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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 5277 | . . . 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 6712 | . . . 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 44712 | . 2 ⊢ (𝜑 → (Σ^‘(𝐹 ↾ 𝑌)) ∈ ℝ*) |
8 | sge0ssre.re | . 2 ⊢ (𝜑 → (Σ^‘𝐹) ∈ ℝ) | |
9 | mnfxr 11217 | . . . 4 ⊢ -∞ ∈ ℝ* | |
10 | 9 | a1i 11 | . . 3 ⊢ (𝜑 → -∞ ∈ ℝ*) |
11 | 0xr 11207 | . . . 4 ⊢ 0 ∈ ℝ* | |
12 | 11 | a1i 11 | . . 3 ⊢ (𝜑 → 0 ∈ ℝ*) |
13 | mnflt0 13051 | . . . 4 ⊢ -∞ < 0 | |
14 | 13 | a1i 11 | . . 3 ⊢ (𝜑 → -∞ < 0) |
15 | 3, 6 | sge0ge0 44711 | . . 3 ⊢ (𝜑 → 0 ≤ (Σ^‘(𝐹 ↾ 𝑌))) |
16 | 10, 12, 7, 14, 15 | xrltletrd 13086 | . 2 ⊢ (𝜑 → -∞ < (Σ^‘(𝐹 ↾ 𝑌))) |
17 | 1, 4 | sge0less 44719 | . 2 ⊢ (𝜑 → (Σ^‘(𝐹 ↾ 𝑌)) ≤ (Σ^‘𝐹)) |
18 | xrre 13094 | . 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 2107 Vcvv 3444 ∩ cin 3910 class class class wbr 5106 ↾ cres 5636 ⟶wf 6493 ‘cfv 6497 (class class class)co 7358 ℝcr 11055 0cc0 11056 +∞cpnf 11191 -∞cmnf 11192 ℝ*cxr 11193 < clt 11194 ≤ cle 11195 [,]cicc 13273 Σ^csumge0 44689 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-inf2 9582 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 ax-pre-sup 11134 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-int 4909 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-se 5590 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-isom 6506 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-er 8651 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-sup 9383 df-oi 9451 df-card 9880 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-div 11818 df-nn 12159 df-2 12221 df-3 12222 df-n0 12419 df-z 12505 df-uz 12769 df-rp 12921 df-ico 13276 df-icc 13277 df-fz 13431 df-fzo 13574 df-seq 13913 df-exp 13974 df-hash 14237 df-cj 14990 df-re 14991 df-im 14992 df-sqrt 15126 df-abs 15127 df-clim 15376 df-sum 15577 df-sumge0 44690 |
This theorem is referenced by: sge0ssrempt 44732 sge0resplit 44733 sge0split 44736 |
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