Theorem elqaalem3 23353

Description: Lemma for elqaa 23357. (Contributed by Mario Carneiro, 23-Jul-2014.) (Revised by AV, 3-Oct-2020.)

Hypotheses

Ref	Expression
elqaa.1	|- ( ph -> A e. CC )
elqaa.2	|- ( ph -> F e. ( (Poly ` QQ ) \ { 0p } ) )
elqaa.3	|- ( ph -> ( F ` A ) = 0 )
elqaa.4	|- B = (coeff ` F )
elqaa.5	|- N = ( k e. NN0 |-> inf ( { n e. NN | ( ( B ` k ) x. n ) e. ZZ } , RR , < ) )
elqaa.6	|- R = ( seq 0 ( x. , N ) ` (deg ` F ) )

Assertion

Ref	Expression
elqaalem3	|- ( ph -> A e. AA )

Distinct variable groups:   k, n, A   B, k, n   ph, k   k, N, n   R, k

Allowed substitution hints:   ph( n)   R( n)   F( k, n)

Proof of Theorem elqaalem3

Dummy variables f m x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elqaa.1	. 2 |- ( ph -> A e. CC )
2		cnex 9638	. . . . . . . 8 |- CC e. _V
3	2	a1i 11	. . . . . . 7 |- ( ph -> CC e. _V )
4		elqaa.6	. . . . . . . . 9 |- R = ( seq 0 ( x. , N ) ` (deg ` F ) )
5		fvex 5889	. . . . . . . . 9 |- ( seq 0 ( x. , N ) ` (deg ` F ) ) e. _V
6	4, 5	eqeltri 2545	. . . . . . . 8 |- R e. _V
7	6	a1i 11	. . . . . . 7 |- ( ( ph /\ z e. CC ) -> R e. _V )
8		fvex 5889	. . . . . . . 8 |- ( F ` z ) e. _V
9	8	a1i 11	. . . . . . 7 |- ( ( ph /\ z e. CC ) -> ( F ` z ) e. _V )
10		fconstmpt 4883	. . . . . . . 8 |- ( CC X. { R } ) = ( z e. CC |-> R )
11	10	a1i 11	. . . . . . 7 |- ( ph -> ( CC X. { R } ) = ( z e. CC |-> R ) )
12		elqaa.2	. . . . . . . . . 10 |- ( ph -> F e. ( (Poly ` QQ ) \ { 0p } ) )
13	12	eldifad 3402	. . . . . . . . 9 |- ( ph -> F e. (Poly ` QQ ) )
14		plyf 23231	. . . . . . . . 9 |- ( F e. (Poly ` QQ ) -> F : CC --> CC )
15	13, 14	syl 17	. . . . . . . 8 |- ( ph -> F : CC --> CC )
16	15	feqmptd 5932	. . . . . . 7 |- ( ph -> F = ( z e. CC |-> ( F ` z ) ) )
17	3, 7, 9, 11, 16	offval2 6567	. . . . . 6 |- ( ph -> ( ( CC X. { R } ) oF x. F ) = ( z e. CC |-> ( R x. ( F ` z ) ) ) )
18		fzfid 12224	. . . . . . . . 9 |- ( ( ph /\ z e. CC ) -> ( 0 ... (deg ` F ) ) e. Fin )
19		nn0uz 11217	. . . . . . . . . . . . . 14 |- NN0 = ( ZZ>= ` 0 )
20		0zd 10973	. . . . . . . . . . . . . 14 |- ( ph -> 0 e. ZZ )
21		ssrab2 3500	. . . . . . . . . . . . . . 15 |- { n e. NN | ( ( B ` m ) x. n ) e. ZZ } C_ NN
22		fveq2 5879	. . . . . . . . . . . . . . . . . . . . . 22 |- ( k = m -> ( B ` k ) = ( B ` m ) )
23	22	oveq1d 6323	. . . . . . . . . . . . . . . . . . . . 21 |- ( k = m -> ( ( B ` k ) x. n ) = ( ( B ` m ) x. n ) )
24	23	eleq1d 2533	. . . . . . . . . . . . . . . . . . . 20 |- ( k = m -> ( ( ( B ` k ) x. n ) e. ZZ <-> ( ( B ` m ) x. n ) e. ZZ ) )
25	24	rabbidv 3022	. . . . . . . . . . . . . . . . . . 19 |- ( k = m -> { n e. NN | ( ( B ` k ) x. n ) e. ZZ } = { n e. NN | ( ( B ` m ) x. n ) e. ZZ } )
26	25	infeq1d 8011	. . . . . . . . . . . . . . . . . 18 |- ( k = m -> inf ( { n e. NN | ( ( B ` k ) x. n ) e. ZZ } , RR , < ) = inf ( { n e. NN | ( ( B ` m ) x. n ) e. ZZ } , RR , < ) )
27		elqaa.5	. . . . . . . . . . . . . . . . . 18 |- N = ( k e. NN0 |-> inf ( { n e. NN | ( ( B ` k ) x. n ) e. ZZ } , RR , < ) )
28		ltso 9732	. . . . . . . . . . . . . . . . . . 19 |- < Or RR
29	28	infex 8027	. . . . . . . . . . . . . . . . . 18 |- inf ( { n e. NN | ( ( B ` m ) x. n ) e. ZZ } , RR , < ) e. _V
30	26, 27, 29	fvmpt 5963	. . . . . . . . . . . . . . . . 17 |- ( m e. NN0 -> ( N ` m ) = inf ( { n e. NN | ( ( B ` m ) x. n ) e. ZZ } , RR , < ) )
31	30	adantl 473	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ m e. NN0 ) -> ( N ` m ) = inf ( { n e. NN | ( ( B ` m ) x. n ) e. ZZ } , RR , < ) )
32		nnuz 11218	. . . . . . . . . . . . . . . . . 18 |- NN = ( ZZ>= ` 1 )
33	21, 32	sseqtri 3450	. . . . . . . . . . . . . . . . 17 |- { n e. NN | ( ( B ` m ) x. n ) e. ZZ } C_ ( ZZ>= ` 1 )
34		0z 10972	. . . . . . . . . . . . . . . . . . . . . 22 |- 0 e. ZZ
35		zq 11293	. . . . . . . . . . . . . . . . . . . . . 22 |- ( 0 e. ZZ -> 0 e. QQ )
36	34, 35	ax-mp 5	. . . . . . . . . . . . . . . . . . . . 21 |- 0 e. QQ
37		elqaa.4	. . . . . . . . . . . . . . . . . . . . . 22 |- B = (coeff ` F )
38	37	coef2 23264	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( F e. (Poly ` QQ ) /\ 0 e. QQ ) -> B : NN0 --> QQ )
39	13, 36, 38	sylancl 675	. . . . . . . . . . . . . . . . . . . 20 |- ( ph -> B : NN0 --> QQ )
40	39	ffvelrnda 6037	. . . . . . . . . . . . . . . . . . 19 |- ( ( ph /\ m e. NN0 ) -> ( B ` m ) e. QQ )
41		qmulz 11290	. . . . . . . . . . . . . . . . . . 19 |- ( ( B ` m ) e. QQ -> E. n e. NN ( ( B ` m ) x. n ) e. ZZ )
42	40, 41	syl 17	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ m e. NN0 ) -> E. n e. NN ( ( B ` m ) x. n ) e. ZZ )
43		rabn0 3755	. . . . . . . . . . . . . . . . . 18 |- ( { n e. NN | ( ( B ` m ) x. n ) e. ZZ } =/= (/) <-> E. n e. NN ( ( B ` m ) x. n ) e. ZZ )
44	42, 43	sylibr 217	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ m e. NN0 ) -> { n e. NN | ( ( B ` m ) x. n ) e. ZZ } =/= (/) )
45		infssuzcl 11268	. . . . . . . . . . . . . . . . 17 |- ( ( { n e. NN | ( ( B ` m ) x. n ) e. ZZ } C_ ( ZZ>= ` 1 ) /\ { n e. NN | ( ( B ` m ) x. n ) e. ZZ } =/= (/) ) -> inf ( { n e. NN | ( ( B ` m ) x. n ) e. ZZ } , RR , < ) e. { n e. NN | ( ( B ` m ) x. n ) e. ZZ } )
46	33, 44, 45	sylancr 676	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ m e. NN0 ) -> inf ( { n e. NN | ( ( B ` m ) x. n ) e. ZZ } , RR , < ) e. { n e. NN | ( ( B ` m ) x. n ) e. ZZ } )
47	31, 46	eqeltrd 2549	. . . . . . . . . . . . . . 15 |- ( ( ph /\ m e. NN0 ) -> ( N ` m ) e. { n e. NN | ( ( B ` m ) x. n ) e. ZZ } )
48	21, 47	sseldi 3416	. . . . . . . . . . . . . 14 |- ( ( ph /\ m e. NN0 ) -> ( N ` m ) e. NN )
49		nnmulcl 10654	. . . . . . . . . . . . . . 15 |- ( ( m e. NN /\ k e. NN ) -> ( m x. k ) e. NN )
50	49	adantl 473	. . . . . . . . . . . . . 14 |- ( ( ph /\ ( m e. NN /\ k e. NN ) ) -> ( m x. k ) e. NN )
51	19, 20, 48, 50	seqf 12272	. . . . . . . . . . . . 13 |- ( ph -> seq 0 ( x. , N ) : NN0 --> NN )
52		dgrcl 23266	. . . . . . . . . . . . . 14 |- ( F e. (Poly ` QQ ) -> (deg ` F ) e. NN0 )
53	13, 52	syl 17	. . . . . . . . . . . . 13 |- ( ph -> (deg ` F ) e. NN0 )
54	51, 53	ffvelrnd 6038	. . . . . . . . . . . 12 |- ( ph -> ( seq 0 ( x. , N ) ` (deg ` F ) ) e. NN )
55	4, 54	syl5eqel 2553	. . . . . . . . . . 11 |- ( ph -> R e. NN )
56	55	nncnd 10647	. . . . . . . . . 10 |- ( ph -> R e. CC )
57	56	adantr 472	. . . . . . . . 9 |- ( ( ph /\ z e. CC ) -> R e. CC )
58		elfznn0 11913	. . . . . . . . . 10 |- ( m e. ( 0 ... (deg ` F ) ) -> m e. NN0 )
59	37	coef3 23265	. . . . . . . . . . . . . 14 |- ( F e. (Poly ` QQ ) -> B : NN0 --> CC )
60	13, 59	syl 17	. . . . . . . . . . . . 13 |- ( ph -> B : NN0 --> CC )
61	60	adantr 472	. . . . . . . . . . . 12 |- ( ( ph /\ z e. CC ) -> B : NN0 --> CC )
62	61	ffvelrnda 6037	. . . . . . . . . . 11 |- ( ( ( ph /\ z e. CC ) /\ m e. NN0 ) -> ( B ` m ) e. CC )
63		expcl 12328	. . . . . . . . . . . 12 |- ( ( z e. CC /\ m e. NN0 ) -> ( z ^ m ) e. CC )
64	63	adantll 728	. . . . . . . . . . 11 |- ( ( ( ph /\ z e. CC ) /\ m e. NN0 ) -> ( z ^ m ) e. CC )
65	62, 64	mulcld 9681	. . . . . . . . . 10 |- ( ( ( ph /\ z e. CC ) /\ m e. NN0 ) -> ( ( B ` m ) x. ( z ^ m ) ) e. CC )
66	58, 65	sylan2 482	. . . . . . . . 9 |- ( ( ( ph /\ z e. CC ) /\ m e. ( 0 ... (deg ` F ) ) ) -> ( ( B ` m ) x. ( z ^ m ) ) e. CC )
67	18, 57, 66	fsummulc2 13922	. . . . . . . 8 |- ( ( ph /\ z e. CC ) -> ( R x. sum_ m e. ( 0 ... (deg ` F ) ) ( ( B ` m ) x. ( z ^ m ) ) ) = sum_ m e. ( 0 ... (deg ` F ) ) ( R x. ( ( B ` m ) x. ( z ^ m ) ) ) )
68		eqid 2471	. . . . . . . . . . 11 |- (deg ` F ) = (deg ` F )
69	37, 68	coeid2 23272	. . . . . . . . . 10 |- ( ( F e. (Poly ` QQ ) /\ z e. CC ) -> ( F ` z ) = sum_ m e. ( 0 ... (deg ` F ) ) ( ( B ` m ) x. ( z ^ m ) ) )
70	13, 69	sylan 479	. . . . . . . . 9 |- ( ( ph /\ z e. CC ) -> ( F ` z ) = sum_ m e. ( 0 ... (deg ` F ) ) ( ( B ` m ) x. ( z ^ m ) ) )
71	70	oveq2d 6324	. . . . . . . 8 |- ( ( ph /\ z e. CC ) -> ( R x. ( F ` z ) ) = ( R x. sum_ m e. ( 0 ... (deg ` F ) ) ( ( B ` m ) x. ( z ^ m ) ) ) )
72	57	adantr 472	. . . . . . . . . . 11 |- ( ( ( ph /\ z e. CC ) /\ m e. NN0 ) -> R e. CC )
73	72, 62, 64	mulassd 9684	. . . . . . . . . 10 |- ( ( ( ph /\ z e. CC ) /\ m e. NN0 ) -> ( ( R x. ( B ` m ) ) x. ( z ^ m ) ) = ( R x. ( ( B ` m ) x. ( z ^ m ) ) ) )
74	58, 73	sylan2 482	. . . . . . . . 9 |- ( ( ( ph /\ z e. CC ) /\ m e. ( 0 ... (deg ` F ) ) ) -> ( ( R x. ( B ` m ) ) x. ( z ^ m ) ) = ( R x. ( ( B ` m ) x. ( z ^ m ) ) ) )
75	74	sumeq2dv 13846	. . . . . . . 8 |- ( ( ph /\ z e. CC ) -> sum_ m e. ( 0 ... (deg ` F ) ) ( ( R x. ( B ` m ) ) x. ( z ^ m ) ) = sum_ m e. ( 0 ... (deg ` F ) ) ( R x. ( ( B ` m ) x. ( z ^ m ) ) ) )
76	67, 71, 75	3eqtr4d 2515	. . . . . . 7 |- ( ( ph /\ z e. CC ) -> ( R x. ( F ` z ) ) = sum_ m e. ( 0 ... (deg ` F ) ) ( ( R x. ( B ` m ) ) x. ( z ^ m ) ) )
77	76	mpteq2dva 4482	. . . . . 6 |- ( ph -> ( z e. CC |-> ( R x. ( F ` z ) ) ) = ( z e. CC |-> sum_ m e. ( 0 ... (deg ` F ) ) ( ( R x. ( B ` m ) ) x. ( z ^ m ) ) ) )
78	17, 77	eqtrd 2505	. . . . 5 |- ( ph -> ( ( CC X. { R } ) oF x. F ) = ( z e. CC |-> sum_ m e. ( 0 ... (deg ` F ) ) ( ( R x. ( B ` m ) ) x. ( z ^ m ) ) ) )
79		zsscn 10969	. . . . . . 7 |- ZZ C_ CC
80	79	a1i 11	. . . . . 6 |- ( ph -> ZZ C_ CC )
81	56	adantr 472	. . . . . . . . . . 11 |- ( ( ph /\ m e. NN0 ) -> R e. CC )
82	48	nncnd 10647	. . . . . . . . . . 11 |- ( ( ph /\ m e. NN0 ) -> ( N ` m ) e. CC )
83	48	nnne0d 10676	. . . . . . . . . . 11 |- ( ( ph /\ m e. NN0 ) -> ( N ` m ) =/= 0 )
84	81, 82, 83	divcan2d 10407	. . . . . . . . . 10 |- ( ( ph /\ m e. NN0 ) -> ( ( N ` m ) x. ( R / ( N ` m ) ) ) = R )
85	84	oveq2d 6324	. . . . . . . . 9 |- ( ( ph /\ m e. NN0 ) -> ( ( B ` m ) x. ( ( N ` m ) x. ( R / ( N ` m ) ) ) ) = ( ( B ` m ) x. R ) )
86	60	ffvelrnda 6037	. . . . . . . . . 10 |- ( ( ph /\ m e. NN0 ) -> ( B ` m ) e. CC )
87	81, 82, 83	divcld 10405	. . . . . . . . . 10 |- ( ( ph /\ m e. NN0 ) -> ( R / ( N ` m ) ) e. CC )
88	86, 82, 87	mulassd 9684	. . . . . . . . 9 |- ( ( ph /\ m e. NN0 ) -> ( ( ( B ` m ) x. ( N ` m ) ) x. ( R / ( N ` m ) ) ) = ( ( B ` m ) x. ( ( N ` m ) x. ( R / ( N ` m ) ) ) ) )
89	81, 86	mulcomd 9682	. . . . . . . . 9 |- ( ( ph /\ m e. NN0 ) -> ( R x. ( B ` m ) ) = ( ( B ` m ) x. R ) )
90	85, 88, 89	3eqtr4rd 2516	. . . . . . . 8 |- ( ( ph /\ m e. NN0 ) -> ( R x. ( B ` m ) ) = ( ( ( B ` m ) x. ( N ` m ) ) x. ( R / ( N ` m ) ) ) )
91	58, 90	sylan2 482	. . . . . . 7 |- ( ( ph /\ m e. ( 0 ... (deg ` F ) ) ) -> ( R x. ( B ` m ) ) = ( ( ( B ` m ) x. ( N ` m ) ) x. ( R / ( N ` m ) ) ) )
92		oveq2 6316	. . . . . . . . . . . . 13 |- ( n = ( N ` m ) -> ( ( B ` m ) x. n ) = ( ( B ` m ) x. ( N ` m ) ) )
93	92	eleq1d 2533	. . . . . . . . . . . 12 |- ( n = ( N ` m ) -> ( ( ( B ` m ) x. n ) e. ZZ <-> ( ( B ` m ) x. ( N ` m ) ) e. ZZ ) )
94	93	elrab 3184	. . . . . . . . . . 11 |- ( ( N ` m ) e. { n e. NN | ( ( B ` m ) x. n ) e. ZZ } <-> ( ( N ` m ) e. NN /\ ( ( B ` m ) x. ( N ` m ) ) e. ZZ ) )
95	94	simprbi 471	. . . . . . . . . 10 |- ( ( N ` m ) e. { n e. NN | ( ( B ` m ) x. n ) e. ZZ } -> ( ( B ` m ) x. ( N ` m ) ) e. ZZ )
96	47, 95	syl 17	. . . . . . . . 9 |- ( ( ph /\ m e. NN0 ) -> ( ( B ` m ) x. ( N ` m ) ) e. ZZ )
97	58, 96	sylan2 482	. . . . . . . 8 |- ( ( ph /\ m e. ( 0 ... (deg ` F ) ) ) -> ( ( B ` m ) x. ( N ` m ) ) e. ZZ )
98		elqaa.3	. . . . . . . . . 10 |- ( ph -> ( F ` A ) = 0 )
99		eqid 2471	. . . . . . . . . 10 |- ( x e. _V , y e. _V |-> ( ( x x. y ) mod ( N ` m ) ) ) = ( x e. _V , y e. _V |-> ( ( x x. y ) mod ( N ` m ) ) )
100	1, 12, 98, 37, 27, 4, 99	elqaalem2 23352	. . . . . . . . 9 |- ( ( ph /\ m e. ( 0 ... (deg ` F ) ) ) -> ( R mod ( N ` m ) ) = 0 )
101	55	adantr 472	. . . . . . . . . 10 |- ( ( ph /\ m e. ( 0 ... (deg ` F ) ) ) -> R e. NN )
102	58, 48	sylan2 482	. . . . . . . . . 10 |- ( ( ph /\ m e. ( 0 ... (deg ` F ) ) ) -> ( N ` m ) e. NN )
103		nnre 10638	. . . . . . . . . . 11 |- ( R e. NN -> R e. RR )
104		nnrp 11334	. . . . . . . . . . 11 |- ( ( N ` m ) e. NN -> ( N ` m ) e. RR+ )
105		mod0 12136	. . . . . . . . . . 11 |- ( ( R e. RR /\ ( N ` m ) e. RR+ ) -> ( ( R mod ( N ` m ) ) = 0 <-> ( R / ( N ` m ) ) e. ZZ ) )
106	103, 104, 105	syl2an 485	. . . . . . . . . 10 |- ( ( R e. NN /\ ( N ` m ) e. NN ) -> ( ( R mod ( N ` m ) ) = 0 <-> ( R / ( N ` m ) ) e. ZZ ) )
107	101, 102, 106	syl2anc 673	. . . . . . . . 9 |- ( ( ph /\ m e. ( 0 ... (deg ` F ) ) ) -> ( ( R mod ( N ` m ) ) = 0 <-> ( R / ( N ` m ) ) e. ZZ ) )
108	100, 107	mpbid 215	. . . . . . . 8 |- ( ( ph /\ m e. ( 0 ... (deg ` F ) ) ) -> ( R / ( N ` m ) ) e. ZZ )
109	97, 108	zmulcld 11069	. . . . . . 7 |- ( ( ph /\ m e. ( 0 ... (deg ` F ) ) ) -> ( ( ( B ` m ) x. ( N ` m ) ) x. ( R / ( N ` m ) ) ) e. ZZ )
110	91, 109	eqeltrd 2549	. . . . . 6 |- ( ( ph /\ m e. ( 0 ... (deg ` F ) ) ) -> ( R x. ( B ` m ) ) e. ZZ )
111	80, 53, 110	elplyd 23235	. . . . 5 |- ( ph -> ( z e. CC |-> sum_ m e. ( 0 ... (deg ` F ) ) ( ( R x. ( B ` m ) ) x. ( z ^ m ) ) ) e. (Poly ` ZZ ) )
112	78, 111	eqeltrd 2549	. . . 4 |- ( ph -> ( ( CC X. { R } ) oF x. F ) e. (Poly ` ZZ ) )
113		eldifsn 4088	. . . . . . 7 |- ( F e. ( (Poly ` QQ ) \ { 0p } ) <-> ( F e. (Poly ` QQ ) /\ F =/= 0p ) )
114	12, 113	sylib 201	. . . . . 6 |- ( ph -> ( F e. (Poly ` QQ ) /\ F =/= 0p ) )
115	114	simprd 470	. . . . 5 |- ( ph -> F =/= 0p )
116		oveq1 6315	. . . . . . 7 |- ( ( ( CC X. { R } ) oF x. F ) = 0p -> ( ( ( CC X. { R } ) oF x. F ) oF / ( CC X. { R } ) ) = ( 0p oF / ( CC X. { R } ) ) )
117	15	ffvelrnda 6037	. . . . . . . . . . 11 |- ( ( ph /\ z e. CC ) -> ( F ` z ) e. CC )
118	55	nnne0d 10676	. . . . . . . . . . . 12 |- ( ph -> R =/= 0 )
119	118	adantr 472	. . . . . . . . . . 11 |- ( ( ph /\ z e. CC ) -> R =/= 0 )
120	117, 57, 119	divcan3d 10410	. . . . . . . . . 10 |- ( ( ph /\ z e. CC ) -> ( ( R x. ( F ` z ) ) / R ) = ( F ` z ) )
121	120	mpteq2dva 4482	. . . . . . . . 9 |- ( ph -> ( z e. CC |-> ( ( R x. ( F ` z ) ) / R ) ) = ( z e. CC |-> ( F ` z ) ) )
122		ovex 6336	. . . . . . . . . . 11 |- ( R x. ( F ` z ) ) e. _V
123	122	a1i 11	. . . . . . . . . 10 |- ( ( ph /\ z e. CC ) -> ( R x. ( F ` z ) ) e. _V )
124	3, 123, 7, 17, 11	offval2 6567	. . . . . . . . 9 |- ( ph -> ( ( ( CC X. { R } ) oF x. F ) oF / ( CC X. { R } ) ) = ( z e. CC |-> ( ( R x. ( F ` z ) ) / R ) ) )
125	121, 124, 16	3eqtr4d 2515	. . . . . . . 8 |- ( ph -> ( ( ( CC X. { R } ) oF x. F ) oF / ( CC X. { R } ) ) = F )
126	56, 118	div0d 10404	. . . . . . . . . 10 |- ( ph -> ( 0 / R ) = 0 )
127	126	mpteq2dv 4483	. . . . . . . . 9 |- ( ph -> ( z e. CC |-> ( 0 / R ) ) = ( z e. CC |-> 0 ) )
128		0cnd 9654	. . . . . . . . . 10 |- ( ( ph /\ z e. CC ) -> 0 e. CC )
129		df-0p 22707	. . . . . . . . . . . 12 |- 0p = ( CC X. { 0 } )
130		fconstmpt 4883	. . . . . . . . . . . 12 |- ( CC X. { 0 } ) = ( z e. CC |-> 0 )
131	129, 130	eqtri 2493	. . . . . . . . . . 11 |- 0p = ( z e. CC |-> 0 )
132	131	a1i 11	. . . . . . . . . 10 |- ( ph -> 0p = ( z e. CC |-> 0 ) )
133	3, 128, 7, 132, 11	offval2 6567	. . . . . . . . 9 |- ( ph -> ( 0p oF / ( CC X. { R } ) ) = ( z e. CC |-> ( 0 / R ) ) )
134	127, 133, 132	3eqtr4d 2515	. . . . . . . 8 |- ( ph -> ( 0p oF / ( CC X. { R } ) ) = 0p )
135	125, 134	eqeq12d 2486	. . . . . . 7 |- ( ph -> ( ( ( ( CC X. { R } ) oF x. F ) oF / ( CC X. { R } ) ) = ( 0p oF / ( CC X. { R } ) ) <-> F = 0p ) )
136	116, 135	syl5ib 227	. . . . . 6 |- ( ph -> ( ( ( CC X. { R } ) oF x. F ) = 0p -> F = 0p ) )
137	136	necon3d 2664	. . . . 5 |- ( ph -> ( F =/= 0p -> ( ( CC X. { R } ) oF x. F ) =/= 0p ) )
138	115, 137	mpd 15	. . . 4 |- ( ph -> ( ( CC X. { R } ) oF x. F ) =/= 0p )
139		eldifsn 4088	. . . 4 |- ( ( ( CC X. { R } ) oF x. F ) e. ( (Poly ` ZZ ) \ { 0p } ) <-> ( ( ( CC X. { R } ) oF x. F ) e. (Poly ` ZZ ) /\ ( ( CC X. { R } ) oF x. F ) =/= 0p ) )
140	112, 138, 139	sylanbrc 677	. . 3 |- ( ph -> ( ( CC X. { R } ) oF x. F ) e. ( (Poly ` ZZ ) \ { 0p } ) )
141	6	fconst 5782	. . . . . . 7 |- ( CC X. { R } ) : CC --> { R }
142		ffn 5739	. . . . . . 7 |- ( ( CC X. { R } ) : CC --> { R } -> ( CC X. { R } ) Fn CC )
143	141, 142	mp1i 13	. . . . . 6 |- ( ph -> ( CC X. { R } ) Fn CC )
144		ffn 5739	. . . . . . 7 |- ( F : CC --> CC -> F Fn CC )
145	15, 144	syl 17	. . . . . 6 |- ( ph -> F Fn CC )
146		inidm 3632	. . . . . 6 |- ( CC i^i CC ) = CC
147	6	fvconst2 6136	. . . . . . 7 |- ( A e. CC -> ( ( CC X. { R } ) ` A ) = R )
148	147	adantl 473	. . . . . 6 |- ( ( ph /\ A e. CC ) -> ( ( CC X. { R } ) ` A ) = R )
149	98	adantr 472	. . . . . 6 |- ( ( ph /\ A e. CC ) -> ( F ` A ) = 0 )
150	143, 145, 3, 3, 146, 148, 149	ofval 6559	. . . . 5 |- ( ( ph /\ A e. CC ) -> ( ( ( CC X. { R } ) oF x. F ) ` A ) = ( R x. 0 ) )
151	1, 150	mpdan 681	. . . 4 |- ( ph -> ( ( ( CC X. { R } ) oF x. F ) ` A ) = ( R x. 0 ) )
152	56	mul01d 9850	. . . 4 |- ( ph -> ( R x. 0 ) = 0 )
153	151, 152	eqtrd 2505	. . 3 |- ( ph -> ( ( ( CC X. { R } ) oF x. F ) ` A ) = 0 )
154		fveq1 5878	. . . . 5 |- ( f = ( ( CC X. { R } ) oF x. F ) -> ( f ` A ) = ( ( ( CC X. { R } ) oF x. F ) ` A ) )
155	154	eqeq1d 2473	. . . 4 |- ( f = ( ( CC X. { R } ) oF x. F ) -> ( ( f ` A ) = 0 <-> ( ( ( CC X. { R } ) oF x. F ) ` A ) = 0 ) )
156	155	rspcev 3136	. . 3 |- ( ( ( ( CC X. { R } ) oF x. F ) e. ( (Poly ` ZZ ) \ { 0p } ) /\ ( ( ( CC X. { R } ) oF x. F ) ` A ) = 0 ) -> E. f e. ( (Poly ` ZZ ) \ { 0p } ) ( f ` A ) = 0 )
157	140, 153, 156	syl2anc 673	. 2 |- ( ph -> E. f e. ( (Poly ` ZZ ) \ { 0p } ) ( f ` A ) = 0 )
158		elaa 23348	. 2 |- ( A e. AA <-> ( A e. CC /\ E. f e. ( (Poly ` ZZ ) \ { 0p } ) ( f ` A ) = 0 ) )
159	1, 157, 158	sylanbrc 677	1 |- ( ph -> A e. AA )

Syntax hints:   -> wi 4   <-> wb 189   /\ wa 376   = wceq 1452   e. wcel 1904   =/= wne 2641   E.wrex 2757   {crab 2760   _Vcvv 3031   \ cdif 3387   C_ wss 3390   (/)c0 3722   {csn 3959   |-> cmpt 4454   X. cxp 4837   Fn wfn 5584   -->wf 5585   ` cfv 5589  (class class class)co 6308   |-> cmpt2 6310   oFcof 6548  infcinf 7973   CCcc 9555   RRcr 9556   0cc0 9557   1c1 9558   x. cmul 9562   < clt 9693   / cdiv 10291   NNcn 10631   NN0cn0 10893   ZZcz 10961   ZZ>=cuz 11182   QQcq 11287   RR+crp 11325   ...cfz 11810   mod cmo 12129   seqcseq 12251   ^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-q 11288  df-rp 11326  df-fz 11811  df-fzo 11943  df-fl 12061  df-mod 12130  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:  elqaa  23357