Theorem etransclem37 40488

Description: ( P - 1 ) factorial divides the N-th derivative of F applied to J. (Contributed by Glauco Siliprandi, 5-Apr-2020.)

Hypotheses

Ref	Expression
etransclem37.s	|- ( ph -> S e. { RR , CC } )
etransclem37.x	|- ( ph -> X e. ( ( TopOpen ` ℂfld ) ↾t S ) )
etransclem37.p	|- ( ph -> P e. NN )
etransclem37.m	|- ( ph -> M e. NN0 )
etransclem37.f	|- F = ( x e. X |-> ( ( x ^ ( P - 1 ) ) x. prod_ j e. ( 1 ... M ) ( ( x - j ) ^ P ) ) )
etransclem37.n	|- ( ph -> N e. NN0 )
etransclem37.h	|- H = ( j e. ( 0 ... M ) |-> ( x e. X |-> ( ( x - j ) ^ if ( j = 0 , ( P - 1 ) , P ) ) ) )
etransclem37.c	|- C = ( n e. NN0 |-> { c e. ( ( 0 ... n ) ^m ( 0 ... M ) ) | sum_ j e. ( 0 ... M ) ( c ` j ) = n } )
etransclem37.9	|- ( ph -> J e. ( 0 ... M ) )
etransclem37.j	|- ( ph -> J e. X )

Assertion

Ref	Expression
etransclem37	|- ( ph -> ( ! ` ( P - 1 ) ) || ( ( ( S Dn F ) ` N ) ` J ) )

Distinct variable groups:   C, c, j, x   H, c, j, n, x   J, c, j, x   M, c, j, n, x   N, c, j, n, x   P, c, j, x   S, c, j, n, x   j, X, n, x   ph, c, j, n, x

Allowed substitution hints:   C( n)   P( n)   F( x, j, n, c)   J( n)   X( c)

Proof of Theorem etransclem37

Dummy variables k m d are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		etransclem37.c	. . . 4 |- C = ( n e. NN0 |-> { c e. ( ( 0 ... n ) ^m ( 0 ... M ) ) | sum_ j e. ( 0 ... M ) ( c ` j ) = n } )
2		etransclem37.n	. . . 4 |- ( ph -> N e. NN0 )
3	1, 2	etransclem16 40467	. . 3 |- ( ph -> ( C ` N ) e. Fin )
4		etransclem37.p	. . . . . 6 |- ( ph -> P e. NN )
5		nnm1nn0 11334	. . . . . 6 |- ( P e. NN -> ( P - 1 ) e. NN0 )
6	4, 5	syl 17	. . . . 5 |- ( ph -> ( P - 1 ) e. NN0 )
7	6	faccld 13071	. . . 4 |- ( ph -> ( ! ` ( P - 1 ) ) e. NN )
8	7	nnzd 11481	. . 3 |- ( ph -> ( ! ` ( P - 1 ) ) e. ZZ )
9	1, 2	etransclem12 40463	. . . . . . . . . . . 12 |- ( ph -> ( C ` N ) = { c e. ( ( 0 ... N ) ^m ( 0 ... M ) ) | sum_ j e. ( 0 ... M ) ( c ` j ) = N } )
10	9	eleq2d 2687	. . . . . . . . . . 11 |- ( ph -> ( c e. ( C ` N ) <-> c e. { c e. ( ( 0 ... N ) ^m ( 0 ... M ) ) | sum_ j e. ( 0 ... M ) ( c ` j ) = N } ) )
11	10	biimpa 501	. . . . . . . . . 10 |- ( ( ph /\ c e. ( C ` N ) ) -> c e. { c e. ( ( 0 ... N ) ^m ( 0 ... M ) ) | sum_ j e. ( 0 ... M ) ( c ` j ) = N } )
12		rabid 3116	. . . . . . . . . . . 12 |- ( c e. { c e. ( ( 0 ... N ) ^m ( 0 ... M ) ) | sum_ j e. ( 0 ... M ) ( c ` j ) = N } <-> ( c e. ( ( 0 ... N ) ^m ( 0 ... M ) ) /\ sum_ j e. ( 0 ... M ) ( c ` j ) = N ) )
13	12	biimpi 206	. . . . . . . . . . 11 |- ( c e. { c e. ( ( 0 ... N ) ^m ( 0 ... M ) ) | sum_ j e. ( 0 ... M ) ( c ` j ) = N } -> ( c e. ( ( 0 ... N ) ^m ( 0 ... M ) ) /\ sum_ j e. ( 0 ... M ) ( c ` j ) = N ) )
14	13	simprd 479	. . . . . . . . . 10 |- ( c e. { c e. ( ( 0 ... N ) ^m ( 0 ... M ) ) | sum_ j e. ( 0 ... M ) ( c ` j ) = N } -> sum_ j e. ( 0 ... M ) ( c ` j ) = N )
15	11, 14	syl 17	. . . . . . . . 9 |- ( ( ph /\ c e. ( C ` N ) ) -> sum_ j e. ( 0 ... M ) ( c ` j ) = N )
16	15	eqcomd 2628	. . . . . . . 8 |- ( ( ph /\ c e. ( C ` N ) ) -> N = sum_ j e. ( 0 ... M ) ( c ` j ) )
17	16	fveq2d 6195	. . . . . . 7 |- ( ( ph /\ c e. ( C ` N ) ) -> ( ! ` N ) = ( ! ` sum_ j e. ( 0 ... M ) ( c ` j ) ) )
18	17	oveq1d 6665	. . . . . 6 |- ( ( ph /\ c e. ( C ` N ) ) -> ( ( ! ` N ) / prod_ j e. ( 0 ... M ) ( ! ` ( c ` j ) ) ) = ( ( ! ` sum_ j e. ( 0 ... M ) ( c ` j ) ) / prod_ j e. ( 0 ... M ) ( ! ` ( c ` j ) ) ) )
19		nfcv 2764	. . . . . . 7 |- F/_ j c
20		fzfid 12772	. . . . . . 7 |- ( ( ph /\ c e. ( C ` N ) ) -> ( 0 ... M ) e. Fin )
21		nn0ex 11298	. . . . . . . . . . 11 |- NN0 e. _V
22	21	a1i 11	. . . . . . . . . 10 |- ( c e. { c e. ( ( 0 ... N ) ^m ( 0 ... M ) ) | sum_ j e. ( 0 ... M ) ( c ` j ) = N } -> NN0 e. _V )
23		fzssnn0 39533	. . . . . . . . . 10 |- ( 0 ... N ) C_ NN0
24		mapss 7900	. . . . . . . . . 10 |- ( ( NN0 e. _V /\ ( 0 ... N ) C_ NN0 ) -> ( ( 0 ... N ) ^m ( 0 ... M ) ) C_ ( NN0 ^m ( 0 ... M ) ) )
25	22, 23, 24	sylancl 694	. . . . . . . . 9 |- ( c e. { c e. ( ( 0 ... N ) ^m ( 0 ... M ) ) | sum_ j e. ( 0 ... M ) ( c ` j ) = N } -> ( ( 0 ... N ) ^m ( 0 ... M ) ) C_ ( NN0 ^m ( 0 ... M ) ) )
26	13	simpld 475	. . . . . . . . 9 |- ( c e. { c e. ( ( 0 ... N ) ^m ( 0 ... M ) ) | sum_ j e. ( 0 ... M ) ( c ` j ) = N } -> c e. ( ( 0 ... N ) ^m ( 0 ... M ) ) )
27	25, 26	sseldd 3604	. . . . . . . 8 |- ( c e. { c e. ( ( 0 ... N ) ^m ( 0 ... M ) ) | sum_ j e. ( 0 ... M ) ( c ` j ) = N } -> c e. ( NN0 ^m ( 0 ... M ) ) )
28	11, 27	syl 17	. . . . . . 7 |- ( ( ph /\ c e. ( C ` N ) ) -> c e. ( NN0 ^m ( 0 ... M ) ) )
29	19, 20, 28	mccl 39830	. . . . . 6 |- ( ( ph /\ c e. ( C ` N ) ) -> ( ( ! ` sum_ j e. ( 0 ... M ) ( c ` j ) ) / prod_ j e. ( 0 ... M ) ( ! ` ( c ` j ) ) ) e. NN )
30	18, 29	eqeltrd 2701	. . . . 5 |- ( ( ph /\ c e. ( C ` N ) ) -> ( ( ! ` N ) / prod_ j e. ( 0 ... M ) ( ! ` ( c ` j ) ) ) e. NN )
31	30	nnzd 11481	. . . 4 |- ( ( ph /\ c e. ( C ` N ) ) -> ( ( ! ` N ) / prod_ j e. ( 0 ... M ) ( ! ` ( c ` j ) ) ) e. ZZ )
32	4	adantr 481	. . . . . 6 |- ( ( ph /\ c e. ( C ` N ) ) -> P e. NN )
33		etransclem37.m	. . . . . . 7 |- ( ph -> M e. NN0 )
34	33	adantr 481	. . . . . 6 |- ( ( ph /\ c e. ( C ` N ) ) -> M e. NN0 )
35		elmapi 7879	. . . . . . 7 |- ( c e. ( ( 0 ... N ) ^m ( 0 ... M ) ) -> c : ( 0 ... M ) --> ( 0 ... N ) )
36	11, 26, 35	3syl 18	. . . . . 6 |- ( ( ph /\ c e. ( C ` N ) ) -> c : ( 0 ... M ) --> ( 0 ... N ) )
37		etransclem37.9	. . . . . . . 8 |- ( ph -> J e. ( 0 ... M ) )
38	37	elfzelzd 39530	. . . . . . 7 |- ( ph -> J e. ZZ )
39	38	adantr 481	. . . . . 6 |- ( ( ph /\ c e. ( C ` N ) ) -> J e. ZZ )
40	32, 34, 36, 39	etransclem10 40461	. . . . 5 |- ( ( ph /\ c e. ( C ` N ) ) -> if ( ( P - 1 ) < ( c ` 0 ) , 0 , ( ( ( ! ` ( P - 1 ) ) / ( ! ` ( ( P - 1 ) - ( c ` 0 ) ) ) ) x. ( J ^ ( ( P - 1 ) - ( c ` 0 ) ) ) ) ) e. ZZ )
41		fzfid 12772	. . . . . 6 |- ( ( ph /\ c e. ( C ` N ) ) -> ( 1 ... M ) e. Fin )
42	32	adantr 481	. . . . . . 7 |- ( ( ( ph /\ c e. ( C ` N ) ) /\ j e. ( 1 ... M ) ) -> P e. NN )
43	36	adantr 481	. . . . . . 7 |- ( ( ( ph /\ c e. ( C ` N ) ) /\ j e. ( 1 ... M ) ) -> c : ( 0 ... M ) --> ( 0 ... N ) )
44		0z 11388	. . . . . . . . . . 11 |- 0 e. ZZ
45		fzp1ss 12392	. . . . . . . . . . 11 |- ( 0 e. ZZ -> ( ( 0 + 1 ) ... M ) C_ ( 0 ... M ) )
46	44, 45	ax-mp 5	. . . . . . . . . 10 |- ( ( 0 + 1 ) ... M ) C_ ( 0 ... M )
47	46	sseli 3599	. . . . . . . . 9 |- ( j e. ( ( 0 + 1 ) ... M ) -> j e. ( 0 ... M ) )
48		1e0p1 11552	. . . . . . . . . 10 |- 1 = ( 0 + 1 )
49	48	oveq1i 6660	. . . . . . . . 9 |- ( 1 ... M ) = ( ( 0 + 1 ) ... M )
50	47, 49	eleq2s 2719	. . . . . . . 8 |- ( j e. ( 1 ... M ) -> j e. ( 0 ... M ) )
51	50	adantl 482	. . . . . . 7 |- ( ( ( ph /\ c e. ( C ` N ) ) /\ j e. ( 1 ... M ) ) -> j e. ( 0 ... M ) )
52	39	adantr 481	. . . . . . 7 |- ( ( ( ph /\ c e. ( C ` N ) ) /\ j e. ( 1 ... M ) ) -> J e. ZZ )
53	42, 43, 51, 52	etransclem3 40454	. . . . . 6 |- ( ( ( ph /\ c e. ( C ` N ) ) /\ j e. ( 1 ... M ) ) -> if ( P < ( c ` j ) , 0 , ( ( ( ! ` P ) / ( ! ` ( P - ( c ` j ) ) ) ) x. ( ( J - j ) ^ ( P - ( c ` j ) ) ) ) ) e. ZZ )
54	41, 53	fprodzcl 14684	. . . . 5 |- ( ( ph /\ c e. ( C ` N ) ) -> prod_ j e. ( 1 ... M ) if ( P < ( c ` j ) , 0 , ( ( ( ! ` P ) / ( ! ` ( P - ( c ` j ) ) ) ) x. ( ( J - j ) ^ ( P - ( c ` j ) ) ) ) ) e. ZZ )
55	40, 54	zmulcld 11488	. . . 4 |- ( ( ph /\ c e. ( C ` N ) ) -> ( if ( ( P - 1 ) < ( c ` 0 ) , 0 , ( ( ( ! ` ( P - 1 ) ) / ( ! ` ( ( P - 1 ) - ( c ` 0 ) ) ) ) x. ( J ^ ( ( P - 1 ) - ( c ` 0 ) ) ) ) ) x. prod_ j e. ( 1 ... M ) if ( P < ( c ` j ) , 0 , ( ( ( ! ` P ) / ( ! ` ( P - ( c ` j ) ) ) ) x. ( ( J - j ) ^ ( P - ( c ` j ) ) ) ) ) ) e. ZZ )
56	31, 55	zmulcld 11488	. . 3 |- ( ( ph /\ c e. ( C ` N ) ) -> ( ( ( ! ` N ) / prod_ j e. ( 0 ... M ) ( ! ` ( c ` j ) ) ) x. ( if ( ( P - 1 ) < ( c ` 0 ) , 0 , ( ( ( ! ` ( P - 1 ) ) / ( ! ` ( ( P - 1 ) - ( c ` 0 ) ) ) ) x. ( J ^ ( ( P - 1 ) - ( c ` 0 ) ) ) ) ) x. prod_ j e. ( 1 ... M ) if ( P < ( c ` j ) , 0 , ( ( ( ! ` P ) / ( ! ` ( P - ( c ` j ) ) ) ) x. ( ( J - j ) ^ ( P - ( c ` j ) ) ) ) ) ) ) e. ZZ )
57	2	adantr 481	. . . 4 |- ( ( ph /\ c e. ( C ` N ) ) -> N e. NN0 )
58		etransclem11 40462	. . . . 5 |- ( n e. NN0 |-> { c e. ( ( 0 ... n ) ^m ( 0 ... M ) ) | sum_ j e. ( 0 ... M ) ( c ` j ) = n } ) = ( m e. NN0 |-> { d e. ( ( 0 ... m ) ^m ( 0 ... M ) ) | sum_ k e. ( 0 ... M ) ( d ` k ) = m } )
59	1, 58	eqtri 2644	. . . 4 |- C = ( m e. NN0 |-> { d e. ( ( 0 ... m ) ^m ( 0 ... M ) ) | sum_ k e. ( 0 ... M ) ( d ` k ) = m } )
60		simpr 477	. . . 4 |- ( ( ph /\ c e. ( C ` N ) ) -> c e. ( C ` N ) )
61	37	adantr 481	. . . 4 |- ( ( ph /\ c e. ( C ` N ) ) -> J e. ( 0 ... M ) )
62		fveq2 6191	. . . . . . . 8 |- ( j = k -> ( c ` j ) = ( c ` k ) )
63	62	fveq2d 6195	. . . . . . 7 |- ( j = k -> ( ! ` ( c ` j ) ) = ( ! ` ( c ` k ) ) )
64	63	cbvprodv 14646	. . . . . 6 |- prod_ j e. ( 0 ... M ) ( ! ` ( c ` j ) ) = prod_ k e. ( 0 ... M ) ( ! ` ( c ` k ) )
65	64	oveq2i 6661	. . . . 5 |- ( ( ! ` N ) / prod_ j e. ( 0 ... M ) ( ! ` ( c ` j ) ) ) = ( ( ! ` N ) / prod_ k e. ( 0 ... M ) ( ! ` ( c ` k ) ) )
66	62	breq2d 4665	. . . . . . . 8 |- ( j = k -> ( P < ( c ` j ) <-> P < ( c ` k ) ) )
67	62	oveq2d 6666	. . . . . . . . . . 11 |- ( j = k -> ( P - ( c ` j ) ) = ( P - ( c ` k ) ) )
68	67	fveq2d 6195	. . . . . . . . . 10 |- ( j = k -> ( ! ` ( P - ( c ` j ) ) ) = ( ! ` ( P - ( c ` k ) ) ) )
69	68	oveq2d 6666	. . . . . . . . 9 |- ( j = k -> ( ( ! ` P ) / ( ! ` ( P - ( c ` j ) ) ) ) = ( ( ! ` P ) / ( ! ` ( P - ( c ` k ) ) ) ) )
70		oveq2 6658	. . . . . . . . . 10 |- ( j = k -> ( J - j ) = ( J - k ) )
71	70, 67	oveq12d 6668	. . . . . . . . 9 |- ( j = k -> ( ( J - j ) ^ ( P - ( c ` j ) ) ) = ( ( J - k ) ^ ( P - ( c ` k ) ) ) )
72	69, 71	oveq12d 6668	. . . . . . . 8 |- ( j = k -> ( ( ( ! ` P ) / ( ! ` ( P - ( c ` j ) ) ) ) x. ( ( J - j ) ^ ( P - ( c ` j ) ) ) ) = ( ( ( ! ` P ) / ( ! ` ( P - ( c ` k ) ) ) ) x. ( ( J - k ) ^ ( P - ( c ` k ) ) ) ) )
73	66, 72	ifbieq2d 4111	. . . . . . 7 |- ( j = k -> if ( P < ( c ` j ) , 0 , ( ( ( ! ` P ) / ( ! ` ( P - ( c ` j ) ) ) ) x. ( ( J - j ) ^ ( P - ( c ` j ) ) ) ) ) = if ( P < ( c ` k ) , 0 , ( ( ( ! ` P ) / ( ! ` ( P - ( c ` k ) ) ) ) x. ( ( J - k ) ^ ( P - ( c ` k ) ) ) ) ) )
74	73	cbvprodv 14646	. . . . . 6 |- prod_ j e. ( 1 ... M ) if ( P < ( c ` j ) , 0 , ( ( ( ! ` P ) / ( ! ` ( P - ( c ` j ) ) ) ) x. ( ( J - j ) ^ ( P - ( c ` j ) ) ) ) ) = prod_ k e. ( 1 ... M ) if ( P < ( c ` k ) , 0 , ( ( ( ! ` P ) / ( ! ` ( P - ( c ` k ) ) ) ) x. ( ( J - k ) ^ ( P - ( c ` k ) ) ) ) )
75	74	oveq2i 6661	. . . . 5 |- ( if ( ( P - 1 ) < ( c ` 0 ) , 0 , ( ( ( ! ` ( P - 1 ) ) / ( ! ` ( ( P - 1 ) - ( c ` 0 ) ) ) ) x. ( J ^ ( ( P - 1 ) - ( c ` 0 ) ) ) ) ) x. prod_ j e. ( 1 ... M ) if ( P < ( c ` j ) , 0 , ( ( ( ! ` P ) / ( ! ` ( P - ( c ` j ) ) ) ) x. ( ( J - j ) ^ ( P - ( c ` j ) ) ) ) ) ) = ( if ( ( P - 1 ) < ( c ` 0 ) , 0 , ( ( ( ! ` ( P - 1 ) ) / ( ! ` ( ( P - 1 ) - ( c ` 0 ) ) ) ) x. ( J ^ ( ( P - 1 ) - ( c ` 0 ) ) ) ) ) x. prod_ k e. ( 1 ... M ) if ( P < ( c ` k ) , 0 , ( ( ( ! ` P ) / ( ! ` ( P - ( c ` k ) ) ) ) x. ( ( J - k ) ^ ( P - ( c ` k ) ) ) ) ) )
76	65, 75	oveq12i 6662	. . . 4 |- ( ( ( ! ` N ) / prod_ j e. ( 0 ... M ) ( ! ` ( c ` j ) ) ) x. ( if ( ( P - 1 ) < ( c ` 0 ) , 0 , ( ( ( ! ` ( P - 1 ) ) / ( ! ` ( ( P - 1 ) - ( c ` 0 ) ) ) ) x. ( J ^ ( ( P - 1 ) - ( c ` 0 ) ) ) ) ) x. prod_ j e. ( 1 ... M ) if ( P < ( c ` j ) , 0 , ( ( ( ! ` P ) / ( ! ` ( P - ( c ` j ) ) ) ) x. ( ( J - j ) ^ ( P - ( c ` j ) ) ) ) ) ) ) = ( ( ( ! ` N ) / prod_ k e. ( 0 ... M ) ( ! ` ( c ` k ) ) ) x. ( if ( ( P - 1 ) < ( c ` 0 ) , 0 , ( ( ( ! ` ( P - 1 ) ) / ( ! ` ( ( P - 1 ) - ( c ` 0 ) ) ) ) x. ( J ^ ( ( P - 1 ) - ( c ` 0 ) ) ) ) ) x. prod_ k e. ( 1 ... M ) if ( P < ( c ` k ) , 0 , ( ( ( ! ` P ) / ( ! ` ( P - ( c ` k ) ) ) ) x. ( ( J - k ) ^ ( P - ( c ` k ) ) ) ) ) ) )
77	32, 34, 57, 59, 60, 61, 76	etransclem28 40479	. . 3 |- ( ( ph /\ c e. ( C ` N ) ) -> ( ! ` ( P - 1 ) ) || ( ( ( ! ` N ) / prod_ j e. ( 0 ... M ) ( ! ` ( c ` j ) ) ) x. ( if ( ( P - 1 ) < ( c ` 0 ) , 0 , ( ( ( ! ` ( P - 1 ) ) / ( ! ` ( ( P - 1 ) - ( c ` 0 ) ) ) ) x. ( J ^ ( ( P - 1 ) - ( c ` 0 ) ) ) ) ) x. prod_ j e. ( 1 ... M ) if ( P < ( c ` j ) , 0 , ( ( ( ! ` P ) / ( ! ` ( P - ( c ` j ) ) ) ) x. ( ( J - j ) ^ ( P - ( c ` j ) ) ) ) ) ) ) )
78	3, 8, 56, 77	fsumdvds 15030	. 2 |- ( ph -> ( ! ` ( P - 1 ) ) || sum_ c e. ( C ` N ) ( ( ( ! ` N ) / prod_ j e. ( 0 ... M ) ( ! ` ( c ` j ) ) ) x. ( if ( ( P - 1 ) < ( c ` 0 ) , 0 , ( ( ( ! ` ( P - 1 ) ) / ( ! ` ( ( P - 1 ) - ( c ` 0 ) ) ) ) x. ( J ^ ( ( P - 1 ) - ( c ` 0 ) ) ) ) ) x. prod_ j e. ( 1 ... M ) if ( P < ( c ` j ) , 0 , ( ( ( ! ` P ) / ( ! ` ( P - ( c ` j ) ) ) ) x. ( ( J - j ) ^ ( P - ( c ` j ) ) ) ) ) ) ) )
79		etransclem37.s	. . 3 |- ( ph -> S e. { RR , CC } )
80		etransclem37.x	. . 3 |- ( ph -> X e. ( ( TopOpen ` ℂfld ) ↾t S ) )
81		etransclem37.f	. . 3 |- F = ( x e. X |-> ( ( x ^ ( P - 1 ) ) x. prod_ j e. ( 1 ... M ) ( ( x - j ) ^ P ) ) )
82		etransclem37.h	. . 3 |- H = ( j e. ( 0 ... M ) |-> ( x e. X |-> ( ( x - j ) ^ if ( j = 0 , ( P - 1 ) , P ) ) ) )
83		etransclem37.j	. . 3 |- ( ph -> J e. X )
84	79, 80, 4, 33, 81, 2, 82, 1, 83	etransclem31 40482	. 2 |- ( ph -> ( ( ( S Dn F ) ` N ) ` J ) = sum_ c e. ( C ` N ) ( ( ( ! ` N ) / prod_ j e. ( 0 ... M ) ( ! ` ( c ` j ) ) ) x. ( if ( ( P - 1 ) < ( c ` 0 ) , 0 , ( ( ( ! ` ( P - 1 ) ) / ( ! ` ( ( P - 1 ) - ( c ` 0 ) ) ) ) x. ( J ^ ( ( P - 1 ) - ( c ` 0 ) ) ) ) ) x. prod_ j e. ( 1 ... M ) if ( P < ( c ` j ) , 0 , ( ( ( ! ` P ) / ( ! ` ( P - ( c ` j ) ) ) ) x. ( ( J - j ) ^ ( P - ( c ` j ) ) ) ) ) ) ) )
85	78, 84	breqtrrd 4681	1 |- ( ph -> ( ! ` ( P - 1 ) ) || ( ( ( S Dn F ) ` N ) ` J ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   {crab 2916   _Vcvv 3200   C_ wss 3574   ifcif 4086   {cpr 4179   class class class wbr 4653   |-> cmpt 4729   -->wf 5884   ` cfv 5888  (class class class)co 6650   ^m cmap 7857   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   < clt 10074   - cmin 10266   / cdiv 10684   NNcn 11020   NN0cn0 11292   ZZcz 11377   ...cfz 12326   ^cexp 12860   !cfa 13060   sum_csu 14416   prod_cprod 14635   || cdvds 14983   ↾_t crest 16081   TopOpenctopn 16082  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-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  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-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:  etransclem44  40495  etransclem45  40496