Theorem sge0uzfsumgt 40661

Description: If a real number is smaller than a generalized sum of nonnegative reals, then it is smaller than some finite subsum. (Contributed by Glauco Siliprandi, 21-Nov-2020.)

Hypotheses

Ref	Expression
sge0uzfsumgt.p	|- F/ k ph
sge0uzfsumgt.h	|- ( ph -> K e. ZZ )
sge0uzfsumgt.z	|- Z = ( ZZ>= ` K )
sge0uzfsumgt.b	|- ( ( ph /\ k e. Z ) -> B e. ( 0 [,) +oo ) )
sge0uzfsumgt.c	|- ( ph -> C e. RR )
sge0uzfsumgt.l	|- ( ph -> C < (Σ^ ` ( k e. Z |-> B ) ) )

Assertion

Ref	Expression
sge0uzfsumgt	|- ( ph -> E. m e. Z C < sum_ k e. ( K ... m ) B )

Distinct variable groups:   B, m   C, m   k, K, m   k, Z, m   ph, m

Allowed substitution hints:   ph( k)   B( k)   C( k)

Proof of Theorem sge0uzfsumgt

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sge0uzfsumgt.p	. . 3 |- F/ k ph
2		sge0uzfsumgt.z	. . . . 5 |- Z = ( ZZ>= ` K )
3		fvex 6201	. . . . 5 |- ( ZZ>= ` K ) e. _V
4	2, 3	eqeltri 2697	. . . 4 |- Z e. _V
5	4	a1i 11	. . 3 |- ( ph -> Z e. _V )
6		sge0uzfsumgt.b	. . 3 |- ( ( ph /\ k e. Z ) -> B e. ( 0 [,) +oo ) )
7		sge0uzfsumgt.c	. . 3 |- ( ph -> C e. RR )
8		sge0uzfsumgt.l	. . 3 |- ( ph -> C < (Σ^ ` ( k e. Z |-> B ) ) )
9	1, 5, 6, 7, 8	sge0gtfsumgt 40660	. 2 |- ( ph -> E. x e. ( ~P Z i^i Fin ) C < sum_ k e. x B )
10		sge0uzfsumgt.h	. . . . . . 7 |- ( ph -> K e. ZZ )
11	10	3ad2ant1 1082	. . . . . 6 |- ( ( ph /\ x e. ( ~P Z i^i Fin ) /\ C < sum_ k e. x B ) -> K e. ZZ )
12		elpwinss 39216	. . . . . . 7 |- ( x e. ( ~P Z i^i Fin ) -> x C_ Z )
13	12	3ad2ant2 1083	. . . . . 6 |- ( ( ph /\ x e. ( ~P Z i^i Fin ) /\ C < sum_ k e. x B ) -> x C_ Z )
14		elinel2 3800	. . . . . . 7 |- ( x e. ( ~P Z i^i Fin ) -> x e. Fin )
15	14	3ad2ant2 1083	. . . . . 6 |- ( ( ph /\ x e. ( ~P Z i^i Fin ) /\ C < sum_ k e. x B ) -> x e. Fin )
16	11, 2, 13, 15	uzfissfz 39542	. . . . 5 |- ( ( ph /\ x e. ( ~P Z i^i Fin ) /\ C < sum_ k e. x B ) -> E. m e. Z x C_ ( K ... m ) )
17	7	ad2antrr 762	. . . . . . . . . 10 |- ( ( ( ph /\ C < sum_ k e. x B ) /\ x C_ ( K ... m ) ) -> C e. RR )
18		nfv 1843	. . . . . . . . . . . . 13 |- F/ k x C_ ( K ... m )
19	1, 18	nfan 1828	. . . . . . . . . . . 12 |- F/ k ( ph /\ x C_ ( K ... m ) )
20		fzfid 12772	. . . . . . . . . . . . 13 |- ( ( ph /\ x C_ ( K ... m ) ) -> ( K ... m ) e. Fin )
21		simpr 477	. . . . . . . . . . . . 13 |- ( ( ph /\ x C_ ( K ... m ) ) -> x C_ ( K ... m ) )
22	20, 21	ssfid 8183	. . . . . . . . . . . 12 |- ( ( ph /\ x C_ ( K ... m ) ) -> x e. Fin )
23		simpll 790	. . . . . . . . . . . . 13 |- ( ( ( ph /\ x C_ ( K ... m ) ) /\ k e. x ) -> ph )
24	21	sselda 3603	. . . . . . . . . . . . 13 |- ( ( ( ph /\ x C_ ( K ... m ) ) /\ k e. x ) -> k e. ( K ... m ) )
25		rge0ssre 12280	. . . . . . . . . . . . . 14 |- ( 0 [,) +oo ) C_ RR
26		fzssuz 12382	. . . . . . . . . . . . . . . . 17 |- ( K ... m ) C_ ( ZZ>= ` K )
27	26, 2	sseqtr4i 3638	. . . . . . . . . . . . . . . 16 |- ( K ... m ) C_ Z
28		id 22	. . . . . . . . . . . . . . . 16 |- ( k e. ( K ... m ) -> k e. ( K ... m ) )
29	27, 28	sseldi 3601	. . . . . . . . . . . . . . 15 |- ( k e. ( K ... m ) -> k e. Z )
30	29, 6	sylan2 491	. . . . . . . . . . . . . 14 |- ( ( ph /\ k e. ( K ... m ) ) -> B e. ( 0 [,) +oo ) )
31	25, 30	sseldi 3601	. . . . . . . . . . . . 13 |- ( ( ph /\ k e. ( K ... m ) ) -> B e. RR )
32	23, 24, 31	syl2anc 693	. . . . . . . . . . . 12 |- ( ( ( ph /\ x C_ ( K ... m ) ) /\ k e. x ) -> B e. RR )
33	19, 22, 32	fsumreclf 39808	. . . . . . . . . . 11 |- ( ( ph /\ x C_ ( K ... m ) ) -> sum_ k e. x B e. RR )
34	33	adantlr 751	. . . . . . . . . 10 |- ( ( ( ph /\ C < sum_ k e. x B ) /\ x C_ ( K ... m ) ) -> sum_ k e. x B e. RR )
35		fzfid 12772	. . . . . . . . . . . 12 |- ( ph -> ( K ... m ) e. Fin )
36	1, 35, 31	fsumreclf 39808	. . . . . . . . . . 11 |- ( ph -> sum_ k e. ( K ... m ) B e. RR )
37	36	ad2antrr 762	. . . . . . . . . 10 |- ( ( ( ph /\ C < sum_ k e. x B ) /\ x C_ ( K ... m ) ) -> sum_ k e. ( K ... m ) B e. RR )
38		simplr 792	. . . . . . . . . 10 |- ( ( ( ph /\ C < sum_ k e. x B ) /\ x C_ ( K ... m ) ) -> C < sum_ k e. x B )
39	31	adantlr 751	. . . . . . . . . . . 12 |- ( ( ( ph /\ x C_ ( K ... m ) ) /\ k e. ( K ... m ) ) -> B e. RR )
40		0xr 10086	. . . . . . . . . . . . . . 15 |- 0 e. RR*
41	40	a1i 11	. . . . . . . . . . . . . 14 |- ( ( ph /\ k e. ( K ... m ) ) -> 0 e. RR* )
42		pnfxr 10092	. . . . . . . . . . . . . . 15 |- +oo e. RR*
43	42	a1i 11	. . . . . . . . . . . . . 14 |- ( ( ph /\ k e. ( K ... m ) ) -> +oo e. RR* )
44		icogelb 12225	. . . . . . . . . . . . . 14 |- ( ( 0 e. RR* /\ +oo e. RR* /\ B e. ( 0 [,) +oo ) ) -> 0 <_ B )
45	41, 43, 30, 44	syl3anc 1326	. . . . . . . . . . . . 13 |- ( ( ph /\ k e. ( K ... m ) ) -> 0 <_ B )
46	45	adantlr 751	. . . . . . . . . . . 12 |- ( ( ( ph /\ x C_ ( K ... m ) ) /\ k e. ( K ... m ) ) -> 0 <_ B )
47	19, 20, 39, 46, 21	fsumlessf 39809	. . . . . . . . . . 11 |- ( ( ph /\ x C_ ( K ... m ) ) -> sum_ k e. x B <_ sum_ k e. ( K ... m ) B )
48	47	adantlr 751	. . . . . . . . . 10 |- ( ( ( ph /\ C < sum_ k e. x B ) /\ x C_ ( K ... m ) ) -> sum_ k e. x B <_ sum_ k e. ( K ... m ) B )
49	17, 34, 37, 38, 48	ltletrd 10197	. . . . . . . . 9 |- ( ( ( ph /\ C < sum_ k e. x B ) /\ x C_ ( K ... m ) ) -> C < sum_ k e. ( K ... m ) B )
50	49	ex 450	. . . . . . . 8 |- ( ( ph /\ C < sum_ k e. x B ) -> ( x C_ ( K ... m ) -> C < sum_ k e. ( K ... m ) B ) )
51	50	adantr 481	. . . . . . 7 |- ( ( ( ph /\ C < sum_ k e. x B ) /\ m e. Z ) -> ( x C_ ( K ... m ) -> C < sum_ k e. ( K ... m ) B ) )
52	51	3adantl2 1218	. . . . . 6 |- ( ( ( ph /\ x e. ( ~P Z i^i Fin ) /\ C < sum_ k e. x B ) /\ m e. Z ) -> ( x C_ ( K ... m ) -> C < sum_ k e. ( K ... m ) B ) )
53	52	reximdva 3017	. . . . 5 |- ( ( ph /\ x e. ( ~P Z i^i Fin ) /\ C < sum_ k e. x B ) -> ( E. m e. Z x C_ ( K ... m ) -> E. m e. Z C < sum_ k e. ( K ... m ) B ) )
54	16, 53	mpd 15	. . . 4 |- ( ( ph /\ x e. ( ~P Z i^i Fin ) /\ C < sum_ k e. x B ) -> E. m e. Z C < sum_ k e. ( K ... m ) B )
55	54	3exp 1264	. . 3 |- ( ph -> ( x e. ( ~P Z i^i Fin ) -> ( C < sum_ k e. x B -> E. m e. Z C < sum_ k e. ( K ... m ) B ) ) )
56	55	rexlimdv 3030	. 2 |- ( ph -> ( E. x e. ( ~P Z i^i Fin ) C < sum_ k e. x B -> E. m e. Z C < sum_ k e. ( K ... m ) B ) )
57	9, 56	mpd 15	1 |- ( ph -> E. m e. Z C < sum_ k e. ( K ... m ) B )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   F/wnf 1708   e. wcel 1990   E.wrex 2913   _Vcvv 3200   i^i cin 3573   C_ wss 3574   ~Pcpw 4158   class class class wbr 4653   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   Fincfn 7955   RRcr 9935   0cc0 9936   +oocpnf 10071   RR*cxr 10073   < clt 10074   <_ cle 10075   ZZcz 11377   ZZ>=cuz 11687   [,)cico 12177   ...cfz 12326   sum_csu 14416  Σ^{^}csumge0 40579

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-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-sum 14417  df-sumge0 40580

This theorem is referenced by:  hoidmvlelem3  40811