Theorem etransclem7 40458

Description: The given product is an integer. (Contributed by Glauco Siliprandi, 5-Apr-2020.)

Hypotheses

Ref	Expression
etransclem7.n	|- ( ph -> P e. NN )
etransclem7.c	|- ( ph -> C : ( 0 ... M ) --> ( 0 ... N ) )
etransclem7.j	|- ( ph -> J e. ( 0 ... M ) )

Assertion

Ref	Expression
etransclem7	|- ( ph -> prod_ j e. ( 1 ... M ) if ( P < ( C ` j ) , 0 , ( ( ( ! ` P ) / ( ! ` ( P - ( C ` j ) ) ) ) x. ( ( J - j ) ^ ( P - ( C ` j ) ) ) ) ) e. ZZ )

Distinct variable groups:   j, M   ph, j

Allowed substitution hints:   C( j)   P( j)   J( j)   N( j)

Proof of Theorem etransclem7

Step	Hyp	Ref	Expression
1		fzfid 12772	. 2 |- ( ph -> ( 1 ... M ) e. Fin )
2		0zd 11389	. . 3 |- ( ( ( ph /\ j e. ( 1 ... M ) ) /\ P < ( C ` j ) ) -> 0 e. ZZ )
3		0zd 11389	. . . . . . . . 9 |- ( ( ( ph /\ j e. ( 1 ... M ) ) /\ -. P < ( C ` j ) ) -> 0 e. ZZ )
4		etransclem7.n	. . . . . . . . . . 11 |- ( ph -> P e. NN )
5	4	nnzd 11481	. . . . . . . . . 10 |- ( ph -> P e. ZZ )
6	5	ad2antrr 762	. . . . . . . . 9 |- ( ( ( ph /\ j e. ( 1 ... M ) ) /\ -. P < ( C ` j ) ) -> P e. ZZ )
7	5	adantr 481	. . . . . . . . . . 11 |- ( ( ph /\ j e. ( 1 ... M ) ) -> P e. ZZ )
8		etransclem7.c	. . . . . . . . . . . . . 14 |- ( ph -> C : ( 0 ... M ) --> ( 0 ... N ) )
9	8	adantr 481	. . . . . . . . . . . . 13 |- ( ( ph /\ j e. ( 1 ... M ) ) -> C : ( 0 ... M ) --> ( 0 ... N ) )
10		0zd 11389	. . . . . . . . . . . . . . . 16 |- ( j e. ( 1 ... M ) -> 0 e. ZZ )
11		fzp1ss 12392	. . . . . . . . . . . . . . . 16 |- ( 0 e. ZZ -> ( ( 0 + 1 ) ... M ) C_ ( 0 ... M ) )
12	10, 11	syl 17	. . . . . . . . . . . . . . 15 |- ( j e. ( 1 ... M ) -> ( ( 0 + 1 ) ... M ) C_ ( 0 ... M ) )
13		id 22	. . . . . . . . . . . . . . . 16 |- ( j e. ( 1 ... M ) -> j e. ( 1 ... M ) )
14		1e0p1 11552	. . . . . . . . . . . . . . . . 17 |- 1 = ( 0 + 1 )
15	14	oveq1i 6660	. . . . . . . . . . . . . . . 16 |- ( 1 ... M ) = ( ( 0 + 1 ) ... M )
16	13, 15	syl6eleq 2711	. . . . . . . . . . . . . . 15 |- ( j e. ( 1 ... M ) -> j e. ( ( 0 + 1 ) ... M ) )
17	12, 16	sseldd 3604	. . . . . . . . . . . . . 14 |- ( j e. ( 1 ... M ) -> j e. ( 0 ... M ) )
18	17	adantl 482	. . . . . . . . . . . . 13 |- ( ( ph /\ j e. ( 1 ... M ) ) -> j e. ( 0 ... M ) )
19	9, 18	ffvelrnd 6360	. . . . . . . . . . . 12 |- ( ( ph /\ j e. ( 1 ... M ) ) -> ( C ` j ) e. ( 0 ... N ) )
20	19	elfzelzd 39530	. . . . . . . . . . 11 |- ( ( ph /\ j e. ( 1 ... M ) ) -> ( C ` j ) e. ZZ )
21	7, 20	zsubcld 11487	. . . . . . . . . 10 |- ( ( ph /\ j e. ( 1 ... M ) ) -> ( P - ( C ` j ) ) e. ZZ )
22	21	adantr 481	. . . . . . . . 9 |- ( ( ( ph /\ j e. ( 1 ... M ) ) /\ -. P < ( C ` j ) ) -> ( P - ( C ` j ) ) e. ZZ )
23	3, 6, 22	3jca 1242	. . . . . . . 8 |- ( ( ( ph /\ j e. ( 1 ... M ) ) /\ -. P < ( C ` j ) ) -> ( 0 e. ZZ /\ P e. ZZ /\ ( P - ( C ` j ) ) e. ZZ ) )
24	20	zred 11482	. . . . . . . . . . 11 |- ( ( ph /\ j e. ( 1 ... M ) ) -> ( C ` j ) e. RR )
25	24	adantr 481	. . . . . . . . . 10 |- ( ( ( ph /\ j e. ( 1 ... M ) ) /\ -. P < ( C ` j ) ) -> ( C ` j ) e. RR )
26	6	zred 11482	. . . . . . . . . 10 |- ( ( ( ph /\ j e. ( 1 ... M ) ) /\ -. P < ( C ` j ) ) -> P e. RR )
27		simpr 477	. . . . . . . . . 10 |- ( ( ( ph /\ j e. ( 1 ... M ) ) /\ -. P < ( C ` j ) ) -> -. P < ( C ` j ) )
28	25, 26, 27	nltled 10187	. . . . . . . . 9 |- ( ( ( ph /\ j e. ( 1 ... M ) ) /\ -. P < ( C ` j ) ) -> ( C ` j ) <_ P )
29	26, 25	subge0d 10617	. . . . . . . . 9 |- ( ( ( ph /\ j e. ( 1 ... M ) ) /\ -. P < ( C ` j ) ) -> ( 0 <_ ( P - ( C ` j ) ) <-> ( C ` j ) <_ P ) )
30	28, 29	mpbird 247	. . . . . . . 8 |- ( ( ( ph /\ j e. ( 1 ... M ) ) /\ -. P < ( C ` j ) ) -> 0 <_ ( P - ( C ` j ) ) )
31		elfzle1 12344	. . . . . . . . . . 11 |- ( ( C ` j ) e. ( 0 ... N ) -> 0 <_ ( C ` j ) )
32	19, 31	syl 17	. . . . . . . . . 10 |- ( ( ph /\ j e. ( 1 ... M ) ) -> 0 <_ ( C ` j ) )
33	32	adantr 481	. . . . . . . . 9 |- ( ( ( ph /\ j e. ( 1 ... M ) ) /\ -. P < ( C ` j ) ) -> 0 <_ ( C ` j ) )
34	26, 25	subge02d 10619	. . . . . . . . 9 |- ( ( ( ph /\ j e. ( 1 ... M ) ) /\ -. P < ( C ` j ) ) -> ( 0 <_ ( C ` j ) <-> ( P - ( C ` j ) ) <_ P ) )
35	33, 34	mpbid 222	. . . . . . . 8 |- ( ( ( ph /\ j e. ( 1 ... M ) ) /\ -. P < ( C ` j ) ) -> ( P - ( C ` j ) ) <_ P )
36	23, 30, 35	jca32 558	. . . . . . 7 |- ( ( ( ph /\ j e. ( 1 ... M ) ) /\ -. P < ( C ` j ) ) -> ( ( 0 e. ZZ /\ P e. ZZ /\ ( P - ( C ` j ) ) e. ZZ ) /\ ( 0 <_ ( P - ( C ` j ) ) /\ ( P - ( C ` j ) ) <_ P ) ) )
37		elfz2 12333	. . . . . . 7 |- ( ( P - ( C ` j ) ) e. ( 0 ... P ) <-> ( ( 0 e. ZZ /\ P e. ZZ /\ ( P - ( C ` j ) ) e. ZZ ) /\ ( 0 <_ ( P - ( C ` j ) ) /\ ( P - ( C ` j ) ) <_ P ) ) )
38	36, 37	sylibr 224	. . . . . 6 |- ( ( ( ph /\ j e. ( 1 ... M ) ) /\ -. P < ( C ` j ) ) -> ( P - ( C ` j ) ) e. ( 0 ... P ) )
39		permnn 13113	. . . . . 6 |- ( ( P - ( C ` j ) ) e. ( 0 ... P ) -> ( ( ! ` P ) / ( ! ` ( P - ( C ` j ) ) ) ) e. NN )
40	38, 39	syl 17	. . . . 5 |- ( ( ( ph /\ j e. ( 1 ... M ) ) /\ -. P < ( C ` j ) ) -> ( ( ! ` P ) / ( ! ` ( P - ( C ` j ) ) ) ) e. NN )
41	40	nnzd 11481	. . . 4 |- ( ( ( ph /\ j e. ( 1 ... M ) ) /\ -. P < ( C ` j ) ) -> ( ( ! ` P ) / ( ! ` ( P - ( C ` j ) ) ) ) e. ZZ )
42		etransclem7.j	. . . . . . . . 9 |- ( ph -> J e. ( 0 ... M ) )
43	42	elfzelzd 39530	. . . . . . . 8 |- ( ph -> J e. ZZ )
44	43	adantr 481	. . . . . . 7 |- ( ( ph /\ j e. ( 1 ... M ) ) -> J e. ZZ )
45		elfzelz 12342	. . . . . . . 8 |- ( j e. ( 1 ... M ) -> j e. ZZ )
46	45	adantl 482	. . . . . . 7 |- ( ( ph /\ j e. ( 1 ... M ) ) -> j e. ZZ )
47	44, 46	zsubcld 11487	. . . . . 6 |- ( ( ph /\ j e. ( 1 ... M ) ) -> ( J - j ) e. ZZ )
48	47	adantr 481	. . . . 5 |- ( ( ( ph /\ j e. ( 1 ... M ) ) /\ -. P < ( C ` j ) ) -> ( J - j ) e. ZZ )
49		elnn0z 11390	. . . . . 6 |- ( ( P - ( C ` j ) ) e. NN0 <-> ( ( P - ( C ` j ) ) e. ZZ /\ 0 <_ ( P - ( C ` j ) ) ) )
50	22, 30, 49	sylanbrc 698	. . . . 5 |- ( ( ( ph /\ j e. ( 1 ... M ) ) /\ -. P < ( C ` j ) ) -> ( P - ( C ` j ) ) e. NN0 )
51		zexpcl 12875	. . . . 5 |- ( ( ( J - j ) e. ZZ /\ ( P - ( C ` j ) ) e. NN0 ) -> ( ( J - j ) ^ ( P - ( C ` j ) ) ) e. ZZ )
52	48, 50, 51	syl2anc 693	. . . 4 |- ( ( ( ph /\ j e. ( 1 ... M ) ) /\ -. P < ( C ` j ) ) -> ( ( J - j ) ^ ( P - ( C ` j ) ) ) e. ZZ )
53	41, 52	zmulcld 11488	. . 3 |- ( ( ( ph /\ j e. ( 1 ... M ) ) /\ -. P < ( C ` j ) ) -> ( ( ( ! ` P ) / ( ! ` ( P - ( C ` j ) ) ) ) x. ( ( J - j ) ^ ( P - ( C ` j ) ) ) ) e. ZZ )
54	2, 53	ifclda 4120	. 2 |- ( ( ph /\ j e. ( 1 ... M ) ) -> if ( P < ( C ` j ) , 0 , ( ( ( ! ` P ) / ( ! ` ( P - ( C ` j ) ) ) ) x. ( ( J - j ) ^ ( P - ( C ` j ) ) ) ) ) e. ZZ )
55	1, 54	fprodzcl 14684	1 |- ( ph -> prod_ j e. ( 1 ... M ) if ( P < ( C ` j ) , 0 , ( ( ( ! ` P ) / ( ! ` ( P - ( C ` j ) ) ) ) x. ( ( J - j ) ^ ( P - ( C ` j ) ) ) ) ) e. ZZ )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   /\ w3a 1037   e. wcel 1990   C_ wss 3574   ifcif 4086   class class class wbr 4653   -->wf 5884   ` cfv 5888  (class class class)co 6650   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   ...cfz 12326   ^cexp 12860   !cfa 13060   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-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-prod 14636

This theorem is referenced by:  etransclem15  40466  etransclem28  40479