Theorem sge0val 39259

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

Assertion

Ref	Expression
sge0val	⊢ ((𝑋 ∈ 𝑉 ∧ 𝐹:𝑋⟶(0[,]+∞)) → (Σ^‘𝐹) = if(+∞ ∈ ran 𝐹, +∞, sup(ran (𝑦 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑤 ∈ 𝑦 (𝐹‘𝑤)), ℝ*, < )))

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 39256	. . 3 ⊢ Σ^ = (𝑥 ∈ V ↦ if(+∞ ∈ ran 𝑥, +∞, sup(ran (𝑦 ∈ (𝒫 dom 𝑥 ∩ Fin) ↦ Σ𝑤 ∈ 𝑦 (𝑥‘𝑤)), ℝ*, < )))
2	1	a1i 11	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐹:𝑋⟶(0[,]+∞)) → Σ^ = (𝑥 ∈ V ↦ if(+∞ ∈ ran 𝑥, +∞, sup(ran (𝑦 ∈ (𝒫 dom 𝑥 ∩ Fin) ↦ Σ𝑤 ∈ 𝑦 (𝑥‘𝑤)), ℝ*, < ))))
3		rneq 5272	. . . . 5 ⊢ (𝑥 = 𝐹 → ran 𝑥 = ran 𝐹)
4	3	eleq2d 2673	. . . 4 ⊢ (𝑥 = 𝐹 → (+∞ ∈ ran 𝑥 ↔ +∞ ∈ ran 𝐹))
5	4	adantl 481	. . 3 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝐹:𝑋⟶(0[,]+∞)) ∧ 𝑥 = 𝐹) → (+∞ ∈ ran 𝑥 ↔ +∞ ∈ ran 𝐹))
6		dmeq 5246	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝐹 → dom 𝑥 = dom 𝐹)
7	6	adantl 481	. . . . . . . . . . 11 ⊢ ((𝐹:𝑋⟶(0[,]+∞) ∧ 𝑥 = 𝐹) → dom 𝑥 = dom 𝐹)
8		fdm 5964	. . . . . . . . . . . 12 ⊢ (𝐹:𝑋⟶(0[,]+∞) → dom 𝐹 = 𝑋)
9	8	adantr 480	. . . . . . . . . . 11 ⊢ ((𝐹:𝑋⟶(0[,]+∞) ∧ 𝑥 = 𝐹) → dom 𝐹 = 𝑋)
10	7, 9	eqtrd 2644	. . . . . . . . . 10 ⊢ ((𝐹:𝑋⟶(0[,]+∞) ∧ 𝑥 = 𝐹) → dom 𝑥 = 𝑋)
11	10	pweqd 4113	. . . . . . . . 9 ⊢ ((𝐹:𝑋⟶(0[,]+∞) ∧ 𝑥 = 𝐹) → 𝒫 dom 𝑥 = 𝒫 𝑋)
12	11	ineq1d 3775	. . . . . . . 8 ⊢ ((𝐹:𝑋⟶(0[,]+∞) ∧ 𝑥 = 𝐹) → (𝒫 dom 𝑥 ∩ Fin) = (𝒫 𝑋 ∩ Fin))
13	12	mpteq1d 4666	. . . . . . 7 ⊢ ((𝐹:𝑋⟶(0[,]+∞) ∧ 𝑥 = 𝐹) → (𝑦 ∈ (𝒫 dom 𝑥 ∩ Fin) ↦ Σ𝑤 ∈ 𝑦 (𝑥‘𝑤)) = (𝑦 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑤 ∈ 𝑦 (𝑥‘𝑤)))
14	13	adantll 746	. . . . . 6 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝐹:𝑋⟶(0[,]+∞)) ∧ 𝑥 = 𝐹) → (𝑦 ∈ (𝒫 dom 𝑥 ∩ Fin) ↦ Σ𝑤 ∈ 𝑦 (𝑥‘𝑤)) = (𝑦 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑤 ∈ 𝑦 (𝑥‘𝑤)))
15		fveq1 6102	. . . . . . . . 9 ⊢ (𝑥 = 𝐹 → (𝑥‘𝑤) = (𝐹‘𝑤))
16	15	sumeq2ad 38632	. . . . . . . 8 ⊢ (𝑥 = 𝐹 → Σ𝑤 ∈ 𝑦 (𝑥‘𝑤) = Σ𝑤 ∈ 𝑦 (𝐹‘𝑤))
17	16	mpteq2dv 4673	. . . . . . 7 ⊢ (𝑥 = 𝐹 → (𝑦 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑤 ∈ 𝑦 (𝑥‘𝑤)) = (𝑦 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑤 ∈ 𝑦 (𝐹‘𝑤)))
18	17	adantl 481	. . . . . 6 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝐹:𝑋⟶(0[,]+∞)) ∧ 𝑥 = 𝐹) → (𝑦 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑤 ∈ 𝑦 (𝑥‘𝑤)) = (𝑦 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑤 ∈ 𝑦 (𝐹‘𝑤)))
19	14, 18	eqtrd 2644	. . . . 5 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝐹:𝑋⟶(0[,]+∞)) ∧ 𝑥 = 𝐹) → (𝑦 ∈ (𝒫 dom 𝑥 ∩ Fin) ↦ Σ𝑤 ∈ 𝑦 (𝑥‘𝑤)) = (𝑦 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑤 ∈ 𝑦 (𝐹‘𝑤)))
20	19	rneqd 5274	. . . 4 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝐹:𝑋⟶(0[,]+∞)) ∧ 𝑥 = 𝐹) → ran (𝑦 ∈ (𝒫 dom 𝑥 ∩ Fin) ↦ Σ𝑤 ∈ 𝑦 (𝑥‘𝑤)) = ran (𝑦 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑤 ∈ 𝑦 (𝐹‘𝑤)))
21	20	supeq1d 8235	. . 3 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝐹:𝑋⟶(0[,]+∞)) ∧ 𝑥 = 𝐹) → sup(ran (𝑦 ∈ (𝒫 dom 𝑥 ∩ Fin) ↦ Σ𝑤 ∈ 𝑦 (𝑥‘𝑤)), ℝ*, < ) = sup(ran (𝑦 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑤 ∈ 𝑦 (𝐹‘𝑤)), ℝ*, < ))
22	5, 21	ifbieq2d 4061	. 2 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝐹:𝑋⟶(0[,]+∞)) ∧ 𝑥 = 𝐹) → if(+∞ ∈ ran 𝑥, +∞, sup(ran (𝑦 ∈ (𝒫 dom 𝑥 ∩ Fin) ↦ Σ𝑤 ∈ 𝑦 (𝑥‘𝑤)), ℝ*, < )) = if(+∞ ∈ ran 𝐹, +∞, sup(ran (𝑦 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑤 ∈ 𝑦 (𝐹‘𝑤)), ℝ*, < )))
23		simpr 476	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐹:𝑋⟶(0[,]+∞)) → 𝐹:𝑋⟶(0[,]+∞))
24		simpl 472	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐹:𝑋⟶(0[,]+∞)) → 𝑋 ∈ 𝑉)
25		fex 6394	. . 3 ⊢ ((𝐹:𝑋⟶(0[,]+∞) ∧ 𝑋 ∈ 𝑉) → 𝐹 ∈ V)
26	23, 24, 25	syl2anc 691	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐹:𝑋⟶(0[,]+∞)) → 𝐹 ∈ V)
27		pnfxr 9971	. . . 4 ⊢ +∞ ∈ ℝ*
28	27	a1i 11	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐹:𝑋⟶(0[,]+∞)) → +∞ ∈ ℝ*)
29		xrltso 11850	. . . . 5 ⊢ < Or ℝ*
30	29	supex 8252	. . . 4 ⊢ sup(ran (𝑦 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑤 ∈ 𝑦 (𝐹‘𝑤)), ℝ*, < ) ∈ V
31	30	a1i 11	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐹:𝑋⟶(0[,]+∞)) → sup(ran (𝑦 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑤 ∈ 𝑦 (𝐹‘𝑤)), ℝ*, < ) ∈ V)
32		ifexg 4107	. . 3 ⊢ ((+∞ ∈ ℝ* ∧ sup(ran (𝑦 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑤 ∈ 𝑦 (𝐹‘𝑤)), ℝ*, < ) ∈ V) → if(+∞ ∈ ran 𝐹, +∞, sup(ran (𝑦 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑤 ∈ 𝑦 (𝐹‘𝑤)), ℝ*, < )) ∈ V)
33	28, 31, 32	syl2anc 691	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐹:𝑋⟶(0[,]+∞)) → if(+∞ ∈ ran 𝐹, +∞, sup(ran (𝑦 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑤 ∈ 𝑦 (𝐹‘𝑤)), ℝ*, < )) ∈ V)
34	2, 22, 26, 33	fvmptd 6197	1 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐹:𝑋⟶(0[,]+∞)) → (Σ^‘𝐹) = if(+∞ ∈ ran 𝐹, +∞, sup(ran (𝑦 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑤 ∈ 𝑦 (𝐹‘𝑤)), ℝ*, < )))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ∩ cin 3539  ifcif 4036  𝒫 cpw 4108   ↦ cmpt 4643  dom cdm 5038  ran crn 5039  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  Fincfn 7841  supcsup 8229  0cc0 9815  +∞cpnf 9950  ℝ^*cxr 9952   < clt 9953  [,]cicc 12049  Σcsu 14264  Σ^{^}csumge0 39255

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-sup 8231  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-seq 12664  df-sum 14265  df-sumge0 39256

This theorem is referenced by:  sge0vald  39262