Theorem etransclem18 40469

Description: The given function is integrable. (Contributed by Glauco Siliprandi, 5-Apr-2020.)

Hypotheses

Ref	Expression
etransclem18.s	|- ( ph -> RR e. { RR , CC } )
etransclem18.x	|- ( ph -> RR e. ( ( TopOpen ` ℂfld ) ↾t RR ) )
etransclem18.p	|- ( ph -> P e. NN )
etransclem18.m	|- ( ph -> M e. NN0 )
etransclem18.f	|- F = ( x e. RR |-> ( ( x ^ ( P - 1 ) ) x. prod_ j e. ( 1 ... M ) ( ( x - j ) ^ P ) ) )
etransclem18.a	|- ( ph -> A e. RR )
etransclem18.b	|- ( ph -> B e. RR )

Assertion

Ref	Expression
etransclem18	|- ( ph -> ( x e. ( A (,) B ) |-> ( ( _e ^c -u x ) x. ( F ` x ) ) ) e. L^1 )

Distinct variable groups:   x, A   x, B   j, M, x   P, j, x   ph, j, x

Allowed substitution hints:   A( j)   B( j)   F( x, j)

Proof of Theorem etransclem18

Dummy variables k y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ioossicc 12259	. . 3 |- ( A (,) B ) C_ ( A [,] B )
2	1	a1i 11	. 2 |- ( ph -> ( A (,) B ) C_ ( A [,] B ) )
3		ioombl 23333	. . 3 |- ( A (,) B ) e. dom vol
4	3	a1i 11	. 2 |- ( ph -> ( A (,) B ) e. dom vol )
5		ere 14819	. . . . . 6 |- _e e. RR
6	5	recni 10052	. . . . 5 |- _e e. CC
7	6	a1i 11	. . . 4 |- ( ( ph /\ x e. ( A [,] B ) ) -> _e e. CC )
8		etransclem18.a	. . . . . . . 8 |- ( ph -> A e. RR )
9		etransclem18.b	. . . . . . . 8 |- ( ph -> B e. RR )
10	8, 9	iccssred 39727	. . . . . . 7 |- ( ph -> ( A [,] B ) C_ RR )
11	10	sselda 3603	. . . . . 6 |- ( ( ph /\ x e. ( A [,] B ) ) -> x e. RR )
12	11	recnd 10068	. . . . 5 |- ( ( ph /\ x e. ( A [,] B ) ) -> x e. CC )
13	12	negcld 10379	. . . 4 |- ( ( ph /\ x e. ( A [,] B ) ) -> -u x e. CC )
14	7, 13	cxpcld 24454	. . 3 |- ( ( ph /\ x e. ( A [,] B ) ) -> ( _e ^c -u x ) e. CC )
15		etransclem18.s	. . . . . . 7 |- ( ph -> RR e. { RR , CC } )
16		etransclem18.x	. . . . . . 7 |- ( ph -> RR e. ( ( TopOpen ` ℂfld ) ↾t RR ) )
17	15, 16	dvdmsscn 40151	. . . . . 6 |- ( ph -> RR C_ CC )
18		etransclem18.p	. . . . . 6 |- ( ph -> P e. NN )
19		etransclem18.f	. . . . . 6 |- F = ( x e. RR |-> ( ( x ^ ( P - 1 ) ) x. prod_ j e. ( 1 ... M ) ( ( x - j ) ^ P ) ) )
20	17, 18, 19	etransclem8 40459	. . . . 5 |- ( ph -> F : RR --> CC )
21	20	adantr 481	. . . 4 |- ( ( ph /\ x e. ( A [,] B ) ) -> F : RR --> CC )
22	21, 11	ffvelrnd 6360	. . 3 |- ( ( ph /\ x e. ( A [,] B ) ) -> ( F ` x ) e. CC )
23	14, 22	mulcld 10060	. 2 |- ( ( ph /\ x e. ( A [,] B ) ) -> ( ( _e ^c -u x ) x. ( F ` x ) ) e. CC )
24		eqidd 2623	. . . . . . . 8 |- ( ( ph /\ x e. ( A [,] B ) ) -> ( y e. CC |-> ( _e ^c y ) ) = ( y e. CC |-> ( _e ^c y ) ) )
25		oveq2 6658	. . . . . . . . 9 |- ( y = -u x -> ( _e ^c y ) = ( _e ^c -u x ) )
26	25	adantl 482	. . . . . . . 8 |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ y = -u x ) -> ( _e ^c y ) = ( _e ^c -u x ) )
27	10, 17	sstrd 3613	. . . . . . . . . 10 |- ( ph -> ( A [,] B ) C_ CC )
28	27	sselda 3603	. . . . . . . . 9 |- ( ( ph /\ x e. ( A [,] B ) ) -> x e. CC )
29	28	negcld 10379	. . . . . . . 8 |- ( ( ph /\ x e. ( A [,] B ) ) -> -u x e. CC )
30	6	a1i 11	. . . . . . . . . 10 |- ( x e. CC -> _e e. CC )
31		negcl 10281	. . . . . . . . . 10 |- ( x e. CC -> -u x e. CC )
32	30, 31	cxpcld 24454	. . . . . . . . 9 |- ( x e. CC -> ( _e ^c -u x ) e. CC )
33	28, 32	syl 17	. . . . . . . 8 |- ( ( ph /\ x e. ( A [,] B ) ) -> ( _e ^c -u x ) e. CC )
34	24, 26, 29, 33	fvmptd 6288	. . . . . . 7 |- ( ( ph /\ x e. ( A [,] B ) ) -> ( ( y e. CC |-> ( _e ^c y ) ) ` -u x ) = ( _e ^c -u x ) )
35	34	eqcomd 2628	. . . . . 6 |- ( ( ph /\ x e. ( A [,] B ) ) -> ( _e ^c -u x ) = ( ( y e. CC |-> ( _e ^c y ) ) ` -u x ) )
36	35	mpteq2dva 4744	. . . . 5 |- ( ph -> ( x e. ( A [,] B ) |-> ( _e ^c -u x ) ) = ( x e. ( A [,] B ) |-> ( ( y e. CC |-> ( _e ^c y ) ) ` -u x ) ) )
37		epr 14936	. . . . . . . . 9 |- _e e. RR+
38		mnfxr 10096	. . . . . . . . . . 11 |- -oo e. RR*
39	38	a1i 11	. . . . . . . . . 10 |- ( _e e. RR+ -> -oo e. RR* )
40		0red 10041	. . . . . . . . . 10 |- ( _e e. RR+ -> 0 e. RR )
41		rpxr 11840	. . . . . . . . . 10 |- ( _e e. RR+ -> _e e. RR* )
42		rpgt0 11844	. . . . . . . . . 10 |- ( _e e. RR+ -> 0 < _e )
43	39, 40, 41, 42	gtnelioc 39712	. . . . . . . . 9 |- ( _e e. RR+ -> -. _e e. ( -oo (,] 0 ) )
44	37, 43	ax-mp 5	. . . . . . . 8 |- -. _e e. ( -oo (,] 0 )
45		eldif 3584	. . . . . . . 8 |- ( _e e. ( CC \ ( -oo (,] 0 ) ) <-> ( _e e. CC /\ -. _e e. ( -oo (,] 0 ) ) )
46	6, 44, 45	mpbir2an 955	. . . . . . 7 |- _e e. ( CC \ ( -oo (,] 0 ) )
47		cxpcncf2 40113	. . . . . . 7 |- ( _e e. ( CC \ ( -oo (,] 0 ) ) -> ( y e. CC |-> ( _e ^c y ) ) e. ( CC -cn-> CC ) )
48	46, 47	mp1i 13	. . . . . 6 |- ( ph -> ( y e. CC |-> ( _e ^c y ) ) e. ( CC -cn-> CC ) )
49		eqid 2622	. . . . . . . 8 |- ( x e. ( A [,] B ) |-> -u x ) = ( x e. ( A [,] B ) |-> -u x )
50	49	negcncf 22721	. . . . . . 7 |- ( ( A [,] B ) C_ CC -> ( x e. ( A [,] B ) |-> -u x ) e. ( ( A [,] B ) -cn-> CC ) )
51	27, 50	syl 17	. . . . . 6 |- ( ph -> ( x e. ( A [,] B ) |-> -u x ) e. ( ( A [,] B ) -cn-> CC ) )
52	48, 51	cncfmpt1f 22716	. . . . 5 |- ( ph -> ( x e. ( A [,] B ) |-> ( ( y e. CC |-> ( _e ^c y ) ) ` -u x ) ) e. ( ( A [,] B ) -cn-> CC ) )
53	36, 52	eqeltrd 2701	. . . 4 |- ( ph -> ( x e. ( A [,] B ) |-> ( _e ^c -u x ) ) e. ( ( A [,] B ) -cn-> CC ) )
54		ax-resscn 9993	. . . . . . . 8 |- RR C_ CC
55	54	a1i 11	. . . . . . 7 |- ( ( ph /\ x e. ( A [,] B ) ) -> RR C_ CC )
56	18	adantr 481	. . . . . . 7 |- ( ( ph /\ x e. ( A [,] B ) ) -> P e. NN )
57		etransclem18.m	. . . . . . . 8 |- ( ph -> M e. NN0 )
58	57	adantr 481	. . . . . . 7 |- ( ( ph /\ x e. ( A [,] B ) ) -> M e. NN0 )
59		etransclem6 40457	. . . . . . . 8 |- ( x e. RR |-> ( ( x ^ ( P - 1 ) ) x. prod_ j e. ( 1 ... M ) ( ( x - j ) ^ P ) ) ) = ( y e. RR |-> ( ( y ^ ( P - 1 ) ) x. prod_ k e. ( 1 ... M ) ( ( y - k ) ^ P ) ) )
60	19, 59	eqtri 2644	. . . . . . 7 |- F = ( y e. RR |-> ( ( y ^ ( P - 1 ) ) x. prod_ k e. ( 1 ... M ) ( ( y - k ) ^ P ) ) )
61	55, 56, 58, 60, 11	etransclem13 40464	. . . . . 6 |- ( ( ph /\ x e. ( A [,] B ) ) -> ( F ` x ) = prod_ k e. ( 0 ... M ) ( ( x - k ) ^ if ( k = 0 , ( P - 1 ) , P ) ) )
62	61	mpteq2dva 4744	. . . . 5 |- ( ph -> ( x e. ( A [,] B ) |-> ( F ` x ) ) = ( x e. ( A [,] B ) |-> prod_ k e. ( 0 ... M ) ( ( x - k ) ^ if ( k = 0 , ( P - 1 ) , P ) ) ) )
63		fzfid 12772	. . . . . 6 |- ( ph -> ( 0 ... M ) e. Fin )
64	12	3adant3 1081	. . . . . . . 8 |- ( ( ph /\ x e. ( A [,] B ) /\ k e. ( 0 ... M ) ) -> x e. CC )
65		elfzelz 12342	. . . . . . . . . 10 |- ( k e. ( 0 ... M ) -> k e. ZZ )
66	65	zcnd 11483	. . . . . . . . 9 |- ( k e. ( 0 ... M ) -> k e. CC )
67	66	3ad2ant3 1084	. . . . . . . 8 |- ( ( ph /\ x e. ( A [,] B ) /\ k e. ( 0 ... M ) ) -> k e. CC )
68	64, 67	subcld 10392	. . . . . . 7 |- ( ( ph /\ x e. ( A [,] B ) /\ k e. ( 0 ... M ) ) -> ( x - k ) e. CC )
69		nnm1nn0 11334	. . . . . . . . . 10 |- ( P e. NN -> ( P - 1 ) e. NN0 )
70	18, 69	syl 17	. . . . . . . . 9 |- ( ph -> ( P - 1 ) e. NN0 )
71	18	nnnn0d 11351	. . . . . . . . 9 |- ( ph -> P e. NN0 )
72	70, 71	ifcld 4131	. . . . . . . 8 |- ( ph -> if ( k = 0 , ( P - 1 ) , P ) e. NN0 )
73	72	3ad2ant1 1082	. . . . . . 7 |- ( ( ph /\ x e. ( A [,] B ) /\ k e. ( 0 ... M ) ) -> if ( k = 0 , ( P - 1 ) , P ) e. NN0 )
74	68, 73	expcld 13008	. . . . . 6 |- ( ( ph /\ x e. ( A [,] B ) /\ k e. ( 0 ... M ) ) -> ( ( x - k ) ^ if ( k = 0 , ( P - 1 ) , P ) ) e. CC )
75		nfv 1843	. . . . . . 7 |- F/ x ( ph /\ k e. ( 0 ... M ) )
76		ssid 3624	. . . . . . . . . . 11 |- CC C_ CC
77	76	a1i 11	. . . . . . . . . 10 |- ( ph -> CC C_ CC )
78	27, 77	idcncfg 40085	. . . . . . . . 9 |- ( ph -> ( x e. ( A [,] B ) |-> x ) e. ( ( A [,] B ) -cn-> CC ) )
79	78	adantr 481	. . . . . . . 8 |- ( ( ph /\ k e. ( 0 ... M ) ) -> ( x e. ( A [,] B ) |-> x ) e. ( ( A [,] B ) -cn-> CC ) )
80	27	adantr 481	. . . . . . . . 9 |- ( ( ph /\ k e. ( 0 ... M ) ) -> ( A [,] B ) C_ CC )
81	66	adantl 482	. . . . . . . . 9 |- ( ( ph /\ k e. ( 0 ... M ) ) -> k e. CC )
82	76	a1i 11	. . . . . . . . 9 |- ( ( ph /\ k e. ( 0 ... M ) ) -> CC C_ CC )
83	80, 81, 82	constcncfg 40084	. . . . . . . 8 |- ( ( ph /\ k e. ( 0 ... M ) ) -> ( x e. ( A [,] B ) |-> k ) e. ( ( A [,] B ) -cn-> CC ) )
84	79, 83	subcncf 40082	. . . . . . 7 |- ( ( ph /\ k e. ( 0 ... M ) ) -> ( x e. ( A [,] B ) |-> ( x - k ) ) e. ( ( A [,] B ) -cn-> CC ) )
85		expcncf 22725	. . . . . . . . 9 |- ( if ( k = 0 , ( P - 1 ) , P ) e. NN0 -> ( y e. CC |-> ( y ^ if ( k = 0 , ( P - 1 ) , P ) ) ) e. ( CC -cn-> CC ) )
86	72, 85	syl 17	. . . . . . . 8 |- ( ph -> ( y e. CC |-> ( y ^ if ( k = 0 , ( P - 1 ) , P ) ) ) e. ( CC -cn-> CC ) )
87	86	adantr 481	. . . . . . 7 |- ( ( ph /\ k e. ( 0 ... M ) ) -> ( y e. CC |-> ( y ^ if ( k = 0 , ( P - 1 ) , P ) ) ) e. ( CC -cn-> CC ) )
88		oveq1 6657	. . . . . . 7 |- ( y = ( x - k ) -> ( y ^ if ( k = 0 , ( P - 1 ) , P ) ) = ( ( x - k ) ^ if ( k = 0 , ( P - 1 ) , P ) ) )
89	75, 84, 87, 82, 88	cncfcompt2 40112	. . . . . 6 |- ( ( ph /\ k e. ( 0 ... M ) ) -> ( x e. ( A [,] B ) |-> ( ( x - k ) ^ if ( k = 0 , ( P - 1 ) , P ) ) ) e. ( ( A [,] B ) -cn-> CC ) )
90	27, 63, 74, 89	fprodcncf 40114	. . . . 5 |- ( ph -> ( x e. ( A [,] B ) |-> prod_ k e. ( 0 ... M ) ( ( x - k ) ^ if ( k = 0 , ( P - 1 ) , P ) ) ) e. ( ( A [,] B ) -cn-> CC ) )
91	62, 90	eqeltrd 2701	. . . 4 |- ( ph -> ( x e. ( A [,] B ) |-> ( F ` x ) ) e. ( ( A [,] B ) -cn-> CC ) )
92	53, 91	mulcncf 23215	. . 3 |- ( ph -> ( x e. ( A [,] B ) |-> ( ( _e ^c -u x ) x. ( F ` x ) ) ) e. ( ( A [,] B ) -cn-> CC ) )
93		cniccibl 23607	. . 3 |- ( ( A e. RR /\ B e. RR /\ ( x e. ( A [,] B ) |-> ( ( _e ^c -u x ) x. ( F ` x ) ) ) e. ( ( A [,] B ) -cn-> CC ) ) -> ( x e. ( A [,] B ) |-> ( ( _e ^c -u x ) x. ( F ` x ) ) ) e. L^1 )
94	8, 9, 92, 93	syl3anc 1326	. 2 |- ( ph -> ( x e. ( A [,] B ) |-> ( ( _e ^c -u x ) x. ( F ` x ) ) ) e. L^1 )
95	2, 4, 23, 94	iblss 23571	1 |- ( ph -> ( x e. ( A (,) B ) |-> ( ( _e ^c -u x ) x. ( F ` x ) ) ) e. L^1 )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   \ cdif 3571   C_ wss 3574   ifcif 4086   {cpr 4179   |-> cmpt 4729   dom cdm 5114   -->wf 5884   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   x. cmul 9941   -oocmnf 10072   RR*cxr 10073   - cmin 10266   -ucneg 10267   NNcn 11020   NN0cn0 11292   RR+crp 11832   (,)cioo 12175   (,]cioc 12176   [,]cicc 12178   ...cfz 12326   ^cexp 12860   prod_cprod 14635   _eceu 14793   ↾_t crest 16081   TopOpenctopn 16082  ℂ_fldccnfld 19746   -cn->ccncf 22679   volcvol 23232   L^1cibl 23386   ^c ccxp 24302

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-cc 9257  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-disj 4621  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-ofr 6898  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-omul 7565  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-acn 8768  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-ioc 12180  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-shft 13807  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-limsup 14202  df-clim 14219  df-rlim 14220  df-sum 14417  df-prod 14636  df-ef 14798  df-e 14799  df-sin 14800  df-cos 14801  df-tan 14802  df-pi 14803  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-cmp 21190  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-ovol 23233  df-vol 23234  df-mbf 23388  df-itg1 23389  df-itg2 23390  df-ibl 23391  df-0p 23437  df-limc 23630  df-dv 23631  df-log 24303  df-cxp 24304

This theorem is referenced by:  etransclem23  40474  etransclem46  40497