Theorem etransclem18 38229

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 11745	. . 3 |- ( A (,) B ) C_ ( A [,] B )
2	1	a1i 11	. 2 |- ( ph -> ( A (,) B ) C_ ( A [,] B ) )
3		ioombl 22597	. . 3 |- ( A (,) B ) e. dom vol
4	3	a1i 11	. 2 |- ( ph -> ( A (,) B ) e. dom vol )
5		ere 14220	. . . . . 6 |- _e e. RR
6	5	recni 9673	. . . . 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 37698	. . . . . . 7 |- ( ph -> ( A [,] B ) C_ RR )
11	10	sselda 3418	. . . . . 6 |- ( ( ph /\ x e. ( A [,] B ) ) -> x e. RR )
12	11	recnd 9687	. . . . 5 |- ( ( ph /\ x e. ( A [,] B ) ) -> x e. CC )
13	12	negcld 9992	. . . 4 |- ( ( ph /\ x e. ( A [,] B ) ) -> -u x e. CC )
14	7, 13	cxpcld 23732	. . 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 37908	. . . . . 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 38219	. . . . 5 |- ( ph -> F : RR --> CC )
21	20	adantr 472	. . . 4 |- ( ( ph /\ x e. ( A [,] B ) ) -> F : RR --> CC )
22	21, 11	ffvelrnd 6038	. . 3 |- ( ( ph /\ x e. ( A [,] B ) ) -> ( F ` x ) e. CC )
23	14, 22	mulcld 9681	. 2 |- ( ( ph /\ x e. ( A [,] B ) ) -> ( ( _e ^c -u x ) x. ( F ` x ) ) e. CC )
24		eqidd 2472	. . . . . . . 8 |- ( ( ph /\ x e. ( A [,] B ) ) -> ( y e. CC |-> ( _e ^c y ) ) = ( y e. CC |-> ( _e ^c y ) ) )
25		oveq2 6316	. . . . . . . . 9 |- ( y = -u x -> ( _e ^c y ) = ( _e ^c -u x ) )
26	25	adantl 473	. . . . . . . 8 |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ y = -u x ) -> ( _e ^c y ) = ( _e ^c -u x ) )
27	10, 17	sstrd 3428	. . . . . . . . . 10 |- ( ph -> ( A [,] B ) C_ CC )
28	27	sselda 3418	. . . . . . . . 9 |- ( ( ph /\ x e. ( A [,] B ) ) -> x e. CC )
29	28	negcld 9992	. . . . . . . 8 |- ( ( ph /\ x e. ( A [,] B ) ) -> -u x e. CC )
30	6	a1i 11	. . . . . . . . . 10 |- ( x e. CC -> _e e. CC )
31		negcl 9895	. . . . . . . . . 10 |- ( x e. CC -> -u x e. CC )
32	30, 31	cxpcld 23732	. . . . . . . . 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 5969	. . . . . . 7 |- ( ( ph /\ x e. ( A [,] B ) ) -> ( ( y e. CC |-> ( _e ^c y ) ) ` -u x ) = ( _e ^c -u x ) )
35	34	eqcomd 2477	. . . . . 6 |- ( ( ph /\ x e. ( A [,] B ) ) -> ( _e ^c -u x ) = ( ( y e. CC |-> ( _e ^c y ) ) ` -u x ) )
36	35	mpteq2dva 4482	. . . . 5 |- ( ph -> ( x e. ( A [,] B ) |-> ( _e ^c -u x ) ) = ( x e. ( A [,] B ) |-> ( ( y e. CC |-> ( _e ^c y ) ) ` -u x ) ) )
37		epr 14337	. . . . . . . . 9 |- _e e. RR+
38		mnfxr 11437	. . . . . . . . . . 11 |- -oo e. RR*
39	38	a1i 11	. . . . . . . . . 10 |- ( _e e. RR+ -> -oo e. RR* )
40		0red 9662	. . . . . . . . . 10 |- ( _e e. RR+ -> 0 e. RR )
41		rpxr 11332	. . . . . . . . . 10 |- ( _e e. RR+ -> _e e. RR* )
42		rpgt0 11336	. . . . . . . . . 10 |- ( _e e. RR+ -> 0 < _e )
43	39, 40, 41, 42	gtnelioc 37683	. . . . . . . . 9 |- ( _e e. RR+ -> -. _e e. ( -oo (,] 0 ) )
44	37, 43	ax-mp 5	. . . . . . . 8 |- -. _e e. ( -oo (,] 0 )
45		eldif 3400	. . . . . . . 8 |- ( _e e. ( CC \ ( -oo (,] 0 ) ) <-> ( _e e. CC /\ -. _e e. ( -oo (,] 0 ) ) )
46	6, 44, 45	mpbir2an 934	. . . . . . 7 |- _e e. ( CC \ ( -oo (,] 0 ) )
47		cxpcncf2 37875	. . . . . . 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 2471	. . . . . . . 8 |- ( x e. ( A [,] B ) |-> -u x ) = ( x e. ( A [,] B ) |-> -u x )
50	49	negcncf 22028	. . . . . . 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 22023	. . . . 5 |- ( ph -> ( x e. ( A [,] B ) |-> ( ( y e. CC |-> ( _e ^c y ) ) ` -u x ) ) e. ( ( A [,] B ) -cn-> CC ) )
53	36, 52	eqeltrd 2549	. . . 4 |- ( ph -> ( x e. ( A [,] B ) |-> ( _e ^c -u x ) ) e. ( ( A [,] B ) -cn-> CC ) )
54		ax-resscn 9614	. . . . . . . 8 |- RR C_ CC
55	54	a1i 11	. . . . . . 7 |- ( ( ph /\ x e. ( A [,] B ) ) -> RR C_ CC )
56	18	adantr 472	. . . . . . 7 |- ( ( ph /\ x e. ( A [,] B ) ) -> P e. NN )
57		etransclem18.m	. . . . . . . 8 |- ( ph -> M e. NN0 )
58	57	adantr 472	. . . . . . 7 |- ( ( ph /\ x e. ( A [,] B ) ) -> M e. NN0 )
59		etransclem6 38217	. . . . . . . 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 2493	. . . . . . 7 |- F = ( y e. RR |-> ( ( y ^ ( P - 1 ) ) x. prod_ k e. ( 1 ... M ) ( ( y - k ) ^ P ) ) )
61	55, 56, 58, 60, 11	etransclem13 38224	. . . . . 6 |- ( ( ph /\ x e. ( A [,] B ) ) -> ( F ` x ) = prod_ k e. ( 0 ... M ) ( ( x - k ) ^ if ( k = 0 , ( P - 1 ) , P ) ) )
62	61	mpteq2dva 4482	. . . . 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 12224	. . . . . 6 |- ( ph -> ( 0 ... M ) e. Fin )
64	12	3adant3 1050	. . . . . . . 8 |- ( ( ph /\ x e. ( A [,] B ) /\ k e. ( 0 ... M ) ) -> x e. CC )
65		elfzelz 11826	. . . . . . . . . 10 |- ( k e. ( 0 ... M ) -> k e. ZZ )
66	65	zcnd 11064	. . . . . . . . 9 |- ( k e. ( 0 ... M ) -> k e. CC )
67	66	3ad2ant3 1053	. . . . . . . 8 |- ( ( ph /\ x e. ( A [,] B ) /\ k e. ( 0 ... M ) ) -> k e. CC )
68	64, 67	subcld 10005	. . . . . . 7 |- ( ( ph /\ x e. ( A [,] B ) /\ k e. ( 0 ... M ) ) -> ( x - k ) e. CC )
69		nnm1nn0 10935	. . . . . . . . . 10 |- ( P e. NN -> ( P - 1 ) e. NN0 )
70	18, 69	syl 17	. . . . . . . . 9 |- ( ph -> ( P - 1 ) e. NN0 )
71	18	nnnn0d 10949	. . . . . . . . 9 |- ( ph -> P e. NN0 )
72	70, 71	ifcld 3915	. . . . . . . 8 |- ( ph -> if ( k = 0 , ( P - 1 ) , P ) e. NN0 )
73	72	3ad2ant1 1051	. . . . . . 7 |- ( ( ph /\ x e. ( A [,] B ) /\ k e. ( 0 ... M ) ) -> if ( k = 0 , ( P - 1 ) , P ) e. NN0 )
74	68, 73	expcld 12454	. . . . . 6 |- ( ( ph /\ x e. ( A [,] B ) /\ k e. ( 0 ... M ) ) -> ( ( x - k ) ^ if ( k = 0 , ( P - 1 ) , P ) ) e. CC )
75		nfv 1769	. . . . . . 7 |- F/ x ( ph /\ k e. ( 0 ... M ) )
76		ssid 3437	. . . . . . . . . . 11 |- CC C_ CC
77	76	a1i 11	. . . . . . . . . 10 |- ( ph -> CC C_ CC )
78	27, 77	idcncfg 37846	. . . . . . . . 9 |- ( ph -> ( x e. ( A [,] B ) |-> x ) e. ( ( A [,] B ) -cn-> CC ) )
79	78	adantr 472	. . . . . . . 8 |- ( ( ph /\ k e. ( 0 ... M ) ) -> ( x e. ( A [,] B ) |-> x ) e. ( ( A [,] B ) -cn-> CC ) )
80	27	adantr 472	. . . . . . . . 9 |- ( ( ph /\ k e. ( 0 ... M ) ) -> ( A [,] B ) C_ CC )
81	66	adantl 473	. . . . . . . . 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 37845	. . . . . . . 8 |- ( ( ph /\ k e. ( 0 ... M ) ) -> ( x e. ( A [,] B ) |-> k ) e. ( ( A [,] B ) -cn-> CC ) )
84	79, 83	subcncf 37843	. . . . . . 7 |- ( ( ph /\ k e. ( 0 ... M ) ) -> ( x e. ( A [,] B ) |-> ( x - k ) ) e. ( ( A [,] B ) -cn-> CC ) )
85		expcncf 22032	. . . . . . . . 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 472	. . . . . . 7 |- ( ( ph /\ k e. ( 0 ... M ) ) -> ( y e. CC |-> ( y ^ if ( k = 0 , ( P - 1 ) , P ) ) ) e. ( CC -cn-> CC ) )
88		oveq1 6315	. . . . . . 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 37874	. . . . . 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 37876	. . . . 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 2549	. . . 4 |- ( ph -> ( x e. ( A [,] B ) |-> ( F ` x ) ) e. ( ( A [,] B ) -cn-> CC ) )
92	53, 91	mulcncf 22476	. . 3 |- ( ph -> ( x e. ( A [,] B ) |-> ( ( _e ^c -u x ) x. ( F ` x ) ) ) e. ( ( A [,] B ) -cn-> CC ) )
93		cniccibl 22877	. . 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 1292	. 2 |- ( ph -> ( x e. ( A [,] B ) |-> ( ( _e ^c -u x ) x. ( F ` x ) ) ) e. L^1 )
95	2, 4, 23, 94	iblss 22841	1 |- ( ph -> ( x e. ( A (,) B ) |-> ( ( _e ^c -u x ) x. ( F ` x ) ) ) e. L^1 )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 376   /\ w3a 1007   = wceq 1452   e. wcel 1904   \ cdif 3387   C_ wss 3390   ifcif 3872   {cpr 3961   |-> cmpt 4454   dom cdm 4839   -->wf 5585   ` cfv 5589  (class class class)co 6308   CCcc 9555   RRcr 9556   0cc0 9557   1c1 9558   x. cmul 9562   -oocmnf 9691   RR*cxr 9692   - cmin 9880   -ucneg 9881   NNcn 10631   NN0cn0 10893   RR+crp 11325   (,)cioo 11660   (,]cioc 11661   [,]cicc 11663   ...cfz 11810   ^cexp 12310   prod_cprod 14036   _eceu 14192   ↾_t crest 15397   TopOpenctopn 15398  ℂ_fldccnfld 19047   -cn->ccncf 21986   volcvol 22493   L^1cibl 22654   ^c ccxp 23584

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164  ax-cc 8883  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635  ax-addf 9636  ax-mulf 9637

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-iin 4272  df-disj 4367  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-of 6550  df-ofr 6551  df-om 6712  df-1st 6812  df-2nd 6813  df-supp 6934  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-2o 7201  df-oadd 7204  df-omul 7205  df-er 7381  df-map 7492  df-pm 7493  df-ixp 7541  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-fsupp 7902  df-fi 7943  df-sup 7974  df-inf 7975  df-oi 8043  df-card 8391  df-acn 8394  df-cda 8616  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-4 10692  df-5 10693  df-6 10694  df-7 10695  df-8 10696  df-9 10697  df-10 10698  df-n0 10894  df-z 10962  df-dec 11075  df-uz 11183  df-q 11288  df-rp 11326  df-xneg 11432  df-xadd 11433  df-xmul 11434  df-ioo 11664  df-ioc 11665  df-ico 11666  df-icc 11667  df-fz 11811  df-fzo 11943  df-fl 12061  df-mod 12130  df-seq 12252  df-exp 12311  df-fac 12498  df-bc 12526  df-hash 12554  df-shft 13207  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-limsup 13603  df-clim 13629  df-rlim 13630  df-sum 13830  df-prod 14037  df-ef 14198  df-e 14199  df-sin 14200  df-cos 14201  df-tan 14202  df-pi 14203  df-struct 15201  df-ndx 15202  df-slot 15203  df-base 15204  df-sets 15205  df-ress 15206  df-plusg 15281  df-mulr 15282  df-starv 15283  df-sca 15284  df-vsca 15285  df-ip 15286  df-tset 15287  df-ple 15288  df-ds 15290  df-unif 15291  df-hom 15292  df-cco 15293  df-rest 15399  df-topn 15400  df-0g 15418  df-gsum 15419  df-topgen 15420  df-pt 15421  df-prds 15424  df-xrs 15478  df-qtop 15484  df-imas 15485  df-xps 15488  df-mre 15570  df-mrc 15571  df-acs 15573  df-mgm 16566  df-sgrp 16605  df-mnd 16615  df-submnd 16661  df-mulg 16754  df-cntz 17049  df-cmn 17510  df-psmet 19039  df-xmet 19040  df-met 19041  df-bl 19042  df-mopn 19043  df-fbas 19044  df-fg 19045  df-cnfld 19048  df-top 19998  df-bases 19999  df-topon 20000  df-topsp 20001  df-cld 20111  df-ntr 20112  df-cls 20113  df-nei 20191  df-lp 20229  df-perf 20230  df-cn 20320  df-cnp 20321  df-haus 20408  df-cmp 20479  df-tx 20654  df-hmeo 20847  df-fil 20939  df-fm 21031  df-flim 21032  df-flf 21033  df-xms 21413  df-ms 21414  df-tms 21415  df-cncf 21988  df-ovol 22494  df-vol 22496  df-mbf 22656  df-itg1 22657  df-itg2 22658  df-ibl 22659  df-0p 22707  df-limc 22900  df-dv 22901  df-log 23585  df-cxp 23586

This theorem is referenced by:  etransclem23  38234  etransclem46  38257