Theorem etransclem44 37960

Description: The given finite sum is nonzero. This is the claim proved after equation (7) in [Juillerat] p. 12 . (Contributed by Glauco Siliprandi, 5-Apr-2020.)

Hypotheses

Ref	Expression
etransclem44.a	|- ( ph -> A : NN0 --> ZZ )
etransclem44.a0	|- ( ph -> ( A ` 0 ) =/= 0 )
etransclem44.m	|- ( ph -> M e. NN0 )
etransclem44.p	|- ( ph -> P e. Prime )
etransclem44.ap	|- ( ph -> ( abs ` ( A ` 0 ) ) < P )
etransclem44.mp	|- ( ph -> ( ! ` M ) < P )
etransclem44.f	|- F = ( x e. RR |-> ( ( x ^ ( P - 1 ) ) x. prod_ j e. ( 1 ... M ) ( ( x - j ) ^ P ) ) )
etransclem44.k	|- K = ( 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 ) ) )

Assertion

Ref	Expression
etransclem44	|- ( ph -> K =/= 0 )

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

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

Proof of Theorem etransclem44

Dummy variables c d n m y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		etransclem44.k	. . . 4 |- K = ( 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 ) ) )
2	1	a1i 11	. . 3 |- ( ph -> K = ( 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 ) ) ) )
3		nfv 1751	. . . . 5 |- F/ k ph
4		nfcv 2584	. . . . 5 |- F/_ k ( ( A ` 0 ) x. ( ( ( RR Dn F ) ` ( P - 1 ) ) ` 0 ) )
5		fzfi 12184	. . . . . . 7 |- ( 0 ... M ) e. Fin
6		fzfi 12184	. . . . . . 7 |- ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) e. Fin
7		xpfi 7844	. . . . . . 7 |- ( ( ( 0 ... M ) e. Fin /\ ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) e. Fin ) -> ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) e. Fin )
8	5, 6, 7	mp2an 676	. . . . . 6 |- ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) e. Fin
9	8	a1i 11	. . . . 5 |- ( ph -> ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) e. Fin )
10		etransclem44.a	. . . . . . . . 9 |- ( ph -> A : NN0 --> ZZ )
11	10	adantr 466	. . . . . . . 8 |- ( ( ph /\ k e. ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) ) -> A : NN0 --> ZZ )
12		fzssnn0 37378	. . . . . . . . . 10 |- ( 0 ... M ) C_ NN0
13		xp1st 6833	. . . . . . . . . 10 |- ( k e. ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) -> ( 1st ` k ) e. ( 0 ... M ) )
14	12, 13	sseldi 3462	. . . . . . . . 9 |- ( k e. ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) -> ( 1st ` k ) e. NN0 )
15	14	adantl 467	. . . . . . . 8 |- ( ( ph /\ k e. ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) ) -> ( 1st ` k ) e. NN0 )
16	11, 15	ffvelrnd 6034	. . . . . . 7 |- ( ( ph /\ k e. ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) ) -> ( A ` ( 1st ` k ) ) e. ZZ )
17		reelprrecn 9631	. . . . . . . . 9 |- RR e. { RR , CC }
18	17	a1i 11	. . . . . . . 8 |- ( ( ph /\ k e. ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) ) -> RR e. { RR , CC } )
19		reopn 37341	. . . . . . . . . 10 |- RR e. ( topGen ` ran (,) )
20		eqid 2422	. . . . . . . . . . 11 |- ( TopOpen ` ℂfld ) = ( TopOpen ` ℂfld )
21	20	tgioo2 21805	. . . . . . . . . 10 |- ( topGen ` ran (,) ) = ( ( TopOpen ` ℂfld ) ↾t RR )
22	19, 21	eleqtri 2508	. . . . . . . . 9 |- RR e. ( ( TopOpen ` ℂfld ) ↾t RR )
23	22	a1i 11	. . . . . . . 8 |- ( ( ph /\ k e. ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) ) -> RR e. ( ( TopOpen ` ℂfld ) ↾t RR ) )
24		etransclem44.p	. . . . . . . . . 10 |- ( ph -> P e. Prime )
25		prmnn 14610	. . . . . . . . . 10 |- ( P e. Prime -> P e. NN )
26	24, 25	syl 17	. . . . . . . . 9 |- ( ph -> P e. NN )
27	26	adantr 466	. . . . . . . 8 |- ( ( ph /\ k e. ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) ) -> P e. NN )
28		etransclem44.m	. . . . . . . . 9 |- ( ph -> M e. NN0 )
29	28	adantr 466	. . . . . . . 8 |- ( ( ph /\ k e. ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) ) -> M e. NN0 )
30		etransclem44.f	. . . . . . . 8 |- F = ( x e. RR |-> ( ( x ^ ( P - 1 ) ) x. prod_ j e. ( 1 ... M ) ( ( x - j ) ^ P ) ) )
31		xp2nd 6834	. . . . . . . . . 10 |- ( k e. ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) -> ( 2nd ` k ) e. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) )
32		elfznn0 11887	. . . . . . . . . 10 |- ( ( 2nd ` k ) e. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) -> ( 2nd ` k ) e. NN0 )
33	31, 32	syl 17	. . . . . . . . 9 |- ( k e. ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) -> ( 2nd ` k ) e. NN0 )
34	33	adantl 467	. . . . . . . 8 |- ( ( ph /\ k e. ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) ) -> ( 2nd ` k ) e. NN0 )
35	15	nn0red 10926	. . . . . . . 8 |- ( ( ph /\ k e. ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) ) -> ( 1st ` k ) e. RR )
36	15	nn0zd 11038	. . . . . . . 8 |- ( ( ph /\ k e. ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) ) -> ( 1st ` k ) e. ZZ )
37	18, 23, 27, 29, 30, 34, 35, 36	etransclem42 37958	. . . . . . 7 |- ( ( ph /\ k e. ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) ) -> ( ( ( RR Dn F ) ` ( 2nd ` k ) ) ` ( 1st ` k ) ) e. ZZ )
38	16, 37	zmulcld 11046	. . . . . 6 |- ( ( ph /\ k e. ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) ) -> ( ( A ` ( 1st ` k ) ) x. ( ( ( RR Dn F ) ` ( 2nd ` k ) ) ` ( 1st ` k ) ) ) e. ZZ )
39	38	zcnd 11041	. . . . 5 |- ( ( 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 )
40		nn0uz 11193	. . . . . . . 8 |- NN0 = ( ZZ>= ` 0 )
41	28, 40	syl6eleq 2520	. . . . . . 7 |- ( ph -> M e. ( ZZ>= ` 0 ) )
42		eluzfz1 11806	. . . . . . 7 |- ( M e. ( ZZ>= ` 0 ) -> 0 e. ( 0 ... M ) )
43	41, 42	syl 17	. . . . . 6 |- ( ph -> 0 e. ( 0 ... M ) )
44		0zd 10949	. . . . . . . . 9 |- ( ph -> 0 e. ZZ )
45	28	nn0zd 11038	. . . . . . . . . . 11 |- ( ph -> M e. ZZ )
46	26	nnzd 11039	. . . . . . . . . . 11 |- ( ph -> P e. ZZ )
47	45, 46	zmulcld 11046	. . . . . . . . . 10 |- ( ph -> ( M x. P ) e. ZZ )
48		nnm1nn0 10911	. . . . . . . . . . . 12 |- ( P e. NN -> ( P - 1 ) e. NN0 )
49	26, 48	syl 17	. . . . . . . . . . 11 |- ( ph -> ( P - 1 ) e. NN0 )
50	49	nn0zd 11038	. . . . . . . . . 10 |- ( ph -> ( P - 1 ) e. ZZ )
51	47, 50	zaddcld 11044	. . . . . . . . 9 |- ( ph -> ( ( M x. P ) + ( P - 1 ) ) e. ZZ )
52	44, 51, 50	3jca 1185	. . . . . . . 8 |- ( ph -> ( 0 e. ZZ /\ ( ( M x. P ) + ( P - 1 ) ) e. ZZ /\ ( P - 1 ) e. ZZ ) )
53	49	nn0ge0d 10928	. . . . . . . 8 |- ( ph -> 0 <_ ( P - 1 ) )
54	26	nnnn0d 10925	. . . . . . . . . . 11 |- ( ph -> P e. NN0 )
55	28, 54	nn0mulcld 10930	. . . . . . . . . 10 |- ( ph -> ( M x. P ) e. NN0 )
56	55	nn0ge0d 10928	. . . . . . . . 9 |- ( ph -> 0 <_ ( M x. P ) )
57	49	nn0red 10926	. . . . . . . . . 10 |- ( ph -> ( P - 1 ) e. RR )
58	47	zred 11040	. . . . . . . . . 10 |- ( ph -> ( M x. P ) e. RR )
59	57, 58	addge02d 10202	. . . . . . . . 9 |- ( ph -> ( 0 <_ ( M x. P ) <-> ( P - 1 ) <_ ( ( M x. P ) + ( P - 1 ) ) ) )
60	56, 59	mpbid 213	. . . . . . . 8 |- ( ph -> ( P - 1 ) <_ ( ( M x. P ) + ( P - 1 ) ) )
61	52, 53, 60	jca32 537	. . . . . . 7 |- ( ph -> ( ( 0 e. ZZ /\ ( ( M x. P ) + ( P - 1 ) ) e. ZZ /\ ( P - 1 ) e. ZZ ) /\ ( 0 <_ ( P - 1 ) /\ ( P - 1 ) <_ ( ( M x. P ) + ( P - 1 ) ) ) ) )
62		elfz2 11791	. . . . . . 7 |- ( ( P - 1 ) e. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) <-> ( ( 0 e. ZZ /\ ( ( M x. P ) + ( P - 1 ) ) e. ZZ /\ ( P - 1 ) e. ZZ ) /\ ( 0 <_ ( P - 1 ) /\ ( P - 1 ) <_ ( ( M x. P ) + ( P - 1 ) ) ) ) )
63	61, 62	sylibr 215	. . . . . 6 |- ( ph -> ( P - 1 ) e. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) )
64		opelxp 4879	. . . . . 6 |- ( <. 0 , ( P - 1 ) >. e. ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) <-> ( 0 e. ( 0 ... M ) /\ ( P - 1 ) e. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) )
65	43, 63, 64	sylanbrc 668	. . . . 5 |- ( ph -> <. 0 , ( P - 1 ) >. e. ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) )
66		fveq2 5877	. . . . . . . 8 |- ( k = <. 0 , ( P - 1 ) >. -> ( 1st ` k ) = ( 1st ` <. 0 , ( P - 1 ) >. ) )
67		0re 9643	. . . . . . . . 9 |- 0 e. RR
68		ovex 6329	. . . . . . . . 9 |- ( P - 1 ) e. _V
69		op1stg 6815	. . . . . . . . 9 |- ( ( 0 e. RR /\ ( P - 1 ) e. _V ) -> ( 1st ` <. 0 , ( P - 1 ) >. ) = 0 )
70	67, 68, 69	mp2an 676	. . . . . . . 8 |- ( 1st ` <. 0 , ( P - 1 ) >. ) = 0
71	66, 70	syl6eq 2479	. . . . . . 7 |- ( k = <. 0 , ( P - 1 ) >. -> ( 1st ` k ) = 0 )
72	71	fveq2d 5881	. . . . . 6 |- ( k = <. 0 , ( P - 1 ) >. -> ( A ` ( 1st ` k ) ) = ( A ` 0 ) )
73		fveq2 5877	. . . . . . . . 9 |- ( k = <. 0 , ( P - 1 ) >. -> ( 2nd ` k ) = ( 2nd ` <. 0 , ( P - 1 ) >. ) )
74		op2ndg 6816	. . . . . . . . . 10 |- ( ( 0 e. RR /\ ( P - 1 ) e. _V ) -> ( 2nd ` <. 0 , ( P - 1 ) >. ) = ( P - 1 ) )
75	67, 68, 74	mp2an 676	. . . . . . . . 9 |- ( 2nd ` <. 0 , ( P - 1 ) >. ) = ( P - 1 )
76	73, 75	syl6eq 2479	. . . . . . . 8 |- ( k = <. 0 , ( P - 1 ) >. -> ( 2nd ` k ) = ( P - 1 ) )
77	76	fveq2d 5881	. . . . . . 7 |- ( k = <. 0 , ( P - 1 ) >. -> ( ( RR Dn F ) ` ( 2nd ` k ) ) = ( ( RR Dn F ) ` ( P - 1 ) ) )
78	77, 71	fveq12d 5883	. . . . . 6 |- ( k = <. 0 , ( P - 1 ) >. -> ( ( ( RR Dn F ) ` ( 2nd ` k ) ) ` ( 1st ` k ) ) = ( ( ( RR Dn F ) ` ( P - 1 ) ) ` 0 ) )
79	72, 78	oveq12d 6319	. . . . 5 |- ( k = <. 0 , ( P - 1 ) >. -> ( ( A ` ( 1st ` k ) ) x. ( ( ( RR Dn F ) ` ( 2nd ` k ) ) ` ( 1st ` k ) ) ) = ( ( A ` 0 ) x. ( ( ( RR Dn F ) ` ( P - 1 ) ) ` 0 ) ) )
80	3, 4, 9, 39, 65, 79	fsumsplit1 37471	. . . 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 ) ) ) = ( ( ( A ` 0 ) x. ( ( ( RR Dn F ) ` ( P - 1 ) ) ` 0 ) ) + sum_ k e. ( ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) \ { <. 0 , ( P - 1 ) >. } ) ( ( A ` ( 1st ` k ) ) x. ( ( ( RR Dn F ) ` ( 2nd ` k ) ) ` ( 1st ` k ) ) ) ) )
81	80	oveq1d 6316	. . 3 |- ( 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 ) ) ) = ( ( ( ( A ` 0 ) x. ( ( ( RR Dn F ) ` ( P - 1 ) ) ` 0 ) ) + sum_ k e. ( ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) \ { <. 0 , ( P - 1 ) >. } ) ( ( A ` ( 1st ` k ) ) x. ( ( ( RR Dn F ) ` ( 2nd ` k ) ) ` ( 1st ` k ) ) ) ) / ( ! ` ( P - 1 ) ) ) )
82	12, 43	sseldi 3462	. . . . . . 7 |- ( ph -> 0 e. NN0 )
83	10, 82	ffvelrnd 6034	. . . . . 6 |- ( ph -> ( A ` 0 ) e. ZZ )
84	17	a1i 11	. . . . . . 7 |- ( ph -> RR e. { RR , CC } )
85	22	a1i 11	. . . . . . 7 |- ( ph -> RR e. ( ( TopOpen ` ℂfld ) ↾t RR ) )
86	67	a1i 11	. . . . . . 7 |- ( ph -> 0 e. RR )
87	84, 85, 26, 28, 30, 49, 86, 44	etransclem42 37958	. . . . . 6 |- ( ph -> ( ( ( RR Dn F ) ` ( P - 1 ) ) ` 0 ) e. ZZ )
88	83, 87	zmulcld 11046	. . . . 5 |- ( ph -> ( ( A ` 0 ) x. ( ( ( RR Dn F ) ` ( P - 1 ) ) ` 0 ) ) e. ZZ )
89	88	zcnd 11041	. . . 4 |- ( ph -> ( ( A ` 0 ) x. ( ( ( RR Dn F ) ` ( P - 1 ) ) ` 0 ) ) e. CC )
90		difss 3592	. . . . . . . 8 |- ( ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) \ { <. 0 , ( P - 1 ) >. } ) C_ ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) )
91		ssfi 7794	. . . . . . . 8 |- ( ( ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) e. Fin /\ ( ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) \ { <. 0 , ( P - 1 ) >. } ) C_ ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) ) -> ( ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) \ { <. 0 , ( P - 1 ) >. } ) e. Fin )
92	8, 90, 91	mp2an 676	. . . . . . 7 |- ( ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) \ { <. 0 , ( P - 1 ) >. } ) e. Fin
93	92	a1i 11	. . . . . 6 |- ( ph -> ( ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) \ { <. 0 , ( P - 1 ) >. } ) e. Fin )
94		eldifi 3587	. . . . . . 7 |- ( k e. ( ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) \ { <. 0 , ( P - 1 ) >. } ) -> k e. ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) )
95	94, 38	sylan2 476	. . . . . 6 |- ( ( ph /\ k e. ( ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) \ { <. 0 , ( P - 1 ) >. } ) ) -> ( ( A ` ( 1st ` k ) ) x. ( ( ( RR Dn F ) ` ( 2nd ` k ) ) ` ( 1st ` k ) ) ) e. ZZ )
96	93, 95	fsumzcl 13786	. . . . 5 |- ( ph -> sum_ k e. ( ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) \ { <. 0 , ( P - 1 ) >. } ) ( ( A ` ( 1st ` k ) ) x. ( ( ( RR Dn F ) ` ( 2nd ` k ) ) ` ( 1st ` k ) ) ) e. ZZ )
97	96	zcnd 11041	. . . 4 |- ( ph -> sum_ k e. ( ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) \ { <. 0 , ( P - 1 ) >. } ) ( ( A ` ( 1st ` k ) ) x. ( ( ( RR Dn F ) ` ( 2nd ` k ) ) ` ( 1st ` k ) ) ) e. CC )
98	49	faccld 37372	. . . . 5 |- ( ph -> ( ! ` ( P - 1 ) ) e. NN )
99	98	nncnd 10625	. . . 4 |- ( ph -> ( ! ` ( P - 1 ) ) e. CC )
100	98	nnne0d 10654	. . . 4 |- ( ph -> ( ! ` ( P - 1 ) ) =/= 0 )
101	89, 97, 99, 100	divdird 10421	. . 3 |- ( ph -> ( ( ( ( A ` 0 ) x. ( ( ( RR Dn F ) ` ( P - 1 ) ) ` 0 ) ) + sum_ k e. ( ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) \ { <. 0 , ( P - 1 ) >. } ) ( ( A ` ( 1st ` k ) ) x. ( ( ( RR Dn F ) ` ( 2nd ` k ) ) ` ( 1st ` k ) ) ) ) / ( ! ` ( P - 1 ) ) ) = ( ( ( ( A ` 0 ) x. ( ( ( RR Dn F ) ` ( P - 1 ) ) ` 0 ) ) / ( ! ` ( P - 1 ) ) ) + ( sum_ k e. ( ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) \ { <. 0 , ( P - 1 ) >. } ) ( ( A ` ( 1st ` k ) ) x. ( ( ( RR Dn F ) ` ( 2nd ` k ) ) ` ( 1st ` k ) ) ) / ( ! ` ( P - 1 ) ) ) ) )
102	2, 81, 101	3eqtrd 2467	. 2 |- ( ph -> K = ( ( ( ( A ` 0 ) x. ( ( ( RR Dn F ) ` ( P - 1 ) ) ` 0 ) ) / ( ! ` ( P - 1 ) ) ) + ( sum_ k e. ( ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) \ { <. 0 , ( P - 1 ) >. } ) ( ( A ` ( 1st ` k ) ) x. ( ( ( RR Dn F ) ` ( 2nd ` k ) ) ` ( 1st ` k ) ) ) / ( ! ` ( P - 1 ) ) ) ) )
103	26	nnne0d 10654	. . 3 |- ( ph -> P =/= 0 )
104	83	zcnd 11041	. . . . 5 |- ( ph -> ( A ` 0 ) e. CC )
105	87	zcnd 11041	. . . . 5 |- ( ph -> ( ( ( RR Dn F ) ` ( P - 1 ) ) ` 0 ) e. CC )
106	104, 105, 99, 100	divassd 10418	. . . 4 |- ( ph -> ( ( ( A ` 0 ) x. ( ( ( RR Dn F ) ` ( P - 1 ) ) ` 0 ) ) / ( ! ` ( P - 1 ) ) ) = ( ( A ` 0 ) x. ( ( ( ( RR Dn F ) ` ( P - 1 ) ) ` 0 ) / ( ! ` ( P - 1 ) ) ) ) )
107		etransclem5 37921	. . . . . . 7 |- ( k e. ( 0 ... M ) |-> ( y e. RR |-> ( ( y - k ) ^ if ( k = 0 , ( P - 1 ) , P ) ) ) ) = ( j e. ( 0 ... M ) |-> ( x e. RR |-> ( ( x - j ) ^ if ( j = 0 , ( P - 1 ) , P ) ) ) )
108		etransclem11 37927	. . . . . . 7 |- ( m e. NN0 |-> { d e. ( ( 0 ... m ) ^m ( 0 ... M ) ) | sum_ k e. ( 0 ... M ) ( d ` k ) = m } ) = ( n e. NN0 |-> { c e. ( ( 0 ... n ) ^m ( 0 ... M ) ) | sum_ j e. ( 0 ... M ) ( c ` j ) = n } )
109	84, 85, 26, 28, 30, 49, 107, 108, 43, 86	etransclem37 37953	. . . . . 6 |- ( ph -> ( ! ` ( P - 1 ) ) || ( ( ( RR Dn F ) ` ( P - 1 ) ) ` 0 ) )
110	98	nnzd 11039	. . . . . . 7 |- ( ph -> ( ! ` ( P - 1 ) ) e. ZZ )
111		dvdsval2 14293	. . . . . . 7 |- ( ( ( ! ` ( P - 1 ) ) e. ZZ /\ ( ! ` ( P - 1 ) ) =/= 0 /\ ( ( ( RR Dn F ) ` ( P - 1 ) ) ` 0 ) e. ZZ ) -> ( ( ! ` ( P - 1 ) ) || ( ( ( RR Dn F ) ` ( P - 1 ) ) ` 0 ) <-> ( ( ( ( RR Dn F ) ` ( P - 1 ) ) ` 0 ) / ( ! ` ( P - 1 ) ) ) e. ZZ ) )
112	110, 100, 87, 111	syl3anc 1264	. . . . . 6 |- ( ph -> ( ( ! ` ( P - 1 ) ) || ( ( ( RR Dn F ) ` ( P - 1 ) ) ` 0 ) <-> ( ( ( ( RR Dn F ) ` ( P - 1 ) ) ` 0 ) / ( ! ` ( P - 1 ) ) ) e. ZZ ) )
113	109, 112	mpbid 213	. . . . 5 |- ( ph -> ( ( ( ( RR Dn F ) ` ( P - 1 ) ) ` 0 ) / ( ! ` ( P - 1 ) ) ) e. ZZ )
114	83, 113	zmulcld 11046	. . . 4 |- ( ph -> ( ( A ` 0 ) x. ( ( ( ( RR Dn F ) ` ( P - 1 ) ) ` 0 ) / ( ! ` ( P - 1 ) ) ) ) e. ZZ )
115	106, 114	eqeltrd 2510	. . 3 |- ( ph -> ( ( ( A ` 0 ) x. ( ( ( RR Dn F ) ` ( P - 1 ) ) ` 0 ) ) / ( ! ` ( P - 1 ) ) ) e. ZZ )
116	94, 39	sylan2 476	. . . . 5 |- ( ( ph /\ k e. ( ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) \ { <. 0 , ( P - 1 ) >. } ) ) -> ( ( A ` ( 1st ` k ) ) x. ( ( ( RR Dn F ) ` ( 2nd ` k ) ) ` ( 1st ` k ) ) ) e. CC )
117	93, 99, 116, 100	fsumdivc 13832	. . . 4 |- ( ph -> ( sum_ k e. ( ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) \ { <. 0 , ( 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 ) ) ) ) \ { <. 0 , ( P - 1 ) >. } ) ( ( ( A ` ( 1st ` k ) ) x. ( ( ( RR Dn F ) ` ( 2nd ` k ) ) ` ( 1st ` k ) ) ) / ( ! ` ( P - 1 ) ) ) )
118	16	zcnd 11041	. . . . . . . 8 |- ( ( ph /\ k e. ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) ) -> ( A ` ( 1st ` k ) ) e. CC )
119	94, 118	sylan2 476	. . . . . . 7 |- ( ( ph /\ k e. ( ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) \ { <. 0 , ( P - 1 ) >. } ) ) -> ( A ` ( 1st ` k ) ) e. CC )
120	94, 37	sylan2 476	. . . . . . . 8 |- ( ( ph /\ k e. ( ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) \ { <. 0 , ( P - 1 ) >. } ) ) -> ( ( ( RR Dn F ) ` ( 2nd ` k ) ) ` ( 1st ` k ) ) e. ZZ )
121	120	zcnd 11041	. . . . . . 7 |- ( ( ph /\ k e. ( ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) \ { <. 0 , ( P - 1 ) >. } ) ) -> ( ( ( RR Dn F ) ` ( 2nd ` k ) ) ` ( 1st ` k ) ) e. CC )
122	99	adantr 466	. . . . . . 7 |- ( ( ph /\ k e. ( ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) \ { <. 0 , ( P - 1 ) >. } ) ) -> ( ! ` ( P - 1 ) ) e. CC )
123	100	adantr 466	. . . . . . 7 |- ( ( ph /\ k e. ( ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) \ { <. 0 , ( P - 1 ) >. } ) ) -> ( ! ` ( P - 1 ) ) =/= 0 )
124	119, 121, 122, 123	divassd 10418	. . . . . 6 |- ( ( ph /\ k e. ( ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) \ { <. 0 , ( P - 1 ) >. } ) ) -> ( ( ( A ` ( 1st ` k ) ) x. ( ( ( RR Dn F ) ` ( 2nd ` k ) ) ` ( 1st ` k ) ) ) / ( ! ` ( P - 1 ) ) ) = ( ( A ` ( 1st ` k ) ) x. ( ( ( ( RR Dn F ) ` ( 2nd ` k ) ) ` ( 1st ` k ) ) / ( ! ` ( P - 1 ) ) ) ) )
125	94, 16	sylan2 476	. . . . . . 7 |- ( ( ph /\ k e. ( ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) \ { <. 0 , ( P - 1 ) >. } ) ) -> ( A ` ( 1st ` k ) ) e. ZZ )
126	17	a1i 11	. . . . . . . . 9 |- ( ( ph /\ k e. ( ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) \ { <. 0 , ( P - 1 ) >. } ) ) -> RR e. { RR , CC } )
127	22	a1i 11	. . . . . . . . 9 |- ( ( ph /\ k e. ( ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) \ { <. 0 , ( P - 1 ) >. } ) ) -> RR e. ( ( TopOpen ` ℂfld ) ↾t RR ) )
128	26	adantr 466	. . . . . . . . 9 |- ( ( ph /\ k e. ( ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) \ { <. 0 , ( P - 1 ) >. } ) ) -> P e. NN )
129	28	adantr 466	. . . . . . . . 9 |- ( ( ph /\ k e. ( ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) \ { <. 0 , ( P - 1 ) >. } ) ) -> M e. NN0 )
130	94	adantl 467	. . . . . . . . . 10 |- ( ( ph /\ k e. ( ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) \ { <. 0 , ( P - 1 ) >. } ) ) -> k e. ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) )
131	130, 33	syl 17	. . . . . . . . 9 |- ( ( ph /\ k e. ( ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) \ { <. 0 , ( P - 1 ) >. } ) ) -> ( 2nd ` k ) e. NN0 )
132	130, 13	syl 17	. . . . . . . . 9 |- ( ( ph /\ k e. ( ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) \ { <. 0 , ( P - 1 ) >. } ) ) -> ( 1st ` k ) e. ( 0 ... M ) )
133	94, 35	sylan2 476	. . . . . . . . 9 |- ( ( ph /\ k e. ( ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) \ { <. 0 , ( P - 1 ) >. } ) ) -> ( 1st ` k ) e. RR )
134	126, 127, 128, 129, 30, 131, 107, 108, 132, 133	etransclem37 37953	. . . . . . . 8 |- ( ( ph /\ k e. ( ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) \ { <. 0 , ( P - 1 ) >. } ) ) -> ( ! ` ( P - 1 ) ) || ( ( ( RR Dn F ) ` ( 2nd ` k ) ) ` ( 1st ` k ) ) )
135	110	adantr 466	. . . . . . . . 9 |- ( ( ph /\ k e. ( ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) \ { <. 0 , ( P - 1 ) >. } ) ) -> ( ! ` ( P - 1 ) ) e. ZZ )
136		dvdsval2 14293	. . . . . . . . 9 |- ( ( ( ! ` ( P - 1 ) ) e. ZZ /\ ( ! ` ( P - 1 ) ) =/= 0 /\ ( ( ( RR Dn F ) ` ( 2nd ` k ) ) ` ( 1st ` k ) ) e. ZZ ) -> ( ( ! ` ( P - 1 ) ) || ( ( ( RR Dn F ) ` ( 2nd ` k ) ) ` ( 1st ` k ) ) <-> ( ( ( ( RR Dn F ) ` ( 2nd ` k ) ) ` ( 1st ` k ) ) / ( ! ` ( P - 1 ) ) ) e. ZZ ) )
137	135, 123, 120, 136	syl3anc 1264	. . . . . . . 8 |- ( ( ph /\ k e. ( ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) \ { <. 0 , ( P - 1 ) >. } ) ) -> ( ( ! ` ( P - 1 ) ) || ( ( ( RR Dn F ) ` ( 2nd ` k ) ) ` ( 1st ` k ) ) <-> ( ( ( ( RR Dn F ) ` ( 2nd ` k ) ) ` ( 1st ` k ) ) / ( ! ` ( P - 1 ) ) ) e. ZZ ) )
138	134, 137	mpbid 213	. . . . . . 7 |- ( ( ph /\ k e. ( ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) \ { <. 0 , ( P - 1 ) >. } ) ) -> ( ( ( ( RR Dn F ) ` ( 2nd ` k ) ) ` ( 1st ` k ) ) / ( ! ` ( P - 1 ) ) ) e. ZZ )
139	125, 138	zmulcld 11046	. . . . . 6 |- ( ( ph /\ k e. ( ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) \ { <. 0 , ( P - 1 ) >. } ) ) -> ( ( A ` ( 1st ` k ) ) x. ( ( ( ( RR Dn F ) ` ( 2nd ` k ) ) ` ( 1st ` k ) ) / ( ! ` ( P - 1 ) ) ) ) e. ZZ )
140	124, 139	eqeltrd 2510	. . . . 5 |- ( ( ph /\ k e. ( ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) \ { <. 0 , ( P - 1 ) >. } ) ) -> ( ( ( A ` ( 1st ` k ) ) x. ( ( ( RR Dn F ) ` ( 2nd ` k ) ) ` ( 1st ` k ) ) ) / ( ! ` ( P - 1 ) ) ) e. ZZ )
141	93, 140	fsumzcl 13786	. . . 4 |- ( ph -> sum_ k e. ( ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) \ { <. 0 , ( P - 1 ) >. } ) ( ( ( A ` ( 1st ` k ) ) x. ( ( ( RR Dn F ) ` ( 2nd ` k ) ) ` ( 1st ` k ) ) ) / ( ! ` ( P - 1 ) ) ) e. ZZ )
142	117, 141	eqeltrd 2510	. . 3 |- ( ph -> ( sum_ k e. ( ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) \ { <. 0 , ( P - 1 ) >. } ) ( ( A ` ( 1st ` k ) ) x. ( ( ( RR Dn F ) ` ( 2nd ` k ) ) ` ( 1st ` k ) ) ) / ( ! ` ( P - 1 ) ) ) e. ZZ )
143		1zzd 10968	. . . . . . . . . . . 12 |- ( ph -> 1 e. ZZ )
144		zabscl 13362	. . . . . . . . . . . . 13 |- ( ( A ` 0 ) e. ZZ -> ( abs ` ( A ` 0 ) ) e. ZZ )
145	83, 144	syl 17	. . . . . . . . . . . 12 |- ( ph -> ( abs ` ( A ` 0 ) ) e. ZZ )
146	143, 50, 145	3jca 1185	. . . . . . . . . . 11 |- ( ph -> ( 1 e. ZZ /\ ( P - 1 ) e. ZZ /\ ( abs ` ( A ` 0 ) ) e. ZZ ) )
147		nn0abscl 13361	. . . . . . . . . . . . . 14 |- ( ( A ` 0 ) e. ZZ -> ( abs ` ( A ` 0 ) ) e. NN0 )
148	83, 147	syl 17	. . . . . . . . . . . . 13 |- ( ph -> ( abs ` ( A ` 0 ) ) e. NN0 )
149		etransclem44.a0	. . . . . . . . . . . . . 14 |- ( ph -> ( A ` 0 ) =/= 0 )
150	104, 149	absne0d 13494	. . . . . . . . . . . . 13 |- ( ph -> ( abs ` ( A ` 0 ) ) =/= 0 )
151		elnnne0 10883	. . . . . . . . . . . . 13 |- ( ( abs ` ( A ` 0 ) ) e. NN <-> ( ( abs ` ( A ` 0 ) ) e. NN0 /\ ( abs ` ( A ` 0 ) ) =/= 0 ) )
152	148, 150, 151	sylanbrc 668	. . . . . . . . . . . 12 |- ( ph -> ( abs ` ( A ` 0 ) ) e. NN )
153	152	nnge1d 10652	. . . . . . . . . . 11 |- ( ph -> 1 <_ ( abs ` ( A ` 0 ) ) )
154		etransclem44.ap	. . . . . . . . . . . 12 |- ( ph -> ( abs ` ( A ` 0 ) ) < P )
155		zltlem1 10989	. . . . . . . . . . . . 13 |- ( ( ( abs ` ( A ` 0 ) ) e. ZZ /\ P e. ZZ ) -> ( ( abs ` ( A ` 0 ) ) < P <-> ( abs ` ( A ` 0 ) ) <_ ( P - 1 ) ) )
156	145, 46, 155	syl2anc 665	. . . . . . . . . . . 12 |- ( ph -> ( ( abs ` ( A ` 0 ) ) < P <-> ( abs ` ( A ` 0 ) ) <_ ( P - 1 ) ) )
157	154, 156	mpbid 213	. . . . . . . . . . 11 |- ( ph -> ( abs ` ( A ` 0 ) ) <_ ( P - 1 ) )
158	146, 153, 157	jca32 537	. . . . . . . . . 10 |- ( ph -> ( ( 1 e. ZZ /\ ( P - 1 ) e. ZZ /\ ( abs ` ( A ` 0 ) ) e. ZZ ) /\ ( 1 <_ ( abs ` ( A ` 0 ) ) /\ ( abs ` ( A ` 0 ) ) <_ ( P - 1 ) ) ) )
159		elfz2 11791	. . . . . . . . . 10 |- ( ( abs ` ( A ` 0 ) ) e. ( 1 ... ( P - 1 ) ) <-> ( ( 1 e. ZZ /\ ( P - 1 ) e. ZZ /\ ( abs ` ( A ` 0 ) ) e. ZZ ) /\ ( 1 <_ ( abs ` ( A ` 0 ) ) /\ ( abs ` ( A ` 0 ) ) <_ ( P - 1 ) ) ) )
160	158, 159	sylibr 215	. . . . . . . . 9 |- ( ph -> ( abs ` ( A ` 0 ) ) e. ( 1 ... ( P - 1 ) ) )
161		fzm1ndvds 14342	. . . . . . . . 9 |- ( ( P e. NN /\ ( abs ` ( A ` 0 ) ) e. ( 1 ... ( P - 1 ) ) ) -> -. P || ( abs ` ( A ` 0 ) ) )
162	26, 160, 161	syl2anc 665	. . . . . . . 8 |- ( ph -> -. P || ( abs ` ( A ` 0 ) ) )
163		dvdsabsb 14307	. . . . . . . . 9 |- ( ( P e. ZZ /\ ( A ` 0 ) e. ZZ ) -> ( P || ( A ` 0 ) <-> P || ( abs ` ( A ` 0 ) ) ) )
164	46, 83, 163	syl2anc 665	. . . . . . . 8 |- ( ph -> ( P || ( A ` 0 ) <-> P || ( abs ` ( A ` 0 ) ) ) )
165	162, 164	mtbird 302	. . . . . . 7 |- ( ph -> -. P || ( A ` 0 ) )
166		etransclem44.mp	. . . . . . . 8 |- ( ph -> ( ! ` M ) < P )
167	28, 24, 166, 30	etransclem41 37957	. . . . . . 7 |- ( ph -> -. P || ( ( ( ( RR Dn F ) ` ( P - 1 ) ) ` 0 ) / ( ! ` ( P - 1 ) ) ) )
168	165, 167	jca 534	. . . . . 6 |- ( ph -> ( -. P || ( A ` 0 ) /\ -. P || ( ( ( ( RR Dn F ) ` ( P - 1 ) ) ` 0 ) / ( ! ` ( P - 1 ) ) ) ) )
169		pm4.56 497	. . . . . 6 |- ( ( -. P || ( A ` 0 ) /\ -. P || ( ( ( ( RR Dn F ) ` ( P - 1 ) ) ` 0 ) / ( ! ` ( P - 1 ) ) ) ) <-> -. ( P || ( A ` 0 ) \/ P || ( ( ( ( RR Dn F ) ` ( P - 1 ) ) ` 0 ) / ( ! ` ( P - 1 ) ) ) ) )
170	168, 169	sylib 199	. . . . 5 |- ( ph -> -. ( P || ( A ` 0 ) \/ P || ( ( ( ( RR Dn F ) ` ( P - 1 ) ) ` 0 ) / ( ! ` ( P - 1 ) ) ) ) )
171		euclemma 14650	. . . . . 6 |- ( ( P e. Prime /\ ( A ` 0 ) e. ZZ /\ ( ( ( ( RR Dn F ) ` ( P - 1 ) ) ` 0 ) / ( ! ` ( P - 1 ) ) ) e. ZZ ) -> ( P || ( ( A ` 0 ) x. ( ( ( ( RR Dn F ) ` ( P - 1 ) ) ` 0 ) / ( ! ` ( P - 1 ) ) ) ) <-> ( P || ( A ` 0 ) \/ P || ( ( ( ( RR Dn F ) ` ( P - 1 ) ) ` 0 ) / ( ! ` ( P - 1 ) ) ) ) ) )
172	24, 83, 113, 171	syl3anc 1264	. . . . 5 |- ( ph -> ( P || ( ( A ` 0 ) x. ( ( ( ( RR Dn F ) ` ( P - 1 ) ) ` 0 ) / ( ! ` ( P - 1 ) ) ) ) <-> ( P || ( A ` 0 ) \/ P || ( ( ( ( RR Dn F ) ` ( P - 1 ) ) ` 0 ) / ( ! ` ( P - 1 ) ) ) ) ) )
173	170, 172	mtbird 302	. . . 4 |- ( ph -> -. P || ( ( A ` 0 ) x. ( ( ( ( RR Dn F ) ` ( P - 1 ) ) ` 0 ) / ( ! ` ( P - 1 ) ) ) ) )
174	106	breq2d 4432	. . . 4 |- ( ph -> ( P || ( ( ( A ` 0 ) x. ( ( ( RR Dn F ) ` ( P - 1 ) ) ` 0 ) ) / ( ! ` ( P - 1 ) ) ) <-> P || ( ( A ` 0 ) x. ( ( ( ( RR Dn F ) ` ( P - 1 ) ) ` 0 ) / ( ! ` ( P - 1 ) ) ) ) ) )
175	173, 174	mtbird 302	. . 3 |- ( ph -> -. P || ( ( ( A ` 0 ) x. ( ( ( RR Dn F ) ` ( P - 1 ) ) ` 0 ) ) / ( ! ` ( P - 1 ) ) ) )
176	46	adantr 466	. . . . . . . 8 |- ( ( ph /\ k e. ( ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) \ { <. 0 , ( P - 1 ) >. } ) ) -> P e. ZZ )
177	176, 125, 138	3jca 1185	. . . . . . 7 |- ( ( ph /\ k e. ( ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) \ { <. 0 , ( P - 1 ) >. } ) ) -> ( P e. ZZ /\ ( A ` ( 1st ` k ) ) e. ZZ /\ ( ( ( ( RR Dn F ) ` ( 2nd ` k ) ) ` ( 1st ` k ) ) / ( ! ` ( P - 1 ) ) ) e. ZZ ) )
178		eldifn 3588	. . . . . . . . . 10 |- ( k e. ( ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) \ { <. 0 , ( P - 1 ) >. } ) -> -. k e. { <. 0 , ( P - 1 ) >. } )
179	94	adantr 466	. . . . . . . . . . . . 13 |- ( ( k e. ( ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) \ { <. 0 , ( P - 1 ) >. } ) /\ ( ( 2nd ` k ) = ( P - 1 ) /\ ( 1st ` k ) = 0 ) ) -> k e. ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) )
180		1st2nd2 6840	. . . . . . . . . . . . 13 |- ( k e. ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) -> k = <. ( 1st ` k ) , ( 2nd ` k ) >. )
181	179, 180	syl 17	. . . . . . . . . . . 12 |- ( ( k e. ( ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) \ { <. 0 , ( P - 1 ) >. } ) /\ ( ( 2nd ` k ) = ( P - 1 ) /\ ( 1st ` k ) = 0 ) ) -> k = <. ( 1st ` k ) , ( 2nd ` k ) >. )
182		simpr 462	. . . . . . . . . . . . . 14 |- ( ( ( 2nd ` k ) = ( P - 1 ) /\ ( 1st ` k ) = 0 ) -> ( 1st ` k ) = 0 )
183		simpl 458	. . . . . . . . . . . . . 14 |- ( ( ( 2nd ` k ) = ( P - 1 ) /\ ( 1st ` k ) = 0 ) -> ( 2nd ` k ) = ( P - 1 ) )
184	182, 183	opeq12d 4192	. . . . . . . . . . . . 13 |- ( ( ( 2nd ` k ) = ( P - 1 ) /\ ( 1st ` k ) = 0 ) -> <. ( 1st ` k ) , ( 2nd ` k ) >. = <. 0 , ( P - 1 ) >. )
185	184	adantl 467	. . . . . . . . . . . 12 |- ( ( k e. ( ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) \ { <. 0 , ( P - 1 ) >. } ) /\ ( ( 2nd ` k ) = ( P - 1 ) /\ ( 1st ` k ) = 0 ) ) -> <. ( 1st ` k ) , ( 2nd ` k ) >. = <. 0 , ( P - 1 ) >. )
186	181, 185	eqtrd 2463	. . . . . . . . . . 11 |- ( ( k e. ( ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) \ { <. 0 , ( P - 1 ) >. } ) /\ ( ( 2nd ` k ) = ( P - 1 ) /\ ( 1st ` k ) = 0 ) ) -> k = <. 0 , ( P - 1 ) >. )
187		elsn 4010	. . . . . . . . . . 11 |- ( k e. { <. 0 , ( P - 1 ) >. } <-> k = <. 0 , ( P - 1 ) >. )
188	186, 187	sylibr 215	. . . . . . . . . 10 |- ( ( k e. ( ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) \ { <. 0 , ( P - 1 ) >. } ) /\ ( ( 2nd ` k ) = ( P - 1 ) /\ ( 1st ` k ) = 0 ) ) -> k e. { <. 0 , ( P - 1 ) >. } )
189	178, 188	mtand 663	. . . . . . . . 9 |- ( k e. ( ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) \ { <. 0 , ( P - 1 ) >. } ) -> -. ( ( 2nd ` k ) = ( P - 1 ) /\ ( 1st ` k ) = 0 ) )
190	189	adantl 467	. . . . . . . 8 |- ( ( ph /\ k e. ( ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) \ { <. 0 , ( P - 1 ) >. } ) ) -> -. ( ( 2nd ` k ) = ( P - 1 ) /\ ( 1st ` k ) = 0 ) )
191	128, 129, 30, 131, 132, 190, 108	etransclem38 37954	. . . . . . 7 |- ( ( ph /\ k e. ( ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) \ { <. 0 , ( P - 1 ) >. } ) ) -> P || ( ( ( ( RR Dn F ) ` ( 2nd ` k ) ) ` ( 1st ` k ) ) / ( ! ` ( P - 1 ) ) ) )
192		dvdsmultr2 14325	. . . . . . 7 |- ( ( P e. ZZ /\ ( A ` ( 1st ` k ) ) e. ZZ /\ ( ( ( ( RR Dn F ) ` ( 2nd ` k ) ) ` ( 1st ` k ) ) / ( ! ` ( P - 1 ) ) ) e. ZZ ) -> ( P || ( ( ( ( RR Dn F ) ` ( 2nd ` k ) ) ` ( 1st ` k ) ) / ( ! ` ( P - 1 ) ) ) -> P || ( ( A ` ( 1st ` k ) ) x. ( ( ( ( RR Dn F ) ` ( 2nd ` k ) ) ` ( 1st ` k ) ) / ( ! ` ( P - 1 ) ) ) ) ) )
193	177, 191, 192	sylc 62	. . . . . 6 |- ( ( ph /\ k e. ( ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) \ { <. 0 , ( P - 1 ) >. } ) ) -> P || ( ( A ` ( 1st ` k ) ) x. ( ( ( ( RR Dn F ) ` ( 2nd ` k ) ) ` ( 1st ` k ) ) / ( ! ` ( P - 1 ) ) ) ) )
194	193, 124	breqtrrd 4447	. . . . 5 |- ( ( ph /\ k e. ( ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) \ { <. 0 , ( P - 1 ) >. } ) ) -> P || ( ( ( A ` ( 1st ` k ) ) x. ( ( ( RR Dn F ) ` ( 2nd ` k ) ) ` ( 1st ` k ) ) ) / ( ! ` ( P - 1 ) ) ) )
195	93, 46, 140, 194	fsumdvds 14333	. . . 4 |- ( ph -> P || sum_ k e. ( ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) \ { <. 0 , ( P - 1 ) >. } ) ( ( ( A ` ( 1st ` k ) ) x. ( ( ( RR Dn F ) ` ( 2nd ` k ) ) ` ( 1st ` k ) ) ) / ( ! ` ( P - 1 ) ) ) )
196	195, 117	breqtrrd 4447	. . 3 |- ( ph -> P || ( sum_ k e. ( ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) \ { <. 0 , ( P - 1 ) >. } ) ( ( A ` ( 1st ` k ) ) x. ( ( ( RR Dn F ) ` ( 2nd ` k ) ) ` ( 1st ` k ) ) ) / ( ! ` ( P - 1 ) ) ) )
197	46, 103, 115, 142, 175, 196	etransclem9 37925	. 2 |- ( ph -> ( ( ( ( A ` 0 ) x. ( ( ( RR Dn F ) ` ( P - 1 ) ) ` 0 ) ) / ( ! ` ( P - 1 ) ) ) + ( sum_ k e. ( ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) \ { <. 0 , ( P - 1 ) >. } ) ( ( A ` ( 1st ` k ) ) x. ( ( ( RR Dn F ) ` ( 2nd ` k ) ) ` ( 1st ` k ) ) ) / ( ! ` ( P - 1 ) ) ) ) =/= 0 )
198	102, 197	eqnetrd 2717	1 |- ( ph -> K =/= 0 )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 187   \/ wo 369   /\ wa 370   /\ w3a 982   = wceq 1437   e. wcel 1868   =/= wne 2618   {crab 2779   _Vcvv 3081   \ cdif 3433   C_ wss 3436   ifcif 3909   {csn 3996   {cpr 3998   <.cop 4002   class class class wbr 4420   |-> cmpt 4479   X. cxp 4847   ran crn 4850   -->wf 5593   ` cfv 5597  (class class class)co 6301   1stc1st 6801   2ndc2nd 6802   ^m cmap 7476   Fincfn 7573   CCcc 9537   RRcr 9538   0cc0 9539   1c1 9540   + caddc 9542   x. cmul 9544   < clt 9675   <_ cle 9676   - cmin 9860   / cdiv 10269   NNcn 10609   NN0cn0 10869   ZZcz 10937   ZZ>=cuz 11159   (,)cioo 11635   ...cfz 11784   ^cexp 12271   !cfa 12458   abscabs 13283   sum_csu 13737   prod_cprod 13944   || cdvds 14290   Primecprime 14607   ↾_t crest 15304   TopOpenctopn 15305   topGenctg 15321  ℂ_fldccnfld 18955   Dncdvn 22803

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 4551  ax-pow 4598  ax-pr 4656  ax-un 6593  ax-inf2 8148  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  ax-mulf 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-iin 4299  df-br 4421  df-opab 4480  df-mpt 4481  df-tr 4516  df-eprel 4760  df-id 4764  df-po 4770  df-so 4771  df-fr 4808  df-se 4809  df-we 4810  df-xp 4855  df-rel 4856  df-cnv 4857  df-co 4858  df-dm 4859  df-rn 4860  df-res 4861  df-ima 4862  df-pred 5395  df-ord 5441  df-on 5442  df-lim 5443  df-suc 5444  df-iota 5561  df-fun 5599  df-fn 5600  df-f 5601  df-f1 5602  df-fo 5603  df-f1o 5604  df-fv 5605  df-isom 5606  df-riota 6263  df-ov 6304  df-oprab 6305  df-mpt2 6306  df-of 6541  df-om 6703  df-1st 6803  df-2nd 6804  df-supp 6922  df-wrecs 7032  df-recs 7094  df-rdg 7132  df-1o 7186  df-2o 7187  df-oadd 7190  df-er 7367  df-map 7478  df-pm 7479  df-ixp 7527  df-en 7574  df-dom 7575  df-sdom 7576  df-fin 7577  df-fsupp 7886  df-fi 7927  df-sup 7958  df-inf 7959  df-oi 8027  df-card 8374  df-cda 8598  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-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-10 10676  df-n0 10870  df-z 10938  df-dec 11052  df-uz 11160  df-q 11265  df-rp 11303  df-xneg 11409  df-xadd 11410  df-xmul 11411  df-ioo 11639  df-ico 11641  df-icc 11642  df-fz 11785  df-fzo 11916  df-fl 12027  df-mod 12096  df-seq 12213  df-exp 12272  df-fac 12459  df-bc 12487  df-hash 12515  df-cj 13148  df-re 13149  df-im 13150  df-sqrt 13284  df-abs 13285  df-clim 13537  df-sum 13738  df-prod 13945  df-dvds 14291  df-gcd 14454  df-prm 14608  df-struct 15108  df-ndx 15109  df-slot 15110  df-base 15111  df-sets 15112  df-ress 15113  df-plusg 15188  df-mulr 15189  df-starv 15190  df-sca 15191  df-vsca 15192  df-ip 15193  df-tset 15194  df-ple 15195  df-ds 15197  df-unif 15198  df-hom 15199  df-cco 15200  df-rest 15306  df-topn 15307  df-0g 15325  df-gsum 15326  df-topgen 15327  df-pt 15328  df-prds 15331  df-xrs 15385  df-qtop 15391  df-imas 15392  df-xps 15395  df-mre 15477  df-mrc 15478  df-acs 15480  df-mgm 16473  df-sgrp 16512  df-mnd 16522  df-submnd 16568  df-mulg 16661  df-cntz 16956  df-cmn 17417  df-psmet 18947  df-xmet 18948  df-met 18949  df-bl 18950  df-mopn 18951  df-fbas 18952  df-fg 18953  df-cnfld 18956  df-top 19905  df-bases 19906  df-topon 19907  df-topsp 19908  df-cld 20018  df-ntr 20019  df-cls 20020  df-nei 20098  df-lp 20136  df-perf 20137  df-cn 20227  df-cnp 20228  df-haus 20315  df-tx 20561  df-hmeo 20754  df-fil 20845  df-fm 20937  df-flim 20938  df-flf 20939  df-xms 21319  df-ms 21320  df-tms 21321  df-cncf 21894  df-limc 22805  df-dv 22806  df-dvn 22807

This theorem is referenced by:  etransclem47  37963