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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > fsumlesge0 | Structured version Visualization version GIF version |
Description: Every finite subsum of nonnegative reals is less than or equal to the extended sum over the whole (possibly infinite) domain. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
Ref | Expression |
---|---|
fsumlesge0.x | ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
fsumlesge0.f | ⊢ (𝜑 → 𝐹:𝑋⟶(0[,)+∞)) |
fsumlesge0.y | ⊢ (𝜑 → 𝑌 ⊆ 𝑋) |
fsumlesge0.fi | ⊢ (𝜑 → 𝑌 ∈ Fin) |
Ref | Expression |
---|---|
fsumlesge0 | ⊢ (𝜑 → Σ𝑥 ∈ 𝑌 (𝐹‘𝑥) ≤ (Σ^‘𝐹)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fsumlesge0.f | . . . . 5 ⊢ (𝜑 → 𝐹:𝑋⟶(0[,)+∞)) | |
2 | 1 | sge0rnre 45657 | . . . 4 ⊢ (𝜑 → ran (𝑦 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑧 ∈ 𝑦 (𝐹‘𝑧)) ⊆ ℝ) |
3 | ressxr 11262 | . . . . 5 ⊢ ℝ ⊆ ℝ* | |
4 | 3 | a1i 11 | . . . 4 ⊢ (𝜑 → ℝ ⊆ ℝ*) |
5 | 2, 4 | sstrd 3987 | . . 3 ⊢ (𝜑 → ran (𝑦 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑧 ∈ 𝑦 (𝐹‘𝑧)) ⊆ ℝ*) |
6 | fsumlesge0.y | . . . . . . 7 ⊢ (𝜑 → 𝑌 ⊆ 𝑋) | |
7 | fsumlesge0.x | . . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
8 | 7, 6 | ssexd 5317 | . . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ V) |
9 | elpwg 4600 | . . . . . . . 8 ⊢ (𝑌 ∈ V → (𝑌 ∈ 𝒫 𝑋 ↔ 𝑌 ⊆ 𝑋)) | |
10 | 8, 9 | syl 17 | . . . . . . 7 ⊢ (𝜑 → (𝑌 ∈ 𝒫 𝑋 ↔ 𝑌 ⊆ 𝑋)) |
11 | 6, 10 | mpbird 257 | . . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝒫 𝑋) |
12 | fsumlesge0.fi | . . . . . 6 ⊢ (𝜑 → 𝑌 ∈ Fin) | |
13 | 11, 12 | elind 4189 | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ (𝒫 𝑋 ∩ Fin)) |
14 | fveq2 6885 | . . . . . . 7 ⊢ (𝑥 = 𝑧 → (𝐹‘𝑥) = (𝐹‘𝑧)) | |
15 | 14 | cbvsumv 15648 | . . . . . 6 ⊢ Σ𝑥 ∈ 𝑌 (𝐹‘𝑥) = Σ𝑧 ∈ 𝑌 (𝐹‘𝑧) |
16 | 15 | a1i 11 | . . . . 5 ⊢ (𝜑 → Σ𝑥 ∈ 𝑌 (𝐹‘𝑥) = Σ𝑧 ∈ 𝑌 (𝐹‘𝑧)) |
17 | sumeq1 15641 | . . . . . 6 ⊢ (𝑦 = 𝑌 → Σ𝑧 ∈ 𝑦 (𝐹‘𝑧) = Σ𝑧 ∈ 𝑌 (𝐹‘𝑧)) | |
18 | 17 | rspceeqv 3628 | . . . . 5 ⊢ ((𝑌 ∈ (𝒫 𝑋 ∩ Fin) ∧ Σ𝑥 ∈ 𝑌 (𝐹‘𝑥) = Σ𝑧 ∈ 𝑌 (𝐹‘𝑧)) → ∃𝑦 ∈ (𝒫 𝑋 ∩ Fin)Σ𝑥 ∈ 𝑌 (𝐹‘𝑥) = Σ𝑧 ∈ 𝑦 (𝐹‘𝑧)) |
19 | 13, 16, 18 | syl2anc 583 | . . . 4 ⊢ (𝜑 → ∃𝑦 ∈ (𝒫 𝑋 ∩ Fin)Σ𝑥 ∈ 𝑌 (𝐹‘𝑥) = Σ𝑧 ∈ 𝑦 (𝐹‘𝑧)) |
20 | sumex 15640 | . . . . . 6 ⊢ Σ𝑥 ∈ 𝑌 (𝐹‘𝑥) ∈ V | |
21 | 20 | a1i 11 | . . . . 5 ⊢ (𝜑 → Σ𝑥 ∈ 𝑌 (𝐹‘𝑥) ∈ V) |
22 | eqid 2726 | . . . . . 6 ⊢ (𝑦 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑧 ∈ 𝑦 (𝐹‘𝑧)) = (𝑦 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑧 ∈ 𝑦 (𝐹‘𝑧)) | |
23 | 22 | elrnmpt 5949 | . . . . 5 ⊢ (Σ𝑥 ∈ 𝑌 (𝐹‘𝑥) ∈ V → (Σ𝑥 ∈ 𝑌 (𝐹‘𝑥) ∈ ran (𝑦 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑧 ∈ 𝑦 (𝐹‘𝑧)) ↔ ∃𝑦 ∈ (𝒫 𝑋 ∩ Fin)Σ𝑥 ∈ 𝑌 (𝐹‘𝑥) = Σ𝑧 ∈ 𝑦 (𝐹‘𝑧))) |
24 | 21, 23 | syl 17 | . . . 4 ⊢ (𝜑 → (Σ𝑥 ∈ 𝑌 (𝐹‘𝑥) ∈ ran (𝑦 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑧 ∈ 𝑦 (𝐹‘𝑧)) ↔ ∃𝑦 ∈ (𝒫 𝑋 ∩ Fin)Σ𝑥 ∈ 𝑌 (𝐹‘𝑥) = Σ𝑧 ∈ 𝑦 (𝐹‘𝑧))) |
25 | 19, 24 | mpbird 257 | . . 3 ⊢ (𝜑 → Σ𝑥 ∈ 𝑌 (𝐹‘𝑥) ∈ ran (𝑦 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑧 ∈ 𝑦 (𝐹‘𝑧))) |
26 | supxrub 13309 | . . 3 ⊢ ((ran (𝑦 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑧 ∈ 𝑦 (𝐹‘𝑧)) ⊆ ℝ* ∧ Σ𝑥 ∈ 𝑌 (𝐹‘𝑥) ∈ ran (𝑦 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑧 ∈ 𝑦 (𝐹‘𝑧))) → Σ𝑥 ∈ 𝑌 (𝐹‘𝑥) ≤ sup(ran (𝑦 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑧 ∈ 𝑦 (𝐹‘𝑧)), ℝ*, < )) | |
27 | 5, 25, 26 | syl2anc 583 | . 2 ⊢ (𝜑 → Σ𝑥 ∈ 𝑌 (𝐹‘𝑥) ≤ sup(ran (𝑦 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑧 ∈ 𝑦 (𝐹‘𝑧)), ℝ*, < )) |
28 | 7, 1 | sge0reval 45665 | . . 3 ⊢ (𝜑 → (Σ^‘𝐹) = sup(ran (𝑦 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑧 ∈ 𝑦 (𝐹‘𝑧)), ℝ*, < )) |
29 | 28 | eqcomd 2732 | . 2 ⊢ (𝜑 → sup(ran (𝑦 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑧 ∈ 𝑦 (𝐹‘𝑧)), ℝ*, < ) = (Σ^‘𝐹)) |
30 | 27, 29 | breqtrd 5167 | 1 ⊢ (𝜑 → Σ𝑥 ∈ 𝑌 (𝐹‘𝑥) ≤ (Σ^‘𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1533 ∈ wcel 2098 ∃wrex 3064 Vcvv 3468 ∩ cin 3942 ⊆ wss 3943 𝒫 cpw 4597 class class class wbr 5141 ↦ cmpt 5224 ran crn 5670 ⟶wf 6533 ‘cfv 6537 (class class class)co 7405 Fincfn 8941 supcsup 9437 ℝcr 11111 0cc0 11112 +∞cpnf 11249 ℝ*cxr 11251 < clt 11252 ≤ cle 11253 [,)cico 13332 Σcsu 15638 Σ^csumge0 45655 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-inf2 9638 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-sup 9439 df-oi 9507 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-n0 12477 df-z 12563 df-uz 12827 df-rp 12981 df-ico 13336 df-icc 13337 df-fz 13491 df-fzo 13634 df-seq 13973 df-exp 14033 df-hash 14296 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-clim 15438 df-sum 15639 df-sumge0 45656 |
This theorem is referenced by: sge0fsum 45680 sge0rnbnd 45686 sge0split 45702 |
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