Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > sge0ssre | 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 |
---|---|
sge0less.x | ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
sge0less.f | ⊢ (𝜑 → 𝐹:𝑋⟶(0[,]+∞)) |
sge0ssre.re | ⊢ (𝜑 → (Σ^‘𝐹) ∈ ℝ) |
Ref | Expression |
---|---|
sge0ssre | ⊢ (𝜑 → (Σ^‘(𝐹 ↾ 𝑌)) ∈ ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sge0less.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
2 | inex1g 5188 | . . . 4 ⊢ (𝑋 ∈ 𝑉 → (𝑋 ∩ 𝑌) ∈ V) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ (𝜑 → (𝑋 ∩ 𝑌) ∈ V) |
4 | sge0less.f | . . . 4 ⊢ (𝜑 → 𝐹:𝑋⟶(0[,]+∞)) | |
5 | fresin 6548 | . . . 4 ⊢ (𝐹:𝑋⟶(0[,]+∞) → (𝐹 ↾ 𝑌):(𝑋 ∩ 𝑌)⟶(0[,]+∞)) | |
6 | 4, 5 | syl 17 | . . 3 ⊢ (𝜑 → (𝐹 ↾ 𝑌):(𝑋 ∩ 𝑌)⟶(0[,]+∞)) |
7 | 3, 6 | sge0xrcl 43488 | . 2 ⊢ (𝜑 → (Σ^‘(𝐹 ↾ 𝑌)) ∈ ℝ*) |
8 | sge0ssre.re | . 2 ⊢ (𝜑 → (Σ^‘𝐹) ∈ ℝ) | |
9 | mnfxr 10779 | . . . 4 ⊢ -∞ ∈ ℝ* | |
10 | 9 | a1i 11 | . . 3 ⊢ (𝜑 → -∞ ∈ ℝ*) |
11 | 0xr 10769 | . . . 4 ⊢ 0 ∈ ℝ* | |
12 | 11 | a1i 11 | . . 3 ⊢ (𝜑 → 0 ∈ ℝ*) |
13 | mnflt0 12606 | . . . 4 ⊢ -∞ < 0 | |
14 | 13 | a1i 11 | . . 3 ⊢ (𝜑 → -∞ < 0) |
15 | 3, 6 | sge0ge0 43487 | . . 3 ⊢ (𝜑 → 0 ≤ (Σ^‘(𝐹 ↾ 𝑌))) |
16 | 10, 12, 7, 14, 15 | xrltletrd 12640 | . 2 ⊢ (𝜑 → -∞ < (Σ^‘(𝐹 ↾ 𝑌))) |
17 | 1, 4 | sge0less 43495 | . 2 ⊢ (𝜑 → (Σ^‘(𝐹 ↾ 𝑌)) ≤ (Σ^‘𝐹)) |
18 | xrre 12648 | . 2 ⊢ ((((Σ^‘(𝐹 ↾ 𝑌)) ∈ ℝ* ∧ (Σ^‘𝐹) ∈ ℝ) ∧ (-∞ < (Σ^‘(𝐹 ↾ 𝑌)) ∧ (Σ^‘(𝐹 ↾ 𝑌)) ≤ (Σ^‘𝐹))) → (Σ^‘(𝐹 ↾ 𝑌)) ∈ ℝ) | |
19 | 7, 8, 16, 17, 18 | syl22anc 838 | 1 ⊢ (𝜑 → (Σ^‘(𝐹 ↾ 𝑌)) ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2114 Vcvv 3399 ∩ cin 3843 class class class wbr 5031 ↾ cres 5528 ⟶wf 6336 ‘cfv 6340 (class class class)co 7173 ℝcr 10617 0cc0 10618 +∞cpnf 10753 -∞cmnf 10754 ℝ*cxr 10755 < clt 10756 ≤ cle 10757 [,]cicc 12827 Σ^csumge0 43465 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-rep 5155 ax-sep 5168 ax-nul 5175 ax-pow 5233 ax-pr 5297 ax-un 7482 ax-inf2 9180 ax-cnex 10674 ax-resscn 10675 ax-1cn 10676 ax-icn 10677 ax-addcl 10678 ax-addrcl 10679 ax-mulcl 10680 ax-mulrcl 10681 ax-mulcom 10682 ax-addass 10683 ax-mulass 10684 ax-distr 10685 ax-i2m1 10686 ax-1ne0 10687 ax-1rid 10688 ax-rnegex 10689 ax-rrecex 10690 ax-cnre 10691 ax-pre-lttri 10692 ax-pre-lttrn 10693 ax-pre-ltadd 10694 ax-pre-mulgt0 10695 ax-pre-sup 10696 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3401 df-sbc 3682 df-csb 3792 df-dif 3847 df-un 3849 df-in 3851 df-ss 3861 df-pss 3863 df-nul 4213 df-if 4416 df-pw 4491 df-sn 4518 df-pr 4520 df-tp 4522 df-op 4524 df-uni 4798 df-int 4838 df-iun 4884 df-br 5032 df-opab 5094 df-mpt 5112 df-tr 5138 df-id 5430 df-eprel 5435 df-po 5443 df-so 5444 df-fr 5484 df-se 5485 df-we 5486 df-xp 5532 df-rel 5533 df-cnv 5534 df-co 5535 df-dm 5536 df-rn 5537 df-res 5538 df-ima 5539 df-pred 6130 df-ord 6176 df-on 6177 df-lim 6178 df-suc 6179 df-iota 6298 df-fun 6342 df-fn 6343 df-f 6344 df-f1 6345 df-fo 6346 df-f1o 6347 df-fv 6348 df-isom 6349 df-riota 7130 df-ov 7176 df-oprab 7177 df-mpo 7178 df-om 7603 df-1st 7717 df-2nd 7718 df-wrecs 7979 df-recs 8040 df-rdg 8078 df-1o 8134 df-er 8323 df-en 8559 df-dom 8560 df-sdom 8561 df-fin 8562 df-sup 8982 df-oi 9050 df-card 9444 df-pnf 10758 df-mnf 10759 df-xr 10760 df-ltxr 10761 df-le 10762 df-sub 10953 df-neg 10954 df-div 11379 df-nn 11720 df-2 11782 df-3 11783 df-n0 11980 df-z 12066 df-uz 12328 df-rp 12476 df-ico 12830 df-icc 12831 df-fz 12985 df-fzo 13128 df-seq 13464 df-exp 13525 df-hash 13786 df-cj 14551 df-re 14552 df-im 14553 df-sqrt 14687 df-abs 14688 df-clim 14938 df-sum 15139 df-sumge0 43466 |
This theorem is referenced by: sge0ssrempt 43508 sge0resplit 43509 sge0split 43512 |
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