Theorem sge0val 40583

Description: The value of the sum of nonnegative extended reals. (Contributed by Glauco Siliprandi, 17-Aug-2020.)

Assertion

Ref	Expression
sge0val	|- ( ( X e. V /\ F : X --> ( 0 [,] +oo ) ) -> (Σ^ ` F ) = if ( +oo e. ran F , +oo , sup ( ran ( y e. ( ~P X i^i Fin ) |-> sum_ w e. y ( F ` w ) ) , RR* , < ) ) )

Distinct variable groups:   w, F, y   y, X

Allowed substitution hints:   V( y, w)   X( w)

Proof of Theorem sge0val

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-sumge0 40580	. . 3 |- Σ^ = ( x e. _V |-> if ( +oo e. ran x , +oo , sup ( ran ( y e. ( ~P dom x i^i Fin ) |-> sum_ w e. y ( x ` w ) ) , RR* , < ) ) )
2	1	a1i 11	. 2 |- ( ( X e. V /\ F : X --> ( 0 [,] +oo ) ) -> Σ^ = ( x e. _V |-> if ( +oo e. ran x , +oo , sup ( ran ( y e. ( ~P dom x i^i Fin ) |-> sum_ w e. y ( x ` w ) ) , RR* , < ) ) ) )
3		rneq 5351	. . . . 5 |- ( x = F -> ran x = ran F )
4	3	eleq2d 2687	. . . 4 |- ( x = F -> ( +oo e. ran x <-> +oo e. ran F ) )
5	4	adantl 482	. . 3 |- ( ( ( X e. V /\ F : X --> ( 0 [,] +oo ) ) /\ x = F ) -> ( +oo e. ran x <-> +oo e. ran F ) )
6		dmeq 5324	. . . . . . . . . . . 12 |- ( x = F -> dom x = dom F )
7	6	adantl 482	. . . . . . . . . . 11 |- ( ( F : X --> ( 0 [,] +oo ) /\ x = F ) -> dom x = dom F )
8		fdm 6051	. . . . . . . . . . . 12 |- ( F : X --> ( 0 [,] +oo ) -> dom F = X )
9	8	adantr 481	. . . . . . . . . . 11 |- ( ( F : X --> ( 0 [,] +oo ) /\ x = F ) -> dom F = X )
10	7, 9	eqtrd 2656	. . . . . . . . . 10 |- ( ( F : X --> ( 0 [,] +oo ) /\ x = F ) -> dom x = X )
11	10	pweqd 4163	. . . . . . . . 9 |- ( ( F : X --> ( 0 [,] +oo ) /\ x = F ) -> ~P dom x = ~P X )
12	11	ineq1d 3813	. . . . . . . 8 |- ( ( F : X --> ( 0 [,] +oo ) /\ x = F ) -> ( ~P dom x i^i Fin ) = ( ~P X i^i Fin ) )
13	12	mpteq1d 4738	. . . . . . 7 |- ( ( F : X --> ( 0 [,] +oo ) /\ x = F ) -> ( y e. ( ~P dom x i^i Fin ) |-> sum_ w e. y ( x ` w ) ) = ( y e. ( ~P X i^i Fin ) |-> sum_ w e. y ( x ` w ) ) )
14	13	adantll 750	. . . . . 6 |- ( ( ( X e. V /\ F : X --> ( 0 [,] +oo ) ) /\ x = F ) -> ( y e. ( ~P dom x i^i Fin ) |-> sum_ w e. y ( x ` w ) ) = ( y e. ( ~P X i^i Fin ) |-> sum_ w e. y ( x ` w ) ) )
15		fveq1 6190	. . . . . . . . 9 |- ( x = F -> ( x ` w ) = ( F ` w ) )
16	15	sumeq2ad 14434	. . . . . . . 8 |- ( x = F -> sum_ w e. y ( x ` w ) = sum_ w e. y ( F ` w ) )
17	16	mpteq2dv 4745	. . . . . . 7 |- ( x = F -> ( y e. ( ~P X i^i Fin ) |-> sum_ w e. y ( x ` w ) ) = ( y e. ( ~P X i^i Fin ) |-> sum_ w e. y ( F ` w ) ) )
18	17	adantl 482	. . . . . 6 |- ( ( ( X e. V /\ F : X --> ( 0 [,] +oo ) ) /\ x = F ) -> ( y e. ( ~P X i^i Fin ) |-> sum_ w e. y ( x ` w ) ) = ( y e. ( ~P X i^i Fin ) |-> sum_ w e. y ( F ` w ) ) )
19	14, 18	eqtrd 2656	. . . . 5 |- ( ( ( X e. V /\ F : X --> ( 0 [,] +oo ) ) /\ x = F ) -> ( y e. ( ~P dom x i^i Fin ) |-> sum_ w e. y ( x ` w ) ) = ( y e. ( ~P X i^i Fin ) |-> sum_ w e. y ( F ` w ) ) )
20	19	rneqd 5353	. . . 4 |- ( ( ( X e. V /\ F : X --> ( 0 [,] +oo ) ) /\ x = F ) -> ran ( y e. ( ~P dom x i^i Fin ) |-> sum_ w e. y ( x ` w ) ) = ran ( y e. ( ~P X i^i Fin ) |-> sum_ w e. y ( F ` w ) ) )
21	20	supeq1d 8352	. . 3 |- ( ( ( X e. V /\ F : X --> ( 0 [,] +oo ) ) /\ x = F ) -> sup ( ran ( y e. ( ~P dom x i^i Fin ) |-> sum_ w e. y ( x ` w ) ) , RR* , < ) = sup ( ran ( y e. ( ~P X i^i Fin ) |-> sum_ w e. y ( F ` w ) ) , RR* , < ) )
22	5, 21	ifbieq2d 4111	. 2 |- ( ( ( X e. V /\ F : X --> ( 0 [,] +oo ) ) /\ x = F ) -> if ( +oo e. ran x , +oo , sup ( ran ( y e. ( ~P dom x i^i Fin ) |-> sum_ w e. y ( x ` w ) ) , RR* , < ) ) = if ( +oo e. ran F , +oo , sup ( ran ( y e. ( ~P X i^i Fin ) |-> sum_ w e. y ( F ` w ) ) , RR* , < ) ) )
23		simpr 477	. . 3 |- ( ( X e. V /\ F : X --> ( 0 [,] +oo ) ) -> F : X --> ( 0 [,] +oo ) )
24		simpl 473	. . 3 |- ( ( X e. V /\ F : X --> ( 0 [,] +oo ) ) -> X e. V )
25		fex 6490	. . 3 |- ( ( F : X --> ( 0 [,] +oo ) /\ X e. V ) -> F e. _V )
26	23, 24, 25	syl2anc 693	. 2 |- ( ( X e. V /\ F : X --> ( 0 [,] +oo ) ) -> F e. _V )
27		pnfxr 10092	. . . 4 |- +oo e. RR*
28	27	a1i 11	. . 3 |- ( ( X e. V /\ F : X --> ( 0 [,] +oo ) ) -> +oo e. RR* )
29		xrltso 11974	. . . . 5 |- < Or RR*
30	29	supex 8369	. . . 4 |- sup ( ran ( y e. ( ~P X i^i Fin ) |-> sum_ w e. y ( F ` w ) ) , RR* , < ) e. _V
31	30	a1i 11	. . 3 |- ( ( X e. V /\ F : X --> ( 0 [,] +oo ) ) -> sup ( ran ( y e. ( ~P X i^i Fin ) |-> sum_ w e. y ( F ` w ) ) , RR* , < ) e. _V )
32		ifexg 4157	. . 3 |- ( ( +oo e. RR* /\ sup ( ran ( y e. ( ~P X i^i Fin ) |-> sum_ w e. y ( F ` w ) ) , RR* , < ) e. _V ) -> if ( +oo e. ran F , +oo , sup ( ran ( y e. ( ~P X i^i Fin ) |-> sum_ w e. y ( F ` w ) ) , RR* , < ) ) e. _V )
33	28, 31, 32	syl2anc 693	. 2 |- ( ( X e. V /\ F : X --> ( 0 [,] +oo ) ) -> if ( +oo e. ran F , +oo , sup ( ran ( y e. ( ~P X i^i Fin ) |-> sum_ w e. y ( F ` w ) ) , RR* , < ) ) e. _V )
34	2, 22, 26, 33	fvmptd 6288	1 |- ( ( X e. V /\ F : X --> ( 0 [,] +oo ) ) -> (Σ^ ` F ) = if ( +oo e. ran F , +oo , sup ( ran ( y e. ( ~P X i^i Fin ) |-> sum_ w e. y ( F ` w ) ) , RR* , < ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   i^i cin 3573   ifcif 4086   ~Pcpw 4158   |-> cmpt 4729   dom cdm 5114   ran crn 5115   -->wf 5884   ` cfv 5888  (class class class)co 6650   Fincfn 7955   supcsup 8346   0cc0 9936   +oocpnf 10071   RR*cxr 10073   < clt 10074   [,]cicc 12178   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-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

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-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-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-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-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-seq 12802  df-sum 14417  df-sumge0 40580

This theorem is referenced by:  sge0vald  40586