Theorem etransclem18 38117

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 11720	. . 3 |- ( A (,) B ) C_ ( A [,] B )
2	1	a1i 11	. 2 |- ( ph -> ( A (,) B ) C_ ( A [,] B ) )
3		ioombl 22518	. . 3 |- ( A (,) B ) e. dom vol
4	3	a1i 11	. 2 |- ( ph -> ( A (,) B ) e. dom vol )
5		ere 14143	. . . . . 6 |- _e e. RR
6	5	recni 9655	. . . . 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 37602	. . . . . . 7 |- ( ph -> ( A [,] B ) C_ RR )
11	10	sselda 3432	. . . . . 6 |- ( ( ph /\ x e. ( A [,] B ) ) -> x e. RR )
12	11	recnd 9669	. . . . 5 |- ( ( ph /\ x e. ( A [,] B ) ) -> x e. CC )
13	12	negcld 9973	. . . 4 |- ( ( ph /\ x e. ( A [,] B ) ) -> -u x e. CC )
14	7, 13	cxpcld 23653	. . 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 37811	. . . . . 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 38107	. . . . 5 |- ( ph -> F : RR --> CC )
21	20	adantr 467	. . . 4 |- ( ( ph /\ x e. ( A [,] B ) ) -> F : RR --> CC )
22	21, 11	ffvelrnd 6023	. . 3 |- ( ( ph /\ x e. ( A [,] B ) ) -> ( F ` x ) e. CC )
23	14, 22	mulcld 9663	. 2 |- ( ( ph /\ x e. ( A [,] B ) ) -> ( ( _e ^c -u x ) x. ( F ` x ) ) e. CC )
24		eqidd 2452	. . . . . . . 8 |- ( ( ph /\ x e. ( A [,] B ) ) -> ( y e. CC |-> ( _e ^c y ) ) = ( y e. CC |-> ( _e ^c y ) ) )
25		oveq2 6298	. . . . . . . . 9 |- ( y = -u x -> ( _e ^c y ) = ( _e ^c -u x ) )
26	25	adantl 468	. . . . . . . 8 |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ y = -u x ) -> ( _e ^c y ) = ( _e ^c -u x ) )
27	10, 17	sstrd 3442	. . . . . . . . . 10 |- ( ph -> ( A [,] B ) C_ CC )
28	27	sselda 3432	. . . . . . . . 9 |- ( ( ph /\ x e. ( A [,] B ) ) -> x e. CC )
29	28	negcld 9973	. . . . . . . 8 |- ( ( ph /\ x e. ( A [,] B ) ) -> -u x e. CC )
30	6	a1i 11	. . . . . . . . . 10 |- ( x e. CC -> _e e. CC )
31		negcl 9875	. . . . . . . . . 10 |- ( x e. CC -> -u x e. CC )
32	30, 31	cxpcld 23653	. . . . . . . . 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 5954	. . . . . . 7 |- ( ( ph /\ x e. ( A [,] B ) ) -> ( ( y e. CC |-> ( _e ^c y ) ) ` -u x ) = ( _e ^c -u x ) )
35	34	eqcomd 2457	. . . . . 6 |- ( ( ph /\ x e. ( A [,] B ) ) -> ( _e ^c -u x ) = ( ( y e. CC |-> ( _e ^c y ) ) ` -u x ) )
36	35	mpteq2dva 4489	. . . . 5 |- ( ph -> ( x e. ( A [,] B ) |-> ( _e ^c -u x ) ) = ( x e. ( A [,] B ) |-> ( ( y e. CC |-> ( _e ^c y ) ) ` -u x ) ) )
37		epr 14260	. . . . . . . . 9 |- _e e. RR+
38		mnfxr 11414	. . . . . . . . . . 11 |- -oo e. RR*
39	38	a1i 11	. . . . . . . . . 10 |- ( _e e. RR+ -> -oo e. RR* )
40		0red 9644	. . . . . . . . . 10 |- ( _e e. RR+ -> 0 e. RR )
41		rpxr 11309	. . . . . . . . . 10 |- ( _e e. RR+ -> _e e. RR* )
42		rpgt0 11313	. . . . . . . . . 10 |- ( _e e. RR+ -> 0 < _e )
43	39, 40, 41, 42	gtnelioc 37587	. . . . . . . . 9 |- ( _e e. RR+ -> -. _e e. ( -oo (,] 0 ) )
44	37, 43	ax-mp 5	. . . . . . . 8 |- -. _e e. ( -oo (,] 0 )
45		eldif 3414	. . . . . . . 8 |- ( _e e. ( CC \ ( -oo (,] 0 ) ) <-> ( _e e. CC /\ -. _e e. ( -oo (,] 0 ) ) )
46	6, 44, 45	mpbir2an 931	. . . . . . 7 |- _e e. ( CC \ ( -oo (,] 0 ) )
47		cxpcncf2 37778	. . . . . . 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 2451	. . . . . . . 8 |- ( x e. ( A [,] B ) |-> -u x ) = ( x e. ( A [,] B ) |-> -u x )
50	49	negcncf 21950	. . . . . . 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 21945	. . . . 5 |- ( ph -> ( x e. ( A [,] B ) |-> ( ( y e. CC |-> ( _e ^c y ) ) ` -u x ) ) e. ( ( A [,] B ) -cn-> CC ) )
53	36, 52	eqeltrd 2529	. . . 4 |- ( ph -> ( x e. ( A [,] B ) |-> ( _e ^c -u x ) ) e. ( ( A [,] B ) -cn-> CC ) )
54		ax-resscn 9596	. . . . . . . 8 |- RR C_ CC
55	54	a1i 11	. . . . . . 7 |- ( ( ph /\ x e. ( A [,] B ) ) -> RR C_ CC )
56	18	adantr 467	. . . . . . 7 |- ( ( ph /\ x e. ( A [,] B ) ) -> P e. NN )
57		etransclem18.m	. . . . . . . 8 |- ( ph -> M e. NN0 )
58	57	adantr 467	. . . . . . 7 |- ( ( ph /\ x e. ( A [,] B ) ) -> M e. NN0 )
59		etransclem6 38105	. . . . . . . 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 2473	. . . . . . 7 |- F = ( y e. RR |-> ( ( y ^ ( P - 1 ) ) x. prod_ k e. ( 1 ... M ) ( ( y - k ) ^ P ) ) )
61	55, 56, 58, 60, 11	etransclem13 38112	. . . . . 6 |- ( ( ph /\ x e. ( A [,] B ) ) -> ( F ` x ) = prod_ k e. ( 0 ... M ) ( ( x - k ) ^ if ( k = 0 , ( P - 1 ) , P ) ) )
62	61	mpteq2dva 4489	. . . . 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 12186	. . . . . 6 |- ( ph -> ( 0 ... M ) e. Fin )
64	12	3adant3 1028	. . . . . . . 8 |- ( ( ph /\ x e. ( A [,] B ) /\ k e. ( 0 ... M ) ) -> x e. CC )
65		elfzelz 11800	. . . . . . . . . 10 |- ( k e. ( 0 ... M ) -> k e. ZZ )
66	65	zcnd 11041	. . . . . . . . 9 |- ( k e. ( 0 ... M ) -> k e. CC )
67	66	3ad2ant3 1031	. . . . . . . 8 |- ( ( ph /\ x e. ( A [,] B ) /\ k e. ( 0 ... M ) ) -> k e. CC )
68	64, 67	subcld 9986	. . . . . . 7 |- ( ( ph /\ x e. ( A [,] B ) /\ k e. ( 0 ... M ) ) -> ( x - k ) e. CC )
69		nnm1nn0 10911	. . . . . . . . . 10 |- ( P e. NN -> ( P - 1 ) e. NN0 )
70	18, 69	syl 17	. . . . . . . . 9 |- ( ph -> ( P - 1 ) e. NN0 )
71	18	nnnn0d 10925	. . . . . . . . 9 |- ( ph -> P e. NN0 )
72	70, 71	ifcld 3924	. . . . . . . 8 |- ( ph -> if ( k = 0 , ( P - 1 ) , P ) e. NN0 )
73	72	3ad2ant1 1029	. . . . . . 7 |- ( ( ph /\ x e. ( A [,] B ) /\ k e. ( 0 ... M ) ) -> if ( k = 0 , ( P - 1 ) , P ) e. NN0 )
74	68, 73	expcld 12416	. . . . . 6 |- ( ( ph /\ x e. ( A [,] B ) /\ k e. ( 0 ... M ) ) -> ( ( x - k ) ^ if ( k = 0 , ( P - 1 ) , P ) ) e. CC )
75		nfv 1761	. . . . . . 7 |- F/ x ( ph /\ k e. ( 0 ... M ) )
76		ssid 3451	. . . . . . . . . . 11 |- CC C_ CC
77	76	a1i 11	. . . . . . . . . 10 |- ( ph -> CC C_ CC )
78	27, 77	idcncfg 37749	. . . . . . . . 9 |- ( ph -> ( x e. ( A [,] B ) |-> x ) e. ( ( A [,] B ) -cn-> CC ) )
79	78	adantr 467	. . . . . . . 8 |- ( ( ph /\ k e. ( 0 ... M ) ) -> ( x e. ( A [,] B ) |-> x ) e. ( ( A [,] B ) -cn-> CC ) )
80	27	adantr 467	. . . . . . . . 9 |- ( ( ph /\ k e. ( 0 ... M ) ) -> ( A [,] B ) C_ CC )
81	66	adantl 468	. . . . . . . . 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 37748	. . . . . . . 8 |- ( ( ph /\ k e. ( 0 ... M ) ) -> ( x e. ( A [,] B ) |-> k ) e. ( ( A [,] B ) -cn-> CC ) )
84	79, 83	subcncf 37746	. . . . . . 7 |- ( ( ph /\ k e. ( 0 ... M ) ) -> ( x e. ( A [,] B ) |-> ( x - k ) ) e. ( ( A [,] B ) -cn-> CC ) )
85		expcncf 21954	. . . . . . . . 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 467	. . . . . . 7 |- ( ( ph /\ k e. ( 0 ... M ) ) -> ( y e. CC |-> ( y ^ if ( k = 0 , ( P - 1 ) , P ) ) ) e. ( CC -cn-> CC ) )
88		oveq1 6297	. . . . . . 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 37777	. . . . . 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 37779	. . . . 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 2529	. . . 4 |- ( ph -> ( x e. ( A [,] B ) |-> ( F ` x ) ) e. ( ( A [,] B ) -cn-> CC ) )
92	53, 91	mulcncf 22398	. . 3 |- ( ph -> ( x e. ( A [,] B ) |-> ( ( _e ^c -u x ) x. ( F ` x ) ) ) e. ( ( A [,] B ) -cn-> CC ) )
93		cniccibl 22798	. . 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 1268	. 2 |- ( ph -> ( x e. ( A [,] B ) |-> ( ( _e ^c -u x ) x. ( F ` x ) ) ) e. L^1 )
95	2, 4, 23, 94	iblss 22762	1 |- ( ph -> ( x e. ( A (,) B ) |-> ( ( _e ^c -u x ) x. ( F ` x ) ) ) e. L^1 )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 371   /\ w3a 985   = wceq 1444   e. wcel 1887   \ cdif 3401   C_ wss 3404   ifcif 3881   {cpr 3970   |-> cmpt 4461   dom cdm 4834   -->wf 5578   ` cfv 5582  (class class class)co 6290   CCcc 9537   RRcr 9538   0cc0 9539   1c1 9540   x. cmul 9544   -oocmnf 9673   RR*cxr 9674   - cmin 9860   -ucneg 9861   NNcn 10609   NN0cn0 10869   RR+crp 11302   (,)cioo 11635   (,]cioc 11636   [,]cicc 11638   ...cfz 11784   ^cexp 12272   prod_cprod 13959   _eceu 14115   ↾_t crest 15319   TopOpenctopn 15320  ℂ_fldccnfld 18970   -cn->ccncf 21908   volcvol 22415   L^1cibl 22575   ^c ccxp 23505

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-inf2 8146  ax-cc 8865  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-pre-sup 9617  ax-addf 9618  ax-mulf 9619

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-fal 1450  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-iin 4281  df-disj 4374  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-se 4794  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-isom 5591  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-of 6531  df-ofr 6532  df-om 6693  df-1st 6793  df-2nd 6794  df-supp 6915  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-2o 7183  df-oadd 7186  df-omul 7187  df-er 7363  df-map 7474  df-pm 7475  df-ixp 7523  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-fsupp 7884  df-fi 7925  df-sup 7956  df-inf 7957  df-oi 8025  df-card 8373  df-acn 8376  df-cda 8598  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-10 10676  df-n0 10870  df-z 10938  df-dec 11052  df-uz 11160  df-q 11265  df-rp 11303  df-xneg 11409  df-xadd 11410  df-xmul 11411  df-ioo 11639  df-ioc 11640  df-ico 11641  df-icc 11642  df-fz 11785  df-fzo 11916  df-fl 12028  df-mod 12097  df-seq 12214  df-exp 12273  df-fac 12460  df-bc 12488  df-hash 12516  df-shft 13130  df-cj 13162  df-re 13163  df-im 13164  df-sqrt 13298  df-abs 13299  df-limsup 13526  df-clim 13552  df-rlim 13553  df-sum 13753  df-prod 13960  df-ef 14121  df-e 14122  df-sin 14123  df-cos 14124  df-tan 14125  df-pi 14126  df-struct 15123  df-ndx 15124  df-slot 15125  df-base 15126  df-sets 15127  df-ress 15128  df-plusg 15203  df-mulr 15204  df-starv 15205  df-sca 15206  df-vsca 15207  df-ip 15208  df-tset 15209  df-ple 15210  df-ds 15212  df-unif 15213  df-hom 15214  df-cco 15215  df-rest 15321  df-topn 15322  df-0g 15340  df-gsum 15341  df-topgen 15342  df-pt 15343  df-prds 15346  df-xrs 15400  df-qtop 15406  df-imas 15407  df-xps 15410  df-mre 15492  df-mrc 15493  df-acs 15495  df-mgm 16488  df-sgrp 16527  df-mnd 16537  df-submnd 16583  df-mulg 16676  df-cntz 16971  df-cmn 17432  df-psmet 18962  df-xmet 18963  df-met 18964  df-bl 18965  df-mopn 18966  df-fbas 18967  df-fg 18968  df-cnfld 18971  df-top 19921  df-bases 19922  df-topon 19923  df-topsp 19924  df-cld 20034  df-ntr 20035  df-cls 20036  df-nei 20114  df-lp 20152  df-perf 20153  df-cn 20243  df-cnp 20244  df-haus 20331  df-cmp 20402  df-tx 20577  df-hmeo 20770  df-fil 20861  df-fm 20953  df-flim 20954  df-flf 20955  df-xms 21335  df-ms 21336  df-tms 21337  df-cncf 21910  df-ovol 22416  df-vol 22418  df-mbf 22577  df-itg1 22578  df-itg2 22579  df-ibl 22580  df-0p 22628  df-limc 22821  df-dv 22822  df-log 23506  df-cxp 23507

This theorem is referenced by:  etransclem23  38122  etransclem46  38145