Theorem elaa2 38211

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 23350	. . . 4 |- AA C_ CC
2		eldifi 3544	. . . 4 |- ( A e. ( AA \ { 0 } ) -> A e. AA )
3	1, 2	sseldi 3416	. . 3 |- ( A e. ( AA \ { 0 } ) -> A e. CC )
4		elaa 23348	. . . . . 6 |- ( A e. AA <-> ( A e. CC /\ E. g e. ( (Poly ` ZZ ) \ { 0p } ) ( g ` A ) = 0 ) )
5	2, 4	sylib 201	. . . . 5 |- ( A e. ( AA \ { 0 } ) -> ( A e. CC /\ E. g e. ( (Poly ` ZZ ) \ { 0p } ) ( g ` A ) = 0 ) )
6	5	simprd 470	. . . 4 |- ( A e. ( AA \ { 0 } ) -> E. g e. ( (Poly ` ZZ ) \ { 0p } ) ( g ` A ) = 0 )
7	2	3ad2ant1 1051	. . . . . 6 |- ( ( A e. ( AA \ { 0 } ) /\ g e. ( (Poly ` ZZ ) \ { 0p } ) /\ ( g ` A ) = 0 ) -> A e. AA )
8		eldifsni 4089	. . . . . . 7 |- ( A e. ( AA \ { 0 } ) -> A =/= 0 )
9	8	3ad2ant1 1051	. . . . . 6 |- ( ( A e. ( AA \ { 0 } ) /\ g e. ( (Poly ` ZZ ) \ { 0p } ) /\ ( g ` A ) = 0 ) -> A =/= 0 )
10		eldifi 3544	. . . . . . 7 |- ( g e. ( (Poly ` ZZ ) \ { 0p } ) -> g e. (Poly ` ZZ ) )
11	10	3ad2ant2 1052	. . . . . 6 |- ( ( A e. ( AA \ { 0 } ) /\ g e. ( (Poly ` ZZ ) \ { 0p } ) /\ ( g ` A ) = 0 ) -> g e. (Poly ` ZZ ) )
12		eldifsni 4089	. . . . . . 7 |- ( g e. ( (Poly ` ZZ ) \ { 0p } ) -> g =/= 0p )
13	12	3ad2ant2 1052	. . . . . 6 |- ( ( A e. ( AA \ { 0 } ) /\ g e. ( (Poly ` ZZ ) \ { 0p } ) /\ ( g ` A ) = 0 ) -> g =/= 0p )
14		simp3 1032	. . . . . 6 |- ( ( A e. ( AA \ { 0 } ) /\ g e. ( (Poly ` ZZ ) \ { 0p } ) /\ ( g ` A ) = 0 ) -> ( g ` A ) = 0 )
15		fveq2 5879	. . . . . . . . 9 |- ( m = n -> ( (coeff ` g ) ` m ) = ( (coeff ` g ) ` n ) )
16	15	neeq1d 2702	. . . . . . . 8 |- ( m = n -> ( ( (coeff ` g ) ` m ) =/= 0 <-> ( (coeff ` g ) ` n ) =/= 0 ) )
17	16	cbvrabv 3030	. . . . . . 7 |- { m e. NN0 | ( (coeff ` g ) ` m ) =/= 0 } = { n e. NN0 | ( (coeff ` g ) ` n ) =/= 0 }
18	17	infeq1i 8012	. . . . . 6 |- inf ( { m e. NN0 | ( (coeff ` g ) ` m ) =/= 0 } , RR , < ) = inf ( { n e. NN0 | ( (coeff ` g ) ` n ) =/= 0 } , RR , < )
19		oveq1 6315	. . . . . . . 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 5883	. . . . . . 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 4488	. . . . . 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 2471	. . . . . 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 38209	. . . . 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 2873	. . . 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 541	. 2 |- ( A e. ( AA \ { 0 } ) -> ( A e. CC /\ E. f e. (Poly ` ZZ ) ( ( (coeff ` f ) ` 0 ) =/= 0 /\ ( f ` A ) = 0 ) ) )
27		simpl 464	. . . . . . . . 9 |- ( ( f e. (Poly ` ZZ ) /\ ( (coeff ` f ) ` 0 ) =/= 0 ) -> f e. (Poly ` ZZ ) )
28		fveq2 5879	. . . . . . . . . . . . . . 15 |- ( f = 0p -> (coeff ` f ) = (coeff ` 0p ) )
29		coe0 23289	. . . . . . . . . . . . . . 15 |- (coeff ` 0p ) = ( NN0 X. { 0 } )
30	28, 29	syl6eq 2521	. . . . . . . . . . . . . 14 |- ( f = 0p -> (coeff ` f ) = ( NN0 X. { 0 } ) )
31	30	fveq1d 5881	. . . . . . . . . . . . 13 |- ( f = 0p -> ( (coeff ` f ) ` 0 ) = ( ( NN0 X. { 0 } ) ` 0 ) )
32		0nn0 10908	. . . . . . . . . . . . . 14 |- 0 e. NN0
33		fvconst2g 6134	. . . . . . . . . . . . . 14 |- ( ( 0 e. NN0 /\ 0 e. NN0 ) -> ( ( NN0 X. { 0 } ) ` 0 ) = 0 )
34	32, 32, 33	mp2an 686	. . . . . . . . . . . . 13 |- ( ( NN0 X. { 0 } ) ` 0 ) = 0
35	31, 34	syl6eq 2521	. . . . . . . . . . . 12 |- ( f = 0p -> ( (coeff ` f ) ` 0 ) = 0 )
36	35	adantl 473	. . . . . . . . . . 11 |- ( ( ( f e. (Poly ` ZZ ) /\ ( (coeff ` f ) ` 0 ) =/= 0 ) /\ f = 0p ) -> ( (coeff ` f ) ` 0 ) = 0 )
37		neneq 2649	. . . . . . . . . . . 12 |- ( ( (coeff ` f ) ` 0 ) =/= 0 -> -. ( (coeff ` f ) ` 0 ) = 0 )
38	37	ad2antlr 741	. . . . . . . . . . 11 |- ( ( ( f e. (Poly ` ZZ ) /\ ( (coeff ` f ) ` 0 ) =/= 0 ) /\ f = 0p ) -> -. ( (coeff ` f ) ` 0 ) = 0 )
39	36, 38	pm2.65da 586	. . . . . . . . . 10 |- ( ( f e. (Poly ` ZZ ) /\ ( (coeff ` f ) ` 0 ) =/= 0 ) -> -. f = 0p )
40		elsn 3973	. . . . . . . . . 10 |- ( f e. { 0p } <-> f = 0p )
41	39, 40	sylnibr 312	. . . . . . . . 9 |- ( ( f e. (Poly ` ZZ ) /\ ( (coeff ` f ) ` 0 ) =/= 0 ) -> -. f e. { 0p } )
42	27, 41	eldifd 3401	. . . . . . . 8 |- ( ( f e. (Poly ` ZZ ) /\ ( (coeff ` f ) ` 0 ) =/= 0 ) -> f e. ( (Poly ` ZZ ) \ { 0p } ) )
43	42	adantrr 731	. . . . . . 7 |- ( ( f e. (Poly ` ZZ ) /\ ( ( (coeff ` f ) ` 0 ) =/= 0 /\ ( f ` A ) = 0 ) ) -> f e. ( (Poly ` ZZ ) \ { 0p } ) )
44		simprr 774	. . . . . . 7 |- ( ( f e. (Poly ` ZZ ) /\ ( ( (coeff ` f ) ` 0 ) =/= 0 /\ ( f ` A ) = 0 ) ) -> ( f ` A ) = 0 )
45	43, 44	jca 541	. . . . . 6 |- ( ( f e. (Poly ` ZZ ) /\ ( ( (coeff ` f ) ` 0 ) =/= 0 /\ ( f ` A ) = 0 ) ) -> ( f e. ( (Poly ` ZZ ) \ { 0p } ) /\ ( f ` A ) = 0 ) )
46	45	reximi2 2851	. . . . 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 579	. . . 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 23348	. . . 4 |- ( A e. AA <-> ( A e. CC /\ E. f e. ( (Poly ` ZZ ) \ { 0p } ) ( f ` A ) = 0 ) )
49	47, 48	sylibr 217	. . 3 |- ( ( A e. CC /\ E. f e. (Poly ` ZZ ) ( ( (coeff ` f ) ` 0 ) =/= 0 /\ ( f ` A ) = 0 ) ) -> A e. AA )
50		simpr 468	. . . 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 1769	. . . . . 6 |- F/ f A e. CC
52		nfre1 2846	. . . . . 6 |- F/ f E. f e. (Poly ` ZZ ) ( ( (coeff ` f ) ` 0 ) =/= 0 /\ ( f ` A ) = 0 )
53	51, 52	nfan 2031	. . . . 5 |- F/ f ( A e. CC /\ E. f e. (Poly ` ZZ ) ( ( (coeff ` f ) ` 0 ) =/= 0 /\ ( f ` A ) = 0 ) )
54		nfv 1769	. . . . 5 |- F/ f -. A e. { 0 }
55		simpl3r 1086	. . . . . . . . 9 |- ( ( ( A e. CC /\ f e. (Poly ` ZZ ) /\ ( ( (coeff ` f ) ` 0 ) =/= 0 /\ ( f ` A ) = 0 ) ) /\ A = 0 ) -> ( f ` A ) = 0 )
56		fveq2 5879	. . . . . . . . . . . . . . 15 |- ( A = 0 -> ( f ` A ) = ( f ` 0 ) )
57		eqid 2471	. . . . . . . . . . . . . . . 16 |- (coeff ` f ) = (coeff ` f )
58	57	coefv0 23281	. . . . . . . . . . . . . . 15 |- ( f e. (Poly ` ZZ ) -> ( f ` 0 ) = ( (coeff ` f ) ` 0 ) )
59	56, 58	sylan9eqr 2527	. . . . . . . . . . . . . 14 |- ( ( f e. (Poly ` ZZ ) /\ A = 0 ) -> ( f ` A ) = ( (coeff ` f ) ` 0 ) )
60	59	adantlr 729	. . . . . . . . . . . . 13 |- ( ( ( f e. (Poly ` ZZ ) /\ ( (coeff ` f ) ` 0 ) =/= 0 ) /\ A = 0 ) -> ( f ` A ) = ( (coeff ` f ) ` 0 ) )
61		simplr 770	. . . . . . . . . . . . 13 |- ( ( ( f e. (Poly ` ZZ ) /\ ( (coeff ` f ) ` 0 ) =/= 0 ) /\ A = 0 ) -> ( (coeff ` f ) ` 0 ) =/= 0 )
62	60, 61	eqnetrd 2710	. . . . . . . . . . . 12 |- ( ( ( f e. (Poly ` ZZ ) /\ ( (coeff ` f ) ` 0 ) =/= 0 ) /\ A = 0 ) -> ( f ` A ) =/= 0 )
63	62	neneqd 2648	. . . . . . . . . . 11 |- ( ( ( f e. (Poly ` ZZ ) /\ ( (coeff ` f ) ` 0 ) =/= 0 ) /\ A = 0 ) -> -. ( f ` A ) = 0 )
64	63	adantlrr 735	. . . . . . . . . 10 |- ( ( ( f e. (Poly ` ZZ ) /\ ( ( (coeff ` f ) ` 0 ) =/= 0 /\ ( f ` A ) = 0 ) ) /\ A = 0 ) -> -. ( f ` A ) = 0 )
65	64	3adantl1 1186	. . . . . . . . 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 586	. . . . . . . 8 |- ( ( A e. CC /\ f e. (Poly ` ZZ ) /\ ( ( (coeff ` f ) ` 0 ) =/= 0 /\ ( f ` A ) = 0 ) ) -> -. A = 0 )
67		elsncg 3983	. . . . . . . . . 10 |- ( A e. CC -> ( A e. { 0 } <-> A = 0 ) )
68	67	biimpa 492	. . . . . . . . 9 |- ( ( A e. CC /\ A e. { 0 } ) -> A = 0 )
69	68	3ad2antl1 1192	. . . . . . . 8 |- ( ( ( A e. CC /\ f e. (Poly ` ZZ ) /\ ( ( (coeff ` f ) ` 0 ) =/= 0 /\ ( f ` A ) = 0 ) ) /\ A e. { 0 } ) -> A = 0 )
70	66, 69	mtand 671	. . . . . . 7 |- ( ( A e. CC /\ f e. (Poly ` ZZ ) /\ ( ( (coeff ` f ) ` 0 ) =/= 0 /\ ( f ` A ) = 0 ) ) -> -. A e. { 0 } )
71	70	3exp 1230	. . . . . 6 |- ( A e. CC -> ( f e. (Poly ` ZZ ) -> ( ( ( (coeff ` f ) ` 0 ) =/= 0 /\ ( f ` A ) = 0 ) -> -. A e. { 0 } ) ) )
72	71	adantr 472	. . . . 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 2866	. . . 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 3401	. 2 |- ( ( A e. CC /\ E. f e. (Poly ` ZZ ) ( ( (coeff ` f ) ` 0 ) =/= 0 /\ ( f ` A ) = 0 ) ) -> A e. ( AA \ { 0 } ) )
76	26, 75	impbii 192	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 189   /\ wa 376   /\ w3a 1007   = wceq 1452   e. wcel 1904   =/= wne 2641   E.wrex 2757   {crab 2760   \ cdif 3387   {csn 3959   |-> cmpt 4454   X. cxp 4837   ` cfv 5589  (class class class)co 6308  infcinf 7973   CCcc 9555   RRcr 9556   0cc0 9557   + caddc 9560   x. cmul 9562   < clt 9693   - cmin 9880   NN0cn0 10893   ZZcz 10961   ...cfz 11810   ^cexp 12310   sum_csu 13829   0pc0p 22706  Polycply 23217  coeffccoe 23219  degcdgr 23220   AAcaa 23346

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635  ax-addf 9636

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-of 6550  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-map 7492  df-pm 7493  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-sup 7974  df-inf 7975  df-oi 8043  df-card 8391  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-n0 10894  df-z 10962  df-uz 11183  df-rp 11326  df-fz 11811  df-fzo 11943  df-fl 12061  df-seq 12252  df-exp 12311  df-hash 12554  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-clim 13629  df-rlim 13630  df-sum 13830  df-0p 22707  df-ply 23221  df-coe 23223  df-dgr 23224  df-aa 23347

This theorem is referenced by:  etransc  38261