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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > sge0isummpt2 | Structured version Visualization version GIF version |
Description: If a series of nonnegative reals is convergent, then it agrees with the generalized sum of nonnegative extended reals. (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
Ref | Expression |
---|---|
sge0isummpt2.kph | ⊢ Ⅎ𝑘𝜑 |
sge0isummpt2.a | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ (0[,)+∞)) |
sge0isummpt2.m | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
sge0isummpt2.z | ⊢ 𝑍 = (ℤ≥‘𝑀) |
sge0isummpt2.b | ⊢ (𝜑 → seq𝑀( + , (𝑘 ∈ 𝑍 ↦ 𝐴)) ⇝ 𝐵) |
Ref | Expression |
---|---|
sge0isummpt2 | ⊢ (𝜑 → (Σ^‘(𝑘 ∈ 𝑍 ↦ 𝐴)) = Σ𝑘 ∈ 𝑍 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sge0isummpt2.z | . . 3 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
2 | sge0isummpt2.m | . . 3 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
3 | simpr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ 𝑍) | |
4 | sge0isummpt2.kph | . . . . . . 7 ⊢ Ⅎ𝑘𝜑 | |
5 | nfv 1911 | . . . . . . 7 ⊢ Ⅎ𝑘 𝑗 ∈ 𝑍 | |
6 | 4, 5 | nfan 1896 | . . . . . 6 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑗 ∈ 𝑍) |
7 | nfcv 2902 | . . . . . . . 8 ⊢ Ⅎ𝑘𝑗 | |
8 | 7 | nfcsb1 3931 | . . . . . . 7 ⊢ Ⅎ𝑘⦋𝑗 / 𝑘⦌𝐴 |
9 | 8 | nfel1 2919 | . . . . . 6 ⊢ Ⅎ𝑘⦋𝑗 / 𝑘⦌𝐴 ∈ (0[,)+∞) |
10 | 6, 9 | nfim 1893 | . . . . 5 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑗 ∈ 𝑍) → ⦋𝑗 / 𝑘⦌𝐴 ∈ (0[,)+∞)) |
11 | eleq1w 2821 | . . . . . . 7 ⊢ (𝑘 = 𝑗 → (𝑘 ∈ 𝑍 ↔ 𝑗 ∈ 𝑍)) | |
12 | 11 | anbi2d 630 | . . . . . 6 ⊢ (𝑘 = 𝑗 → ((𝜑 ∧ 𝑘 ∈ 𝑍) ↔ (𝜑 ∧ 𝑗 ∈ 𝑍))) |
13 | csbeq1a 3921 | . . . . . . 7 ⊢ (𝑘 = 𝑗 → 𝐴 = ⦋𝑗 / 𝑘⦌𝐴) | |
14 | 13 | eleq1d 2823 | . . . . . 6 ⊢ (𝑘 = 𝑗 → (𝐴 ∈ (0[,)+∞) ↔ ⦋𝑗 / 𝑘⦌𝐴 ∈ (0[,)+∞))) |
15 | 12, 14 | imbi12d 344 | . . . . 5 ⊢ (𝑘 = 𝑗 → (((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ (0[,)+∞)) ↔ ((𝜑 ∧ 𝑗 ∈ 𝑍) → ⦋𝑗 / 𝑘⦌𝐴 ∈ (0[,)+∞)))) |
16 | sge0isummpt2.a | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ (0[,)+∞)) | |
17 | 10, 15, 16 | chvarfv 2237 | . . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → ⦋𝑗 / 𝑘⦌𝐴 ∈ (0[,)+∞)) |
18 | nfcv 2902 | . . . . . . 7 ⊢ Ⅎ𝑖𝐴 | |
19 | nfcsb1v 3932 | . . . . . . 7 ⊢ Ⅎ𝑘⦋𝑖 / 𝑘⦌𝐴 | |
20 | csbeq1a 3921 | . . . . . . 7 ⊢ (𝑘 = 𝑖 → 𝐴 = ⦋𝑖 / 𝑘⦌𝐴) | |
21 | 18, 19, 20 | cbvmpt 5258 | . . . . . 6 ⊢ (𝑘 ∈ 𝑍 ↦ 𝐴) = (𝑖 ∈ 𝑍 ↦ ⦋𝑖 / 𝑘⦌𝐴) |
22 | 21 | eqcomi 2743 | . . . . 5 ⊢ (𝑖 ∈ 𝑍 ↦ ⦋𝑖 / 𝑘⦌𝐴) = (𝑘 ∈ 𝑍 ↦ 𝐴) |
23 | 7, 8, 13, 22 | fvmptf 7036 | . . . 4 ⊢ ((𝑗 ∈ 𝑍 ∧ ⦋𝑗 / 𝑘⦌𝐴 ∈ (0[,)+∞)) → ((𝑖 ∈ 𝑍 ↦ ⦋𝑖 / 𝑘⦌𝐴)‘𝑗) = ⦋𝑗 / 𝑘⦌𝐴) |
24 | 3, 17, 23 | syl2anc 584 | . . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → ((𝑖 ∈ 𝑍 ↦ ⦋𝑖 / 𝑘⦌𝐴)‘𝑗) = ⦋𝑗 / 𝑘⦌𝐴) |
25 | rge0ssre 13492 | . . . . 5 ⊢ (0[,)+∞) ⊆ ℝ | |
26 | ax-resscn 11209 | . . . . 5 ⊢ ℝ ⊆ ℂ | |
27 | 25, 26 | sstri 4004 | . . . 4 ⊢ (0[,)+∞) ⊆ ℂ |
28 | 27, 17 | sselid 3992 | . . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → ⦋𝑗 / 𝑘⦌𝐴 ∈ ℂ) |
29 | sge0isummpt2.b | . . . 4 ⊢ (𝜑 → seq𝑀( + , (𝑘 ∈ 𝑍 ↦ 𝐴)) ⇝ 𝐵) | |
30 | 21 | a1i 11 | . . . . . 6 ⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ 𝐴) = (𝑖 ∈ 𝑍 ↦ ⦋𝑖 / 𝑘⦌𝐴)) |
31 | 30 | seqeq3d 14046 | . . . . 5 ⊢ (𝜑 → seq𝑀( + , (𝑘 ∈ 𝑍 ↦ 𝐴)) = seq𝑀( + , (𝑖 ∈ 𝑍 ↦ ⦋𝑖 / 𝑘⦌𝐴))) |
32 | 31 | breq1d 5157 | . . . 4 ⊢ (𝜑 → (seq𝑀( + , (𝑘 ∈ 𝑍 ↦ 𝐴)) ⇝ 𝐵 ↔ seq𝑀( + , (𝑖 ∈ 𝑍 ↦ ⦋𝑖 / 𝑘⦌𝐴)) ⇝ 𝐵)) |
33 | 29, 32 | mpbid 232 | . . 3 ⊢ (𝜑 → seq𝑀( + , (𝑖 ∈ 𝑍 ↦ ⦋𝑖 / 𝑘⦌𝐴)) ⇝ 𝐵) |
34 | 1, 2, 24, 28, 33 | isumclim 15789 | . 2 ⊢ (𝜑 → Σ𝑗 ∈ 𝑍 ⦋𝑗 / 𝑘⦌𝐴 = 𝐵) |
35 | nfcv 2902 | . . . 4 ⊢ Ⅎ𝑗𝐴 | |
36 | 13, 35, 8 | cbvsum 15727 | . . 3 ⊢ Σ𝑘 ∈ 𝑍 𝐴 = Σ𝑗 ∈ 𝑍 ⦋𝑗 / 𝑘⦌𝐴 |
37 | 36 | a1i 11 | . 2 ⊢ (𝜑 → Σ𝑘 ∈ 𝑍 𝐴 = Σ𝑗 ∈ 𝑍 ⦋𝑗 / 𝑘⦌𝐴) |
38 | 4, 16, 2, 1, 29 | sge0isummpt 46385 | . 2 ⊢ (𝜑 → (Σ^‘(𝑘 ∈ 𝑍 ↦ 𝐴)) = 𝐵) |
39 | 34, 37, 38 | 3eqtr4rd 2785 | 1 ⊢ (𝜑 → (Σ^‘(𝑘 ∈ 𝑍 ↦ 𝐴)) = Σ𝑘 ∈ 𝑍 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1536 Ⅎwnf 1779 ∈ wcel 2105 ⦋csb 3907 class class class wbr 5147 ↦ cmpt 5230 ‘cfv 6562 (class class class)co 7430 ℂcc 11150 ℝcr 11151 0cc0 11152 + caddc 11155 +∞cpnf 11289 ℤcz 12610 ℤ≥cuz 12875 [,)cico 13385 seqcseq 14038 ⇝ cli 15516 Σcsu 15718 Σ^csumge0 46317 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-inf2 9678 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 ax-pre-sup 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-se 5641 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-isom 6571 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-1st 8012 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-er 8743 df-pm 8867 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-sup 9479 df-inf 9480 df-oi 9547 df-card 9976 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-div 11918 df-nn 12264 df-2 12326 df-3 12327 df-n0 12524 df-z 12611 df-uz 12876 df-rp 13032 df-ico 13389 df-icc 13390 df-fz 13544 df-fzo 13691 df-fl 13828 df-seq 14039 df-exp 14099 df-hash 14366 df-cj 15134 df-re 15135 df-im 15136 df-sqrt 15270 df-abs 15271 df-clim 15520 df-rlim 15521 df-sum 15719 df-sumge0 46318 |
This theorem is referenced by: (None) |
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