Theorem etransclem47 40498

Description: _e is transcendental. Section *5 of [Juillerat] p. 11 can be used as a reference for this proof. (Contributed by Glauco Siliprandi, 5-Apr-2020.)

Hypotheses

Ref	Expression
etransclem47.q	|- ( ph -> Q e. ( (Poly ` ZZ ) \ { 0p } ) )
etransclem47.qe0	|- ( ph -> ( Q ` _e ) = 0 )
etransclem47.a	|- A = (coeff ` Q )
etransclem47.a0	|- ( ph -> ( A ` 0 ) =/= 0 )
etransclem47.m	|- M = (deg ` Q )
etransclem47.p	|- ( ph -> P e. Prime )
etransclem47.ap	|- ( ph -> ( abs ` ( A ` 0 ) ) < P )
etransclem47.mp	|- ( ph -> ( ! ` M ) < P )
etransclem47.9	|- ( ph -> ( sum_ j e. ( 0 ... M ) ( ( abs ` ( ( A ` j ) x. ( _e ^c j ) ) ) x. ( M x. ( M ^ ( M + 1 ) ) ) ) x. ( ( ( M ^ ( M + 1 ) ) ^ ( P - 1 ) ) / ( ! ` ( P - 1 ) ) ) ) < 1 )
etransclem47.f	|- F = ( x e. RR |-> ( ( x ^ ( P - 1 ) ) x. prod_ j e. ( 1 ... M ) ( ( x - j ) ^ P ) ) )
etransclem47.l	|- L = sum_ j e. ( 0 ... M ) ( ( ( A ` j ) x. ( _e ^c j ) ) x. S. ( 0 (,) j ) ( ( _e ^c -u x ) x. ( F ` x ) ) _d x )
etransclem47.k	|- K = ( L / ( ! ` ( P - 1 ) ) )

Assertion

Ref	Expression
etransclem47	|- ( ph -> E. k e. ZZ ( k =/= 0 /\ ( abs ` k ) < 1 ) )

Distinct variable groups:   x, k   A, j, k   j, F, k, x   k, K   j, M, k, x   P, j, k, x   Q, j   ph, j, k, x

Allowed substitution hints:   A( x)   Q( x, k)   K( x, j)   L( x, j, k)

Proof of Theorem etransclem47

Dummy variables i y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		etransclem47.k	. . . . 5 |- K = ( L / ( ! ` ( P - 1 ) ) )
2	1	a1i 11	. . . 4 |- ( ph -> K = ( L / ( ! ` ( P - 1 ) ) ) )
3		etransclem47.q	. . . . 5 |- ( ph -> Q e. ( (Poly ` ZZ ) \ { 0p } ) )
4		etransclem47.qe0	. . . . 5 |- ( ph -> ( Q ` _e ) = 0 )
5		etransclem47.a	. . . . 5 |- A = (coeff ` Q )
6		etransclem47.m	. . . . 5 |- M = (deg ` Q )
7		ssid 3624	. . . . . 6 |- RR C_ RR
8	7	a1i 11	. . . . 5 |- ( ph -> RR C_ RR )
9		reelprrecn 10028	. . . . . 6 |- RR e. { RR , CC }
10	9	a1i 11	. . . . 5 |- ( ph -> RR e. { RR , CC } )
11		reopn 39501	. . . . . . 7 |- RR e. ( topGen ` ran (,) )
12		eqid 2622	. . . . . . . 8 |- ( TopOpen ` ℂfld ) = ( TopOpen ` ℂfld )
13	12	tgioo2 22606	. . . . . . 7 |- ( topGen ` ran (,) ) = ( ( TopOpen ` ℂfld ) ↾t RR )
14	11, 13	eleqtri 2699	. . . . . 6 |- RR e. ( ( TopOpen ` ℂfld ) ↾t RR )
15	14	a1i 11	. . . . 5 |- ( ph -> RR e. ( ( TopOpen ` ℂfld ) ↾t RR ) )
16		etransclem47.p	. . . . . 6 |- ( ph -> P e. Prime )
17		prmnn 15388	. . . . . 6 |- ( P e. Prime -> P e. NN )
18	16, 17	syl 17	. . . . 5 |- ( ph -> P e. NN )
19		etransclem47.f	. . . . 5 |- F = ( x e. RR |-> ( ( x ^ ( P - 1 ) ) x. prod_ j e. ( 1 ... M ) ( ( x - j ) ^ P ) ) )
20		etransclem47.l	. . . . 5 |- L = sum_ j e. ( 0 ... M ) ( ( ( A ` j ) x. ( _e ^c j ) ) x. S. ( 0 (,) j ) ( ( _e ^c -u x ) x. ( F ` x ) ) _d x )
21		eqid 2622	. . . . 5 |- ( ( M x. P ) + ( P - 1 ) ) = ( ( M x. P ) + ( P - 1 ) )
22		fveq2 6191	. . . . . . 7 |- ( y = x -> ( ( ( RR Dn F ) ` i ) ` y ) = ( ( ( RR Dn F ) ` i ) ` x ) )
23	22	sumeq2ad 14434	. . . . . 6 |- ( y = x -> sum_ i e. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ( ( ( RR Dn F ) ` i ) ` y ) = sum_ i e. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ( ( ( RR Dn F ) ` i ) ` x ) )
24	23	cbvmptv 4750	. . . . 5 |- ( y e. RR |-> sum_ i e. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ( ( ( RR Dn F ) ` i ) ` y ) ) = ( x e. RR |-> sum_ i e. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ( ( ( RR Dn F ) ` i ) ` x ) )
25		negeq 10273	. . . . . . . . 9 |- ( z = x -> -u z = -u x )
26	25	oveq2d 6666	. . . . . . . 8 |- ( z = x -> ( _e ^c -u z ) = ( _e ^c -u x ) )
27		fveq2 6191	. . . . . . . 8 |- ( z = x -> ( ( y e. RR |-> sum_ i e. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ( ( ( RR Dn F ) ` i ) ` y ) ) ` z ) = ( ( y e. RR |-> sum_ i e. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ( ( ( RR Dn F ) ` i ) ` y ) ) ` x ) )
28	26, 27	oveq12d 6668	. . . . . . 7 |- ( z = x -> ( ( _e ^c -u z ) x. ( ( y e. RR |-> sum_ i e. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ( ( ( RR Dn F ) ` i ) ` y ) ) ` z ) ) = ( ( _e ^c -u x ) x. ( ( y e. RR |-> sum_ i e. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ( ( ( RR Dn F ) ` i ) ` y ) ) ` x ) ) )
29	28	negeqd 10275	. . . . . 6 |- ( z = x -> -u ( ( _e ^c -u z ) x. ( ( y e. RR |-> sum_ i e. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ( ( ( RR Dn F ) ` i ) ` y ) ) ` z ) ) = -u ( ( _e ^c -u x ) x. ( ( y e. RR |-> sum_ i e. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ( ( ( RR Dn F ) ` i ) ` y ) ) ` x ) ) )
30	29	cbvmptv 4750	. . . . 5 |- ( z e. ( 0 [,] j ) |-> -u ( ( _e ^c -u z ) x. ( ( y e. RR |-> sum_ i e. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ( ( ( RR Dn F ) ` i ) ` y ) ) ` z ) ) ) = ( x e. ( 0 [,] j ) |-> -u ( ( _e ^c -u x ) x. ( ( y e. RR |-> sum_ i e. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ( ( ( RR Dn F ) ` i ) ` y ) ) ` x ) ) )
31	3, 4, 5, 6, 8, 10, 15, 18, 19, 20, 21, 24, 30	etransclem46 40497	. . . 4 |- ( ph -> ( L / ( ! ` ( P - 1 ) ) ) = ( -u sum_ k e. ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) ( ( A ` ( 1st ` k ) ) x. ( ( ( RR Dn F ) ` ( 2nd ` k ) ) ` ( 1st ` k ) ) ) / ( ! ` ( P - 1 ) ) ) )
32		fzfid 12772	. . . . . . . 8 |- ( ph -> ( 0 ... M ) e. Fin )
33		fzfid 12772	. . . . . . . 8 |- ( ph -> ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) e. Fin )
34		xpfi 8231	. . . . . . . 8 |- ( ( ( 0 ... M ) e. Fin /\ ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) e. Fin ) -> ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) e. Fin )
35	32, 33, 34	syl2anc 693	. . . . . . 7 |- ( ph -> ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) e. Fin )
36	3	eldifad 3586	. . . . . . . . . . . 12 |- ( ph -> Q e. (Poly ` ZZ ) )
37		0zd 11389	. . . . . . . . . . . 12 |- ( ph -> 0 e. ZZ )
38	5	coef2 23987	. . . . . . . . . . . 12 |- ( ( Q e. (Poly ` ZZ ) /\ 0 e. ZZ ) -> A : NN0 --> ZZ )
39	36, 37, 38	syl2anc 693	. . . . . . . . . . 11 |- ( ph -> A : NN0 --> ZZ )
40	39	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ k e. ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) ) -> A : NN0 --> ZZ )
41		xp1st 7198	. . . . . . . . . . . 12 |- ( k e. ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) -> ( 1st ` k ) e. ( 0 ... M ) )
42		elfznn0 12433	. . . . . . . . . . . 12 |- ( ( 1st ` k ) e. ( 0 ... M ) -> ( 1st ` k ) e. NN0 )
43	41, 42	syl 17	. . . . . . . . . . 11 |- ( k e. ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) -> ( 1st ` k ) e. NN0 )
44	43	adantl 482	. . . . . . . . . 10 |- ( ( ph /\ k e. ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) ) -> ( 1st ` k ) e. NN0 )
45	40, 44	ffvelrnd 6360	. . . . . . . . 9 |- ( ( ph /\ k e. ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) ) -> ( A ` ( 1st ` k ) ) e. ZZ )
46	45	zcnd 11483	. . . . . . . 8 |- ( ( ph /\ k e. ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) ) -> ( A ` ( 1st ` k ) ) e. CC )
47	9	a1i 11	. . . . . . . . . 10 |- ( ( ph /\ k e. ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) ) -> RR e. { RR , CC } )
48	14	a1i 11	. . . . . . . . . 10 |- ( ( ph /\ k e. ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) ) -> RR e. ( ( TopOpen ` ℂfld ) ↾t RR ) )
49	18	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ k e. ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) ) -> P e. NN )
50		dgrcl 23989	. . . . . . . . . . . . 13 |- ( Q e. (Poly ` ZZ ) -> (deg ` Q ) e. NN0 )
51	36, 50	syl 17	. . . . . . . . . . . 12 |- ( ph -> (deg ` Q ) e. NN0 )
52	6, 51	syl5eqel 2705	. . . . . . . . . . 11 |- ( ph -> M e. NN0 )
53	52	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ k e. ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) ) -> M e. NN0 )
54		xp2nd 7199	. . . . . . . . . . . 12 |- ( k e. ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) -> ( 2nd ` k ) e. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) )
55		elfznn0 12433	. . . . . . . . . . . 12 |- ( ( 2nd ` k ) e. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) -> ( 2nd ` k ) e. NN0 )
56	54, 55	syl 17	. . . . . . . . . . 11 |- ( k e. ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) -> ( 2nd ` k ) e. NN0 )
57	56	adantl 482	. . . . . . . . . 10 |- ( ( ph /\ k e. ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) ) -> ( 2nd ` k ) e. NN0 )
58	47, 48, 49, 53, 19, 57	etransclem33 40484	. . . . . . . . 9 |- ( ( ph /\ k e. ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) ) -> ( ( RR Dn F ) ` ( 2nd ` k ) ) : RR --> CC )
59	44	nn0red 11352	. . . . . . . . 9 |- ( ( ph /\ k e. ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) ) -> ( 1st ` k ) e. RR )
60	58, 59	ffvelrnd 6360	. . . . . . . 8 |- ( ( ph /\ k e. ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) ) -> ( ( ( RR Dn F ) ` ( 2nd ` k ) ) ` ( 1st ` k ) ) e. CC )
61	46, 60	mulcld 10060	. . . . . . 7 |- ( ( ph /\ k e. ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) ) -> ( ( A ` ( 1st ` k ) ) x. ( ( ( RR Dn F ) ` ( 2nd ` k ) ) ` ( 1st ` k ) ) ) e. CC )
62	35, 61	fsumcl 14464	. . . . . 6 |- ( ph -> sum_ k e. ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) ( ( A ` ( 1st ` k ) ) x. ( ( ( RR Dn F ) ` ( 2nd ` k ) ) ` ( 1st ` k ) ) ) e. CC )
63		nnm1nn0 11334	. . . . . . . . 9 |- ( P e. NN -> ( P - 1 ) e. NN0 )
64	18, 63	syl 17	. . . . . . . 8 |- ( ph -> ( P - 1 ) e. NN0 )
65	64	faccld 13071	. . . . . . 7 |- ( ph -> ( ! ` ( P - 1 ) ) e. NN )
66	65	nncnd 11036	. . . . . 6 |- ( ph -> ( ! ` ( P - 1 ) ) e. CC )
67	65	nnne0d 11065	. . . . . 6 |- ( ph -> ( ! ` ( P - 1 ) ) =/= 0 )
68	62, 66, 67	divnegd 10814	. . . . 5 |- ( ph -> -u ( sum_ k e. ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) ( ( A ` ( 1st ` k ) ) x. ( ( ( RR Dn F ) ` ( 2nd ` k ) ) ` ( 1st ` k ) ) ) / ( ! ` ( P - 1 ) ) ) = ( -u sum_ k e. ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) ( ( A ` ( 1st ` k ) ) x. ( ( ( RR Dn F ) ` ( 2nd ` k ) ) ` ( 1st ` k ) ) ) / ( ! ` ( P - 1 ) ) ) )
69	68	eqcomd 2628	. . . 4 |- ( ph -> ( -u sum_ k e. ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) ( ( A ` ( 1st ` k ) ) x. ( ( ( RR Dn F ) ` ( 2nd ` k ) ) ` ( 1st ` k ) ) ) / ( ! ` ( P - 1 ) ) ) = -u ( sum_ k e. ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) ( ( A ` ( 1st ` k ) ) x. ( ( ( RR Dn F ) ` ( 2nd ` k ) ) ` ( 1st ` k ) ) ) / ( ! ` ( P - 1 ) ) ) )
70	2, 31, 69	3eqtrd 2660	. . 3 |- ( ph -> K = -u ( sum_ k e. ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) ( ( A ` ( 1st ` k ) ) x. ( ( ( RR Dn F ) ` ( 2nd ` k ) ) ` ( 1st ` k ) ) ) / ( ! ` ( P - 1 ) ) ) )
71		eqid 2622	. . . . 5 |- ( sum_ k e. ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) ( ( A ` ( 1st ` k ) ) x. ( ( ( RR Dn F ) ` ( 2nd ` k ) ) ` ( 1st ` k ) ) ) / ( ! ` ( P - 1 ) ) ) = ( sum_ k e. ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) ( ( A ` ( 1st ` k ) ) x. ( ( ( RR Dn F ) ` ( 2nd ` k ) ) ` ( 1st ` k ) ) ) / ( ! ` ( P - 1 ) ) )
72	18, 52, 19, 39, 71	etransclem45 40496	. . . 4 |- ( ph -> ( sum_ k e. ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) ( ( A ` ( 1st ` k ) ) x. ( ( ( RR Dn F ) ` ( 2nd ` k ) ) ` ( 1st ` k ) ) ) / ( ! ` ( P - 1 ) ) ) e. ZZ )
73	72	znegcld 11484	. . 3 |- ( ph -> -u ( sum_ k e. ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) ( ( A ` ( 1st ` k ) ) x. ( ( ( RR Dn F ) ` ( 2nd ` k ) ) ` ( 1st ` k ) ) ) / ( ! ` ( P - 1 ) ) ) e. ZZ )
74	70, 73	eqeltrd 2701	. 2 |- ( ph -> K e. ZZ )
75	1, 31	syl5eq 2668	. . 3 |- ( ph -> K = ( -u sum_ k e. ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) ( ( A ` ( 1st ` k ) ) x. ( ( ( RR Dn F ) ` ( 2nd ` k ) ) ` ( 1st ` k ) ) ) / ( ! ` ( P - 1 ) ) ) )
76	62, 66, 67	divcld 10801	. . . . 5 |- ( ph -> ( sum_ k e. ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) ( ( A ` ( 1st ` k ) ) x. ( ( ( RR Dn F ) ` ( 2nd ` k ) ) ` ( 1st ` k ) ) ) / ( ! ` ( P - 1 ) ) ) e. CC )
77		etransclem47.a0	. . . . . 6 |- ( ph -> ( A ` 0 ) =/= 0 )
78		etransclem47.ap	. . . . . 6 |- ( ph -> ( abs ` ( A ` 0 ) ) < P )
79		etransclem47.mp	. . . . . 6 |- ( ph -> ( ! ` M ) < P )
80	39, 77, 52, 16, 78, 79, 19, 71	etransclem44 40495	. . . . 5 |- ( ph -> ( sum_ k e. ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) ( ( A ` ( 1st ` k ) ) x. ( ( ( RR Dn F ) ` ( 2nd ` k ) ) ` ( 1st ` k ) ) ) / ( ! ` ( P - 1 ) ) ) =/= 0 )
81	76, 80	negne0d 10390	. . . 4 |- ( ph -> -u ( sum_ k e. ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) ( ( A ` ( 1st ` k ) ) x. ( ( ( RR Dn F ) ` ( 2nd ` k ) ) ` ( 1st ` k ) ) ) / ( ! ` ( P - 1 ) ) ) =/= 0 )
82	69, 81	eqnetrd 2861	. . 3 |- ( ph -> ( -u sum_ k e. ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) ( ( A ` ( 1st ` k ) ) x. ( ( ( RR Dn F ) ` ( 2nd ` k ) ) ` ( 1st ` k ) ) ) / ( ! ` ( P - 1 ) ) ) =/= 0 )
83	75, 82	eqnetrd 2861	. 2 |- ( ph -> K =/= 0 )
84		eldifsni 4320	. . . . . 6 |- ( Q e. ( (Poly ` ZZ ) \ { 0p } ) -> Q =/= 0p )
85	3, 84	syl 17	. . . . 5 |- ( ph -> Q =/= 0p )
86		ere 14819	. . . . . . 7 |- _e e. RR
87	86	recni 10052	. . . . . 6 |- _e e. CC
88	87	a1i 11	. . . . 5 |- ( ph -> _e e. CC )
89		dgrnznn 24003	. . . . 5 |- ( ( ( Q e. (Poly ` ZZ ) /\ Q =/= 0p ) /\ ( _e e. CC /\ ( Q ` _e ) = 0 ) ) -> (deg ` Q ) e. NN )
90	36, 85, 88, 4, 89	syl22anc 1327	. . . 4 |- ( ph -> (deg ` Q ) e. NN )
91	6, 90	syl5eqel 2705	. . 3 |- ( ph -> M e. NN )
92		etransclem47.9	. . 3 |- ( ph -> ( sum_ j e. ( 0 ... M ) ( ( abs ` ( ( A ` j ) x. ( _e ^c j ) ) ) x. ( M x. ( M ^ ( M + 1 ) ) ) ) x. ( ( ( M ^ ( M + 1 ) ) ^ ( P - 1 ) ) / ( ! ` ( P - 1 ) ) ) ) < 1 )
93	39, 20, 1, 18, 91, 19, 92	etransclem23 40474	. 2 |- ( ph -> ( abs ` K ) < 1 )
94		neeq1 2856	. . . 4 |- ( k = K -> ( k =/= 0 <-> K =/= 0 ) )
95		fveq2 6191	. . . . 5 |- ( k = K -> ( abs ` k ) = ( abs ` K ) )
96	95	breq1d 4663	. . . 4 |- ( k = K -> ( ( abs ` k ) < 1 <-> ( abs ` K ) < 1 ) )
97	94, 96	anbi12d 747	. . 3 |- ( k = K -> ( ( k =/= 0 /\ ( abs ` k ) < 1 ) <-> ( K =/= 0 /\ ( abs ` K ) < 1 ) ) )
98	97	rspcev 3309	. 2 |- ( ( K e. ZZ /\ ( K =/= 0 /\ ( abs ` K ) < 1 ) ) -> E. k e. ZZ ( k =/= 0 /\ ( abs ` k ) < 1 ) )
99	74, 83, 93, 98	syl12anc 1324	1 |- ( ph -> E. k e. ZZ ( k =/= 0 /\ ( abs ` k ) < 1 ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   \ cdif 3571   C_ wss 3574   {csn 4177   {cpr 4179   class class class wbr 4653   |-> cmpt 4729   X. cxp 5112   ran crn 5115   -->wf 5884   ` cfv 5888  (class class class)co 6650   1stc1st 7166   2ndc2nd 7167   Fincfn 7955   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   < clt 10074   - cmin 10266   -ucneg 10267   / cdiv 10684   NNcn 11020   NN0cn0 11292   ZZcz 11377   (,)cioo 12175   [,]cicc 12178   ...cfz 12326   ^cexp 12860   !cfa 13060   abscabs 13974   sum_csu 14416   prod_cprod 14635   _eceu 14793   Primecprime 15385   ↾_t crest 16081   TopOpenctopn 16082   topGenctg 16098  ℂ_fldccnfld 19746   S.citg 23387   0pc0p 23436   Dncdvn 23628  Polycply 23940  coeffccoe 23942  degcdgr 23943   ^c ccxp 24302

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cc 9257  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-addf 10015  ax-mulf 10016

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-disj 4621  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-ofr 6898  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-omul 7565  df-er 7742  df-map 7859  df-pm 7860  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-fi 8317  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-acn 8768  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ioo 12179  df-ioc 12180  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-fac 13061  df-bc 13090  df-hash 13118  df-shft 13807  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-limsup 14202  df-clim 14219  df-rlim 14220  df-sum 14417  df-prod 14636  df-ef 14798  df-e 14799  df-sin 14800  df-cos 14801  df-tan 14802  df-pi 14803  df-dvds 14984  df-gcd 15217  df-prm 15386  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-hom 15966  df-cco 15967  df-rest 16083  df-topn 16084  df-0g 16102  df-gsum 16103  df-topgen 16104  df-pt 16105  df-prds 16108  df-xrs 16162  df-qtop 16167  df-imas 16168  df-xps 16170  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-mulg 17541  df-cntz 17750  df-cmn 18195  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-fbas 19743  df-fg 19744  df-cnfld 19747  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cld 20823  df-ntr 20824  df-cls 20825  df-nei 20902  df-lp 20940  df-perf 20941  df-cn 21031  df-cnp 21032  df-haus 21119  df-cmp 21190  df-tx 21365  df-hmeo 21558  df-fil 21650  df-fm 21742  df-flim 21743  df-flf 21744  df-xms 22125  df-ms 22126  df-tms 22127  df-cncf 22681  df-ovol 23233  df-vol 23234  df-mbf 23388  df-itg1 23389  df-itg2 23390  df-ibl 23391  df-itg 23392  df-0p 23437  df-limc 23630  df-dv 23631  df-dvn 23632  df-ply 23944  df-coe 23946  df-dgr 23947  df-log 24303  df-cxp 24304

This theorem is referenced by:  etransclem48  40499