Theorem etransclem44 38143

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 1761	. . . . 5 |- F/ k ph
4		nfcv 2592	. . . . 5 |- F/_ k ( ( A ` 0 ) x. ( ( ( RR Dn F ) ` ( P - 1 ) ) ` 0 ) )
5		fzfi 12185	. . . . . . 7 |- ( 0 ... M ) e. Fin
6		fzfi 12185	. . . . . . 7 |- ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) e. Fin
7		xpfi 7842	. . . . . . 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 678	. . . . . 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 467	. . . . . . . 8 |- ( ( ph /\ k e. ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) ) -> A : NN0 --> ZZ )
12		fzssnn0 37539	. . . . . . . . . 10 |- ( 0 ... M ) C_ NN0
13		xp1st 6823	. . . . . . . . . 10 |- ( k e. ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) -> ( 1st ` k ) e. ( 0 ... M ) )
14	12, 13	sseldi 3430	. . . . . . . . 9 |- ( k e. ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) -> ( 1st ` k ) e. NN0 )
15	14	adantl 468	. . . . . . . 8 |- ( ( ph /\ k e. ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) ) -> ( 1st ` k ) e. NN0 )
16	11, 15	ffvelrnd 6023	. . . . . . 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 37502	. . . . . . . . . 10 |- RR e. ( topGen ` ran (,) )
20		eqid 2451	. . . . . . . . . . 11 |- ( TopOpen ` ℂfld ) = ( TopOpen ` ℂfld )
21	20	tgioo2 21821	. . . . . . . . . 10 |- ( topGen ` ran (,) ) = ( ( TopOpen ` ℂfld ) ↾t RR )
22	19, 21	eleqtri 2527	. . . . . . . . 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 14625	. . . . . . . . . 10 |- ( P e. Prime -> P e. NN )
26	24, 25	syl 17	. . . . . . . . 9 |- ( ph -> P e. NN )
27	26	adantr 467	. . . . . . . 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 467	. . . . . . . 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 6824	. . . . . . . . . 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 468	. . . . . . . 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 38141	. . . . . . 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 2539	. . . . . . 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 1188	. . . . . . . 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 214	. . . . . . . 8 |- ( ph -> ( P - 1 ) <_ ( ( M x. P ) + ( P - 1 ) ) )
61	52, 53, 60	jca32 538	. . . . . . 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 216	. . . . . 6 |- ( ph -> ( P - 1 ) e. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) )
64		opelxp 4864	. . . . . 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 670	. . . . 5 |- ( ph -> <. 0 , ( P - 1 ) >. e. ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) )
66		fveq2 5865	. . . . . . . 8 |- ( k = <. 0 , ( P - 1 ) >. -> ( 1st ` k ) = ( 1st ` <. 0 , ( P - 1 ) >. ) )
67		0re 9643	. . . . . . . . 9 |- 0 e. RR
68		ovex 6318	. . . . . . . . 9 |- ( P - 1 ) e. _V
69		op1stg 6805	. . . . . . . . 9 |- ( ( 0 e. RR /\ ( P - 1 ) e. _V ) -> ( 1st ` <. 0 , ( P - 1 ) >. ) = 0 )
70	67, 68, 69	mp2an 678	. . . . . . . 8 |- ( 1st ` <. 0 , ( P - 1 ) >. ) = 0
71	66, 70	syl6eq 2501	. . . . . . 7 |- ( k = <. 0 , ( P - 1 ) >. -> ( 1st ` k ) = 0 )
72	71	fveq2d 5869	. . . . . 6 |- ( k = <. 0 , ( P - 1 ) >. -> ( A ` ( 1st ` k ) ) = ( A ` 0 ) )
73		fveq2 5865	. . . . . . . . 9 |- ( k = <. 0 , ( P - 1 ) >. -> ( 2nd ` k ) = ( 2nd ` <. 0 , ( P - 1 ) >. ) )
74		op2ndg 6806	. . . . . . . . . 10 |- ( ( 0 e. RR /\ ( P - 1 ) e. _V ) -> ( 2nd ` <. 0 , ( P - 1 ) >. ) = ( P - 1 ) )
75	67, 68, 74	mp2an 678	. . . . . . . . 9 |- ( 2nd ` <. 0 , ( P - 1 ) >. ) = ( P - 1 )
76	73, 75	syl6eq 2501	. . . . . . . 8 |- ( k = <. 0 , ( P - 1 ) >. -> ( 2nd ` k ) = ( P - 1 ) )
77	76	fveq2d 5869	. . . . . . 7 |- ( k = <. 0 , ( P - 1 ) >. -> ( ( RR Dn F ) ` ( 2nd ` k ) ) = ( ( RR Dn F ) ` ( P - 1 ) ) )
78	77, 71	fveq12d 5871	. . . . . 6 |- ( k = <. 0 , ( P - 1 ) >. -> ( ( ( RR Dn F ) ` ( 2nd ` k ) ) ` ( 1st ` k ) ) = ( ( ( RR Dn F ) ` ( P - 1 ) ) ` 0 ) )
79	72, 78	oveq12d 6308	. . . . 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 37651	. . . 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 6305	. . 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 3430	. . . . . . 7 |- ( ph -> 0 e. NN0 )
83	10, 82	ffvelrnd 6023	. . . . . 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 38141	. . . . . 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 3560	. . . . . . . 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 7792	. . . . . . . 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 678	. . . . . . 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 3555	. . . . . . 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 477	. . . . . 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 13801	. . . . 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 37533	. . . . 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 2489	. 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 38104	. . . . . . 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 38110	. . . . . . 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 38136	. . . . . 6 |- ( ph -> ( ! ` ( P - 1 ) ) || ( ( ( RR Dn F ) ` ( P - 1 ) ) ` 0 ) )
110	98	nnzd 11039	. . . . . . 7 |- ( ph -> ( ! ` ( P - 1 ) ) e. ZZ )
111		dvdsval2 14308	. . . . . . 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 1268	. . . . . 6 |- ( ph -> ( ( ! ` ( P - 1 ) ) || ( ( ( RR Dn F ) ` ( P - 1 ) ) ` 0 ) <-> ( ( ( ( RR Dn F ) ` ( P - 1 ) ) ` 0 ) / ( ! ` ( P - 1 ) ) ) e. ZZ ) )
113	109, 112	mpbid 214	. . . . 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 2529	. . 3 |- ( ph -> ( ( ( A ` 0 ) x. ( ( ( RR Dn F ) ` ( P - 1 ) ) ` 0 ) ) / ( ! ` ( P - 1 ) ) ) e. ZZ )
116	94, 39	sylan2 477	. . . . 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 13847	. . . 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 477	. . . . . . 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 477	. . . . . . . 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 467	. . . . . . 7 |- ( ( ph /\ k e. ( ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) \ { <. 0 , ( P - 1 ) >. } ) ) -> ( ! ` ( P - 1 ) ) e. CC )
123	100	adantr 467	. . . . . . 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 477	. . . . . . 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 467	. . . . . . . . 9 |- ( ( ph /\ k e. ( ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) \ { <. 0 , ( P - 1 ) >. } ) ) -> P e. NN )
129	28	adantr 467	. . . . . . . . 9 |- ( ( ph /\ k e. ( ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) \ { <. 0 , ( P - 1 ) >. } ) ) -> M e. NN0 )
130	94	adantl 468	. . . . . . . . . 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 477	. . . . . . . . 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 38136	. . . . . . . 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 467	. . . . . . . . 9 |- ( ( ph /\ k e. ( ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) \ { <. 0 , ( P - 1 ) >. } ) ) -> ( ! ` ( P - 1 ) ) e. ZZ )
136		dvdsval2 14308	. . . . . . . . 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 1268	. . . . . . . 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 214	. . . . . . 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 2529	. . . . 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 13801	. . . 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 2529	. . 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 13376	. . . . . . . . . . . . 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 1188	. . . . . . . . . . 11 |- ( ph -> ( 1 e. ZZ /\ ( P - 1 ) e. ZZ /\ ( abs ` ( A ` 0 ) ) e. ZZ ) )
147		nn0abscl 13375	. . . . . . . . . . . . . 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 13509	. . . . . . . . . . . . 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 670	. . . . . . . . . . . 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 667	. . . . . . . . . . . 12 |- ( ph -> ( ( abs ` ( A ` 0 ) ) < P <-> ( abs ` ( A ` 0 ) ) <_ ( P - 1 ) ) )
157	154, 156	mpbid 214	. . . . . . . . . . 11 |- ( ph -> ( abs ` ( A ` 0 ) ) <_ ( P - 1 ) )
158	146, 153, 157	jca32 538	. . . . . . . . . 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 216	. . . . . . . . 9 |- ( ph -> ( abs ` ( A ` 0 ) ) e. ( 1 ... ( P - 1 ) ) )
161		fzm1ndvds 14357	. . . . . . . . 9 |- ( ( P e. NN /\ ( abs ` ( A ` 0 ) ) e. ( 1 ... ( P - 1 ) ) ) -> -. P || ( abs ` ( A ` 0 ) ) )
162	26, 160, 161	syl2anc 667	. . . . . . . 8 |- ( ph -> -. P || ( abs ` ( A ` 0 ) ) )
163		dvdsabsb 14322	. . . . . . . . 9 |- ( ( P e. ZZ /\ ( A ` 0 ) e. ZZ ) -> ( P || ( A ` 0 ) <-> P || ( abs ` ( A ` 0 ) ) ) )
164	46, 83, 163	syl2anc 667	. . . . . . . 8 |- ( ph -> ( P || ( A ` 0 ) <-> P || ( abs ` ( A ` 0 ) ) ) )
165	162, 164	mtbird 303	. . . . . . 7 |- ( ph -> -. P || ( A ` 0 ) )
166		etransclem44.mp	. . . . . . . 8 |- ( ph -> ( ! ` M ) < P )
167	28, 24, 166, 30	etransclem41 38140	. . . . . . 7 |- ( ph -> -. P || ( ( ( ( RR Dn F ) ` ( P - 1 ) ) ` 0 ) / ( ! ` ( P - 1 ) ) ) )
168	165, 167	jca 535	. . . . . 6 |- ( ph -> ( -. P || ( A ` 0 ) /\ -. P || ( ( ( ( RR Dn F ) ` ( P - 1 ) ) ` 0 ) / ( ! ` ( P - 1 ) ) ) ) )
169		pm4.56 498	. . . . . 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 200	. . . . 5 |- ( ph -> -. ( P || ( A ` 0 ) \/ P || ( ( ( ( RR Dn F ) ` ( P - 1 ) ) ` 0 ) / ( ! ` ( P - 1 ) ) ) ) )
171		euclemma 14665	. . . . . 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 1268	. . . . 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 303	. . . 4 |- ( ph -> -. P || ( ( A ` 0 ) x. ( ( ( ( RR Dn F ) ` ( P - 1 ) ) ` 0 ) / ( ! ` ( P - 1 ) ) ) ) )
174	106	breq2d 4414	. . . 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 303	. . 3 |- ( ph -> -. P || ( ( ( A ` 0 ) x. ( ( ( RR Dn F ) ` ( P - 1 ) ) ` 0 ) ) / ( ! ` ( P - 1 ) ) ) )
176	46	adantr 467	. . . . . . . 8 |- ( ( ph /\ k e. ( ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) \ { <. 0 , ( P - 1 ) >. } ) ) -> P e. ZZ )
177	176, 125, 138	3jca 1188	. . . . . . 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 3556	. . . . . . . . . 10 |- ( k e. ( ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) \ { <. 0 , ( P - 1 ) >. } ) -> -. k e. { <. 0 , ( P - 1 ) >. } )
179	94	adantr 467	. . . . . . . . . . . . 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 6830	. . . . . . . . . . . . 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 463	. . . . . . . . . . . . . 14 |- ( ( ( 2nd ` k ) = ( P - 1 ) /\ ( 1st ` k ) = 0 ) -> ( 1st ` k ) = 0 )
183		simpl 459	. . . . . . . . . . . . . 14 |- ( ( ( 2nd ` k ) = ( P - 1 ) /\ ( 1st ` k ) = 0 ) -> ( 2nd ` k ) = ( P - 1 ) )
184	182, 183	opeq12d 4174	. . . . . . . . . . . . 13 |- ( ( ( 2nd ` k ) = ( P - 1 ) /\ ( 1st ` k ) = 0 ) -> <. ( 1st ` k ) , ( 2nd ` k ) >. = <. 0 , ( P - 1 ) >. )
185	184	adantl 468	. . . . . . . . . . . 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 2485	. . . . . . . . . . 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 3982	. . . . . . . . . . 11 |- ( k e. { <. 0 , ( P - 1 ) >. } <-> k = <. 0 , ( P - 1 ) >. )
188	186, 187	sylibr 216	. . . . . . . . . 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 665	. . . . . . . . 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 468	. . . . . . . 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 38137	. . . . . . 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 14340	. . . . . . 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 4429	. . . . 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 14348	. . . 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 4429	. . 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 38108	. 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 2691	1 |- ( ph -> K =/= 0 )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 188   \/ wo 370   /\ wa 371   /\ w3a 985   = wceq 1444   e. wcel 1887   =/= wne 2622   {crab 2741   _Vcvv 3045   \ cdif 3401   C_ wss 3404   ifcif 3881   {csn 3968   {cpr 3970   <.cop 3974   class class class wbr 4402   |-> cmpt 4461   X. cxp 4832   ran crn 4835   -->wf 5578   ` cfv 5582  (class class class)co 6290   1stc1st 6791   2ndc2nd 6792   ^m cmap 7472   Fincfn 7569   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 12272   !cfa 12459   abscabs 13297   sum_csu 13752   prod_cprod 13959   || cdvds 14305   Primecprime 14622   ↾_t crest 15319   TopOpenctopn 15320   topGenctg 15336  ℂ_fldccnfld 18970   Dncdvn 22819

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  ax-mulf 9619

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-iin 4281  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-supp 6915  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-2o 7183  df-oadd 7186  df-er 7363  df-map 7474  df-pm 7475  df-ixp 7523  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-fsupp 7884  df-fi 7925  df-sup 7956  df-inf 7957  df-oi 8025  df-card 8373  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 12028  df-mod 12097  df-seq 12214  df-exp 12273  df-fac 12460  df-bc 12488  df-hash 12516  df-cj 13162  df-re 13163  df-im 13164  df-sqrt 13298  df-abs 13299  df-clim 13552  df-sum 13753  df-prod 13960  df-dvds 14306  df-gcd 14469  df-prm 14623  df-struct 15123  df-ndx 15124  df-slot 15125  df-base 15126  df-sets 15127  df-ress 15128  df-plusg 15203  df-mulr 15204  df-starv 15205  df-sca 15206  df-vsca 15207  df-ip 15208  df-tset 15209  df-ple 15210  df-ds 15212  df-unif 15213  df-hom 15214  df-cco 15215  df-rest 15321  df-topn 15322  df-0g 15340  df-gsum 15341  df-topgen 15342  df-pt 15343  df-prds 15346  df-xrs 15400  df-qtop 15406  df-imas 15407  df-xps 15410  df-mre 15492  df-mrc 15493  df-acs 15495  df-mgm 16488  df-sgrp 16527  df-mnd 16537  df-submnd 16583  df-mulg 16676  df-cntz 16971  df-cmn 17432  df-psmet 18962  df-xmet 18963  df-met 18964  df-bl 18965  df-mopn 18966  df-fbas 18967  df-fg 18968  df-cnfld 18971  df-top 19921  df-bases 19922  df-topon 19923  df-topsp 19924  df-cld 20034  df-ntr 20035  df-cls 20036  df-nei 20114  df-lp 20152  df-perf 20153  df-cn 20243  df-cnp 20244  df-haus 20331  df-tx 20577  df-hmeo 20770  df-fil 20861  df-fm 20953  df-flim 20954  df-flf 20955  df-xms 21335  df-ms 21336  df-tms 21337  df-cncf 21910  df-limc 22821  df-dv 22822  df-dvn 22823

This theorem is referenced by:  etransclem47  38146