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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 486 | . . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ 𝑍) | |
4 | sge0isummpt2.kph | . . . . . . 7 ⊢ Ⅎ𝑘𝜑 | |
5 | nfv 1918 | . . . . . . 7 ⊢ Ⅎ𝑘 𝑗 ∈ 𝑍 | |
6 | 4, 5 | nfan 1903 | . . . . . 6 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑗 ∈ 𝑍) |
7 | nfcv 2904 | . . . . . . . 8 ⊢ Ⅎ𝑘𝑗 | |
8 | 7 | nfcsb1 3880 | . . . . . . 7 ⊢ Ⅎ𝑘⦋𝑗 / 𝑘⦌𝐴 |
9 | 8 | nfel1 2920 | . . . . . 6 ⊢ Ⅎ𝑘⦋𝑗 / 𝑘⦌𝐴 ∈ (0[,)+∞) |
10 | 6, 9 | nfim 1900 | . . . . 5 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑗 ∈ 𝑍) → ⦋𝑗 / 𝑘⦌𝐴 ∈ (0[,)+∞)) |
11 | eleq1w 2817 | . . . . . . 7 ⊢ (𝑘 = 𝑗 → (𝑘 ∈ 𝑍 ↔ 𝑗 ∈ 𝑍)) | |
12 | 11 | anbi2d 630 | . . . . . 6 ⊢ (𝑘 = 𝑗 → ((𝜑 ∧ 𝑘 ∈ 𝑍) ↔ (𝜑 ∧ 𝑗 ∈ 𝑍))) |
13 | csbeq1a 3870 | . . . . . . 7 ⊢ (𝑘 = 𝑗 → 𝐴 = ⦋𝑗 / 𝑘⦌𝐴) | |
14 | 13 | eleq1d 2819 | . . . . . 6 ⊢ (𝑘 = 𝑗 → (𝐴 ∈ (0[,)+∞) ↔ ⦋𝑗 / 𝑘⦌𝐴 ∈ (0[,)+∞))) |
15 | 12, 14 | imbi12d 345 | . . . . 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 2234 | . . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → ⦋𝑗 / 𝑘⦌𝐴 ∈ (0[,)+∞)) |
18 | nfcv 2904 | . . . . . . 7 ⊢ Ⅎ𝑖𝐴 | |
19 | nfcsb1v 3881 | . . . . . . 7 ⊢ Ⅎ𝑘⦋𝑖 / 𝑘⦌𝐴 | |
20 | csbeq1a 3870 | . . . . . . 7 ⊢ (𝑘 = 𝑖 → 𝐴 = ⦋𝑖 / 𝑘⦌𝐴) | |
21 | 18, 19, 20 | cbvmpt 5217 | . . . . . 6 ⊢ (𝑘 ∈ 𝑍 ↦ 𝐴) = (𝑖 ∈ 𝑍 ↦ ⦋𝑖 / 𝑘⦌𝐴) |
22 | 21 | eqcomi 2742 | . . . . 5 ⊢ (𝑖 ∈ 𝑍 ↦ ⦋𝑖 / 𝑘⦌𝐴) = (𝑘 ∈ 𝑍 ↦ 𝐴) |
23 | 7, 8, 13, 22 | fvmptf 6970 | . . . 4 ⊢ ((𝑗 ∈ 𝑍 ∧ ⦋𝑗 / 𝑘⦌𝐴 ∈ (0[,)+∞)) → ((𝑖 ∈ 𝑍 ↦ ⦋𝑖 / 𝑘⦌𝐴)‘𝑗) = ⦋𝑗 / 𝑘⦌𝐴) |
24 | 3, 17, 23 | syl2anc 585 | . . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → ((𝑖 ∈ 𝑍 ↦ ⦋𝑖 / 𝑘⦌𝐴)‘𝑗) = ⦋𝑗 / 𝑘⦌𝐴) |
25 | rge0ssre 13379 | . . . . 5 ⊢ (0[,)+∞) ⊆ ℝ | |
26 | ax-resscn 11113 | . . . . 5 ⊢ ℝ ⊆ ℂ | |
27 | 25, 26 | sstri 3954 | . . . 4 ⊢ (0[,)+∞) ⊆ ℂ |
28 | 27, 17 | sselid 3943 | . . 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 13920 | . . . . 5 ⊢ (𝜑 → seq𝑀( + , (𝑘 ∈ 𝑍 ↦ 𝐴)) = seq𝑀( + , (𝑖 ∈ 𝑍 ↦ ⦋𝑖 / 𝑘⦌𝐴))) |
32 | 31 | breq1d 5116 | . . . 4 ⊢ (𝜑 → (seq𝑀( + , (𝑘 ∈ 𝑍 ↦ 𝐴)) ⇝ 𝐵 ↔ seq𝑀( + , (𝑖 ∈ 𝑍 ↦ ⦋𝑖 / 𝑘⦌𝐴)) ⇝ 𝐵)) |
33 | 29, 32 | mpbid 231 | . . 3 ⊢ (𝜑 → seq𝑀( + , (𝑖 ∈ 𝑍 ↦ ⦋𝑖 / 𝑘⦌𝐴)) ⇝ 𝐵) |
34 | 1, 2, 24, 28, 33 | isumclim 15647 | . 2 ⊢ (𝜑 → Σ𝑗 ∈ 𝑍 ⦋𝑗 / 𝑘⦌𝐴 = 𝐵) |
35 | nfcv 2904 | . . . 4 ⊢ Ⅎ𝑗𝑍 | |
36 | nfcv 2904 | . . . 4 ⊢ Ⅎ𝑘𝑍 | |
37 | nfcv 2904 | . . . 4 ⊢ Ⅎ𝑗𝐴 | |
38 | 13, 35, 36, 37, 8 | cbvsum 15585 | . . 3 ⊢ Σ𝑘 ∈ 𝑍 𝐴 = Σ𝑗 ∈ 𝑍 ⦋𝑗 / 𝑘⦌𝐴 |
39 | 38 | a1i 11 | . 2 ⊢ (𝜑 → Σ𝑘 ∈ 𝑍 𝐴 = Σ𝑗 ∈ 𝑍 ⦋𝑗 / 𝑘⦌𝐴) |
40 | 4, 16, 2, 1, 29 | sge0isummpt 44757 | . 2 ⊢ (𝜑 → (Σ^‘(𝑘 ∈ 𝑍 ↦ 𝐴)) = 𝐵) |
41 | 34, 39, 40 | 3eqtr4rd 2784 | 1 ⊢ (𝜑 → (Σ^‘(𝑘 ∈ 𝑍 ↦ 𝐴)) = Σ𝑘 ∈ 𝑍 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 Ⅎwnf 1786 ∈ wcel 2107 ⦋csb 3856 class class class wbr 5106 ↦ cmpt 5189 ‘cfv 6497 (class class class)co 7358 ℂcc 11054 ℝcr 11055 0cc0 11056 + caddc 11059 +∞cpnf 11191 ℤcz 12504 ℤ≥cuz 12768 [,)cico 13272 seqcseq 13912 ⇝ cli 15372 Σcsu 15576 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-pm 8771 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-sup 9383 df-inf 9384 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-fl 13703 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-rlim 15377 df-sum 15577 df-sumge0 44690 |
This theorem is referenced by: (None) |
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