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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 9082 | . . . . 5 ⊢ {𝐵} ∈ Fin | |
4 | 3 | a1i 11 | . . . 4 ⊢ (𝜑 → {𝐵} ∈ Fin) |
5 | 4 | elexd 3502 | . . 3 ⊢ (𝜑 → {𝐵} ∈ V) |
6 | sge0splitsn.n | . . . 4 ⊢ (𝜑 → ¬ 𝐵 ∈ 𝐴) | |
7 | disjsn 4716 | . . . 4 ⊢ ((𝐴 ∩ {𝐵}) = ∅ ↔ ¬ 𝐵 ∈ 𝐴) | |
8 | 6, 7 | sylibr 234 | . . 3 ⊢ (𝜑 → (𝐴 ∩ {𝐵}) = ∅) |
9 | sge0splitsn.c | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ (0[,]+∞)) | |
10 | elsni 4648 | . . . . 5 ⊢ (𝑘 ∈ {𝐵} → 𝑘 = 𝐵) | |
11 | sge0splitsn.d | . . . . . 6 ⊢ (𝑘 = 𝐵 → 𝐶 = 𝐷) | |
12 | 11 | adantl 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 = 𝐵) → 𝐶 = 𝐷) |
13 | 10, 12 | sylan2 593 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝐵}) → 𝐶 = 𝐷) |
14 | sge0splitsn.e | . . . . 5 ⊢ (𝜑 → 𝐷 ∈ (0[,]+∞)) | |
15 | 14 | adantr 480 | . . . 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 46367 | . 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 46393 | . . 3 ⊢ (𝜑 → (Σ^‘(𝑘 ∈ {𝐵} ↦ 𝐶)) = 𝐷) |
20 | 19 | oveq2d 7447 | . 2 ⊢ (𝜑 → ((Σ^‘(𝑘 ∈ 𝐴 ↦ 𝐶)) +𝑒 (Σ^‘(𝑘 ∈ {𝐵} ↦ 𝐶))) = ((Σ^‘(𝑘 ∈ 𝐴 ↦ 𝐶)) +𝑒 𝐷)) |
21 | 17, 20 | eqtrd 2775 | 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 395 = wceq 1537 Ⅎwnf 1780 ∈ wcel 2106 Vcvv 3478 ∪ cun 3961 ∩ cin 3962 ∅c0 4339 {csn 4631 ↦ cmpt 5231 ‘cfv 6563 (class class class)co 7431 Fincfn 8984 0cc0 11153 +∞cpnf 11290 +𝑒 cxad 13150 [,]cicc 13387 Σ^csumge0 46318 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-inf2 9679 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-sup 9480 df-oi 9548 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-n0 12525 df-z 12612 df-uz 12877 df-rp 13033 df-xadd 13153 df-ico 13390 df-icc 13391 df-fz 13545 df-fzo 13692 df-seq 14040 df-exp 14100 df-hash 14367 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-clim 15521 df-sum 15720 df-sumge0 46319 |
This theorem is referenced by: hoidmv1lelem2 46548 |
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