Theorem esumlub 30122

Description: The extended sum is the lowest upper bound for the partial sums. (Contributed by Thierry Arnoux, 19-Oct-2017.) (Proof shortened by AV, 12-Dec-2019.)

Hypotheses

Ref	Expression
esumlub.f	|- F/ k ph
esumlub.0	|- ( ph -> A e. V )
esumlub.1	|- ( ( ph /\ k e. A ) -> B e. ( 0 [,] +oo ) )
esumlub.2	|- ( ph -> X e. RR* )
esumlub.3	|- ( ph -> X < Σ* k e. A B )

Assertion

Ref	Expression
esumlub	|- ( ph -> E. a e. ( ~P A i^i Fin ) X < Σ* k e. a B )

Distinct variable groups:   k, a, A   B, a   X, a   ph, a

Allowed substitution hints:   ph( k)   B( k)   V( k, a)   X( k)

Proof of Theorem esumlub

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

Step	Hyp	Ref	Expression
1		esumlub.3	. . . 4 |- ( ph -> X < Σ* k e. A B )
2		esumlub.f	. . . . . . 7 |- F/ k ph
3		nfcv 2764	. . . . . . 7 |- F/_ k A
4		esumlub.0	. . . . . . 7 |- ( ph -> A e. V )
5		esumlub.1	. . . . . . 7 |- ( ( ph /\ k e. A ) -> B e. ( 0 [,] +oo ) )
6		eqidd 2623	. . . . . . 7 |- ( ( ph /\ x e. ( ~P A i^i Fin ) ) -> ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. x |-> B ) ) = ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. x |-> B ) ) )
7	2, 3, 4, 5, 6	esumval 30108	. . . . . 6 |- ( ph -> Σ* k e. A B = sup ( ran ( x e. ( ~P A i^i Fin ) |-> ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. x |-> B ) ) ) , RR* , < ) )
8	7	breq2d 4665	. . . . 5 |- ( ph -> ( X < Σ* k e. A B <-> X < sup ( ran ( x e. ( ~P A i^i Fin ) |-> ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. x |-> B ) ) ) , RR* , < ) ) )
9		iccssxr 12256	. . . . . . . . 9 |- ( 0 [,] +oo ) C_ RR*
10		xrge0base 29685	. . . . . . . . . 10 |- ( 0 [,] +oo ) = ( Base ` ( RR*s ↾s ( 0 [,] +oo ) ) )
11		xrge0cmn 19788	. . . . . . . . . . 11 |- ( RR*s ↾s ( 0 [,] +oo ) ) e. CMnd
12	11	a1i 11	. . . . . . . . . 10 |- ( ( ph /\ x e. ( ~P A i^i Fin ) ) -> ( RR*s ↾s ( 0 [,] +oo ) ) e. CMnd )
13		inss2 3834	. . . . . . . . . . 11 |- ( ~P A i^i Fin ) C_ Fin
14		simpr 477	. . . . . . . . . . 11 |- ( ( ph /\ x e. ( ~P A i^i Fin ) ) -> x e. ( ~P A i^i Fin ) )
15	13, 14	sseldi 3601	. . . . . . . . . 10 |- ( ( ph /\ x e. ( ~P A i^i Fin ) ) -> x e. Fin )
16		nfv 1843	. . . . . . . . . . . 12 |- F/ k x e. ( ~P A i^i Fin )
17	2, 16	nfan 1828	. . . . . . . . . . 11 |- F/ k ( ph /\ x e. ( ~P A i^i Fin ) )
18		simpll 790	. . . . . . . . . . . . 13 |- ( ( ( ph /\ x e. ( ~P A i^i Fin ) ) /\ k e. x ) -> ph )
19		inss1 3833	. . . . . . . . . . . . . . . . 17 |- ( ~P A i^i Fin ) C_ ~P A
20	19	sseli 3599	. . . . . . . . . . . . . . . 16 |- ( x e. ( ~P A i^i Fin ) -> x e. ~P A )
21	20	ad2antlr 763	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ x e. ( ~P A i^i Fin ) ) /\ k e. x ) -> x e. ~P A )
22	21	elpwid 4170	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ x e. ( ~P A i^i Fin ) ) /\ k e. x ) -> x C_ A )
23		simpr 477	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ x e. ( ~P A i^i Fin ) ) /\ k e. x ) -> k e. x )
24	22, 23	sseldd 3604	. . . . . . . . . . . . 13 |- ( ( ( ph /\ x e. ( ~P A i^i Fin ) ) /\ k e. x ) -> k e. A )
25	18, 24, 5	syl2anc 693	. . . . . . . . . . . 12 |- ( ( ( ph /\ x e. ( ~P A i^i Fin ) ) /\ k e. x ) -> B e. ( 0 [,] +oo ) )
26	25	ex 450	. . . . . . . . . . 11 |- ( ( ph /\ x e. ( ~P A i^i Fin ) ) -> ( k e. x -> B e. ( 0 [,] +oo ) ) )
27	17, 26	ralrimi 2957	. . . . . . . . . 10 |- ( ( ph /\ x e. ( ~P A i^i Fin ) ) -> A. k e. x B e. ( 0 [,] +oo ) )
28	10, 12, 15, 27	gsummptcl 18366	. . . . . . . . 9 |- ( ( ph /\ x e. ( ~P A i^i Fin ) ) -> ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. x |-> B ) ) e. ( 0 [,] +oo ) )
29	9, 28	sseldi 3601	. . . . . . . 8 |- ( ( ph /\ x e. ( ~P A i^i Fin ) ) -> ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. x |-> B ) ) e. RR* )
30	29	ralrimiva 2966	. . . . . . 7 |- ( ph -> A. x e. ( ~P A i^i Fin ) ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. x |-> B ) ) e. RR* )
31		eqid 2622	. . . . . . . 8 |- ( x e. ( ~P A i^i Fin ) |-> ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. x |-> B ) ) ) = ( x e. ( ~P A i^i Fin ) |-> ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. x |-> B ) ) )
32	31	rnmptss 6392	. . . . . . 7 |- ( A. x e. ( ~P A i^i Fin ) ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. x |-> B ) ) e. RR* -> ran ( x e. ( ~P A i^i Fin ) |-> ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. x |-> B ) ) ) C_ RR* )
33	30, 32	syl 17	. . . . . 6 |- ( ph -> ran ( x e. ( ~P A i^i Fin ) |-> ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. x |-> B ) ) ) C_ RR* )
34		esumlub.2	. . . . . 6 |- ( ph -> X e. RR* )
35		supxrlub 12155	. . . . . 6 |- ( ( ran ( x e. ( ~P A i^i Fin ) |-> ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. x |-> B ) ) ) C_ RR* /\ X e. RR* ) -> ( X < sup ( ran ( x e. ( ~P A i^i Fin ) |-> ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. x |-> B ) ) ) , RR* , < ) <-> E. y e. ran ( x e. ( ~P A i^i Fin ) |-> ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. x |-> B ) ) ) X < y ) )
36	33, 34, 35	syl2anc 693	. . . . 5 |- ( ph -> ( X < sup ( ran ( x e. ( ~P A i^i Fin ) |-> ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. x |-> B ) ) ) , RR* , < ) <-> E. y e. ran ( x e. ( ~P A i^i Fin ) |-> ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. x |-> B ) ) ) X < y ) )
37	8, 36	bitrd 268	. . . 4 |- ( ph -> ( X < Σ* k e. A B <-> E. y e. ran ( x e. ( ~P A i^i Fin ) |-> ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. x |-> B ) ) ) X < y ) )
38	1, 37	mpbid 222	. . 3 |- ( ph -> E. y e. ran ( x e. ( ~P A i^i Fin ) |-> ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. x |-> B ) ) ) X < y )
39		ovex 6678	. . . . 5 |- ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. a |-> B ) ) e. _V
40	39	a1i 11	. . . 4 |- ( ( ph /\ a e. ( ~P A i^i Fin ) ) -> ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. a |-> B ) ) e. _V )
41		mpteq1 4737	. . . . . . . 8 |- ( x = a -> ( k e. x |-> B ) = ( k e. a |-> B ) )
42	41	oveq2d 6666	. . . . . . 7 |- ( x = a -> ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. x |-> B ) ) = ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. a |-> B ) ) )
43	42	cbvmptv 4750	. . . . . 6 |- ( x e. ( ~P A i^i Fin ) |-> ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. x |-> B ) ) ) = ( a e. ( ~P A i^i Fin ) |-> ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. a |-> B ) ) )
44	43, 39	elrnmpti 5376	. . . . 5 |- ( y e. ran ( x e. ( ~P A i^i Fin ) |-> ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. x |-> B ) ) ) <-> E. a e. ( ~P A i^i Fin ) y = ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. a |-> B ) ) )
45	44	a1i 11	. . . 4 |- ( ph -> ( y e. ran ( x e. ( ~P A i^i Fin ) |-> ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. x |-> B ) ) ) <-> E. a e. ( ~P A i^i Fin ) y = ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. a |-> B ) ) ) )
46		simpr 477	. . . . 5 |- ( ( ph /\ y = ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. a |-> B ) ) ) -> y = ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. a |-> B ) ) )
47	46	breq2d 4665	. . . 4 |- ( ( ph /\ y = ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. a |-> B ) ) ) -> ( X < y <-> X < ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. a |-> B ) ) ) )
48	40, 45, 47	rexxfr2d 4883	. . 3 |- ( ph -> ( E. y e. ran ( x e. ( ~P A i^i Fin ) |-> ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. x |-> B ) ) ) X < y <-> E. a e. ( ~P A i^i Fin ) X < ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. a |-> B ) ) ) )
49	38, 48	mpbid 222	. 2 |- ( ph -> E. a e. ( ~P A i^i Fin ) X < ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. a |-> B ) ) )
50		nfv 1843	. . . . . . 7 |- F/ k a e. ( ~P A i^i Fin )
51	2, 50	nfan 1828	. . . . . 6 |- F/ k ( ph /\ a e. ( ~P A i^i Fin ) )
52		simpr 477	. . . . . . 7 |- ( ( ph /\ a e. ( ~P A i^i Fin ) ) -> a e. ( ~P A i^i Fin ) )
53	13, 52	sseldi 3601	. . . . . 6 |- ( ( ph /\ a e. ( ~P A i^i Fin ) ) -> a e. Fin )
54		simpll 790	. . . . . . 7 |- ( ( ( ph /\ a e. ( ~P A i^i Fin ) ) /\ k e. a ) -> ph )
55	19	sseli 3599	. . . . . . . . . 10 |- ( a e. ( ~P A i^i Fin ) -> a e. ~P A )
56	55	ad2antlr 763	. . . . . . . . 9 |- ( ( ( ph /\ a e. ( ~P A i^i Fin ) ) /\ k e. a ) -> a e. ~P A )
57	56	elpwid 4170	. . . . . . . 8 |- ( ( ( ph /\ a e. ( ~P A i^i Fin ) ) /\ k e. a ) -> a C_ A )
58		simpr 477	. . . . . . . 8 |- ( ( ( ph /\ a e. ( ~P A i^i Fin ) ) /\ k e. a ) -> k e. a )
59	57, 58	sseldd 3604	. . . . . . 7 |- ( ( ( ph /\ a e. ( ~P A i^i Fin ) ) /\ k e. a ) -> k e. A )
60	54, 59, 5	syl2anc 693	. . . . . 6 |- ( ( ( ph /\ a e. ( ~P A i^i Fin ) ) /\ k e. a ) -> B e. ( 0 [,] +oo ) )
61	51, 53, 60	gsumesum 30121	. . . . 5 |- ( ( ph /\ a e. ( ~P A i^i Fin ) ) -> ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. a |-> B ) ) = Σ* k e. a B )
62	61	breq2d 4665	. . . 4 |- ( ( ph /\ a e. ( ~P A i^i Fin ) ) -> ( X < ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. a |-> B ) ) <-> X < Σ* k e. a B ) )
63	62	biimpd 219	. . 3 |- ( ( ph /\ a e. ( ~P A i^i Fin ) ) -> ( X < ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. a |-> B ) ) -> X < Σ* k e. a B ) )
64	63	reximdva 3017	. 2 |- ( ph -> ( E. a e. ( ~P A i^i Fin ) X < ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. a |-> B ) ) -> E. a e. ( ~P A i^i Fin ) X < Σ* k e. a B ) )
65	49, 64	mpd 15	1 |- ( ph -> E. a e. ( ~P A i^i Fin ) X < Σ* k e. a B )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   F/wnf 1708   e. wcel 1990   A.wral 2912   E.wrex 2913   _Vcvv 3200   i^i cin 3573   C_ wss 3574   ~Pcpw 4158   class class class wbr 4653   |-> cmpt 4729   ran crn 5115  (class class class)co 6650   Fincfn 7955   supcsup 8346   0cc0 9936   +oocpnf 10071   RR*cxr 10073   < clt 10074   [,]cicc 12178   ↾_s cress 15858   gsum_g cgsu 16101   RR*scxrs 16160  CMndccmn 18193  Σ^*cesum 30089

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-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-iin 4523  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-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  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-fsupp 8276  df-fi 8317  df-sup 8348  df-inf 8349  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-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-xadd 11947  df-ioo 12179  df-ioc 12180  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-seq 12802  df-hash 13118  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-tset 15960  df-ple 15961  df-ds 15964  df-rest 16083  df-topn 16084  df-0g 16102  df-gsum 16103  df-topgen 16104  df-ordt 16161  df-xrs 16162  df-mre 16246  df-mrc 16247  df-acs 16249  df-ps 17200  df-tsr 17201  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-cntz 17750  df-cmn 18195  df-fbas 19743  df-fg 19744  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-ntr 20824  df-nei 20902  df-cn 21031  df-haus 21119  df-fil 21650  df-fm 21742  df-flim 21743  df-flf 21744  df-tsms 21930  df-esum 30090

This theorem is referenced by:  esumfsup  30132  esum2d  30155