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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > sge0ssrempt | Structured version Visualization version GIF version |
Description: If a sum of nonnegative extended reals is real, than any subsum is real. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
Ref | Expression |
---|---|
sge0ssrempt.xph | ⊢ Ⅎ𝑥𝜑 |
sge0ssrempt.a | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
sge0ssrempt.b | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ (0[,]+∞)) |
sge0ssrempt.re | ⊢ (𝜑 → (Σ^‘(𝑥 ∈ 𝐴 ↦ 𝐵)) ∈ ℝ) |
sge0ssrempt.c | ⊢ (𝜑 → 𝐶 ⊆ 𝐴) |
Ref | Expression |
---|---|
sge0ssrempt | ⊢ (𝜑 → (Σ^‘(𝑥 ∈ 𝐶 ↦ 𝐵)) ∈ ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sge0ssrempt.c | . . . . 5 ⊢ (𝜑 → 𝐶 ⊆ 𝐴) | |
2 | 1 | resmptd 5591 | . . . 4 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵) ↾ 𝐶) = (𝑥 ∈ 𝐶 ↦ 𝐵)) |
3 | 2 | fveq2d 6334 | . . 3 ⊢ (𝜑 → (Σ^‘((𝑥 ∈ 𝐴 ↦ 𝐵) ↾ 𝐶)) = (Σ^‘(𝑥 ∈ 𝐶 ↦ 𝐵))) |
4 | 3 | eqcomd 2777 | . 2 ⊢ (𝜑 → (Σ^‘(𝑥 ∈ 𝐶 ↦ 𝐵)) = (Σ^‘((𝑥 ∈ 𝐴 ↦ 𝐵) ↾ 𝐶))) |
5 | sge0ssrempt.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
6 | sge0ssrempt.xph | . . . 4 ⊢ Ⅎ𝑥𝜑 | |
7 | sge0ssrempt.b | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ (0[,]+∞)) | |
8 | eqid 2771 | . . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
9 | 6, 7, 8 | fmptdf 6527 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶(0[,]+∞)) |
10 | sge0ssrempt.re | . . 3 ⊢ (𝜑 → (Σ^‘(𝑥 ∈ 𝐴 ↦ 𝐵)) ∈ ℝ) | |
11 | 5, 9, 10 | sge0ssre 41124 | . 2 ⊢ (𝜑 → (Σ^‘((𝑥 ∈ 𝐴 ↦ 𝐵) ↾ 𝐶)) ∈ ℝ) |
12 | 4, 11 | eqeltrd 2850 | 1 ⊢ (𝜑 → (Σ^‘(𝑥 ∈ 𝐶 ↦ 𝐵)) ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 382 Ⅎwnf 1856 ∈ wcel 2145 ⊆ wss 3723 ↦ cmpt 4863 ↾ cres 5251 ‘cfv 6029 (class class class)co 6791 ℝcr 10135 0cc0 10136 +∞cpnf 10271 [,]cicc 12376 Σ^csumge0 41089 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4904 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7094 ax-inf2 8700 ax-cnex 10192 ax-resscn 10193 ax-1cn 10194 ax-icn 10195 ax-addcl 10196 ax-addrcl 10197 ax-mulcl 10198 ax-mulrcl 10199 ax-mulcom 10200 ax-addass 10201 ax-mulass 10202 ax-distr 10203 ax-i2m1 10204 ax-1ne0 10205 ax-1rid 10206 ax-rnegex 10207 ax-rrecex 10208 ax-cnre 10209 ax-pre-lttri 10210 ax-pre-lttrn 10211 ax-pre-ltadd 10212 ax-pre-mulgt0 10213 ax-pre-sup 10214 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3or 1072 df-3an 1073 df-tru 1634 df-fal 1637 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-tp 4321 df-op 4323 df-uni 4575 df-int 4612 df-iun 4656 df-br 4787 df-opab 4847 df-mpt 4864 df-tr 4887 df-id 5157 df-eprel 5162 df-po 5170 df-so 5171 df-fr 5208 df-se 5209 df-we 5210 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-pred 5821 df-ord 5867 df-on 5868 df-lim 5869 df-suc 5870 df-iota 5992 df-fun 6031 df-fn 6032 df-f 6033 df-f1 6034 df-fo 6035 df-f1o 6036 df-fv 6037 df-isom 6038 df-riota 6752 df-ov 6794 df-oprab 6795 df-mpt2 6796 df-om 7211 df-1st 7313 df-2nd 7314 df-wrecs 7557 df-recs 7619 df-rdg 7657 df-1o 7711 df-oadd 7715 df-er 7894 df-en 8108 df-dom 8109 df-sdom 8110 df-fin 8111 df-sup 8502 df-oi 8569 df-card 8963 df-pnf 10276 df-mnf 10277 df-xr 10278 df-ltxr 10279 df-le 10280 df-sub 10468 df-neg 10469 df-div 10885 df-nn 11221 df-2 11279 df-3 11280 df-n0 11493 df-z 11578 df-uz 11887 df-rp 12029 df-ico 12379 df-icc 12380 df-fz 12527 df-fzo 12667 df-seq 13002 df-exp 13061 df-hash 13315 df-cj 14040 df-re 14041 df-im 14042 df-sqrt 14176 df-abs 14177 df-clim 14420 df-sum 14618 df-sumge0 41090 |
This theorem is referenced by: sge0iunmptlemre 41142 hoidmv1lelem2 41319 hoidmvlelem2 41323 |
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