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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > sge0splitsn | Structured version Visualization version GIF version |
Description: Separate out a term in a generalized sum of nonnegative extended reals. (Contributed by Glauco Siliprandi, 21-Nov-2020.) |
Ref | Expression |
---|---|
sge0splitsn.ph | ⊢ Ⅎ𝑘𝜑 |
sge0splitsn.a | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
sge0splitsn.b | ⊢ (𝜑 → 𝐵 ∈ 𝑊) |
sge0splitsn.n | ⊢ (𝜑 → ¬ 𝐵 ∈ 𝐴) |
sge0splitsn.c | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ (0[,]+∞)) |
sge0splitsn.d | ⊢ (𝑘 = 𝐵 → 𝐶 = 𝐷) |
sge0splitsn.e | ⊢ (𝜑 → 𝐷 ∈ (0[,]+∞)) |
Ref | Expression |
---|---|
sge0splitsn | ⊢ (𝜑 → (Σ^‘(𝑘 ∈ (𝐴 ∪ {𝐵}) ↦ 𝐶)) = ((Σ^‘(𝑘 ∈ 𝐴 ↦ 𝐶)) +𝑒 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sge0splitsn.ph | . . 3 ⊢ Ⅎ𝑘𝜑 | |
2 | sge0splitsn.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
3 | snfi 8947 | . . . . 5 ⊢ {𝐵} ∈ Fin | |
4 | 3 | a1i 11 | . . . 4 ⊢ (𝜑 → {𝐵} ∈ Fin) |
5 | 4 | elexd 3464 | . . 3 ⊢ (𝜑 → {𝐵} ∈ V) |
6 | sge0splitsn.n | . . . 4 ⊢ (𝜑 → ¬ 𝐵 ∈ 𝐴) | |
7 | disjsn 4671 | . . . 4 ⊢ ((𝐴 ∩ {𝐵}) = ∅ ↔ ¬ 𝐵 ∈ 𝐴) | |
8 | 6, 7 | sylibr 233 | . . 3 ⊢ (𝜑 → (𝐴 ∩ {𝐵}) = ∅) |
9 | sge0splitsn.c | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ (0[,]+∞)) | |
10 | elsni 4602 | . . . . 5 ⊢ (𝑘 ∈ {𝐵} → 𝑘 = 𝐵) | |
11 | sge0splitsn.d | . . . . . 6 ⊢ (𝑘 = 𝐵 → 𝐶 = 𝐷) | |
12 | 11 | adantl 483 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 = 𝐵) → 𝐶 = 𝐷) |
13 | 10, 12 | sylan2 594 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝐵}) → 𝐶 = 𝐷) |
14 | sge0splitsn.e | . . . . 5 ⊢ (𝜑 → 𝐷 ∈ (0[,]+∞)) | |
15 | 14 | adantr 482 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝐵}) → 𝐷 ∈ (0[,]+∞)) |
16 | 13, 15 | eqeltrd 2839 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝐵}) → 𝐶 ∈ (0[,]+∞)) |
17 | 1, 2, 5, 8, 9, 16 | sge0splitmpt 44547 | . 2 ⊢ (𝜑 → (Σ^‘(𝑘 ∈ (𝐴 ∪ {𝐵}) ↦ 𝐶)) = ((Σ^‘(𝑘 ∈ 𝐴 ↦ 𝐶)) +𝑒 (Σ^‘(𝑘 ∈ {𝐵} ↦ 𝐶)))) |
18 | sge0splitsn.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑊) | |
19 | 1, 18, 14, 11 | sge0snmptf 44573 | . . 3 ⊢ (𝜑 → (Σ^‘(𝑘 ∈ {𝐵} ↦ 𝐶)) = 𝐷) |
20 | 19 | oveq2d 7368 | . 2 ⊢ (𝜑 → ((Σ^‘(𝑘 ∈ 𝐴 ↦ 𝐶)) +𝑒 (Σ^‘(𝑘 ∈ {𝐵} ↦ 𝐶))) = ((Σ^‘(𝑘 ∈ 𝐴 ↦ 𝐶)) +𝑒 𝐷)) |
21 | 17, 20 | eqtrd 2778 | 1 ⊢ (𝜑 → (Σ^‘(𝑘 ∈ (𝐴 ∪ {𝐵}) ↦ 𝐶)) = ((Σ^‘(𝑘 ∈ 𝐴 ↦ 𝐶)) +𝑒 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 397 = wceq 1542 Ⅎwnf 1786 ∈ wcel 2107 Vcvv 3444 ∪ cun 3907 ∩ cin 3908 ∅c0 4281 {csn 4585 ↦ cmpt 5187 ‘cfv 6494 (class class class)co 7352 Fincfn 8842 0cc0 11010 +∞cpnf 11145 +𝑒 cxad 12986 [,]cicc 13222 Σ^csumge0 44498 |
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 2709 ax-rep 5241 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7665 ax-inf2 9536 ax-cnex 11066 ax-resscn 11067 ax-1cn 11068 ax-icn 11069 ax-addcl 11070 ax-addrcl 11071 ax-mulcl 11072 ax-mulrcl 11073 ax-mulcom 11074 ax-addass 11075 ax-mulass 11076 ax-distr 11077 ax-i2m1 11078 ax-1ne0 11079 ax-1rid 11080 ax-rnegex 11081 ax-rrecex 11082 ax-cnre 11083 ax-pre-lttri 11084 ax-pre-lttrn 11085 ax-pre-ltadd 11086 ax-pre-mulgt0 11087 ax-pre-sup 11088 |
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 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-pss 3928 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-int 4907 df-iun 4955 df-br 5105 df-opab 5167 df-mpt 5188 df-tr 5222 df-id 5530 df-eprel 5536 df-po 5544 df-so 5545 df-fr 5587 df-se 5588 df-we 5589 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6252 df-ord 6319 df-on 6320 df-lim 6321 df-suc 6322 df-iota 6446 df-fun 6496 df-fn 6497 df-f 6498 df-f1 6499 df-fo 6500 df-f1o 6501 df-fv 6502 df-isom 6503 df-riota 7308 df-ov 7355 df-oprab 7356 df-mpo 7357 df-om 7796 df-1st 7914 df-2nd 7915 df-frecs 8205 df-wrecs 8236 df-recs 8310 df-rdg 8349 df-1o 8405 df-er 8607 df-en 8843 df-dom 8844 df-sdom 8845 df-fin 8846 df-sup 9337 df-oi 9405 df-card 9834 df-pnf 11150 df-mnf 11151 df-xr 11152 df-ltxr 11153 df-le 11154 df-sub 11346 df-neg 11347 df-div 11772 df-nn 12113 df-2 12175 df-3 12176 df-n0 12373 df-z 12459 df-uz 12723 df-rp 12871 df-xadd 12989 df-ico 13225 df-icc 13226 df-fz 13380 df-fzo 13523 df-seq 13862 df-exp 13923 df-hash 14185 df-cj 14944 df-re 14945 df-im 14946 df-sqrt 15080 df-abs 15081 df-clim 15330 df-sum 15531 df-sumge0 44499 |
This theorem is referenced by: hoidmv1lelem2 44728 |
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