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 5216 | . . . 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 6542 | . . . 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 42660 | . 2 ⊢ (𝜑 → (Σ^‘(𝐹 ↾ 𝑌)) ∈ ℝ*) |
8 | sge0ssre.re | . 2 ⊢ (𝜑 → (Σ^‘𝐹) ∈ ℝ) | |
9 | mnfxr 10692 | . . . 4 ⊢ -∞ ∈ ℝ* | |
10 | 9 | a1i 11 | . . 3 ⊢ (𝜑 → -∞ ∈ ℝ*) |
11 | 0xr 10682 | . . . 4 ⊢ 0 ∈ ℝ* | |
12 | 11 | a1i 11 | . . 3 ⊢ (𝜑 → 0 ∈ ℝ*) |
13 | mnflt0 12514 | . . . 4 ⊢ -∞ < 0 | |
14 | 13 | a1i 11 | . . 3 ⊢ (𝜑 → -∞ < 0) |
15 | 3, 6 | sge0ge0 42659 | . . 3 ⊢ (𝜑 → 0 ≤ (Σ^‘(𝐹 ↾ 𝑌))) |
16 | 10, 12, 7, 14, 15 | xrltletrd 12548 | . 2 ⊢ (𝜑 → -∞ < (Σ^‘(𝐹 ↾ 𝑌))) |
17 | 1, 4 | sge0less 42667 | . 2 ⊢ (𝜑 → (Σ^‘(𝐹 ↾ 𝑌)) ≤ (Σ^‘𝐹)) |
18 | xrre 12556 | . 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 836 | 1 ⊢ (𝜑 → (Σ^‘(𝐹 ↾ 𝑌)) ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2110 Vcvv 3495 ∩ cin 3935 class class class wbr 5059 ↾ cres 5552 ⟶wf 6346 ‘cfv 6350 (class class class)co 7150 ℝcr 10530 0cc0 10531 +∞cpnf 10666 -∞cmnf 10667 ℝ*cxr 10668 < clt 10669 ≤ cle 10670 [,]cicc 12735 Σ^csumge0 42637 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-inf2 9098 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-int 4870 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-se 5510 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-isom 6359 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-sup 8900 df-oi 8968 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-2 11694 df-3 11695 df-n0 11892 df-z 11976 df-uz 12238 df-rp 12384 df-ico 12738 df-icc 12739 df-fz 12887 df-fzo 13028 df-seq 13364 df-exp 13424 df-hash 13685 df-cj 14452 df-re 14453 df-im 14454 df-sqrt 14588 df-abs 14589 df-clim 14839 df-sum 15037 df-sumge0 42638 |
This theorem is referenced by: sge0ssrempt 42680 sge0resplit 42681 sge0split 42684 |
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