Theorem sge0val 46794

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 46791	. . 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 5891	. . . . 5 ⊢ (𝑥 = 𝐹 → ran 𝑥 = ran 𝐹)
4	3	eleq2d 2822	. . . 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 5858	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝐹 → dom 𝑥 = dom 𝐹)
7	6	adantl 481	. . . . . . . . . . 11 ⊢ ((𝐹:𝑋⟶(0[,]+∞) ∧ 𝑥 = 𝐹) → dom 𝑥 = dom 𝐹)
8		fdm 6677	. . . . . . . . . . . 12 ⊢ (𝐹:𝑋⟶(0[,]+∞) → dom 𝐹 = 𝑋)
9	8	adantr 480	. . . . . . . . . . 11 ⊢ ((𝐹:𝑋⟶(0[,]+∞) ∧ 𝑥 = 𝐹) → dom 𝐹 = 𝑋)
10	7, 9	eqtrd 2771	. . . . . . . . . 10 ⊢ ((𝐹:𝑋⟶(0[,]+∞) ∧ 𝑥 = 𝐹) → dom 𝑥 = 𝑋)
11	10	pweqd 4558	. . . . . . . . 9 ⊢ ((𝐹:𝑋⟶(0[,]+∞) ∧ 𝑥 = 𝐹) → 𝒫 dom 𝑥 = 𝒫 𝑋)
12	11	ineq1d 4159	. . . . . . . 8 ⊢ ((𝐹:𝑋⟶(0[,]+∞) ∧ 𝑥 = 𝐹) → (𝒫 dom 𝑥 ∩ Fin) = (𝒫 𝑋 ∩ Fin))
13	12	mpteq1d 5175	. . . . . . 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 6839	. . . . . . . . 9 ⊢ (𝑥 = 𝐹 → (𝑥‘𝑤) = (𝐹‘𝑤))
16	15	sumeq2sdv 15665	. . . . . . . 8 ⊢ (𝑥 = 𝐹 → Σ𝑤 ∈ 𝑦 (𝑥‘𝑤) = Σ𝑤 ∈ 𝑦 (𝐹‘𝑤))
17	16	mpteq2dv 5179	. . . . . . 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 2771	. . . . 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 5893	. . . 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 9359	. . 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 4493	. 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 7182	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐹:𝑋⟶(0[,]+∞)) → 𝐹 ∈ V)
26		pnfxr 11199	. . . 4 ⊢ +∞ ∈ ℝ*
27	26	a1i 11	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐹:𝑋⟶(0[,]+∞)) → +∞ ∈ ℝ*)
28		xrltso 13092	. . . . 5 ⊢ < Or ℝ*
29	28	supex 9377	. . . 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 4515	. 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 6955	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 3429   ∩ cin 3888  ifcif 4466  𝒫 cpw 4541   ↦ cmpt 5166  dom cdm 5631  ran crn 5632  ⟶wf 6494  ‘cfv 6498  (class class class)co 7367  Fincfn 8893  supcsup 9353  0cc0 11038  +∞cpnf 11176  ℝ^*cxr 11178   < clt 11179  [,]cicc 13301  Σcsu 15648  Σ^{^}csumge0 46790

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 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-cnex 11094  ax-resscn 11095  ax-pre-lttri 11112  ax-pre-lttrn 11113

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-po 5539  df-so 5540  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-ov 7370  df-oprab 7371  df-mpo 7372  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-er 8643  df-en 8894  df-dom 8895  df-sdom 8896  df-sup 9355  df-pnf 11181  df-mnf 11182  df-xr 11183  df-ltxr 11184  df-seq 13964  df-sum 15649  df-sumge0 46791

This theorem is referenced by:  sge0vald  46797