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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > sge0pnfval | Structured version Visualization version GIF version |
Description: If a term in the sum of nonnegative extended reals is +∞, then the value of the sum is +∞. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
Ref | Expression |
---|---|
sge0pnfval.x | ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
sge0pnfval.f | ⊢ (𝜑 → 𝐹:𝑋⟶(0[,]+∞)) |
sge0pnfval.pnf | ⊢ (𝜑 → +∞ ∈ ran 𝐹) |
Ref | Expression |
---|---|
sge0pnfval | ⊢ (𝜑 → (Σ^‘𝐹) = +∞) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sge0pnfval.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
2 | sge0pnfval.f | . . 3 ⊢ (𝜑 → 𝐹:𝑋⟶(0[,]+∞)) | |
3 | 1, 2 | sge0vald 46325 | . 2 ⊢ (𝜑 → (Σ^‘𝐹) = if(+∞ ∈ ran 𝐹, +∞, sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)), ℝ*, < ))) |
4 | sge0pnfval.pnf | . . 3 ⊢ (𝜑 → +∞ ∈ ran 𝐹) | |
5 | 4 | iftrued 4539 | . 2 ⊢ (𝜑 → if(+∞ ∈ ran 𝐹, +∞, sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)), ℝ*, < )) = +∞) |
6 | 3, 5 | eqtrd 2775 | 1 ⊢ (𝜑 → (Σ^‘𝐹) = +∞) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 ∩ cin 3962 ifcif 4531 𝒫 cpw 4605 ↦ cmpt 5231 ran crn 5690 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 Fincfn 8984 supcsup 9478 0cc0 11153 +∞cpnf 11290 ℝ*cxr 11292 < clt 11293 [,]cicc 13387 Σcsu 15719 Σ^csumge0 46318 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-pre-lttri 11227 ax-pre-lttrn 11228 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-po 5597 df-so 5598 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-sup 9480 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-seq 14040 df-sum 15720 df-sumge0 46319 |
This theorem is referenced by: sge0sn 46335 sge0tsms 46336 sge0cl 46337 sge0f1o 46338 sge0rern 46344 sge0supre 46345 sge0sup 46347 sge0pr 46350 sge0le 46363 sge0split 46365 sge0iunmpt 46374 sge0pnfmpt 46401 |
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