Theorem etransclem26 40477

Description: Every term in the sum of the N-th derivative of F applied to J is an integer. (Contributed by Glauco Siliprandi, 5-Apr-2020.)

Hypotheses

Ref	Expression
etransclem26.p	|- ( ph -> P e. NN )
etransclem26.m	|- ( ph -> M e. NN0 )
etransclem26.n	|- ( ph -> N e. NN0 )
etransclem26.jz	|- ( ph -> J e. ZZ )
etransclem26.c	|- C = ( n e. NN0 |-> { c e. ( ( 0 ... n ) ^m ( 0 ... M ) ) | sum_ j e. ( 0 ... M ) ( c ` j ) = n } )
etransclem26.d	|- ( ph -> D e. ( C ` N ) )

Assertion

Ref	Expression
etransclem26	|- ( ph -> ( ( ( ! ` N ) / prod_ j e. ( 0 ... M ) ( ! ` ( D ` j ) ) ) x. ( if ( ( P - 1 ) < ( D ` 0 ) , 0 , ( ( ( ! ` ( P - 1 ) ) / ( ! ` ( ( P - 1 ) - ( D ` 0 ) ) ) ) x. ( J ^ ( ( P - 1 ) - ( D ` 0 ) ) ) ) ) x. prod_ j e. ( 1 ... M ) if ( P < ( D ` j ) , 0 , ( ( ( ! ` P ) / ( ! ` ( P - ( D ` j ) ) ) ) x. ( ( J - j ) ^ ( P - ( D ` j ) ) ) ) ) ) ) e. ZZ )

Distinct variable groups:   D, c, j   M, c, j, n   N, c, n   ph, j, n

Allowed substitution hints:   ph( c)   C( j, n, c)   D( n)   P( j, n, c)   J( j, n, c)   N( j)

Proof of Theorem etransclem26

Step	Hyp	Ref	Expression
1		etransclem26.d	. . . . . . . . . 10 |- ( ph -> D e. ( C ` N ) )
2		etransclem26.c	. . . . . . . . . . 11 |- C = ( n e. NN0 |-> { c e. ( ( 0 ... n ) ^m ( 0 ... M ) ) | sum_ j e. ( 0 ... M ) ( c ` j ) = n } )
3		etransclem26.n	. . . . . . . . . . 11 |- ( ph -> N e. NN0 )
4	2, 3	etransclem12 40463	. . . . . . . . . 10 |- ( ph -> ( C ` N ) = { c e. ( ( 0 ... N ) ^m ( 0 ... M ) ) | sum_ j e. ( 0 ... M ) ( c ` j ) = N } )
5	1, 4	eleqtrd 2703	. . . . . . . . 9 |- ( ph -> D e. { c e. ( ( 0 ... N ) ^m ( 0 ... M ) ) | sum_ j e. ( 0 ... M ) ( c ` j ) = N } )
6		fveq1 6190	. . . . . . . . . . . 12 |- ( c = D -> ( c ` j ) = ( D ` j ) )
7	6	sumeq2ad 14434	. . . . . . . . . . 11 |- ( c = D -> sum_ j e. ( 0 ... M ) ( c ` j ) = sum_ j e. ( 0 ... M ) ( D ` j ) )
8	7	eqeq1d 2624	. . . . . . . . . 10 |- ( c = D -> ( sum_ j e. ( 0 ... M ) ( c ` j ) = N <-> sum_ j e. ( 0 ... M ) ( D ` j ) = N ) )
9	8	elrab 3363	. . . . . . . . 9 |- ( D e. { c e. ( ( 0 ... N ) ^m ( 0 ... M ) ) | sum_ j e. ( 0 ... M ) ( c ` j ) = N } <-> ( D e. ( ( 0 ... N ) ^m ( 0 ... M ) ) /\ sum_ j e. ( 0 ... M ) ( D ` j ) = N ) )
10	5, 9	sylib 208	. . . . . . . 8 |- ( ph -> ( D e. ( ( 0 ... N ) ^m ( 0 ... M ) ) /\ sum_ j e. ( 0 ... M ) ( D ` j ) = N ) )
11	10	simprd 479	. . . . . . 7 |- ( ph -> sum_ j e. ( 0 ... M ) ( D ` j ) = N )
12	11	eqcomd 2628	. . . . . 6 |- ( ph -> N = sum_ j e. ( 0 ... M ) ( D ` j ) )
13	12	fveq2d 6195	. . . . 5 |- ( ph -> ( ! ` N ) = ( ! ` sum_ j e. ( 0 ... M ) ( D ` j ) ) )
14	13	oveq1d 6665	. . . 4 |- ( ph -> ( ( ! ` N ) / prod_ j e. ( 0 ... M ) ( ! ` ( D ` j ) ) ) = ( ( ! ` sum_ j e. ( 0 ... M ) ( D ` j ) ) / prod_ j e. ( 0 ... M ) ( ! ` ( D ` j ) ) ) )
15		nfcv 2764	. . . . 5 |- F/_ j D
16		fzfid 12772	. . . . 5 |- ( ph -> ( 0 ... M ) e. Fin )
17		nn0ex 11298	. . . . . . 7 |- NN0 e. _V
18		fzssnn0 39533	. . . . . . 7 |- ( 0 ... N ) C_ NN0
19		mapss 7900	. . . . . . 7 |- ( ( NN0 e. _V /\ ( 0 ... N ) C_ NN0 ) -> ( ( 0 ... N ) ^m ( 0 ... M ) ) C_ ( NN0 ^m ( 0 ... M ) ) )
20	17, 18, 19	mp2an 708	. . . . . 6 |- ( ( 0 ... N ) ^m ( 0 ... M ) ) C_ ( NN0 ^m ( 0 ... M ) )
21	10	simpld 475	. . . . . 6 |- ( ph -> D e. ( ( 0 ... N ) ^m ( 0 ... M ) ) )
22	20, 21	sseldi 3601	. . . . 5 |- ( ph -> D e. ( NN0 ^m ( 0 ... M ) ) )
23	15, 16, 22	mccl 39830	. . . 4 |- ( ph -> ( ( ! ` sum_ j e. ( 0 ... M ) ( D ` j ) ) / prod_ j e. ( 0 ... M ) ( ! ` ( D ` j ) ) ) e. NN )
24	14, 23	eqeltrd 2701	. . 3 |- ( ph -> ( ( ! ` N ) / prod_ j e. ( 0 ... M ) ( ! ` ( D ` j ) ) ) e. NN )
25	24	nnzd 11481	. 2 |- ( ph -> ( ( ! ` N ) / prod_ j e. ( 0 ... M ) ( ! ` ( D ` j ) ) ) e. ZZ )
26		etransclem26.p	. . . 4 |- ( ph -> P e. NN )
27		etransclem26.m	. . . 4 |- ( ph -> M e. NN0 )
28		elmapi 7879	. . . . 5 |- ( D e. ( ( 0 ... N ) ^m ( 0 ... M ) ) -> D : ( 0 ... M ) --> ( 0 ... N ) )
29	21, 28	syl 17	. . . 4 |- ( ph -> D : ( 0 ... M ) --> ( 0 ... N ) )
30		etransclem26.jz	. . . 4 |- ( ph -> J e. ZZ )
31	26, 27, 29, 30	etransclem10 40461	. . 3 |- ( ph -> if ( ( P - 1 ) < ( D ` 0 ) , 0 , ( ( ( ! ` ( P - 1 ) ) / ( ! ` ( ( P - 1 ) - ( D ` 0 ) ) ) ) x. ( J ^ ( ( P - 1 ) - ( D ` 0 ) ) ) ) ) e. ZZ )
32		fzfid 12772	. . . 4 |- ( ph -> ( 1 ... M ) e. Fin )
33	26	adantr 481	. . . . 5 |- ( ( ph /\ j e. ( 1 ... M ) ) -> P e. NN )
34	29	adantr 481	. . . . 5 |- ( ( ph /\ j e. ( 1 ... M ) ) -> D : ( 0 ... M ) --> ( 0 ... N ) )
35		0z 11388	. . . . . . . 8 |- 0 e. ZZ
36		fzp1ss 12392	. . . . . . . 8 |- ( 0 e. ZZ -> ( ( 0 + 1 ) ... M ) C_ ( 0 ... M ) )
37	35, 36	ax-mp 5	. . . . . . 7 |- ( ( 0 + 1 ) ... M ) C_ ( 0 ... M )
38		1e0p1 11552	. . . . . . . . . 10 |- 1 = ( 0 + 1 )
39	38	oveq1i 6660	. . . . . . . . 9 |- ( 1 ... M ) = ( ( 0 + 1 ) ... M )
40	39	eleq2i 2693	. . . . . . . 8 |- ( j e. ( 1 ... M ) <-> j e. ( ( 0 + 1 ) ... M ) )
41	40	biimpi 206	. . . . . . 7 |- ( j e. ( 1 ... M ) -> j e. ( ( 0 + 1 ) ... M ) )
42	37, 41	sseldi 3601	. . . . . 6 |- ( j e. ( 1 ... M ) -> j e. ( 0 ... M ) )
43	42	adantl 482	. . . . 5 |- ( ( ph /\ j e. ( 1 ... M ) ) -> j e. ( 0 ... M ) )
44	30	adantr 481	. . . . 5 |- ( ( ph /\ j e. ( 1 ... M ) ) -> J e. ZZ )
45	33, 34, 43, 44	etransclem3 40454	. . . 4 |- ( ( ph /\ j e. ( 1 ... M ) ) -> if ( P < ( D ` j ) , 0 , ( ( ( ! ` P ) / ( ! ` ( P - ( D ` j ) ) ) ) x. ( ( J - j ) ^ ( P - ( D ` j ) ) ) ) ) e. ZZ )
46	32, 45	fprodzcl 14684	. . 3 |- ( ph -> prod_ j e. ( 1 ... M ) if ( P < ( D ` j ) , 0 , ( ( ( ! ` P ) / ( ! ` ( P - ( D ` j ) ) ) ) x. ( ( J - j ) ^ ( P - ( D ` j ) ) ) ) ) e. ZZ )
47	31, 46	zmulcld 11488	. 2 |- ( ph -> ( if ( ( P - 1 ) < ( D ` 0 ) , 0 , ( ( ( ! ` ( P - 1 ) ) / ( ! ` ( ( P - 1 ) - ( D ` 0 ) ) ) ) x. ( J ^ ( ( P - 1 ) - ( D ` 0 ) ) ) ) ) x. prod_ j e. ( 1 ... M ) if ( P < ( D ` j ) , 0 , ( ( ( ! ` P ) / ( ! ` ( P - ( D ` j ) ) ) ) x. ( ( J - j ) ^ ( P - ( D ` j ) ) ) ) ) ) e. ZZ )
48	25, 47	zmulcld 11488	1 |- ( ph -> ( ( ( ! ` N ) / prod_ j e. ( 0 ... M ) ( ! ` ( D ` j ) ) ) x. ( if ( ( P - 1 ) < ( D ` 0 ) , 0 , ( ( ( ! ` ( P - 1 ) ) / ( ! ` ( ( P - 1 ) - ( D ` 0 ) ) ) ) x. ( J ^ ( ( P - 1 ) - ( D ` 0 ) ) ) ) ) x. prod_ j e. ( 1 ... M ) if ( P < ( D ` j ) , 0 , ( ( ( ! ` P ) / ( ! ` ( P - ( D ` j ) ) ) ) x. ( ( J - j ) ^ ( P - ( D ` j ) ) ) ) ) ) ) e. ZZ )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   {crab 2916   _Vcvv 3200   C_ wss 3574   ifcif 4086   class class class wbr 4653   |-> cmpt 4729   -->wf 5884   ` cfv 5888  (class class class)co 6650   ^m cmap 7857   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

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

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-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-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-oi 8415  df-card 8765  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-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  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

This theorem is referenced by:  etransclem28  40479  etransclem36  40487  etransclem38  40489