Theorem sge0val 46812

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 46809	. . 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 5885	. . . . 5 ⊢ (𝑥 = 𝐹 → ran 𝑥 = ran 𝐹)
4	3	eleq2d 2823	. . . 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 5852	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝐹 → dom 𝑥 = dom 𝐹)
7	6	adantl 481	. . . . . . . . . . 11 ⊢ ((𝐹:𝑋⟶(0[,]+∞) ∧ 𝑥 = 𝐹) → dom 𝑥 = dom 𝐹)
8		fdm 6671	. . . . . . . . . . . 12 ⊢ (𝐹:𝑋⟶(0[,]+∞) → dom 𝐹 = 𝑋)
9	8	adantr 480	. . . . . . . . . . 11 ⊢ ((𝐹:𝑋⟶(0[,]+∞) ∧ 𝑥 = 𝐹) → dom 𝐹 = 𝑋)
10	7, 9	eqtrd 2772	. . . . . . . . . 10 ⊢ ((𝐹:𝑋⟶(0[,]+∞) ∧ 𝑥 = 𝐹) → dom 𝑥 = 𝑋)
11	10	pweqd 4559	. . . . . . . . 9 ⊢ ((𝐹:𝑋⟶(0[,]+∞) ∧ 𝑥 = 𝐹) → 𝒫 dom 𝑥 = 𝒫 𝑋)
12	11	ineq1d 4160	. . . . . . . 8 ⊢ ((𝐹:𝑋⟶(0[,]+∞) ∧ 𝑥 = 𝐹) → (𝒫 dom 𝑥 ∩ Fin) = (𝒫 𝑋 ∩ Fin))
13	12	mpteq1d 5176	. . . . . . 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 715	. . . . . 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 6833	. . . . . . . . 9 ⊢ (𝑥 = 𝐹 → (𝑥‘𝑤) = (𝐹‘𝑤))
16	15	sumeq2sdv 15656	. . . . . . . 8 ⊢ (𝑥 = 𝐹 → Σ𝑤 ∈ 𝑦 (𝑥‘𝑤) = Σ𝑤 ∈ 𝑦 (𝐹‘𝑤))
17	16	mpteq2dv 5180	. . . . . . 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 2772	. . . . 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 5887	. . . 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 9352	. . 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 4494	. 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 484	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐹:𝑋⟶(0[,]+∞)) → 𝐹:𝑋⟶(0[,]+∞))
24		simpl 482	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐹:𝑋⟶(0[,]+∞)) → 𝑋 ∈ 𝑉)
25	23, 24	fexd 7175	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐹:𝑋⟶(0[,]+∞)) → 𝐹 ∈ V)
26		pnfxr 11190	. . . 4 ⊢ +∞ ∈ ℝ*
27	26	a1i 11	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐹:𝑋⟶(0[,]+∞)) → +∞ ∈ ℝ*)
28		xrltso 13083	. . . . 5 ⊢ < Or ℝ*
29	28	supex 9370	. . . 4 ⊢ sup(ran (𝑦 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑤 ∈ 𝑦 (𝐹‘𝑤)), ℝ*, < ) ∈ V
30	29	a1i 11	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐹:𝑋⟶(0[,]+∞)) → sup(ran (𝑦 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑤 ∈ 𝑦 (𝐹‘𝑤)), ℝ*, < ) ∈ V)
31	27, 30	ifexd 4516	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐹:𝑋⟶(0[,]+∞)) → if(+∞ ∈ ran 𝐹, +∞, sup(ran (𝑦 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑤 ∈ 𝑦 (𝐹‘𝑤)), ℝ*, < )) ∈ V)
32	2, 22, 25, 31	fvmptd 6949	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 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  Vcvv 3430   ∩ cin 3889  ifcif 4467  𝒫 cpw 4542   ↦ cmpt 5167  dom cdm 5624  ran crn 5625  ⟶wf 6488  ‘cfv 6492  (class class class)co 7360  Fincfn 8886  supcsup 9346  0cc0 11029  +∞cpnf 11167  ℝ^*cxr 11169   < clt 11170  [,]cicc 13292  Σcsu 15639  Σ^{^}csumge0 46808

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682  ax-cnex 11085  ax-resscn 11086  ax-pre-lttri 11103  ax-pre-lttrn 11104

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5519  df-po 5532  df-so 5533  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7363  df-oprab 7364  df-mpo 7365  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-rdg 8342  df-er 8636  df-en 8887  df-dom 8888  df-sdom 8889  df-sup 9348  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-seq 13955  df-sum 15640  df-sumge0 46809

This theorem is referenced by:  sge0vald  46815