Theorem etransclem44 38255

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 1769	. . . . 5 |- F/ k ph
4		nfcv 2612	. . . . 5 |- F/_ k ( ( A ` 0 ) x. ( ( ( RR Dn F ) ` ( P - 1 ) ) ` 0 ) )
5		fzfi 12223	. . . . . . 7 |- ( 0 ... M ) e. Fin
6		fzfi 12223	. . . . . . 7 |- ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) e. Fin
7		xpfi 7860	. . . . . . 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 686	. . . . . 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 472	. . . . . . . 8 |- ( ( ph /\ k e. ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) ) -> A : NN0 --> ZZ )
12		fzssnn0 37627	. . . . . . . . . 10 |- ( 0 ... M ) C_ NN0
13		xp1st 6842	. . . . . . . . . 10 |- ( k e. ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) -> ( 1st ` k ) e. ( 0 ... M ) )
14	12, 13	sseldi 3416	. . . . . . . . 9 |- ( k e. ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) -> ( 1st ` k ) e. NN0 )
15	14	adantl 473	. . . . . . . 8 |- ( ( ph /\ k e. ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) ) -> ( 1st ` k ) e. NN0 )
16	11, 15	ffvelrnd 6038	. . . . . . 7 |- ( ( ph /\ k e. ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) ) -> ( A ` ( 1st ` k ) ) e. ZZ )
17		reelprrecn 9649	. . . . . . . . 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 37591	. . . . . . . . . 10 |- RR e. ( topGen ` ran (,) )
20		eqid 2471	. . . . . . . . . . 11 |- ( TopOpen ` ℂfld ) = ( TopOpen ` ℂfld )
21	20	tgioo2 21899	. . . . . . . . . 10 |- ( topGen ` ran (,) ) = ( ( TopOpen ` ℂfld ) ↾t RR )
22	19, 21	eleqtri 2547	. . . . . . . . 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 14704	. . . . . . . . . 10 |- ( P e. Prime -> P e. NN )
26	24, 25	syl 17	. . . . . . . . 9 |- ( ph -> P e. NN )
27	26	adantr 472	. . . . . . . 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 472	. . . . . . . 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 6843	. . . . . . . . . 10 |- ( k e. ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) -> ( 2nd ` k ) e. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) )
32		elfznn0 11913	. . . . . . . . . 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 473	. . . . . . . 8 |- ( ( ph /\ k e. ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) ) -> ( 2nd ` k ) e. NN0 )
35	15	nn0red 10950	. . . . . . . 8 |- ( ( ph /\ k e. ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) ) -> ( 1st ` k ) e. RR )
36	15	nn0zd 11061	. . . . . . . 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 38253	. . . . . . 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 11069	. . . . . 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 11064	. . . . 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 11217	. . . . . . . 8 |- NN0 = ( ZZ>= ` 0 )
41	28, 40	syl6eleq 2559	. . . . . . 7 |- ( ph -> M e. ( ZZ>= ` 0 ) )
42		eluzfz1 11832	. . . . . . 7 |- ( M e. ( ZZ>= ` 0 ) -> 0 e. ( 0 ... M ) )
43	41, 42	syl 17	. . . . . 6 |- ( ph -> 0 e. ( 0 ... M ) )
44		0zd 10973	. . . . . . . . 9 |- ( ph -> 0 e. ZZ )
45	28	nn0zd 11061	. . . . . . . . . . 11 |- ( ph -> M e. ZZ )
46	26	nnzd 11062	. . . . . . . . . . 11 |- ( ph -> P e. ZZ )
47	45, 46	zmulcld 11069	. . . . . . . . . 10 |- ( ph -> ( M x. P ) e. ZZ )
48		nnm1nn0 10935	. . . . . . . . . . . 12 |- ( P e. NN -> ( P - 1 ) e. NN0 )
49	26, 48	syl 17	. . . . . . . . . . 11 |- ( ph -> ( P - 1 ) e. NN0 )
50	49	nn0zd 11061	. . . . . . . . . 10 |- ( ph -> ( P - 1 ) e. ZZ )
51	47, 50	zaddcld 11067	. . . . . . . . 9 |- ( ph -> ( ( M x. P ) + ( P - 1 ) ) e. ZZ )
52	44, 51, 50	3jca 1210	. . . . . . . 8 |- ( ph -> ( 0 e. ZZ /\ ( ( M x. P ) + ( P - 1 ) ) e. ZZ /\ ( P - 1 ) e. ZZ ) )
53	49	nn0ge0d 10952	. . . . . . . 8 |- ( ph -> 0 <_ ( P - 1 ) )
54	26	nnnn0d 10949	. . . . . . . . . . 11 |- ( ph -> P e. NN0 )
55	28, 54	nn0mulcld 10954	. . . . . . . . . 10 |- ( ph -> ( M x. P ) e. NN0 )
56	55	nn0ge0d 10952	. . . . . . . . 9 |- ( ph -> 0 <_ ( M x. P ) )
57	49	nn0red 10950	. . . . . . . . . 10 |- ( ph -> ( P - 1 ) e. RR )
58	47	zred 11063	. . . . . . . . . 10 |- ( ph -> ( M x. P ) e. RR )
59	57, 58	addge02d 10223	. . . . . . . . 9 |- ( ph -> ( 0 <_ ( M x. P ) <-> ( P - 1 ) <_ ( ( M x. P ) + ( P - 1 ) ) ) )
60	56, 59	mpbid 215	. . . . . . . 8 |- ( ph -> ( P - 1 ) <_ ( ( M x. P ) + ( P - 1 ) ) )
61	52, 53, 60	jca32 544	. . . . . . 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 11817	. . . . . . 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 217	. . . . . 6 |- ( ph -> ( P - 1 ) e. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) )
64		opelxp 4869	. . . . . 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 677	. . . . 5 |- ( ph -> <. 0 , ( P - 1 ) >. e. ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) )
66		fveq2 5879	. . . . . . . 8 |- ( k = <. 0 , ( P - 1 ) >. -> ( 1st ` k ) = ( 1st ` <. 0 , ( P - 1 ) >. ) )
67		0re 9661	. . . . . . . . 9 |- 0 e. RR
68		ovex 6336	. . . . . . . . 9 |- ( P - 1 ) e. _V
69		op1stg 6824	. . . . . . . . 9 |- ( ( 0 e. RR /\ ( P - 1 ) e. _V ) -> ( 1st ` <. 0 , ( P - 1 ) >. ) = 0 )
70	67, 68, 69	mp2an 686	. . . . . . . 8 |- ( 1st ` <. 0 , ( P - 1 ) >. ) = 0
71	66, 70	syl6eq 2521	. . . . . . 7 |- ( k = <. 0 , ( P - 1 ) >. -> ( 1st ` k ) = 0 )
72	71	fveq2d 5883	. . . . . 6 |- ( k = <. 0 , ( P - 1 ) >. -> ( A ` ( 1st ` k ) ) = ( A ` 0 ) )
73		fveq2 5879	. . . . . . . . 9 |- ( k = <. 0 , ( P - 1 ) >. -> ( 2nd ` k ) = ( 2nd ` <. 0 , ( P - 1 ) >. ) )
74		op2ndg 6825	. . . . . . . . . 10 |- ( ( 0 e. RR /\ ( P - 1 ) e. _V ) -> ( 2nd ` <. 0 , ( P - 1 ) >. ) = ( P - 1 ) )
75	67, 68, 74	mp2an 686	. . . . . . . . 9 |- ( 2nd ` <. 0 , ( P - 1 ) >. ) = ( P - 1 )
76	73, 75	syl6eq 2521	. . . . . . . 8 |- ( k = <. 0 , ( P - 1 ) >. -> ( 2nd ` k ) = ( P - 1 ) )
77	76	fveq2d 5883	. . . . . . 7 |- ( k = <. 0 , ( P - 1 ) >. -> ( ( RR Dn F ) ` ( 2nd ` k ) ) = ( ( RR Dn F ) ` ( P - 1 ) ) )
78	77, 71	fveq12d 5885	. . . . . 6 |- ( k = <. 0 , ( P - 1 ) >. -> ( ( ( RR Dn F ) ` ( 2nd ` k ) ) ` ( 1st ` k ) ) = ( ( ( RR Dn F ) ` ( P - 1 ) ) ` 0 ) )
79	72, 78	oveq12d 6326	. . . . 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 37747	. . . 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 6323	. . 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 3416	. . . . . . 7 |- ( ph -> 0 e. NN0 )
83	10, 82	ffvelrnd 6038	. . . . . 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 38253	. . . . . 6 |- ( ph -> ( ( ( RR Dn F ) ` ( P - 1 ) ) ` 0 ) e. ZZ )
88	83, 87	zmulcld 11069	. . . . 5 |- ( ph -> ( ( A ` 0 ) x. ( ( ( RR Dn F ) ` ( P - 1 ) ) ` 0 ) ) e. ZZ )
89	88	zcnd 11064	. . . 4 |- ( ph -> ( ( A ` 0 ) x. ( ( ( RR Dn F ) ` ( P - 1 ) ) ` 0 ) ) e. CC )
90		difss 3549	. . . . . . . 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 7810	. . . . . . . 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 686	. . . . . . 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 3544	. . . . . . 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 482	. . . . . 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 13878	. . . . 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 11064	. . . 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 37621	. . . . 5 |- ( ph -> ( ! ` ( P - 1 ) ) e. NN )
99	98	nncnd 10647	. . . 4 |- ( ph -> ( ! ` ( P - 1 ) ) e. CC )
100	98	nnne0d 10676	. . . 4 |- ( ph -> ( ! ` ( P - 1 ) ) =/= 0 )
101	89, 97, 99, 100	divdird 10443	. . 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 2509	. 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 10676	. . 3 |- ( ph -> P =/= 0 )
104	83	zcnd 11064	. . . . 5 |- ( ph -> ( A ` 0 ) e. CC )
105	87	zcnd 11064	. . . . 5 |- ( ph -> ( ( ( RR Dn F ) ` ( P - 1 ) ) ` 0 ) e. CC )
106	104, 105, 99, 100	divassd 10440	. . . 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 38216	. . . . . . 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 38222	. . . . . . 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 38248	. . . . . 6 |- ( ph -> ( ! ` ( P - 1 ) ) || ( ( ( RR Dn F ) ` ( P - 1 ) ) ` 0 ) )
110	98	nnzd 11062	. . . . . . 7 |- ( ph -> ( ! ` ( P - 1 ) ) e. ZZ )
111		dvdsval2 14385	. . . . . . 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 1292	. . . . . 6 |- ( ph -> ( ( ! ` ( P - 1 ) ) || ( ( ( RR Dn F ) ` ( P - 1 ) ) ` 0 ) <-> ( ( ( ( RR Dn F ) ` ( P - 1 ) ) ` 0 ) / ( ! ` ( P - 1 ) ) ) e. ZZ ) )
113	109, 112	mpbid 215	. . . . 5 |- ( ph -> ( ( ( ( RR Dn F ) ` ( P - 1 ) ) ` 0 ) / ( ! ` ( P - 1 ) ) ) e. ZZ )
114	83, 113	zmulcld 11069	. . . 4 |- ( ph -> ( ( A ` 0 ) x. ( ( ( ( RR Dn F ) ` ( P - 1 ) ) ` 0 ) / ( ! ` ( P - 1 ) ) ) ) e. ZZ )
115	106, 114	eqeltrd 2549	. . 3 |- ( ph -> ( ( ( A ` 0 ) x. ( ( ( RR Dn F ) ` ( P - 1 ) ) ` 0 ) ) / ( ! ` ( P - 1 ) ) ) e. ZZ )
116	94, 39	sylan2 482	. . . . 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 13924	. . . 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 11064	. . . . . . . 8 |- ( ( ph /\ k e. ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) ) -> ( A ` ( 1st ` k ) ) e. CC )
119	94, 118	sylan2 482	. . . . . . 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 482	. . . . . . . 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 11064	. . . . . . 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 472	. . . . . . 7 |- ( ( ph /\ k e. ( ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) \ { <. 0 , ( P - 1 ) >. } ) ) -> ( ! ` ( P - 1 ) ) e. CC )
123	100	adantr 472	. . . . . . 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 10440	. . . . . 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 482	. . . . . . 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 472	. . . . . . . . 9 |- ( ( ph /\ k e. ( ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) \ { <. 0 , ( P - 1 ) >. } ) ) -> P e. NN )
129	28	adantr 472	. . . . . . . . 9 |- ( ( ph /\ k e. ( ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) \ { <. 0 , ( P - 1 ) >. } ) ) -> M e. NN0 )
130	94	adantl 473	. . . . . . . . . 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 482	. . . . . . . . 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 38248	. . . . . . . 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 472	. . . . . . . . 9 |- ( ( ph /\ k e. ( ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) \ { <. 0 , ( P - 1 ) >. } ) ) -> ( ! ` ( P - 1 ) ) e. ZZ )
136		dvdsval2 14385	. . . . . . . . 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 1292	. . . . . . . 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 215	. . . . . . 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 11069	. . . . . 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 2549	. . . . 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 13878	. . . 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 2549	. . 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 10992	. . . . . . . . . . . 12 |- ( ph -> 1 e. ZZ )
144		zabscl 13453	. . . . . . . . . . . . 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 1210	. . . . . . . . . . 11 |- ( ph -> ( 1 e. ZZ /\ ( P - 1 ) e. ZZ /\ ( abs ` ( A ` 0 ) ) e. ZZ ) )
147		nn0abscl 13452	. . . . . . . . . . . . . 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 13586	. . . . . . . . . . . . 13 |- ( ph -> ( abs ` ( A ` 0 ) ) =/= 0 )
151		elnnne0 10907	. . . . . . . . . . . . 13 |- ( ( abs ` ( A ` 0 ) ) e. NN <-> ( ( abs ` ( A ` 0 ) ) e. NN0 /\ ( abs ` ( A ` 0 ) ) =/= 0 ) )
152	148, 150, 151	sylanbrc 677	. . . . . . . . . . . 12 |- ( ph -> ( abs ` ( A ` 0 ) ) e. NN )
153	152	nnge1d 10674	. . . . . . . . . . 11 |- ( ph -> 1 <_ ( abs ` ( A ` 0 ) ) )
154		etransclem44.ap	. . . . . . . . . . . 12 |- ( ph -> ( abs ` ( A ` 0 ) ) < P )
155		zltlem1 11013	. . . . . . . . . . . . 13 |- ( ( ( abs ` ( A ` 0 ) ) e. ZZ /\ P e. ZZ ) -> ( ( abs ` ( A ` 0 ) ) < P <-> ( abs ` ( A ` 0 ) ) <_ ( P - 1 ) ) )
156	145, 46, 155	syl2anc 673	. . . . . . . . . . . 12 |- ( ph -> ( ( abs ` ( A ` 0 ) ) < P <-> ( abs ` ( A ` 0 ) ) <_ ( P - 1 ) ) )
157	154, 156	mpbid 215	. . . . . . . . . . 11 |- ( ph -> ( abs ` ( A ` 0 ) ) <_ ( P - 1 ) )
158	146, 153, 157	jca32 544	. . . . . . . . . 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 11817	. . . . . . . . . 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 217	. . . . . . . . 9 |- ( ph -> ( abs ` ( A ` 0 ) ) e. ( 1 ... ( P - 1 ) ) )
161		fzm1ndvds 14434	. . . . . . . . 9 |- ( ( P e. NN /\ ( abs ` ( A ` 0 ) ) e. ( 1 ... ( P - 1 ) ) ) -> -. P || ( abs ` ( A ` 0 ) ) )
162	26, 160, 161	syl2anc 673	. . . . . . . 8 |- ( ph -> -. P || ( abs ` ( A ` 0 ) ) )
163		dvdsabsb 14399	. . . . . . . . 9 |- ( ( P e. ZZ /\ ( A ` 0 ) e. ZZ ) -> ( P || ( A ` 0 ) <-> P || ( abs ` ( A ` 0 ) ) ) )
164	46, 83, 163	syl2anc 673	. . . . . . . 8 |- ( ph -> ( P || ( A ` 0 ) <-> P || ( abs ` ( A ` 0 ) ) ) )
165	162, 164	mtbird 308	. . . . . . 7 |- ( ph -> -. P || ( A ` 0 ) )
166		etransclem44.mp	. . . . . . . 8 |- ( ph -> ( ! ` M ) < P )
167	28, 24, 166, 30	etransclem41 38252	. . . . . . 7 |- ( ph -> -. P || ( ( ( ( RR Dn F ) ` ( P - 1 ) ) ` 0 ) / ( ! ` ( P - 1 ) ) ) )
168	165, 167	jca 541	. . . . . 6 |- ( ph -> ( -. P || ( A ` 0 ) /\ -. P || ( ( ( ( RR Dn F ) ` ( P - 1 ) ) ` 0 ) / ( ! ` ( P - 1 ) ) ) ) )
169		pm4.56 503	. . . . . 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 201	. . . . 5 |- ( ph -> -. ( P || ( A ` 0 ) \/ P || ( ( ( ( RR Dn F ) ` ( P - 1 ) ) ` 0 ) / ( ! ` ( P - 1 ) ) ) ) )
171		euclemma 14744	. . . . . 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 1292	. . . . 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 308	. . . 4 |- ( ph -> -. P || ( ( A ` 0 ) x. ( ( ( ( RR Dn F ) ` ( P - 1 ) ) ` 0 ) / ( ! ` ( P - 1 ) ) ) ) )
174	106	breq2d 4407	. . . 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 308	. . 3 |- ( ph -> -. P || ( ( ( A ` 0 ) x. ( ( ( RR Dn F ) ` ( P - 1 ) ) ` 0 ) ) / ( ! ` ( P - 1 ) ) ) )
176	46	adantr 472	. . . . . . . 8 |- ( ( ph /\ k e. ( ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) \ { <. 0 , ( P - 1 ) >. } ) ) -> P e. ZZ )
177	176, 125, 138	3jca 1210	. . . . . . 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 3545	. . . . . . . . . 10 |- ( k e. ( ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) \ { <. 0 , ( P - 1 ) >. } ) -> -. k e. { <. 0 , ( P - 1 ) >. } )
179	94	adantr 472	. . . . . . . . . . . . 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 6849	. . . . . . . . . . . . 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 468	. . . . . . . . . . . . . 14 |- ( ( ( 2nd ` k ) = ( P - 1 ) /\ ( 1st ` k ) = 0 ) -> ( 1st ` k ) = 0 )
183		simpl 464	. . . . . . . . . . . . . 14 |- ( ( ( 2nd ` k ) = ( P - 1 ) /\ ( 1st ` k ) = 0 ) -> ( 2nd ` k ) = ( P - 1 ) )
184	182, 183	opeq12d 4166	. . . . . . . . . . . . 13 |- ( ( ( 2nd ` k ) = ( P - 1 ) /\ ( 1st ` k ) = 0 ) -> <. ( 1st ` k ) , ( 2nd ` k ) >. = <. 0 , ( P - 1 ) >. )
185	184	adantl 473	. . . . . . . . . . . 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 2505	. . . . . . . . . . 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 3973	. . . . . . . . . . 11 |- ( k e. { <. 0 , ( P - 1 ) >. } <-> k = <. 0 , ( P - 1 ) >. )
188	186, 187	sylibr 217	. . . . . . . . . 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 671	. . . . . . . . 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 473	. . . . . . . 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 38249	. . . . . . 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 14417	. . . . . . 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 61	. . . . . 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 4422	. . . . 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 14425	. . . 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 4422	. . 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 38220	. 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 2710	1 |- ( ph -> K =/= 0 )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 189   \/ wo 375   /\ wa 376   /\ w3a 1007   = wceq 1452   e. wcel 1904   =/= wne 2641   {crab 2760   _Vcvv 3031   \ cdif 3387   C_ wss 3390   ifcif 3872   {csn 3959   {cpr 3961   <.cop 3965   class class class wbr 4395   |-> cmpt 4454   X. cxp 4837   ran crn 4840   -->wf 5585   ` cfv 5589  (class class class)co 6308   1stc1st 6810   2ndc2nd 6811   ^m cmap 7490   Fincfn 7587   CCcc 9555   RRcr 9556   0cc0 9557   1c1 9558   + caddc 9560   x. cmul 9562   < clt 9693   <_ cle 9694   - cmin 9880   / cdiv 10291   NNcn 10631   NN0cn0 10893   ZZcz 10961   ZZ>=cuz 11182   (,)cioo 11660   ...cfz 11810   ^cexp 12310   !cfa 12497   abscabs 13374   sum_csu 13829   prod_cprod 14036   || cdvds 14382   Primecprime 14701   ↾_t crest 15397   TopOpenctopn 15398   topGenctg 15414  ℂ_fldccnfld 19047   Dncdvn 22898

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635  ax-addf 9636  ax-mulf 9637

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-iin 4272  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-of 6550  df-om 6712  df-1st 6812  df-2nd 6813  df-supp 6934  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-2o 7201  df-oadd 7204  df-er 7381  df-map 7492  df-pm 7493  df-ixp 7541  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-fsupp 7902  df-fi 7943  df-sup 7974  df-inf 7975  df-oi 8043  df-card 8391  df-cda 8616  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-4 10692  df-5 10693  df-6 10694  df-7 10695  df-8 10696  df-9 10697  df-10 10698  df-n0 10894  df-z 10962  df-dec 11075  df-uz 11183  df-q 11288  df-rp 11326  df-xneg 11432  df-xadd 11433  df-xmul 11434  df-ioo 11664  df-ico 11666  df-icc 11667  df-fz 11811  df-fzo 11943  df-fl 12061  df-mod 12130  df-seq 12252  df-exp 12311  df-fac 12498  df-bc 12526  df-hash 12554  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-clim 13629  df-sum 13830  df-prod 14037  df-dvds 14383  df-gcd 14548  df-prm 14702  df-struct 15201  df-ndx 15202  df-slot 15203  df-base 15204  df-sets 15205  df-ress 15206  df-plusg 15281  df-mulr 15282  df-starv 15283  df-sca 15284  df-vsca 15285  df-ip 15286  df-tset 15287  df-ple 15288  df-ds 15290  df-unif 15291  df-hom 15292  df-cco 15293  df-rest 15399  df-topn 15400  df-0g 15418  df-gsum 15419  df-topgen 15420  df-pt 15421  df-prds 15424  df-xrs 15478  df-qtop 15484  df-imas 15485  df-xps 15488  df-mre 15570  df-mrc 15571  df-acs 15573  df-mgm 16566  df-sgrp 16605  df-mnd 16615  df-submnd 16661  df-mulg 16754  df-cntz 17049  df-cmn 17510  df-psmet 19039  df-xmet 19040  df-met 19041  df-bl 19042  df-mopn 19043  df-fbas 19044  df-fg 19045  df-cnfld 19048  df-top 19998  df-bases 19999  df-topon 20000  df-topsp 20001  df-cld 20111  df-ntr 20112  df-cls 20113  df-nei 20191  df-lp 20229  df-perf 20230  df-cn 20320  df-cnp 20321  df-haus 20408  df-tx 20654  df-hmeo 20847  df-fil 20939  df-fm 21031  df-flim 21032  df-flf 21033  df-xms 21413  df-ms 21414  df-tms 21415  df-cncf 21988  df-limc 22900  df-dv 22901  df-dvn 22902

This theorem is referenced by:  etransclem47  38258