Theorem elaa2 38099

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 23271	. . . 4 |- AA C_ CC
2		eldifi 3555	. . . 4 |- ( A e. ( AA \ { 0 } ) -> A e. AA )
3	1, 2	sseldi 3430	. . 3 |- ( A e. ( AA \ { 0 } ) -> A e. CC )
4		elaa 23269	. . . . . 6 |- ( A e. AA <-> ( A e. CC /\ E. g e. ( (Poly ` ZZ ) \ { 0p } ) ( g ` A ) = 0 ) )
5	2, 4	sylib 200	. . . . 5 |- ( A e. ( AA \ { 0 } ) -> ( A e. CC /\ E. g e. ( (Poly ` ZZ ) \ { 0p } ) ( g ` A ) = 0 ) )
6	5	simprd 465	. . . 4 |- ( A e. ( AA \ { 0 } ) -> E. g e. ( (Poly ` ZZ ) \ { 0p } ) ( g ` A ) = 0 )
7	2	3ad2ant1 1029	. . . . . 6 |- ( ( A e. ( AA \ { 0 } ) /\ g e. ( (Poly ` ZZ ) \ { 0p } ) /\ ( g ` A ) = 0 ) -> A e. AA )
8		eldifsni 4098	. . . . . . 7 |- ( A e. ( AA \ { 0 } ) -> A =/= 0 )
9	8	3ad2ant1 1029	. . . . . 6 |- ( ( A e. ( AA \ { 0 } ) /\ g e. ( (Poly ` ZZ ) \ { 0p } ) /\ ( g ` A ) = 0 ) -> A =/= 0 )
10		eldifi 3555	. . . . . . 7 |- ( g e. ( (Poly ` ZZ ) \ { 0p } ) -> g e. (Poly ` ZZ ) )
11	10	3ad2ant2 1030	. . . . . 6 |- ( ( A e. ( AA \ { 0 } ) /\ g e. ( (Poly ` ZZ ) \ { 0p } ) /\ ( g ` A ) = 0 ) -> g e. (Poly ` ZZ ) )
12		eldifsni 4098	. . . . . . 7 |- ( g e. ( (Poly ` ZZ ) \ { 0p } ) -> g =/= 0p )
13	12	3ad2ant2 1030	. . . . . 6 |- ( ( A e. ( AA \ { 0 } ) /\ g e. ( (Poly ` ZZ ) \ { 0p } ) /\ ( g ` A ) = 0 ) -> g =/= 0p )
14		simp3 1010	. . . . . 6 |- ( ( A e. ( AA \ { 0 } ) /\ g e. ( (Poly ` ZZ ) \ { 0p } ) /\ ( g ` A ) = 0 ) -> ( g ` A ) = 0 )
15		fveq2 5865	. . . . . . . . 9 |- ( m = n -> ( (coeff ` g ) ` m ) = ( (coeff ` g ) ` n ) )
16	15	neeq1d 2683	. . . . . . . 8 |- ( m = n -> ( ( (coeff ` g ) ` m ) =/= 0 <-> ( (coeff ` g ) ` n ) =/= 0 ) )
17	16	cbvrabv 3044	. . . . . . 7 |- { m e. NN0 | ( (coeff ` g ) ` m ) =/= 0 } = { n e. NN0 | ( (coeff ` g ) ` n ) =/= 0 }
18	17	infeq1i 7994	. . . . . 6 |- inf ( { m e. NN0 | ( (coeff ` g ) ` m ) =/= 0 } , RR , < ) = inf ( { n e. NN0 | ( (coeff ` g ) ` n ) =/= 0 } , RR , < )
19		oveq1 6297	. . . . . . . 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 5869	. . . . . . 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 4495	. . . . . 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 2451	. . . . . 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 38097	. . . . 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 2881	. . . 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 535	. 2 |- ( A e. ( AA \ { 0 } ) -> ( A e. CC /\ E. f e. (Poly ` ZZ ) ( ( (coeff ` f ) ` 0 ) =/= 0 /\ ( f ` A ) = 0 ) ) )
27		simpl 459	. . . . . . . . 9 |- ( ( f e. (Poly ` ZZ ) /\ ( (coeff ` f ) ` 0 ) =/= 0 ) -> f e. (Poly ` ZZ ) )
28		fveq2 5865	. . . . . . . . . . . . . . 15 |- ( f = 0p -> (coeff ` f ) = (coeff ` 0p ) )
29		coe0 23210	. . . . . . . . . . . . . . 15 |- (coeff ` 0p ) = ( NN0 X. { 0 } )
30	28, 29	syl6eq 2501	. . . . . . . . . . . . . 14 |- ( f = 0p -> (coeff ` f ) = ( NN0 X. { 0 } ) )
31	30	fveq1d 5867	. . . . . . . . . . . . 13 |- ( f = 0p -> ( (coeff ` f ) ` 0 ) = ( ( NN0 X. { 0 } ) ` 0 ) )
32		0nn0 10884	. . . . . . . . . . . . . 14 |- 0 e. NN0
33		fvconst2g 6118	. . . . . . . . . . . . . 14 |- ( ( 0 e. NN0 /\ 0 e. NN0 ) -> ( ( NN0 X. { 0 } ) ` 0 ) = 0 )
34	32, 32, 33	mp2an 678	. . . . . . . . . . . . 13 |- ( ( NN0 X. { 0 } ) ` 0 ) = 0
35	31, 34	syl6eq 2501	. . . . . . . . . . . 12 |- ( f = 0p -> ( (coeff ` f ) ` 0 ) = 0 )
36	35	adantl 468	. . . . . . . . . . 11 |- ( ( ( f e. (Poly ` ZZ ) /\ ( (coeff ` f ) ` 0 ) =/= 0 ) /\ f = 0p ) -> ( (coeff ` f ) ` 0 ) = 0 )
37		neneq 2630	. . . . . . . . . . . 12 |- ( ( (coeff ` f ) ` 0 ) =/= 0 -> -. ( (coeff ` f ) ` 0 ) = 0 )
38	37	ad2antlr 733	. . . . . . . . . . 11 |- ( ( ( f e. (Poly ` ZZ ) /\ ( (coeff ` f ) ` 0 ) =/= 0 ) /\ f = 0p ) -> -. ( (coeff ` f ) ` 0 ) = 0 )
39	36, 38	pm2.65da 580	. . . . . . . . . 10 |- ( ( f e. (Poly ` ZZ ) /\ ( (coeff ` f ) ` 0 ) =/= 0 ) -> -. f = 0p )
40		elsn 3982	. . . . . . . . . 10 |- ( f e. { 0p } <-> f = 0p )
41	39, 40	sylnibr 307	. . . . . . . . 9 |- ( ( f e. (Poly ` ZZ ) /\ ( (coeff ` f ) ` 0 ) =/= 0 ) -> -. f e. { 0p } )
42	27, 41	eldifd 3415	. . . . . . . 8 |- ( ( f e. (Poly ` ZZ ) /\ ( (coeff ` f ) ` 0 ) =/= 0 ) -> f e. ( (Poly ` ZZ ) \ { 0p } ) )
43	42	adantrr 723	. . . . . . 7 |- ( ( f e. (Poly ` ZZ ) /\ ( ( (coeff ` f ) ` 0 ) =/= 0 /\ ( f ` A ) = 0 ) ) -> f e. ( (Poly ` ZZ ) \ { 0p } ) )
44		simprr 766	. . . . . . 7 |- ( ( f e. (Poly ` ZZ ) /\ ( ( (coeff ` f ) ` 0 ) =/= 0 /\ ( f ` A ) = 0 ) ) -> ( f ` A ) = 0 )
45	43, 44	jca 535	. . . . . 6 |- ( ( f e. (Poly ` ZZ ) /\ ( ( (coeff ` f ) ` 0 ) =/= 0 /\ ( f ` A ) = 0 ) ) -> ( f e. ( (Poly ` ZZ ) \ { 0p } ) /\ ( f ` A ) = 0 ) )
46	45	reximi2 2854	. . . . 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 573	. . . 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 23269	. . . 4 |- ( A e. AA <-> ( A e. CC /\ E. f e. ( (Poly ` ZZ ) \ { 0p } ) ( f ` A ) = 0 ) )
49	47, 48	sylibr 216	. . 3 |- ( ( A e. CC /\ E. f e. (Poly ` ZZ ) ( ( (coeff ` f ) ` 0 ) =/= 0 /\ ( f ` A ) = 0 ) ) -> A e. AA )
50		simpr 463	. . . 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 1761	. . . . . 6 |- F/ f A e. CC
52		nfre1 2848	. . . . . 6 |- F/ f E. f e. (Poly ` ZZ ) ( ( (coeff ` f ) ` 0 ) =/= 0 /\ ( f ` A ) = 0 )
53	51, 52	nfan 2011	. . . . 5 |- F/ f ( A e. CC /\ E. f e. (Poly ` ZZ ) ( ( (coeff ` f ) ` 0 ) =/= 0 /\ ( f ` A ) = 0 ) )
54		nfv 1761	. . . . 5 |- F/ f -. A e. { 0 }
55		simpl3r 1064	. . . . . . . . 9 |- ( ( ( A e. CC /\ f e. (Poly ` ZZ ) /\ ( ( (coeff ` f ) ` 0 ) =/= 0 /\ ( f ` A ) = 0 ) ) /\ A = 0 ) -> ( f ` A ) = 0 )
56		fveq2 5865	. . . . . . . . . . . . . . 15 |- ( A = 0 -> ( f ` A ) = ( f ` 0 ) )
57		eqid 2451	. . . . . . . . . . . . . . . 16 |- (coeff ` f ) = (coeff ` f )
58	57	coefv0 23202	. . . . . . . . . . . . . . 15 |- ( f e. (Poly ` ZZ ) -> ( f ` 0 ) = ( (coeff ` f ) ` 0 ) )
59	56, 58	sylan9eqr 2507	. . . . . . . . . . . . . 14 |- ( ( f e. (Poly ` ZZ ) /\ A = 0 ) -> ( f ` A ) = ( (coeff ` f ) ` 0 ) )
60	59	adantlr 721	. . . . . . . . . . . . 13 |- ( ( ( f e. (Poly ` ZZ ) /\ ( (coeff ` f ) ` 0 ) =/= 0 ) /\ A = 0 ) -> ( f ` A ) = ( (coeff ` f ) ` 0 ) )
61		simplr 762	. . . . . . . . . . . . 13 |- ( ( ( f e. (Poly ` ZZ ) /\ ( (coeff ` f ) ` 0 ) =/= 0 ) /\ A = 0 ) -> ( (coeff ` f ) ` 0 ) =/= 0 )
62	60, 61	eqnetrd 2691	. . . . . . . . . . . 12 |- ( ( ( f e. (Poly ` ZZ ) /\ ( (coeff ` f ) ` 0 ) =/= 0 ) /\ A = 0 ) -> ( f ` A ) =/= 0 )
63	62	neneqd 2629	. . . . . . . . . . 11 |- ( ( ( f e. (Poly ` ZZ ) /\ ( (coeff ` f ) ` 0 ) =/= 0 ) /\ A = 0 ) -> -. ( f ` A ) = 0 )
64	63	adantlrr 727	. . . . . . . . . 10 |- ( ( ( f e. (Poly ` ZZ ) /\ ( ( (coeff ` f ) ` 0 ) =/= 0 /\ ( f ` A ) = 0 ) ) /\ A = 0 ) -> -. ( f ` A ) = 0 )
65	64	3adantl1 1164	. . . . . . . . 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 580	. . . . . . . 8 |- ( ( A e. CC /\ f e. (Poly ` ZZ ) /\ ( ( (coeff ` f ) ` 0 ) =/= 0 /\ ( f ` A ) = 0 ) ) -> -. A = 0 )
67		elsncg 3991	. . . . . . . . . 10 |- ( A e. CC -> ( A e. { 0 } <-> A = 0 ) )
68	67	biimpa 487	. . . . . . . . 9 |- ( ( A e. CC /\ A e. { 0 } ) -> A = 0 )
69	68	3ad2antl1 1170	. . . . . . . 8 |- ( ( ( A e. CC /\ f e. (Poly ` ZZ ) /\ ( ( (coeff ` f ) ` 0 ) =/= 0 /\ ( f ` A ) = 0 ) ) /\ A e. { 0 } ) -> A = 0 )
70	66, 69	mtand 665	. . . . . . 7 |- ( ( A e. CC /\ f e. (Poly ` ZZ ) /\ ( ( (coeff ` f ) ` 0 ) =/= 0 /\ ( f ` A ) = 0 ) ) -> -. A e. { 0 } )
71	70	3exp 1207	. . . . . 6 |- ( A e. CC -> ( f e. (Poly ` ZZ ) -> ( ( ( (coeff ` f ) ` 0 ) =/= 0 /\ ( f ` A ) = 0 ) -> -. A e. { 0 } ) ) )
72	71	adantr 467	. . . . 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 2871	. . . 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 3415	. 2 |- ( ( A e. CC /\ E. f e. (Poly ` ZZ ) ( ( (coeff ` f ) ` 0 ) =/= 0 /\ ( f ` A ) = 0 ) ) -> A e. ( AA \ { 0 } ) )
76	26, 75	impbii 191	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 188   /\ wa 371   /\ w3a 985   = wceq 1444   e. wcel 1887   =/= wne 2622   E.wrex 2738   {crab 2741   \ cdif 3401   {csn 3968   |-> cmpt 4461   X. cxp 4832   ` cfv 5582  (class class class)co 6290  infcinf 7955   CCcc 9537   RRcr 9538   0cc0 9539   + caddc 9542   x. cmul 9544   < clt 9675   - cmin 9860   NN0cn0 10869   ZZcz 10937   ...cfz 11784   ^cexp 12272   sum_csu 13752   0pc0p 22627  Polycply 23138  coeffccoe 23140  degcdgr 23141   AAcaa 23267

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-inf2 8146  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-pre-sup 9617  ax-addf 9618

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-fal 1450  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-se 4794  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-isom 5591  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-of 6531  df-om 6693  df-1st 6793  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-oadd 7186  df-er 7363  df-map 7474  df-pm 7475  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-sup 7956  df-inf 7957  df-oi 8025  df-card 8373  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-nn 10610  df-2 10668  df-3 10669  df-n0 10870  df-z 10938  df-uz 11160  df-rp 11303  df-fz 11785  df-fzo 11916  df-fl 12028  df-seq 12214  df-exp 12273  df-hash 12516  df-cj 13162  df-re 13163  df-im 13164  df-sqrt 13298  df-abs 13299  df-clim 13552  df-rlim 13553  df-sum 13753  df-0p 22628  df-ply 23142  df-coe 23144  df-dgr 23145  df-aa 23268

This theorem is referenced by:  etransc  38149