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| Mirrors > Home > MPE Home > Th. List > Mathboxes > sge0pnfmpt | 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, 3-Mar-2021.) | 
| Ref | Expression | 
|---|---|
| sge0pnfmpt.k | ⊢ Ⅎ𝑘𝜑 | 
| sge0pnfmpt.a | ⊢ (𝜑 → 𝐴 ∈ 𝑉) | 
| sge0pnfmpt.b | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,]+∞)) | 
| sge0pnfmpt.p | ⊢ (𝜑 → ∃𝑘 ∈ 𝐴 𝐵 = +∞) | 
| Ref | Expression | 
|---|---|
| sge0pnfmpt | ⊢ (𝜑 → (Σ^‘(𝑘 ∈ 𝐴 ↦ 𝐵)) = +∞) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | sge0pnfmpt.a | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
| 2 | sge0pnfmpt.k | . . 3 ⊢ Ⅎ𝑘𝜑 | |
| 3 | sge0pnfmpt.b | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,]+∞)) | |
| 4 | eqid 2737 | . . 3 ⊢ (𝑘 ∈ 𝐴 ↦ 𝐵) = (𝑘 ∈ 𝐴 ↦ 𝐵) | |
| 5 | 2, 3, 4 | fmptdf 7137 | . 2 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐵):𝐴⟶(0[,]+∞)) | 
| 6 | sge0pnfmpt.p | . . . 4 ⊢ (𝜑 → ∃𝑘 ∈ 𝐴 𝐵 = +∞) | |
| 7 | eqcom 2744 | . . . . 5 ⊢ (𝐵 = +∞ ↔ +∞ = 𝐵) | |
| 8 | 7 | rexbii 3094 | . . . 4 ⊢ (∃𝑘 ∈ 𝐴 𝐵 = +∞ ↔ ∃𝑘 ∈ 𝐴 +∞ = 𝐵) | 
| 9 | 6, 8 | sylib 218 | . . 3 ⊢ (𝜑 → ∃𝑘 ∈ 𝐴 +∞ = 𝐵) | 
| 10 | pnfex 11314 | . . . 4 ⊢ +∞ ∈ V | |
| 11 | 10 | a1i 11 | . . 3 ⊢ (𝜑 → +∞ ∈ V) | 
| 12 | 4, 9, 11 | elrnmptd 5974 | . 2 ⊢ (𝜑 → +∞ ∈ ran (𝑘 ∈ 𝐴 ↦ 𝐵)) | 
| 13 | 1, 5, 12 | sge0pnfval 46388 | 1 ⊢ (𝜑 → (Σ^‘(𝑘 ∈ 𝐴 ↦ 𝐵)) = +∞) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 Ⅎwnf 1783 ∈ wcel 2108 ∃wrex 3070 Vcvv 3480 ↦ cmpt 5225 ‘cfv 6561 (class class class)co 7431 0cc0 11155 +∞cpnf 11292 [,]cicc 13390 Σ^csumge0 46377 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-pre-lttri 11229 ax-pre-lttrn 11230 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-po 5592 df-so 5593 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-sup 9482 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-seq 14043 df-sum 15723 df-sumge0 46378 | 
| This theorem is referenced by: voliunsge0lem 46487 | 
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