Theorem elaa2 37919

Description: Elementhood in the set of nonzero algebraic numbers: when A is nonzero, the polynomial f can be chosen with a nonzero constant term. (Contributed by Glauco Siliprandi, 5-Apr-2020.) (Proof shortened by AV, 1-Oct-2020.)

Assertion

Ref	Expression
elaa2	|- ( A e. ( AA \ { 0 } ) <-> ( A e. CC /\ E. f e. (Poly ` ZZ ) ( ( (coeff ` f ) ` 0 ) =/= 0 /\ ( f ` A ) = 0 ) ) )

Distinct variable group:   A, f

Proof of Theorem elaa2

Dummy variables g k z j m n are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		aasscn 23258	. . . 4 |- AA C_ CC
2		eldifi 3587	. . . 4 |- ( A e. ( AA \ { 0 } ) -> A e. AA )
3	1, 2	sseldi 3462	. . 3 |- ( A e. ( AA \ { 0 } ) -> A e. CC )
4		elaa 23256	. . . . . 6 |- ( A e. AA <-> ( A e. CC /\ E. g e. ( (Poly ` ZZ ) \ { 0p } ) ( g ` A ) = 0 ) )
5	2, 4	sylib 199	. . . . 5 |- ( A e. ( AA \ { 0 } ) -> ( A e. CC /\ E. g e. ( (Poly ` ZZ ) \ { 0p } ) ( g ` A ) = 0 ) )
6	5	simprd 464	. . . 4 |- ( A e. ( AA \ { 0 } ) -> E. g e. ( (Poly ` ZZ ) \ { 0p } ) ( g ` A ) = 0 )
7	2	3ad2ant1 1026	. . . . . 6 |- ( ( A e. ( AA \ { 0 } ) /\ g e. ( (Poly ` ZZ ) \ { 0p } ) /\ ( g ` A ) = 0 ) -> A e. AA )
8		eldifsni 4123	. . . . . . 7 |- ( A e. ( AA \ { 0 } ) -> A =/= 0 )
9	8	3ad2ant1 1026	. . . . . 6 |- ( ( A e. ( AA \ { 0 } ) /\ g e. ( (Poly ` ZZ ) \ { 0p } ) /\ ( g ` A ) = 0 ) -> A =/= 0 )
10		eldifi 3587	. . . . . . 7 |- ( g e. ( (Poly ` ZZ ) \ { 0p } ) -> g e. (Poly ` ZZ ) )
11	10	3ad2ant2 1027	. . . . . 6 |- ( ( A e. ( AA \ { 0 } ) /\ g e. ( (Poly ` ZZ ) \ { 0p } ) /\ ( g ` A ) = 0 ) -> g e. (Poly ` ZZ ) )
12		eldifsni 4123	. . . . . . 7 |- ( g e. ( (Poly ` ZZ ) \ { 0p } ) -> g =/= 0p )
13	12	3ad2ant2 1027	. . . . . 6 |- ( ( A e. ( AA \ { 0 } ) /\ g e. ( (Poly ` ZZ ) \ { 0p } ) /\ ( g ` A ) = 0 ) -> g =/= 0p )
14		simp3 1007	. . . . . 6 |- ( ( A e. ( AA \ { 0 } ) /\ g e. ( (Poly ` ZZ ) \ { 0p } ) /\ ( g ` A ) = 0 ) -> ( g ` A ) = 0 )
15		fveq2 5878	. . . . . . . . 9 |- ( m = n -> ( (coeff ` g ) ` m ) = ( (coeff ` g ) ` n ) )
16	15	neeq1d 2701	. . . . . . . 8 |- ( m = n -> ( ( (coeff ` g ) ` m ) =/= 0 <-> ( (coeff ` g ) ` n ) =/= 0 ) )
17	16	cbvrabv 3080	. . . . . . 7 |- { m e. NN0 | ( (coeff ` g ) ` m ) =/= 0 } = { n e. NN0 | ( (coeff ` g ) ` n ) =/= 0 }
18	17	infeq1i 7997	. . . . . 6 |- inf ( { m e. NN0 | ( (coeff ` g ) ` m ) =/= 0 } , RR , < ) = inf ( { n e. NN0 | ( (coeff ` g ) ` n ) =/= 0 } , RR , < )
19		oveq1 6309	. . . . . . . 8 |- ( j = k -> ( j + inf ( { m e. NN0 | ( (coeff ` g ) ` m ) =/= 0 } , RR , < ) ) = ( k + inf ( { m e. NN0 | ( (coeff ` g ) ` m ) =/= 0 } , RR , < ) ) )
20	19	fveq2d 5882	. . . . . . 7 |- ( j = k -> ( (coeff ` g ) ` ( j + inf ( { m e. NN0 | ( (coeff ` g ) ` m ) =/= 0 } , RR , < ) ) ) = ( (coeff ` g ) ` ( k + inf ( { m e. NN0 | ( (coeff ` g ) ` m ) =/= 0 } , RR , < ) ) ) )
21	20	cbvmptv 4513	. . . . . 6 |- ( j e. NN0 |-> ( (coeff ` g ) ` ( j + inf ( { m e. NN0 | ( (coeff ` g ) ` m ) =/= 0 } , RR , < ) ) ) ) = ( k e. NN0 |-> ( (coeff ` g ) ` ( k + inf ( { m e. NN0 | ( (coeff ` g ) ` m ) =/= 0 } , RR , < ) ) ) )
22		eqid 2422	. . . . . 6 |- ( z e. CC |-> sum_ k e. ( 0 ... ( (deg ` g ) - inf ( { m e. NN0 | ( (coeff ` g ) ` m ) =/= 0 } , RR , < ) ) ) ( ( ( j e. NN0 |-> ( (coeff ` g ) ` ( j + inf ( { m e. NN0 | ( (coeff ` g ) ` m ) =/= 0 } , RR , < ) ) ) ) ` k ) x. ( z ^ k ) ) ) = ( z e. CC |-> sum_ k e. ( 0 ... ( (deg ` g ) - inf ( { m e. NN0 | ( (coeff ` g ) ` m ) =/= 0 } , RR , < ) ) ) ( ( ( j e. NN0 |-> ( (coeff ` g ) ` ( j + inf ( { m e. NN0 | ( (coeff ` g ) ` m ) =/= 0 } , RR , < ) ) ) ) ` k ) x. ( z ^ k ) ) )
23	7, 9, 11, 13, 14, 18, 21, 22	elaa2lem 37917	. . . . 5 |- ( ( A e. ( AA \ { 0 } ) /\ g e. ( (Poly ` ZZ ) \ { 0p } ) /\ ( g ` A ) = 0 ) -> E. f e. (Poly ` ZZ ) ( ( (coeff ` f ) ` 0 ) =/= 0 /\ ( f ` A ) = 0 ) )
24	23	rexlimdv3a 2919	. . . 4 |- ( A e. ( AA \ { 0 } ) -> ( E. g e. ( (Poly ` ZZ ) \ { 0p } ) ( g ` A ) = 0 -> E. f e. (Poly ` ZZ ) ( ( (coeff ` f ) ` 0 ) =/= 0 /\ ( f ` A ) = 0 ) ) )
25	6, 24	mpd 15	. . 3 |- ( A e. ( AA \ { 0 } ) -> E. f e. (Poly ` ZZ ) ( ( (coeff ` f ) ` 0 ) =/= 0 /\ ( f ` A ) = 0 ) )
26	3, 25	jca 534	. 2 |- ( A e. ( AA \ { 0 } ) -> ( A e. CC /\ E. f e. (Poly ` ZZ ) ( ( (coeff ` f ) ` 0 ) =/= 0 /\ ( f ` A ) = 0 ) ) )
27		simpl 458	. . . . . . . . 9 |- ( ( f e. (Poly ` ZZ ) /\ ( (coeff ` f ) ` 0 ) =/= 0 ) -> f e. (Poly ` ZZ ) )
28		fveq2 5878	. . . . . . . . . . . . . . 15 |- ( f = 0p -> (coeff ` f ) = (coeff ` 0p ) )
29		coe0 23197	. . . . . . . . . . . . . . 15 |- (coeff ` 0p ) = ( NN0 X. { 0 } )
30	28, 29	syl6eq 2479	. . . . . . . . . . . . . 14 |- ( f = 0p -> (coeff ` f ) = ( NN0 X. { 0 } ) )
31	30	fveq1d 5880	. . . . . . . . . . . . 13 |- ( f = 0p -> ( (coeff ` f ) ` 0 ) = ( ( NN0 X. { 0 } ) ` 0 ) )
32		0nn0 10885	. . . . . . . . . . . . . 14 |- 0 e. NN0
33		fvconst2g 6130	. . . . . . . . . . . . . 14 |- ( ( 0 e. NN0 /\ 0 e. NN0 ) -> ( ( NN0 X. { 0 } ) ` 0 ) = 0 )
34	32, 32, 33	mp2an 676	. . . . . . . . . . . . 13 |- ( ( NN0 X. { 0 } ) ` 0 ) = 0
35	31, 34	syl6eq 2479	. . . . . . . . . . . 12 |- ( f = 0p -> ( (coeff ` f ) ` 0 ) = 0 )
36	35	adantl 467	. . . . . . . . . . 11 |- ( ( ( f e. (Poly ` ZZ ) /\ ( (coeff ` f ) ` 0 ) =/= 0 ) /\ f = 0p ) -> ( (coeff ` f ) ` 0 ) = 0 )
37		neneq 2626	. . . . . . . . . . . 12 |- ( ( (coeff ` f ) ` 0 ) =/= 0 -> -. ( (coeff ` f ) ` 0 ) = 0 )
38	37	ad2antlr 731	. . . . . . . . . . 11 |- ( ( ( f e. (Poly ` ZZ ) /\ ( (coeff ` f ) ` 0 ) =/= 0 ) /\ f = 0p ) -> -. ( (coeff ` f ) ` 0 ) = 0 )
39	36, 38	pm2.65da 578	. . . . . . . . . 10 |- ( ( f e. (Poly ` ZZ ) /\ ( (coeff ` f ) ` 0 ) =/= 0 ) -> -. f = 0p )
40		elsn 4010	. . . . . . . . . 10 |- ( f e. { 0p } <-> f = 0p )
41	39, 40	sylnibr 306	. . . . . . . . 9 |- ( ( f e. (Poly ` ZZ ) /\ ( (coeff ` f ) ` 0 ) =/= 0 ) -> -. f e. { 0p } )
42	27, 41	eldifd 3447	. . . . . . . 8 |- ( ( f e. (Poly ` ZZ ) /\ ( (coeff ` f ) ` 0 ) =/= 0 ) -> f e. ( (Poly ` ZZ ) \ { 0p } ) )
43	42	adantrr 721	. . . . . . 7 |- ( ( f e. (Poly ` ZZ ) /\ ( ( (coeff ` f ) ` 0 ) =/= 0 /\ ( f ` A ) = 0 ) ) -> f e. ( (Poly ` ZZ ) \ { 0p } ) )
44		simprr 764	. . . . . . 7 |- ( ( f e. (Poly ` ZZ ) /\ ( ( (coeff ` f ) ` 0 ) =/= 0 /\ ( f ` A ) = 0 ) ) -> ( f ` A ) = 0 )
45	43, 44	jca 534	. . . . . 6 |- ( ( f e. (Poly ` ZZ ) /\ ( ( (coeff ` f ) ` 0 ) =/= 0 /\ ( f ` A ) = 0 ) ) -> ( f e. ( (Poly ` ZZ ) \ { 0p } ) /\ ( f ` A ) = 0 ) )
46	45	reximi2 2892	. . . . 5 |- ( E. f e. (Poly ` ZZ ) ( ( (coeff ` f ) ` 0 ) =/= 0 /\ ( f ` A ) = 0 ) -> E. f e. ( (Poly ` ZZ ) \ { 0p } ) ( f ` A ) = 0 )
47	46	anim2i 571	. . . 4 |- ( ( A e. CC /\ E. f e. (Poly ` ZZ ) ( ( (coeff ` f ) ` 0 ) =/= 0 /\ ( f ` A ) = 0 ) ) -> ( A e. CC /\ E. f e. ( (Poly ` ZZ ) \ { 0p } ) ( f ` A ) = 0 ) )
48		elaa 23256	. . . 4 |- ( A e. AA <-> ( A e. CC /\ E. f e. ( (Poly ` ZZ ) \ { 0p } ) ( f ` A ) = 0 ) )
49	47, 48	sylibr 215	. . 3 |- ( ( A e. CC /\ E. f e. (Poly ` ZZ ) ( ( (coeff ` f ) ` 0 ) =/= 0 /\ ( f ` A ) = 0 ) ) -> A e. AA )
50		simpr 462	. . . 4 |- ( ( A e. CC /\ E. f e. (Poly ` ZZ ) ( ( (coeff ` f ) ` 0 ) =/= 0 /\ ( f ` A ) = 0 ) ) -> E. f e. (Poly ` ZZ ) ( ( (coeff ` f ) ` 0 ) =/= 0 /\ ( f ` A ) = 0 ) )
51		nfv 1751	. . . . . 6 |- F/ f A e. CC
52		nfre1 2886	. . . . . 6 |- F/ f E. f e. (Poly ` ZZ ) ( ( (coeff ` f ) ` 0 ) =/= 0 /\ ( f ` A ) = 0 )
53	51, 52	nfan 1984	. . . . 5 |- F/ f ( A e. CC /\ E. f e. (Poly ` ZZ ) ( ( (coeff ` f ) ` 0 ) =/= 0 /\ ( f ` A ) = 0 ) )
54		nfv 1751	. . . . 5 |- F/ f -. A e. { 0 }
55		simpl3r 1061	. . . . . . . . 9 |- ( ( ( A e. CC /\ f e. (Poly ` ZZ ) /\ ( ( (coeff ` f ) ` 0 ) =/= 0 /\ ( f ` A ) = 0 ) ) /\ A = 0 ) -> ( f ` A ) = 0 )
56		fveq2 5878	. . . . . . . . . . . . . . 15 |- ( A = 0 -> ( f ` A ) = ( f ` 0 ) )
57		eqid 2422	. . . . . . . . . . . . . . . 16 |- (coeff ` f ) = (coeff ` f )
58	57	coefv0 23189	. . . . . . . . . . . . . . 15 |- ( f e. (Poly ` ZZ ) -> ( f ` 0 ) = ( (coeff ` f ) ` 0 ) )
59	56, 58	sylan9eqr 2485	. . . . . . . . . . . . . 14 |- ( ( f e. (Poly ` ZZ ) /\ A = 0 ) -> ( f ` A ) = ( (coeff ` f ) ` 0 ) )
60	59	adantlr 719	. . . . . . . . . . . . 13 |- ( ( ( f e. (Poly ` ZZ ) /\ ( (coeff ` f ) ` 0 ) =/= 0 ) /\ A = 0 ) -> ( f ` A ) = ( (coeff ` f ) ` 0 ) )
61		simplr 760	. . . . . . . . . . . . 13 |- ( ( ( f e. (Poly ` ZZ ) /\ ( (coeff ` f ) ` 0 ) =/= 0 ) /\ A = 0 ) -> ( (coeff ` f ) ` 0 ) =/= 0 )
62	60, 61	eqnetrd 2717	. . . . . . . . . . . 12 |- ( ( ( f e. (Poly ` ZZ ) /\ ( (coeff ` f ) ` 0 ) =/= 0 ) /\ A = 0 ) -> ( f ` A ) =/= 0 )
63	62	neneqd 2625	. . . . . . . . . . 11 |- ( ( ( f e. (Poly ` ZZ ) /\ ( (coeff ` f ) ` 0 ) =/= 0 ) /\ A = 0 ) -> -. ( f ` A ) = 0 )
64	63	adantlrr 725	. . . . . . . . . 10 |- ( ( ( f e. (Poly ` ZZ ) /\ ( ( (coeff ` f ) ` 0 ) =/= 0 /\ ( f ` A ) = 0 ) ) /\ A = 0 ) -> -. ( f ` A ) = 0 )
65	64	3adantl1 1161	. . . . . . . . 9 |- ( ( ( A e. CC /\ f e. (Poly ` ZZ ) /\ ( ( (coeff ` f ) ` 0 ) =/= 0 /\ ( f ` A ) = 0 ) ) /\ A = 0 ) -> -. ( f ` A ) = 0 )
66	55, 65	pm2.65da 578	. . . . . . . 8 |- ( ( A e. CC /\ f e. (Poly ` ZZ ) /\ ( ( (coeff ` f ) ` 0 ) =/= 0 /\ ( f ` A ) = 0 ) ) -> -. A = 0 )
67		elsncg 4019	. . . . . . . . . 10 |- ( A e. CC -> ( A e. { 0 } <-> A = 0 ) )
68	67	biimpa 486	. . . . . . . . 9 |- ( ( A e. CC /\ A e. { 0 } ) -> A = 0 )
69	68	3ad2antl1 1167	. . . . . . . 8 |- ( ( ( A e. CC /\ f e. (Poly ` ZZ ) /\ ( ( (coeff ` f ) ` 0 ) =/= 0 /\ ( f ` A ) = 0 ) ) /\ A e. { 0 } ) -> A = 0 )
70	66, 69	mtand 663	. . . . . . 7 |- ( ( A e. CC /\ f e. (Poly ` ZZ ) /\ ( ( (coeff ` f ) ` 0 ) =/= 0 /\ ( f ` A ) = 0 ) ) -> -. A e. { 0 } )
71	70	3exp 1204	. . . . . 6 |- ( A e. CC -> ( f e. (Poly ` ZZ ) -> ( ( ( (coeff ` f ) ` 0 ) =/= 0 /\ ( f ` A ) = 0 ) -> -. A e. { 0 } ) ) )
72	71	adantr 466	. . . . 5 |- ( ( A e. CC /\ E. f e. (Poly ` ZZ ) ( ( (coeff ` f ) ` 0 ) =/= 0 /\ ( f ` A ) = 0 ) ) -> ( f e. (Poly ` ZZ ) -> ( ( ( (coeff ` f ) ` 0 ) =/= 0 /\ ( f ` A ) = 0 ) -> -. A e. { 0 } ) ) )
73	53, 54, 72	rexlimd 2909	. . . 4 |- ( ( A e. CC /\ E. f e. (Poly ` ZZ ) ( ( (coeff ` f ) ` 0 ) =/= 0 /\ ( f ` A ) = 0 ) ) -> ( E. f e. (Poly ` ZZ ) ( ( (coeff ` f ) ` 0 ) =/= 0 /\ ( f ` A ) = 0 ) -> -. A e. { 0 } ) )
74	50, 73	mpd 15	. . 3 |- ( ( A e. CC /\ E. f e. (Poly ` ZZ ) ( ( (coeff ` f ) ` 0 ) =/= 0 /\ ( f ` A ) = 0 ) ) -> -. A e. { 0 } )
75	49, 74	eldifd 3447	. 2 |- ( ( A e. CC /\ E. f e. (Poly ` ZZ ) ( ( (coeff ` f ) ` 0 ) =/= 0 /\ ( f ` A ) = 0 ) ) -> A e. ( AA \ { 0 } ) )
76	26, 75	impbii 190	1 |- ( A e. ( AA \ { 0 } ) <-> ( A e. CC /\ E. f e. (Poly ` ZZ ) ( ( (coeff ` f ) ` 0 ) =/= 0 /\ ( f ` A ) = 0 ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 187   /\ wa 370   /\ w3a 982   = wceq 1437   e. wcel 1868   =/= wne 2618   E.wrex 2776   {crab 2779   \ cdif 3433   {csn 3996   |-> cmpt 4479   X. cxp 4848   ` cfv 5598  (class class class)co 6302  infcinf 7958   CCcc 9538   RRcr 9539   0cc0 9540   + caddc 9543   x. cmul 9545   < clt 9676   - cmin 9861   NN0cn0 10870   ZZcz 10938   ...cfz 11785   ^cexp 12272   sum_csu 13740   0pc0p 22614  Polycply 23125  coeffccoe 23127  degcdgr 23128   AAcaa 23254

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-rep 4533  ax-sep 4543  ax-nul 4552  ax-pow 4599  ax-pr 4657  ax-un 6594  ax-inf2 8149  ax-cnex 9596  ax-resscn 9597  ax-1cn 9598  ax-icn 9599  ax-addcl 9600  ax-addrcl 9601  ax-mulcl 9602  ax-mulrcl 9603  ax-mulcom 9604  ax-addass 9605  ax-mulass 9606  ax-distr 9607  ax-i2m1 9608  ax-1ne0 9609  ax-1rid 9610  ax-rnegex 9611  ax-rrecex 9612  ax-cnre 9613  ax-pre-lttri 9614  ax-pre-lttrn 9615  ax-pre-ltadd 9616  ax-pre-mulgt0 9617  ax-pre-sup 9618  ax-addf 9619

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-nel 2621  df-ral 2780  df-rex 2781  df-reu 2782  df-rmo 2783  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-tp 4001  df-op 4003  df-uni 4217  df-int 4253  df-iun 4298  df-br 4421  df-opab 4480  df-mpt 4481  df-tr 4516  df-eprel 4761  df-id 4765  df-po 4771  df-so 4772  df-fr 4809  df-se 4810  df-we 4811  df-xp 4856  df-rel 4857  df-cnv 4858  df-co 4859  df-dm 4860  df-rn 4861  df-res 4862  df-ima 4863  df-pred 5396  df-ord 5442  df-on 5443  df-lim 5444  df-suc 5445  df-iota 5562  df-fun 5600  df-fn 5601  df-f 5602  df-f1 5603  df-fo 5604  df-f1o 5605  df-fv 5606  df-isom 5607  df-riota 6264  df-ov 6305  df-oprab 6306  df-mpt2 6307  df-of 6542  df-om 6704  df-1st 6804  df-2nd 6805  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-1o 7187  df-oadd 7191  df-er 7368  df-map 7479  df-pm 7480  df-en 7575  df-dom 7576  df-sdom 7577  df-fin 7578  df-sup 7959  df-inf 7960  df-oi 8028  df-card 8375  df-pnf 9678  df-mnf 9679  df-xr 9680  df-ltxr 9681  df-le 9682  df-sub 9863  df-neg 9864  df-div 10271  df-nn 10611  df-2 10669  df-3 10670  df-n0 10871  df-z 10939  df-uz 11161  df-rp 11304  df-fz 11786  df-fzo 11917  df-fl 12028  df-seq 12214  df-exp 12273  df-hash 12516  df-cj 13151  df-re 13152  df-im 13153  df-sqrt 13287  df-abs 13288  df-clim 13540  df-rlim 13541  df-sum 13741  df-0p 22615  df-ply 23129  df-coe 23131  df-dgr 23132  df-aa 23255

This theorem is referenced by:  etransc  37969