Theorem etransclem44 40495

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:   x, k   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 1843	. . . . 5 |- F/ k ph
4		nfcv 2764	. . . . 5 |- F/_ k ( ( A ` 0 ) x. ( ( ( RR Dn F ) ` ( P - 1 ) ) ` 0 ) )
5		fzfi 12771	. . . . . . 7 |- ( 0 ... M ) e. Fin
6		fzfi 12771	. . . . . . 7 |- ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) e. Fin
7		xpfi 8231	. . . . . . 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 708	. . . . . 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 481	. . . . . . . 8 |- ( ( ph /\ k e. ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) ) -> A : NN0 --> ZZ )
12		fzssnn0 39533	. . . . . . . . . 10 |- ( 0 ... M ) C_ NN0
13		xp1st 7198	. . . . . . . . . 10 |- ( k e. ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) -> ( 1st ` k ) e. ( 0 ... M ) )
14	12, 13	sseldi 3601	. . . . . . . . 9 |- ( k e. ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) -> ( 1st ` k ) e. NN0 )
15	14	adantl 482	. . . . . . . 8 |- ( ( ph /\ k e. ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) ) -> ( 1st ` k ) e. NN0 )
16	11, 15	ffvelrnd 6360	. . . . . . 7 |- ( ( ph /\ k e. ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) ) -> ( A ` ( 1st ` k ) ) e. ZZ )
17		reelprrecn 10028	. . . . . . . . 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 39501	. . . . . . . . . 10 |- RR e. ( topGen ` ran (,) )
20		eqid 2622	. . . . . . . . . . 11 |- ( TopOpen ` ℂfld ) = ( TopOpen ` ℂfld )
21	20	tgioo2 22606	. . . . . . . . . 10 |- ( topGen ` ran (,) ) = ( ( TopOpen ` ℂfld ) ↾t RR )
22	19, 21	eleqtri 2699	. . . . . . . . 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 15388	. . . . . . . . . 10 |- ( P e. Prime -> P e. NN )
26	24, 25	syl 17	. . . . . . . . 9 |- ( ph -> P e. NN )
27	26	adantr 481	. . . . . . . 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 481	. . . . . . . 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 7199	. . . . . . . . . 10 |- ( k e. ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) -> ( 2nd ` k ) e. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) )
32		elfznn0 12433	. . . . . . . . . 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 482	. . . . . . . 8 |- ( ( ph /\ k e. ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) ) -> ( 2nd ` k ) e. NN0 )
35	15	nn0red 11352	. . . . . . . 8 |- ( ( ph /\ k e. ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) ) -> ( 1st ` k ) e. RR )
36	15	nn0zd 11480	. . . . . . . 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 40493	. . . . . . 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 11488	. . . . . 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 11483	. . . . 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 11722	. . . . . . . 8 |- NN0 = ( ZZ>= ` 0 )
41	28, 40	syl6eleq 2711	. . . . . . 7 |- ( ph -> M e. ( ZZ>= ` 0 ) )
42		eluzfz1 12348	. . . . . . 7 |- ( M e. ( ZZ>= ` 0 ) -> 0 e. ( 0 ... M ) )
43	41, 42	syl 17	. . . . . 6 |- ( ph -> 0 e. ( 0 ... M ) )
44		0zd 11389	. . . . . . . . 9 |- ( ph -> 0 e. ZZ )
45	28	nn0zd 11480	. . . . . . . . . . 11 |- ( ph -> M e. ZZ )
46	26	nnzd 11481	. . . . . . . . . . 11 |- ( ph -> P e. ZZ )
47	45, 46	zmulcld 11488	. . . . . . . . . 10 |- ( ph -> ( M x. P ) e. ZZ )
48		nnm1nn0 11334	. . . . . . . . . . . 12 |- ( P e. NN -> ( P - 1 ) e. NN0 )
49	26, 48	syl 17	. . . . . . . . . . 11 |- ( ph -> ( P - 1 ) e. NN0 )
50	49	nn0zd 11480	. . . . . . . . . 10 |- ( ph -> ( P - 1 ) e. ZZ )
51	47, 50	zaddcld 11486	. . . . . . . . 9 |- ( ph -> ( ( M x. P ) + ( P - 1 ) ) e. ZZ )
52	44, 51, 50	3jca 1242	. . . . . . . 8 |- ( ph -> ( 0 e. ZZ /\ ( ( M x. P ) + ( P - 1 ) ) e. ZZ /\ ( P - 1 ) e. ZZ ) )
53	49	nn0ge0d 11354	. . . . . . . 8 |- ( ph -> 0 <_ ( P - 1 ) )
54	26	nnnn0d 11351	. . . . . . . . . . 11 |- ( ph -> P e. NN0 )
55	28, 54	nn0mulcld 11356	. . . . . . . . . 10 |- ( ph -> ( M x. P ) e. NN0 )
56	55	nn0ge0d 11354	. . . . . . . . 9 |- ( ph -> 0 <_ ( M x. P ) )
57	49	nn0red 11352	. . . . . . . . . 10 |- ( ph -> ( P - 1 ) e. RR )
58	47	zred 11482	. . . . . . . . . 10 |- ( ph -> ( M x. P ) e. RR )
59	57, 58	addge02d 10616	. . . . . . . . 9 |- ( ph -> ( 0 <_ ( M x. P ) <-> ( P - 1 ) <_ ( ( M x. P ) + ( P - 1 ) ) ) )
60	56, 59	mpbid 222	. . . . . . . 8 |- ( ph -> ( P - 1 ) <_ ( ( M x. P ) + ( P - 1 ) ) )
61	52, 53, 60	jca32 558	. . . . . . 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 12333	. . . . . . 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 224	. . . . . 6 |- ( ph -> ( P - 1 ) e. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) )
64		opelxp 5146	. . . . . 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 698	. . . . 5 |- ( ph -> <. 0 , ( P - 1 ) >. e. ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) )
66		fveq2 6191	. . . . . . . 8 |- ( k = <. 0 , ( P - 1 ) >. -> ( 1st ` k ) = ( 1st ` <. 0 , ( P - 1 ) >. ) )
67		0re 10040	. . . . . . . . 9 |- 0 e. RR
68		ovex 6678	. . . . . . . . 9 |- ( P - 1 ) e. _V
69		op1stg 7180	. . . . . . . . 9 |- ( ( 0 e. RR /\ ( P - 1 ) e. _V ) -> ( 1st ` <. 0 , ( P - 1 ) >. ) = 0 )
70	67, 68, 69	mp2an 708	. . . . . . . 8 |- ( 1st ` <. 0 , ( P - 1 ) >. ) = 0
71	66, 70	syl6eq 2672	. . . . . . 7 |- ( k = <. 0 , ( P - 1 ) >. -> ( 1st ` k ) = 0 )
72	71	fveq2d 6195	. . . . . 6 |- ( k = <. 0 , ( P - 1 ) >. -> ( A ` ( 1st ` k ) ) = ( A ` 0 ) )
73		fveq2 6191	. . . . . . . . 9 |- ( k = <. 0 , ( P - 1 ) >. -> ( 2nd ` k ) = ( 2nd ` <. 0 , ( P - 1 ) >. ) )
74		op2ndg 7181	. . . . . . . . . 10 |- ( ( 0 e. RR /\ ( P - 1 ) e. _V ) -> ( 2nd ` <. 0 , ( P - 1 ) >. ) = ( P - 1 ) )
75	67, 68, 74	mp2an 708	. . . . . . . . 9 |- ( 2nd ` <. 0 , ( P - 1 ) >. ) = ( P - 1 )
76	73, 75	syl6eq 2672	. . . . . . . 8 |- ( k = <. 0 , ( P - 1 ) >. -> ( 2nd ` k ) = ( P - 1 ) )
77	76	fveq2d 6195	. . . . . . 7 |- ( k = <. 0 , ( P - 1 ) >. -> ( ( RR Dn F ) ` ( 2nd ` k ) ) = ( ( RR Dn F ) ` ( P - 1 ) ) )
78	77, 71	fveq12d 6197	. . . . . 6 |- ( k = <. 0 , ( P - 1 ) >. -> ( ( ( RR Dn F ) ` ( 2nd ` k ) ) ` ( 1st ` k ) ) = ( ( ( RR Dn F ) ` ( P - 1 ) ) ` 0 ) )
79	72, 78	oveq12d 6668	. . . . 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 39804	. . . 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 6665	. . 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 3601	. . . . . . 7 |- ( ph -> 0 e. NN0 )
83	10, 82	ffvelrnd 6360	. . . . . 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 40493	. . . . . 6 |- ( ph -> ( ( ( RR Dn F ) ` ( P - 1 ) ) ` 0 ) e. ZZ )
88	83, 87	zmulcld 11488	. . . . 5 |- ( ph -> ( ( A ` 0 ) x. ( ( ( RR Dn F ) ` ( P - 1 ) ) ` 0 ) ) e. ZZ )
89	88	zcnd 11483	. . . 4 |- ( ph -> ( ( A ` 0 ) x. ( ( ( RR Dn F ) ` ( P - 1 ) ) ` 0 ) ) e. CC )
90		difss 3737	. . . . . . . 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 8180	. . . . . . . 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 708	. . . . . . 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 3732	. . . . . . 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 491	. . . . . 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 14466	. . . . 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 11483	. . . 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 13071	. . . . 5 |- ( ph -> ( ! ` ( P - 1 ) ) e. NN )
99	98	nncnd 11036	. . . 4 |- ( ph -> ( ! ` ( P - 1 ) ) e. CC )
100	98	nnne0d 11065	. . . 4 |- ( ph -> ( ! ` ( P - 1 ) ) =/= 0 )
101	89, 97, 99, 100	divdird 10839	. . 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 2660	. 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 11065	. . 3 |- ( ph -> P =/= 0 )
104	83	zcnd 11483	. . . . 5 |- ( ph -> ( A ` 0 ) e. CC )
105	87	zcnd 11483	. . . . 5 |- ( ph -> ( ( ( RR Dn F ) ` ( P - 1 ) ) ` 0 ) e. CC )
106	104, 105, 99, 100	divassd 10836	. . . 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 40456	. . . . . . 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 40462	. . . . . . 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 40488	. . . . . 6 |- ( ph -> ( ! ` ( P - 1 ) ) || ( ( ( RR Dn F ) ` ( P - 1 ) ) ` 0 ) )
110	98	nnzd 11481	. . . . . . 7 |- ( ph -> ( ! ` ( P - 1 ) ) e. ZZ )
111		dvdsval2 14986	. . . . . . 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 1326	. . . . . 6 |- ( ph -> ( ( ! ` ( P - 1 ) ) || ( ( ( RR Dn F ) ` ( P - 1 ) ) ` 0 ) <-> ( ( ( ( RR Dn F ) ` ( P - 1 ) ) ` 0 ) / ( ! ` ( P - 1 ) ) ) e. ZZ ) )
113	109, 112	mpbid 222	. . . . 5 |- ( ph -> ( ( ( ( RR Dn F ) ` ( P - 1 ) ) ` 0 ) / ( ! ` ( P - 1 ) ) ) e. ZZ )
114	83, 113	zmulcld 11488	. . . 4 |- ( ph -> ( ( A ` 0 ) x. ( ( ( ( RR Dn F ) ` ( P - 1 ) ) ` 0 ) / ( ! ` ( P - 1 ) ) ) ) e. ZZ )
115	106, 114	eqeltrd 2701	. . 3 |- ( ph -> ( ( ( A ` 0 ) x. ( ( ( RR Dn F ) ` ( P - 1 ) ) ` 0 ) ) / ( ! ` ( P - 1 ) ) ) e. ZZ )
116	94, 39	sylan2 491	. . . . 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 14518	. . . 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 11483	. . . . . . . 8 |- ( ( ph /\ k e. ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) ) -> ( A ` ( 1st ` k ) ) e. CC )
119	94, 118	sylan2 491	. . . . . . 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 491	. . . . . . . 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 11483	. . . . . . 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 481	. . . . . . 7 |- ( ( ph /\ k e. ( ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) \ { <. 0 , ( P - 1 ) >. } ) ) -> ( ! ` ( P - 1 ) ) e. CC )
123	100	adantr 481	. . . . . . 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 10836	. . . . . 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 491	. . . . . . 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 481	. . . . . . . . 9 |- ( ( ph /\ k e. ( ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) \ { <. 0 , ( P - 1 ) >. } ) ) -> P e. NN )
129	28	adantr 481	. . . . . . . . 9 |- ( ( ph /\ k e. ( ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) \ { <. 0 , ( P - 1 ) >. } ) ) -> M e. NN0 )
130	94	adantl 482	. . . . . . . . . 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 491	. . . . . . . . 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 40488	. . . . . . . 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 481	. . . . . . . . 9 |- ( ( ph /\ k e. ( ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) \ { <. 0 , ( P - 1 ) >. } ) ) -> ( ! ` ( P - 1 ) ) e. ZZ )
136		dvdsval2 14986	. . . . . . . . 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 1326	. . . . . . . 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 222	. . . . . . 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 11488	. . . . . 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 2701	. . . . 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 14466	. . . 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 2701	. . 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 11408	. . . . . . . . . . . 12 |- ( ph -> 1 e. ZZ )
144		zabscl 14053	. . . . . . . . . . . . 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 1242	. . . . . . . . . . 11 |- ( ph -> ( 1 e. ZZ /\ ( P - 1 ) e. ZZ /\ ( abs ` ( A ` 0 ) ) e. ZZ ) )
147		nn0abscl 14052	. . . . . . . . . . . . . 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 14186	. . . . . . . . . . . . 13 |- ( ph -> ( abs ` ( A ` 0 ) ) =/= 0 )
151		elnnne0 11306	. . . . . . . . . . . . 13 |- ( ( abs ` ( A ` 0 ) ) e. NN <-> ( ( abs ` ( A ` 0 ) ) e. NN0 /\ ( abs ` ( A ` 0 ) ) =/= 0 ) )
152	148, 150, 151	sylanbrc 698	. . . . . . . . . . . 12 |- ( ph -> ( abs ` ( A ` 0 ) ) e. NN )
153	152	nnge1d 11063	. . . . . . . . . . 11 |- ( ph -> 1 <_ ( abs ` ( A ` 0 ) ) )
154		etransclem44.ap	. . . . . . . . . . . 12 |- ( ph -> ( abs ` ( A ` 0 ) ) < P )
155		zltlem1 11430	. . . . . . . . . . . . 13 |- ( ( ( abs ` ( A ` 0 ) ) e. ZZ /\ P e. ZZ ) -> ( ( abs ` ( A ` 0 ) ) < P <-> ( abs ` ( A ` 0 ) ) <_ ( P - 1 ) ) )
156	145, 46, 155	syl2anc 693	. . . . . . . . . . . 12 |- ( ph -> ( ( abs ` ( A ` 0 ) ) < P <-> ( abs ` ( A ` 0 ) ) <_ ( P - 1 ) ) )
157	154, 156	mpbid 222	. . . . . . . . . . 11 |- ( ph -> ( abs ` ( A ` 0 ) ) <_ ( P - 1 ) )
158	146, 153, 157	jca32 558	. . . . . . . . . 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 12333	. . . . . . . . . 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 224	. . . . . . . . 9 |- ( ph -> ( abs ` ( A ` 0 ) ) e. ( 1 ... ( P - 1 ) ) )
161		fzm1ndvds 15044	. . . . . . . . 9 |- ( ( P e. NN /\ ( abs ` ( A ` 0 ) ) e. ( 1 ... ( P - 1 ) ) ) -> -. P || ( abs ` ( A ` 0 ) ) )
162	26, 160, 161	syl2anc 693	. . . . . . . 8 |- ( ph -> -. P || ( abs ` ( A ` 0 ) ) )
163		dvdsabsb 15001	. . . . . . . . 9 |- ( ( P e. ZZ /\ ( A ` 0 ) e. ZZ ) -> ( P || ( A ` 0 ) <-> P || ( abs ` ( A ` 0 ) ) ) )
164	46, 83, 163	syl2anc 693	. . . . . . . 8 |- ( ph -> ( P || ( A ` 0 ) <-> P || ( abs ` ( A ` 0 ) ) ) )
165	162, 164	mtbird 315	. . . . . . 7 |- ( ph -> -. P || ( A ` 0 ) )
166		etransclem44.mp	. . . . . . . 8 |- ( ph -> ( ! ` M ) < P )
167	28, 24, 166, 30	etransclem41 40492	. . . . . . 7 |- ( ph -> -. P || ( ( ( ( RR Dn F ) ` ( P - 1 ) ) ` 0 ) / ( ! ` ( P - 1 ) ) ) )
168	165, 167	jca 554	. . . . . 6 |- ( ph -> ( -. P || ( A ` 0 ) /\ -. P || ( ( ( ( RR Dn F ) ` ( P - 1 ) ) ` 0 ) / ( ! ` ( P - 1 ) ) ) ) )
169		pm4.56 516	. . . . . 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 208	. . . . 5 |- ( ph -> -. ( P || ( A ` 0 ) \/ P || ( ( ( ( RR Dn F ) ` ( P - 1 ) ) ` 0 ) / ( ! ` ( P - 1 ) ) ) ) )
171		euclemma 15425	. . . . . 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 1326	. . . . 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 315	. . . 4 |- ( ph -> -. P || ( ( A ` 0 ) x. ( ( ( ( RR Dn F ) ` ( P - 1 ) ) ` 0 ) / ( ! ` ( P - 1 ) ) ) ) )
174	106	breq2d 4665	. . . 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 315	. . 3 |- ( ph -> -. P || ( ( ( A ` 0 ) x. ( ( ( RR Dn F ) ` ( P - 1 ) ) ` 0 ) ) / ( ! ` ( P - 1 ) ) ) )
176	46	adantr 481	. . . . . . . 8 |- ( ( ph /\ k e. ( ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) \ { <. 0 , ( P - 1 ) >. } ) ) -> P e. ZZ )
177	176, 125, 138	3jca 1242	. . . . . . 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 3733	. . . . . . . . . 10 |- ( k e. ( ( ( 0 ... M ) X. ( 0 ... ( ( M x. P ) + ( P - 1 ) ) ) ) \ { <. 0 , ( P - 1 ) >. } ) -> -. k e. { <. 0 , ( P - 1 ) >. } )
179	94	adantr 481	. . . . . . . . . . . . 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 7205	. . . . . . . . . . . . 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 477	. . . . . . . . . . . . . 14 |- ( ( ( 2nd ` k ) = ( P - 1 ) /\ ( 1st ` k ) = 0 ) -> ( 1st ` k ) = 0 )
183		simpl 473	. . . . . . . . . . . . . 14 |- ( ( ( 2nd ` k ) = ( P - 1 ) /\ ( 1st ` k ) = 0 ) -> ( 2nd ` k ) = ( P - 1 ) )
184	182, 183	opeq12d 4410	. . . . . . . . . . . . 13 |- ( ( ( 2nd ` k ) = ( P - 1 ) /\ ( 1st ` k ) = 0 ) -> <. ( 1st ` k ) , ( 2nd ` k ) >. = <. 0 , ( P - 1 ) >. )
185	184	adantl 482	. . . . . . . . . . . 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 2656	. . . . . . . . . . 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		velsn 4193	. . . . . . . . . . 11 |- ( k e. { <. 0 , ( P - 1 ) >. } <-> k = <. 0 , ( P - 1 ) >. )
188	186, 187	sylibr 224	. . . . . . . . . 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 691	. . . . . . . . 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 482	. . . . . . . 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 40489	. . . . . . 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 15021	. . . . . . 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 65	. . . . . 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 4681	. . . . 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 15030	. . . 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 4681	. . 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 40460	. 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 2861	1 |- ( ph -> K =/= 0 )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   {crab 2916   _Vcvv 3200   \ cdif 3571   C_ wss 3574   ifcif 4086   {csn 4177   {cpr 4179   <.cop 4183   class class class wbr 4653   |-> cmpt 4729   X. cxp 5112   ran crn 5115   -->wf 5884   ` cfv 5888  (class class class)co 6650   1stc1st 7166   2ndc2nd 7167   ^m cmap 7857   Fincfn 7955   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   < clt 10074   <_ cle 10075   - cmin 10266   / cdiv 10684   NNcn 11020   NN0cn0 11292   ZZcz 11377   ZZ>=cuz 11687   (,)cioo 12175   ...cfz 12326   ^cexp 12860   !cfa 13060   abscabs 13974   sum_csu 14416   prod_cprod 14635   || cdvds 14983   Primecprime 15385   ↾_t crest 16081   TopOpenctopn 16082   topGenctg 16098  ℂ_fldccnfld 19746   Dncdvn 23628

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

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

This theorem is referenced by:  etransclem47  40498