (coeff ` 0p ) ` (deg ` 0p ) ) = 0
33		0ne1 10599	. . . . . . . . . . . . 13 |- 0 =/= 1
34	32, 33	eqnetri 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 ) )
37	35, 36	fveq12d 5870	. . . . . . . . . . . . 13 |- ( b = 0p -> ( (coeff ` b ) ` (deg ` b ) ) = ( (coeff ` 0p ) ` (deg ` 0p ) ) )
38	37	neeq1d 2744	. . . . . . . . . . . 12 |- ( b = 0p -> ( ( (coeff ` b ) ` (deg ` b ) ) =/= 1 <-> ( (coeff ` 0p ) ` (deg ` 0p ) ) =/= 1 ) )
39	34, 38	mpbiri 233	. . . . . . . . . . 11 |- ( b = 0p -> ( (coeff ` b ) ` (deg ` b ) ) =/= 1 )
40	39	necon2i 2710	. . . . . . . . . 10 |- ( ( (coeff ` b ) ` (deg ` b ) ) = 1 -> b =/= 0p )
41	40	ad2antll 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 ) )
43	23, 41, 42	sylanbrc 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 )
45	43, 44	jca 532	. . . . . . 7 |- ( ( b e. (Poly ` QQ ) /\ ( ( b ` a ) = 0 /\ ( (coeff ` b ) ` (deg ` b ) ) = 1 ) ) -> ( b e. ( (Poly ` QQ ) \ { 0p } ) /\ ( b ` a ) = 0 ) )
46	45	reximi2 2931	. . . . . 6 |- ( E. b e. (Poly ` QQ ) ( ( b ` a ) = 0 /\ ( (coeff ` b ) ` (deg ` b ) ) = 1 ) -> E. b e. ( (Poly ` QQ ) \ { 0p } ) ( b ` a ) = 0 )
47	46	anim2i 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 ) )
49	47, 48	sylibr 212	. . . 4 |- ( ( a e. CC /\ E. b e. (Poly ` QQ ) ( ( b ` a ) = 0 /\ ( (coeff ` b ) ` (deg ` b ) ) = 1 ) ) -> a e. AA )
50	22, 49	impbii 188	. . 3 |- ( a e. AA <-> ( a e. CC /\ E. b e. (Poly ` QQ ) ( ( b ` a ) = 0 /\ ( (coeff ` b ) ` (deg ` b ) ) = 1 ) ) )
51	1, 5, 50	3bitr4ri 278	. 2 |- ( a e. AA <-> a e. (IntgOver ` QQ ) )
52	51	eqriv 2463	1 |- AA = (IntgOver ` QQ )

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  deg_AAcdgraa 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)