Theorem etransclem44 32300

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 1712	. . . . 5 |- F/ k ph
4		nfcv 2616	. . . . 5 |- F/_ k ( ( A ` 0 ) x. ( ( ( RR Dn F ) ` ( P - 1 ) ) ` 0 ) )
5		fzfi 12064	. . . . . . 7 |- ( 0 ... M ) e. Fin
6		fzfi 12064	. . . . . . 7 |- ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) e. Fin
7		xpfi 7783	. . . . . . 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 670	. . . . . 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 463	. . . . . . . 8 |- ( ( ph /\ k e. ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) ) -> A : NN0 --> ZZ )
12		fzssnn0 31761	. . . . . . . . . 10 |- ( 0 ... M ) C_ NN0
13		xp1st 6803	. . . . . . . . . 10 |- ( k e. ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) -> ( 1st ` k ) e. ( 0 ... M ) )
14	12, 13	sseldi 3487	. . . . . . . . 9 |- ( k e. ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) -> ( 1st ` k ) e. NN0 )
15	14	adantl 464	. . . . . . . 8 |- ( ( ph /\ k e. ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) ) -> ( 1st ` k ) e. NN0 )
16	11, 15	ffvelrnd 6008	. . . . . . 7 |- ( ( ph /\ k e. ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) ) -> ( A ` ( 1st ` k ) ) e. ZZ )
17		reelprrecn 9573	. . . . . . . . 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 31716	. . . . . . . . . 10 |- RR e. ( topGen ` ran (,) )
20		eqid 2454	. . . . . . . . . . 11 |- ( TopOpen ` ℂfld ) = ( TopOpen ` ℂfld )
21	20	tgioo2 21474	. . . . . . . . . 10 |- ( topGen ` ran (,) ) = ( ( TopOpen ` ℂfld ) ↾t RR )
22	19, 21	eleqtri 2540	. . . . . . . . 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 14304	. . . . . . . . . 10 |- ( P e. Prime -> P e. NN )
26	24, 25	syl 16	. . . . . . . . 9 |- ( ph -> P e. NN )
27	26	adantr 463	. . . . . . . 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 463	. . . . . . . 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 6804	. . . . . . . . . 10 |- ( k e. ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) -> ( 2nd ` k ) e. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) )
32		elfznn0 11775	. . . . . . . . . 10 |- ( ( 2nd ` k ) e. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) -> ( 2nd ` k ) e. NN0 )
33	31, 32	syl 16	. . . . . . . . 9 |- ( k e. ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) -> ( 2nd ` k ) e. NN0 )
34	33	adantl 464	. . . . . . . 8 |- ( ( ph /\ k e. ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) ) -> ( 2nd ` k ) e. NN0 )
35	15	nn0red 10849	. . . . . . . 8 |- ( ( ph /\ k e. ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) ) -> ( 1st ` k ) e. RR )
36	15	nn0zd 10963	. . . . . . . 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 32298	. . . . . . 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 10971	. . . . . 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 10966	. . . . 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 11116	. . . . . . . 8 |- NN0 = ( ZZ>= ` 0 )
41	28, 40	syl6eleq 2552	. . . . . . 7 |- ( ph -> M e. ( ZZ>= ` 0 ) )
42		eluzfz1 11696	. . . . . . 7 |- ( M e. ( ZZ>= ` 0 ) -> 0 e. ( 0 ... M ) )
43	41, 42	syl 16	. . . . . 6 |- ( ph -> 0 e. ( 0 ... M ) )
44		0zd 10872	. . . . . . . . 9 |- ( ph -> 0 e. ZZ )
45	28	nn0zd 10963	. . . . . . . . . . 11 |- ( ph -> M e. ZZ )
46	26	nnzd 10964	. . . . . . . . . . 11 |- ( ph -> P e. ZZ )
47	45, 46	zmulcld 10971	. . . . . . . . . 10 |- ( ph -> ( M x. P ) e. ZZ )
48		nnm1nn0 10833	. . . . . . . . . . . 12 |- ( P e. NN -> ( P - 1 ) e. NN0 )
49	26, 48	syl 16	. . . . . . . . . . 11 |- ( ph -> ( P - 1 ) e. NN0 )
50	49	nn0zd 10963	. . . . . . . . . 10 |- ( ph -> ( P - 1 ) e. ZZ )
51	47, 50	zaddcld 10969	. . . . . . . . 9 |- ( ph -> ( ( M x. P ) + ( P - 1 ) ) e. ZZ )
52	44, 51, 50	3jca 1174	. . . . . . . 8 |- ( ph -> ( 0 e. ZZ /\ ( ( M x. P ) + ( P - 1 ) ) e. ZZ /\ ( P - 1 ) e. ZZ ) )
53	49	nn0ge0d 10851	. . . . . . . 8 |- ( ph -> 0 <_ ( P - 1 ) )
54	26	nnnn0d 10848	. . . . . . . . . . 11 |- ( ph -> P e. NN0 )
55	28, 54	nn0mulcld 10853	. . . . . . . . . 10 |- ( ph -> ( M x. P ) e. NN0 )
56	55	nn0ge0d 10851	. . . . . . . . 9 |- ( ph -> 0 <_ ( M x. P ) )
57	49	nn0red 10849	. . . . . . . . . 10 |- ( ph -> ( P - 1 ) e. RR )
58	47	zred 10965	. . . . . . . . . 10 |- ( ph -> ( M x. P ) e. RR )
59	57, 58	addge02d 10137	. . . . . . . . 9 |- ( ph -> ( 0 <_ ( M x. P ) <-> ( P - 1 ) <_ ( ( M x. P ) + ( P - 1 ) ) ) )
60	56, 59	mpbid 210	. . . . . . . 8 |- ( ph -> ( P - 1 ) <_ ( ( M x. P ) + ( P - 1 ) ) )
61	52, 53, 60	jca32 533	. . . . . . 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 11682	. . . . . . 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 212	. . . . . 6 |- ( ph -> ( P - 1 ) e. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) )
64		opelxp 5018	. . . . . 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 662	. . . . 5 |- ( ph -> <. 0 , ( P - 1 ) >. e. ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) )
66		fveq2 5848	. . . . . . . 8 |- ( k = <. 0 , ( P - 1 ) >. -> ( 1st ` k ) = ( 1st ` <. 0 , ( P - 1 ) >. ) )
67		0re 9585	. . . . . . . . 9 |- 0 e. RR
68		ovex 6298	. . . . . . . . 9 |- ( P - 1 ) e. _V
69		op1stg 6785	. . . . . . . . 9 |- ( ( 0 e. RR /\ ( P - 1 ) e. _V ) -> ( 1st ` <. 0 , ( P - 1 ) >. ) = 0 )
70	67, 68, 69	mp2an 670	. . . . . . . 8 |- ( 1st ` <. 0 , ( P - 1 ) >. ) = 0
71	66, 70	syl6eq 2511	. . . . . . 7 |- ( k = <. 0 , ( P - 1 ) >. -> ( 1st ` k ) = 0 )
72	71	fveq2d 5852	. . . . . 6 |- ( k = <. 0 , ( P - 1 ) >. -> ( A ` ( 1st ` k ) ) = ( A ` 0 ) )
73		fveq2 5848	. . . . . . . . 9 |- ( k = <. 0 , ( P - 1 ) >. -> ( 2nd ` k ) = ( 2nd ` <. 0 , ( P - 1 ) >. ) )
74		op2ndg 6786	. . . . . . . . . 10 |- ( ( 0 e. RR /\ ( P - 1 ) e. _V ) -> ( 2nd ` <. 0 , ( P - 1 ) >. ) = ( P - 1 ) )
75	67, 68, 74	mp2an 670	. . . . . . . . 9 |- ( 2nd ` <. 0 , ( P - 1 ) >. ) = ( P - 1 )
76	73, 75	syl6eq 2511	. . . . . . . 8 |- ( k = <. 0 , ( P - 1 ) >. -> ( 2nd ` k ) = ( P - 1 ) )
77	76	fveq2d 5852	. . . . . . 7 |- ( k = <. 0 , ( P - 1 ) >. -> ( ( RR Dn F ) ` ( 2nd ` k ) ) = ( ( RR Dn F ) ` ( P - 1 ) ) )
78	77, 71	fveq12d 5854	. . . . . 6 |- ( k = <. 0 , ( P - 1 ) >. -> ( ( ( RR Dn F ) ` ( 2nd ` k ) ) ` ( 1st ` k ) ) = ( ( ( RR Dn F ) ` ( P - 1 ) ) ` 0 ) )
79	72, 78	oveq12d 6288	. . . . 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 31812	. . . 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 6285	. . 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 3487	. . . . . . 7 |- ( ph -> 0 e. NN0 )
83	10, 82	ffvelrnd 6008	. . . . . 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 32298	. . . . . 6 |- ( ph -> ( ( ( RR Dn F ) ` ( P - 1 ) ) ` 0 ) e. ZZ )
88	83, 87	zmulcld 10971	. . . . 5 |- ( ph -> ( ( A ` 0 ) x. ( ( ( RR Dn F ) ` ( P - 1 ) ) ` 0 ) ) e. ZZ )
89	88	zcnd 10966	. . . 4 |- ( ph -> ( ( A ` 0 ) x. ( ( ( RR Dn F ) ` ( P - 1 ) ) ` 0 ) ) e. CC )
90		difss 3617	. . . . . . . 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 7733	. . . . . . . 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 670	. . . . . . 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 3612	. . . . . . 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 472	. . . . . 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 13639	. . . . 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 10966	. . . 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 31755	. . . . 5 |- ( ph -> ( ! ` ( P - 1 ) ) e. NN )
99	98	nncnd 10547	. . . 4 |- ( ph -> ( ! ` ( P - 1 ) ) e. CC )
100	98	nnne0d 10576	. . . 4 |- ( ph -> ( ! ` ( P - 1 ) ) =/= 0 )
101	89, 97, 99, 100	divdird 10354	. . 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 2499	. 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 10576	. . 3 |- ( ph -> P =/= 0 )
104	83	zcnd 10966	. . . . 5 |- ( ph -> ( A ` 0 ) e. CC )
105	87	zcnd 10966	. . . . 5 |- ( ph -> ( ( ( RR Dn F ) ` ( P - 1 ) ) ` 0 ) e. CC )
106	104, 105, 99, 100	divassd 10351	. . . 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 32261	. . . . . . 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 32267	. . . . . . 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 32293	. . . . . 6 |- ( ph -> ( ! ` ( P - 1 ) ) || ( ( ( RR Dn F ) ` ( P - 1 ) ) ` 0 ) )
110	98	nnzd 10964	. . . . . . 7 |- ( ph -> ( ! ` ( P - 1 ) ) e. ZZ )
111		dvdsval2 14073	. . . . . . 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 1226	. . . . . 6 |- ( ph -> ( ( ! ` ( P - 1 ) ) || ( ( ( RR Dn F ) ` ( P - 1 ) ) ` 0 ) <-> ( ( ( ( RR Dn F ) ` ( P - 1 ) ) ` 0 ) / ( ! ` ( P - 1 ) ) ) e. ZZ ) )
113	109, 112	mpbid 210	. . . . 5 |- ( ph -> ( ( ( ( RR Dn F ) ` ( P - 1 ) ) ` 0 ) / ( ! ` ( P - 1 ) ) ) e. ZZ )
114	83, 113	zmulcld 10971	. . . 4 |- ( ph -> ( ( A ` 0 ) x. ( ( ( ( RR Dn F ) ` ( P - 1 ) ) ` 0 ) / ( ! ` ( P - 1 ) ) ) ) e. ZZ )
115	106, 114	eqeltrd 2542	. . 3 |- ( ph -> ( ( ( A ` 0 ) x. ( ( ( RR Dn F ) ` ( P - 1 ) ) ` 0 ) ) / ( ! ` ( P - 1 ) ) ) e. ZZ )
116	94, 39	sylan2 472	. . . . 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 13683	. . . 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 10966	. . . . . . . 8 |- ( ( ph /\ k e. ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) ) -> ( A ` ( 1st ` k ) ) e. CC )
119	94, 118	sylan2 472	. . . . . . 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 472	. . . . . . . 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 10966	. . . . . . 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 463	. . . . . . 7 |- ( ( ph /\ k e. ( ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) \ { <. 0 , ( P - 1 ) >. } ) ) -> ( ! ` ( P - 1 ) ) e. CC )
123	100	adantr 463	. . . . . . 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 10351	. . . . . 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 472	. . . . . . 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 463	. . . . . . . . 9 |- ( ( ph /\ k e. ( ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) \ { <. 0 , ( P - 1 ) >. } ) ) -> P e. NN )
129	28	adantr 463	. . . . . . . . 9 |- ( ( ph /\ k e. ( ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) \ { <. 0 , ( P - 1 ) >. } ) ) -> M e. NN0 )
130	94	adantl 464	. . . . . . . . . 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 16	. . . . . . . . 9 |- ( ( ph /\ k e. ( ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) \ { <. 0 , ( P - 1 ) >. } ) ) -> ( 2nd ` k ) e. NN0 )
132	130, 13	syl 16	. . . . . . . . 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 472	. . . . . . . . 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 32293	. . . . . . . 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 463	. . . . . . . . 9 |- ( ( ph /\ k e. ( ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) \ { <. 0 , ( P - 1 ) >. } ) ) -> ( ! ` ( P - 1 ) ) e. ZZ )
136		dvdsval2 14073	. . . . . . . . 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 1226	. . . . . . . 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 210	. . . . . . 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 10971	. . . . . 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 2542	. . . . 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 13639	. . . 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 2542	. . 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 10891	. . . . . . . . . . . 12 |- ( ph -> 1 e. ZZ )
144		zabscl 13228	. . . . . . . . . . . . 13 |- ( ( A ` 0 ) e. ZZ -> ( abs ` ( A ` 0 ) ) e. ZZ )
145	83, 144	syl 16	. . . . . . . . . . . 12 |- ( ph -> ( abs ` ( A ` 0 ) ) e. ZZ )
146	143, 50, 145	3jca 1174	. . . . . . . . . . 11 |- ( ph -> ( 1 e. ZZ /\ ( P - 1 ) e. ZZ /\ ( abs ` ( A ` 0 ) ) e. ZZ ) )
147		nn0abscl 13227	. . . . . . . . . . . . . 14 |- ( ( A ` 0 ) e. ZZ -> ( abs ` ( A ` 0 ) ) e. NN0 )
148	83, 147	syl 16	. . . . . . . . . . . . 13 |- ( ph -> ( abs ` ( A ` 0 ) ) e. NN0 )
149		etransclem44.a0	. . . . . . . . . . . . . 14 |- ( ph -> ( A ` 0 ) =/= 0 )
150	104, 149	absne0d 13360	. . . . . . . . . . . . 13 |- ( ph -> ( abs ` ( A ` 0 ) ) =/= 0 )
151		elnnne0 10805	. . . . . . . . . . . . 13 |- ( ( abs ` ( A ` 0 ) ) e. NN <-> ( ( abs ` ( A ` 0 ) ) e. NN0 /\ ( abs ` ( A ` 0 ) ) =/= 0 ) )
152	148, 150, 151	sylanbrc 662	. . . . . . . . . . . 12 |- ( ph -> ( abs ` ( A ` 0 ) ) e. NN )
153	152	nnge1d 10574	. . . . . . . . . . 11 |- ( ph -> 1 <_ ( abs ` ( A ` 0 ) ) )
154		etransclem44.ap	. . . . . . . . . . . 12 |- ( ph -> ( abs ` ( A ` 0 ) ) < P )
155		zltlem1 10912	. . . . . . . . . . . . 13 |- ( ( ( abs ` ( A ` 0 ) ) e. ZZ /\ P e. ZZ ) -> ( ( abs ` ( A ` 0 ) ) < P <-> ( abs ` ( A ` 0 ) ) <_ ( P - 1 ) ) )
156	145, 46, 155	syl2anc 659	. . . . . . . . . . . 12 |- ( ph -> ( ( abs ` ( A ` 0 ) ) < P <-> ( abs ` ( A ` 0 ) ) <_ ( P - 1 ) ) )
157	154, 156	mpbid 210	. . . . . . . . . . 11 |- ( ph -> ( abs ` ( A ` 0 ) ) <_ ( P - 1 ) )
158	146, 153, 157	jca32 533	. . . . . . . . . 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 11682	. . . . . . . . . 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 212	. . . . . . . . 9 |- ( ph -> ( abs ` ( A ` 0 ) ) e. ( 1 ... ( P - 1 ) ) )
161		fzm1ndvds 14122	. . . . . . . . 9 |- ( ( P e. NN /\ ( abs ` ( A ` 0 ) ) e. ( 1 ... ( P - 1 ) ) ) -> -. P || ( abs ` ( A ` 0 ) ) )
162	26, 160, 161	syl2anc 659	. . . . . . . 8 |- ( ph -> -. P || ( abs ` ( A ` 0 ) ) )
163		dvdsabsb 14087	. . . . . . . . 9 |- ( ( P e. ZZ /\ ( A ` 0 ) e. ZZ ) -> ( P || ( A ` 0 ) <-> P || ( abs ` ( A ` 0 ) ) ) )
164	46, 83, 163	syl2anc 659	. . . . . . . 8 |- ( ph -> ( P || ( A ` 0 ) <-> P || ( abs ` ( A ` 0 ) ) ) )
165	162, 164	mtbird 299	. . . . . . 7 |- ( ph -> -. P || ( A ` 0 ) )
166		etransclem44.mp	. . . . . . . 8 |- ( ph -> ( ! ` M ) < P )
167	28, 24, 166, 30	etransclem41 32297	. . . . . . 7 |- ( ph -> -. P || ( ( ( ( RR Dn F ) ` ( P - 1 ) ) ` 0 ) / ( ! ` ( P - 1 ) ) ) )
168	165, 167	jca 530	. . . . . 6 |- ( ph -> ( -. P || ( A ` 0 ) /\ -. P || ( ( ( ( RR Dn F ) ` ( P - 1 ) ) ` 0 ) / ( ! ` ( P - 1 ) ) ) ) )
169		pm4.56 493	. . . . . 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 196	. . . . 5 |- ( ph -> -. ( P || ( A ` 0 ) \/ P || ( ( ( ( RR Dn F ) ` ( P - 1 ) ) ` 0 ) / ( ! ` ( P - 1 ) ) ) ) )
171		euclemma 14333	. . . . . 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 1226	. . . . 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 299	. . . 4 |- ( ph -> -. P || ( ( A ` 0 ) x. ( ( ( ( RR Dn F ) ` ( P - 1 ) ) ` 0 ) / ( ! ` ( P - 1 ) ) ) ) )
174	106	breq2d 4451	. . . 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 299	. . 3 |- ( ph -> -. P || ( ( ( A ` 0 ) x. ( ( ( RR Dn F ) ` ( P - 1 ) ) ` 0 ) ) / ( ! ` ( P - 1 ) ) ) )
176	46	adantr 463	. . . . . . . 8 |- ( ( ph /\ k e. ( ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) \ { <. 0 , ( P - 1 ) >. } ) ) -> P e. ZZ )
177	176, 125, 138	3jca 1174	. . . . . . 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 3613	. . . . . . . . . 10 |- ( k e. ( ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) \ { <. 0 , ( P - 1 ) >. } ) -> -. k e. { <. 0 , ( P - 1 ) >. } )
179	94	adantr 463	. . . . . . . . . . . . 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 6810	. . . . . . . . . . . . 13 |- ( k e. ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) -> k = <. ( 1st ` k ) , ( 2nd ` k ) >. )
181	179, 180	syl 16	. . . . . . . . . . . 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 459	. . . . . . . . . . . . . 14 |- ( ( ( 2nd ` k ) = ( P - 1 ) /\ ( 1st ` k ) = 0 ) -> ( 1st ` k ) = 0 )
183		simpl 455	. . . . . . . . . . . . . 14 |- ( ( ( 2nd ` k ) = ( P - 1 ) /\ ( 1st ` k ) = 0 ) -> ( 2nd ` k ) = ( P - 1 ) )
184	182, 183	opeq12d 4211	. . . . . . . . . . . . 13 |- ( ( ( 2nd ` k ) = ( P - 1 ) /\ ( 1st ` k ) = 0 ) -> <. ( 1st ` k ) , ( 2nd ` k ) >. = <. 0 , ( P - 1 ) >. )
185	184	adantl 464	. . . . . . . . . . . 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 2495	. . . . . . . . . . 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 4030	. . . . . . . . . . 11 |- ( k e. { <. 0 , ( P - 1 ) >. } <-> k = <. 0 , ( P - 1 ) >. )
188	186, 187	sylibr 212	. . . . . . . . . 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 657	. . . . . . . . 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 464	. . . . . . . 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 32294	. . . . . . 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 14105	. . . . . . 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 60	. . . . . 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 4465	. . . . 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 14113	. . . 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 4465	. . 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 32265	. 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 2747	1 |- ( ph -> K =/= 0 )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   \/ wo 366   /\ wa 367   /\ w3a 971   = wceq 1398   e. wcel 1823   =/= wne 2649   {crab 2808   _Vcvv 3106   \ cdif 3458   C_ wss 3461   ifcif 3929   {csn 4016   {cpr 4018   <.cop 4022   class class class wbr 4439   |-> cmpt 4497   X. cxp 4986   ran crn 4989   -->wf 5566   ` cfv 5570  (class class class)co 6270   1stc1st 6771   2ndc2nd 6772   ^m cmap 7412   Fincfn 7509   CCcc 9479   RRcr 9480   0cc0 9481   1c1 9482   + caddc 9484   x. cmul 9486   < clt 9617   <_ cle 9618   - cmin 9796   / cdiv 10202   NNcn 10531   NN0cn0 10791   ZZcz 10860   ZZ>=cuz 11082   (,)cioo 11532   ...cfz 11675   ^cexp 12148   !cfa 12335   abscabs 13149   sum_csu 13590   prod_cprod 13794   || cdvds 14070   Primecprime 14301   ↾_t crest 14910   TopOpenctopn 14911   topGenctg 14927  ℂ_fldccnfld 18615   Dncdvn 22434

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-inf2 8049  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559  ax-addf 9560  ax-mulf 9561

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-fal 1404  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-iin 4318  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-se 4828  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-isom 5579  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-of 6513  df-om 6674  df-1st 6773  df-2nd 6774  df-supp 6892  df-recs 7034  df-rdg 7068  df-1o 7122  df-2o 7123  df-oadd 7126  df-er 7303  df-map 7414  df-pm 7415  df-ixp 7463  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-fsupp 7822  df-fi 7863  df-sup 7893  df-oi 7927  df-card 8311  df-cda 8539  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-nn 10532  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-n0 10792  df-z 10861  df-dec 10977  df-uz 11083  df-q 11184  df-rp 11222  df-xneg 11321  df-xadd 11322  df-xmul 11323  df-ioo 11536  df-ico 11538  df-icc 11539  df-fz 11676  df-fzo 11800  df-fl 11910  df-mod 11979  df-seq 12090  df-exp 12149  df-fac 12336  df-bc 12363  df-hash 12388  df-cj 13014  df-re 13015  df-im 13016  df-sqrt 13150  df-abs 13151  df-clim 13393  df-sum 13591  df-prod 13795  df-dvds 14071  df-gcd 14229  df-prm 14302  df-struct 14718  df-ndx 14719  df-slot 14720  df-base 14721  df-sets 14722  df-ress 14723  df-plusg 14797  df-mulr 14798  df-starv 14799  df-sca 14800  df-vsca 14801  df-ip 14802  df-tset 14803  df-ple 14804  df-ds 14806  df-unif 14807  df-hom 14808  df-cco 14809  df-rest 14912  df-topn 14913  df-0g 14931  df-gsum 14932  df-topgen 14933  df-pt 14934  df-prds 14937  df-xrs 14991  df-qtop 14996  df-imas 14997  df-xps 14999  df-mre 15075  df-mrc 15076  df-acs 15078  df-mgm 16071  df-sgrp 16110  df-mnd 16120  df-submnd 16166  df-mulg 16259  df-cntz 16554  df-cmn 16999  df-psmet 18606  df-xmet 18607  df-met 18608  df-bl 18609  df-mopn 18610  df-fbas 18611  df-fg 18612  df-cnfld 18616  df-top 19566  df-bases 19568  df-topon 19569  df-topsp 19570  df-cld 19687  df-ntr 19688  df-cls 19689  df-nei 19766  df-lp 19804  df-perf 19805  df-cn 19895  df-cnp 19896  df-haus 19983  df-tx 20229  df-hmeo 20422  df-fil 20513  df-fm 20605  df-flim 20606  df-flf 20607  df-xms 20989  df-ms 20990  df-tms 20991  df-cncf 21548  df-limc 22436  df-dv 22437  df-dvn 22438

This theorem is referenced by:  etransclem47  32303