Theorem aaitgo 29444

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 2895	. . 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 10960	. . . . 5 |- QQ C_ CC
3		itgoval 29443	. . . . 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 2505	. . 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 21742	. . . . 5 |- ( a e. AA -> a e. CC )
7		mpaacl 29435	. . . . . 6 |- ( a e. AA -> (minPolyAA ` a ) e. (Poly ` QQ ) )
8		mpaaroot 29437	. . . . . 6 |- ( a e. AA -> ( (minPolyAA ` a ) ` a ) = 0 )
9		mpaadgr 29436	. . . . . . . 8 |- ( a e. AA -> (deg ` (minPolyAA ` a ) ) = (degAA ` a ) )
10	9	fveq2d 5692	. . . . . . 7 |- ( a e. AA -> ( (coeff ` (minPolyAA ` a ) ) ` (deg ` (minPolyAA ` a ) ) ) = ( (coeff ` (minPolyAA ` a ) ) ` (degAA ` a ) ) )
11		mpaamn 29438	. . . . . . 7 |- ( a e. AA -> ( (coeff ` (minPolyAA ` a ) ) ` (degAA ` a ) ) = 1 )
12	10, 11	eqtrd 2473	. . . . . 6 |- ( a e. AA -> ( (coeff ` (minPolyAA ` a ) ) ` (deg ` (minPolyAA ` a ) ) ) = 1 )
13		fveq1 5687	. . . . . . . . 9 |- ( b = (minPolyAA ` a ) -> ( b ` a ) = ( (minPolyAA ` a ) ` a ) )
14	13	eqeq1d 2449	. . . . . . . 8 |- ( b = (minPolyAA ` a ) -> ( ( b ` a ) = 0 <-> ( (minPolyAA ` a ) ` a ) = 0 ) )
15		fveq2 5688	. . . . . . . . . 10 |- ( b = (minPolyAA ` a ) -> (coeff ` b ) = (coeff ` (minPolyAA ` a ) ) )
16		fveq2 5688	. . . . . . . . . 10 |- ( b = (minPolyAA ` a ) -> (deg ` b ) = (deg ` (minPolyAA ` a ) ) )
17	15, 16	fveq12d 5694	. . . . . . . . 9 |- ( b = (minPolyAA ` a ) -> ( (coeff ` b ) ` (deg ` b ) ) = ( (coeff ` (minPolyAA ` a ) ) ` (deg ` (minPolyAA ` a ) ) ) )
18	17	eqeq1d 2449	. . . . . . . 8 |- ( b = (minPolyAA ` a ) -> ( ( (coeff ` b ) ` (deg ` b ) ) = 1 <-> ( (coeff ` (minPolyAA ` a ) ) ` (deg ` (minPolyAA ` a ) ) ) = 1 ) )
19	14, 18	anbi12d 705	. . . . . . 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 3070	. . . . . 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 1211	. . . . 5 |- ( a e. AA -> E. b e. (Poly ` QQ ) ( ( b ` a ) = 0 /\ ( (coeff ` b ) ` (deg ` b ) ) = 1 ) )
22	6, 21	jca 529	. . . 4 |- ( a e. AA -> ( a e. CC /\ E. b e. (Poly ` QQ ) ( ( b ` a ) = 0 /\ ( (coeff ` b ) ` (deg ` b ) ) = 1 ) ) )
23		simpl 454	. . . . . . . . 9 |- ( ( b e. (Poly ` QQ ) /\ ( ( b ` a ) = 0 /\ ( (coeff ` b ) ` (deg ` b ) ) = 1 ) ) -> b e. (Poly ` QQ ) )
24		coe0 21682	. . . . . . . . . . . . . . 15 |- (coeff ` 0p ) = ( NN0 X. { 0 } )
25	24	fveq1i 5689	. . . . . . . . . . . . . 14 |- ( (coeff ` 0p ) ` (deg ` 0p ) ) = ( ( NN0 X. { 0 } ) ` (deg ` 0p ) )
26		dgr0 21688	. . . . . . . . . . . . . . . 16 |- (deg ` 0p ) = 0
27		0nn0 10590	. . . . . . . . . . . . . . . 16 |- 0 e. NN0
28	26, 27	eqeltri 2511	. . . . . . . . . . . . . . 15 |- (deg ` 0p ) e. NN0
29		c0ex 9376	. . . . . . . . . . . . . . . 16 |- 0 e. _V
30	29	fvconst2 5930	. . . . . . . . . . . . . . 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 2461	. . . . . . . . . . . . 13 |- ( (coeff ` 0p ) ` (deg ` 0p ) ) = 0
33		0ne1 10385	. . . . . . . . . . . . 13 |- 0 =/= 1
34	32, 33	eqnetri 2623	. . . . . . . . . . . 12 |- ( (coeff ` 0p ) ` (deg ` 0p ) ) =/= 1
35		fveq2 5688	. . . . . . . . . . . . . 14 |- ( b = 0p -> (coeff ` b ) = (coeff ` 0p ) )
36		fveq2 5688	. . . . . . . . . . . . . 14 |- ( b = 0p -> (deg ` b ) = (deg ` 0p ) )
37	35, 36	fveq12d 5694	. . . . . . . . . . . . 13 |- ( b = 0p -> ( (coeff ` b ) ` (deg ` b ) ) = ( (coeff ` 0p ) ` (deg ` 0p ) ) )
38	37	neeq1d 2619	. . . . . . . . . . . 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 2656	. . . . . . . . . 10 |- ( ( (coeff ` b ) ` (deg ` b ) ) = 1 -> b =/= 0p )
41	40	ad2antll 723	. . . . . . . . 9 |- ( ( b e. (Poly ` QQ ) /\ ( ( b ` a ) = 0 /\ ( (coeff ` b ) ` (deg ` b ) ) = 1 ) ) -> b =/= 0p )
42		eldifsn 3997	. . . . . . . . 9 |- ( b e. ( (Poly ` QQ ) \ { 0p } ) <-> ( b e. (Poly ` QQ ) /\ b =/= 0p ) )
43	23, 41, 42	sylanbrc 659	. . . . . . . 8 |- ( ( b e. (Poly ` QQ ) /\ ( ( b ` a ) = 0 /\ ( (coeff ` b ) ` (deg ` b ) ) = 1 ) ) -> b e. ( (Poly ` QQ ) \ { 0p } ) )
44		simprl 750	. . . . . . . 8 |- ( ( b e. (Poly ` QQ ) /\ ( ( b ` a ) = 0 /\ ( (coeff ` b ) ` (deg ` b ) ) = 1 ) ) -> ( b ` a ) = 0 )
45	43, 44	jca 529	. . . . . . 7 |- ( ( b e. (Poly ` QQ ) /\ ( ( b ` a ) = 0 /\ ( (coeff ` b ) ` (deg ` b ) ) = 1 ) ) -> ( b e. ( (Poly ` QQ ) \ { 0p } ) /\ ( b ` a ) = 0 ) )
46	45	reximi2 2820	. . . . . 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 566	. . . . 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 21747	. . . . 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 2438	1 |- AA = (IntgOver ` QQ )

Syntax hints:   /\ wa 369   = wceq 1364   e. wcel 1761   =/= wne 2604   E.wrex 2714   {crab 2717   \ cdif 3322   C_ wss 3325   {csn 3874   X. cxp 4834   ` cfv 5415   CCcc 9276   0cc0 9278   1c1 9279   NN0cn0 10575   QQcq 10949   0pc0p 21106  Polycply 21611  coeffccoe 21613  degcdgr 21614   AAcaa 21739  deg_AAcdgraa 29422  minPolyAAcmpaa 29423  IntgOvercitgo 29439

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-inf2 7843  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-pre-sup 9356  ax-addf 9357

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-fal 1370  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-se 4676  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-isom 5424  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-of 6319  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-1o 6916  df-oadd 6920  df-er 7097  df-map 7212  df-pm 7213  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-sup 7687  df-oi 7720  df-card 8105  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-nn 10319  df-2 10376  df-3 10377  df-n0 10576  df-z 10643  df-uz 10858  df-q 10950  df-rp 10988  df-fz 11434  df-fzo 11545  df-fl 11638  df-mod 11705  df-seq 11803  df-exp 11862  df-hash 12100  df-cj 12584  df-re 12585  df-im 12586  df-sqr 12720  df-abs 12721  df-clim 12962  df-rlim 12963  df-sum 13160  df-0p 21107  df-ply 21615  df-coe 21617  df-dgr 21618  df-aa 21740  df-dgraa 29424  df-mpaa 29425  df-itgo 29441

This theorem is referenced by: (None)