Theorem sge0gerp 40612

Description: The arbitrary sum of nonnegative extended reals is larger or equal to a given extended real number if this number can be approximated from below by finite subsums. (Contributed by Glauco Siliprandi, 17-Aug-2020.)

Hypotheses

Ref	Expression
sge0gerp.x	|- ( ph -> X e. V )
sge0gerp.f	|- ( ph -> F : X --> ( 0 [,] +oo ) )
sge0gerp.a	|- ( ph -> A e. RR* )
sge0gerp.z	|- ( ( ph /\ x e. RR+ ) -> E. z e. ( ~P X i^i Fin ) A <_ ( (Σ^ ` ( F |` z ) ) +e x ) )

Assertion

Ref	Expression
sge0gerp	|- ( ph -> A <_ (Σ^ ` F ) )

Distinct variable groups:   x, A, z   x, F, z   x, X, z   ph, x, z

Allowed substitution hints:   V( x, z)

Proof of Theorem sge0gerp

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1843	. . 3 |- F/ x ph
2		simpr 477	. . . . . 6 |- ( ( ph /\ z e. ( ~P X i^i Fin ) ) -> z e. ( ~P X i^i Fin ) )
3		sge0gerp.f	. . . . . . . 8 |- ( ph -> F : X --> ( 0 [,] +oo ) )
4	3	adantr 481	. . . . . . 7 |- ( ( ph /\ z e. ( ~P X i^i Fin ) ) -> F : X --> ( 0 [,] +oo ) )
5		elinel1 3799	. . . . . . . . 9 |- ( z e. ( ~P X i^i Fin ) -> z e. ~P X )
6		elpwi 4168	. . . . . . . . 9 |- ( z e. ~P X -> z C_ X )
7	5, 6	syl 17	. . . . . . . 8 |- ( z e. ( ~P X i^i Fin ) -> z C_ X )
8	7	adantl 482	. . . . . . 7 |- ( ( ph /\ z e. ( ~P X i^i Fin ) ) -> z C_ X )
9	4, 8	fssresd 6071	. . . . . 6 |- ( ( ph /\ z e. ( ~P X i^i Fin ) ) -> ( F |` z ) : z --> ( 0 [,] +oo ) )
10	2, 9	sge0xrcl 40602	. . . . 5 |- ( ( ph /\ z e. ( ~P X i^i Fin ) ) -> (Σ^ ` ( F |` z ) ) e. RR* )
11	10	ralrimiva 2966	. . . 4 |- ( ph -> A. z e. ( ~P X i^i Fin ) (Σ^ ` ( F |` z ) ) e. RR* )
12		eqid 2622	. . . . 5 |- ( z e. ( ~P X i^i Fin ) |-> (Σ^ ` ( F |` z ) ) ) = ( z e. ( ~P X i^i Fin ) |-> (Σ^ ` ( F |` z ) ) )
13	12	rnmptss 6392	. . . 4 |- ( A. z e. ( ~P X i^i Fin ) (Σ^ ` ( F |` z ) ) e. RR* -> ran ( z e. ( ~P X i^i Fin ) |-> (Σ^ ` ( F |` z ) ) ) C_ RR* )
14	11, 13	syl 17	. . 3 |- ( ph -> ran ( z e. ( ~P X i^i Fin ) |-> (Σ^ ` ( F |` z ) ) ) C_ RR* )
15		sge0gerp.a	. . 3 |- ( ph -> A e. RR* )
16		sge0gerp.z	. . . 4 |- ( ( ph /\ x e. RR+ ) -> E. z e. ( ~P X i^i Fin ) A <_ ( (Σ^ ` ( F |` z ) ) +e x ) )
17		nfv 1843	. . . . 5 |- F/ z ( ph /\ x e. RR+ )
18		nfmpt1 4747	. . . . . . 7 |- F/_ z ( z e. ( ~P X i^i Fin ) |-> (Σ^ ` ( F |` z ) ) )
19	18	nfrn 5368	. . . . . 6 |- F/_ z ran ( z e. ( ~P X i^i Fin ) |-> (Σ^ ` ( F |` z ) ) )
20		nfv 1843	. . . . . 6 |- F/ z A <_ ( y +e x )
21	19, 20	nfrex 3007	. . . . 5 |- F/ z E. y e. ran ( z e. ( ~P X i^i Fin ) |-> (Σ^ ` ( F |` z ) ) ) A <_ ( y +e x )
22		id 22	. . . . . . . . 9 |- ( z e. ( ~P X i^i Fin ) -> z e. ( ~P X i^i Fin ) )
23		fvexd 6203	. . . . . . . . 9 |- ( z e. ( ~P X i^i Fin ) -> (Σ^ ` ( F |` z ) ) e. _V )
24	12	elrnmpt1 5374	. . . . . . . . 9 |- ( ( z e. ( ~P X i^i Fin ) /\ (Σ^ ` ( F |` z ) ) e. _V ) -> (Σ^ ` ( F |` z ) ) e. ran ( z e. ( ~P X i^i Fin ) |-> (Σ^ ` ( F |` z ) ) ) )
25	22, 23, 24	syl2anc 693	. . . . . . . 8 |- ( z e. ( ~P X i^i Fin ) -> (Σ^ ` ( F |` z ) ) e. ran ( z e. ( ~P X i^i Fin ) |-> (Σ^ ` ( F |` z ) ) ) )
26	25	3ad2ant2 1083	. . . . . . 7 |- ( ( ( ph /\ x e. RR+ ) /\ z e. ( ~P X i^i Fin ) /\ A <_ ( (Σ^ ` ( F |` z ) ) +e x ) ) -> (Σ^ ` ( F |` z ) ) e. ran ( z e. ( ~P X i^i Fin ) |-> (Σ^ ` ( F |` z ) ) ) )
27		simp3 1063	. . . . . . 7 |- ( ( ( ph /\ x e. RR+ ) /\ z e. ( ~P X i^i Fin ) /\ A <_ ( (Σ^ ` ( F |` z ) ) +e x ) ) -> A <_ ( (Σ^ ` ( F |` z ) ) +e x ) )
28		nfv 1843	. . . . . . . 8 |- F/ y A <_ ( (Σ^ ` ( F |` z ) ) +e x )
29		oveq1 6657	. . . . . . . . 9 |- ( y = (Σ^ ` ( F |` z ) ) -> ( y +e x ) = ( (Σ^ ` ( F |` z ) ) +e x ) )
30	29	breq2d 4665	. . . . . . . 8 |- ( y = (Σ^ ` ( F |` z ) ) -> ( A <_ ( y +e x ) <-> A <_ ( (Σ^ ` ( F |` z ) ) +e x ) ) )
31	28, 30	rspce 3304	. . . . . . 7 |- ( ( (Σ^ ` ( F |` z ) ) e. ran ( z e. ( ~P X i^i Fin ) |-> (Σ^ ` ( F |` z ) ) ) /\ A <_ ( (Σ^ ` ( F |` z ) ) +e x ) ) -> E. y e. ran ( z e. ( ~P X i^i Fin ) |-> (Σ^ ` ( F |` z ) ) ) A <_ ( y +e x ) )
32	26, 27, 31	syl2anc 693	. . . . . 6 |- ( ( ( ph /\ x e. RR+ ) /\ z e. ( ~P X i^i Fin ) /\ A <_ ( (Σ^ ` ( F |` z ) ) +e x ) ) -> E. y e. ran ( z e. ( ~P X i^i Fin ) |-> (Σ^ ` ( F |` z ) ) ) A <_ ( y +e x ) )
33	32	3exp 1264	. . . . 5 |- ( ( ph /\ x e. RR+ ) -> ( z e. ( ~P X i^i Fin ) -> ( A <_ ( (Σ^ ` ( F |` z ) ) +e x ) -> E. y e. ran ( z e. ( ~P X i^i Fin ) |-> (Σ^ ` ( F |` z ) ) ) A <_ ( y +e x ) ) ) )
34	17, 21, 33	rexlimd 3026	. . . 4 |- ( ( ph /\ x e. RR+ ) -> ( E. z e. ( ~P X i^i Fin ) A <_ ( (Σ^ ` ( F |` z ) ) +e x ) -> E. y e. ran ( z e. ( ~P X i^i Fin ) |-> (Σ^ ` ( F |` z ) ) ) A <_ ( y +e x ) ) )
35	16, 34	mpd 15	. . 3 |- ( ( ph /\ x e. RR+ ) -> E. y e. ran ( z e. ( ~P X i^i Fin ) |-> (Σ^ ` ( F |` z ) ) ) A <_ ( y +e x ) )
36	1, 14, 15, 35	supxrge 39554	. 2 |- ( ph -> A <_ sup ( ran ( z e. ( ~P X i^i Fin ) |-> (Σ^ ` ( F |` z ) ) ) , RR* , < ) )
37		sge0gerp.x	. . . 4 |- ( ph -> X e. V )
38	37, 3	sge0sup 40608	. . 3 |- ( ph -> (Σ^ ` F ) = sup ( ran ( z e. ( ~P X i^i Fin ) |-> (Σ^ ` ( F |` z ) ) ) , RR* , < ) )
39	38	eqcomd 2628	. 2 |- ( ph -> sup ( ran ( z e. ( ~P X i^i Fin ) |-> (Σ^ ` ( F |` z ) ) ) , RR* , < ) = (Σ^ ` F ) )
40	36, 39	breqtrd 4679	1 |- ( ph -> A <_ (Σ^ ` F ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   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   |` cres 5116   -->wf 5884   ` cfv 5888  (class class class)co 6650   Fincfn 7955   supcsup 8346   0cc0 9936   +oocpnf 10071   RR*cxr 10073   < clt 10074   <_ cle 10075   RR+crp 11832   +ecxad 11944   [,]cicc 12178  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-xadd 11947  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:  sge0gerpmpt  40619