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Mirrors > Home > ILE Home > Th. List > isumclim | GIF version |
Description: An infinite sum equals the value its series converges to. (Contributed by NM, 25-Dec-2005.) (Revised by Mario Carneiro, 23-Apr-2014.) |
Ref | Expression |
---|---|
isumclim.1 | ⊢ 𝑍 = (ℤ≥‘𝑀) |
isumclim.2 | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
isumclim.3 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐴) |
isumclim.4 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ ℂ) |
isumclim.6 | ⊢ (𝜑 → seq𝑀( + , 𝐹) ⇝ 𝐵) |
Ref | Expression |
---|---|
isumclim | ⊢ (𝜑 → Σ𝑘 ∈ 𝑍 𝐴 = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isumclim.1 | . . 3 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
2 | isumclim.2 | . . 3 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
3 | isumclim.3 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐴) | |
4 | isumclim.4 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ ℂ) | |
5 | 1, 2, 3, 4 | isum 11502 | . 2 ⊢ (𝜑 → Σ𝑘 ∈ 𝑍 𝐴 = ( ⇝ ‘seq𝑀( + , 𝐹))) |
6 | fclim 11411 | . . . 4 ⊢ ⇝ :dom ⇝ ⟶ℂ | |
7 | ffun 5394 | . . . 4 ⊢ ( ⇝ :dom ⇝ ⟶ℂ → Fun ⇝ ) | |
8 | 6, 7 | ax-mp 5 | . . 3 ⊢ Fun ⇝ |
9 | isumclim.6 | . . 3 ⊢ (𝜑 → seq𝑀( + , 𝐹) ⇝ 𝐵) | |
10 | funbrfv 5583 | . . 3 ⊢ (Fun ⇝ → (seq𝑀( + , 𝐹) ⇝ 𝐵 → ( ⇝ ‘seq𝑀( + , 𝐹)) = 𝐵)) | |
11 | 8, 9, 10 | mpsyl 65 | . 2 ⊢ (𝜑 → ( ⇝ ‘seq𝑀( + , 𝐹)) = 𝐵) |
12 | 5, 11 | eqtrd 2222 | 1 ⊢ (𝜑 → Σ𝑘 ∈ 𝑍 𝐴 = 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1364 ∈ wcel 2160 class class class wbr 4025 dom cdm 4651 Fun wfun 5236 ⟶wf 5238 ‘cfv 5242 ℂcc 7856 + caddc 7861 ℤcz 9303 ℤ≥cuz 9578 seqcseq 10504 ⇝ cli 11395 Σcsu 11470 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-coll 4140 ax-sep 4143 ax-nul 4151 ax-pow 4199 ax-pr 4234 ax-un 4458 ax-setind 4561 ax-iinf 4612 ax-cnex 7949 ax-resscn 7950 ax-1cn 7951 ax-1re 7952 ax-icn 7953 ax-addcl 7954 ax-addrcl 7955 ax-mulcl 7956 ax-mulrcl 7957 ax-addcom 7958 ax-mulcom 7959 ax-addass 7960 ax-mulass 7961 ax-distr 7962 ax-i2m1 7963 ax-0lt1 7964 ax-1rid 7965 ax-0id 7966 ax-rnegex 7967 ax-precex 7968 ax-cnre 7969 ax-pre-ltirr 7970 ax-pre-ltwlin 7971 ax-pre-lttrn 7972 ax-pre-apti 7973 ax-pre-ltadd 7974 ax-pre-mulgt0 7975 ax-pre-mulext 7976 ax-arch 7977 ax-caucvg 7978 |
This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rmo 2476 df-rab 2477 df-v 2758 df-sbc 2982 df-csb 3077 df-dif 3151 df-un 3153 df-in 3155 df-ss 3162 df-nul 3443 df-if 3554 df-pw 3599 df-sn 3620 df-pr 3621 df-op 3623 df-uni 3832 df-int 3867 df-iun 3910 df-br 4026 df-opab 4087 df-mpt 4088 df-tr 4124 df-id 4318 df-po 4321 df-iso 4322 df-iord 4391 df-on 4393 df-ilim 4394 df-suc 4396 df-iom 4615 df-xp 4657 df-rel 4658 df-cnv 4659 df-co 4660 df-dm 4661 df-rn 4662 df-res 4663 df-ima 4664 df-iota 5203 df-fun 5244 df-fn 5245 df-f 5246 df-f1 5247 df-fo 5248 df-f1o 5249 df-fv 5250 df-isom 5251 df-riota 5861 df-ov 5909 df-oprab 5910 df-mpo 5911 df-1st 6180 df-2nd 6181 df-recs 6345 df-irdg 6410 df-frec 6431 df-1o 6456 df-oadd 6460 df-er 6574 df-en 6782 df-dom 6783 df-fin 6784 df-pnf 8042 df-mnf 8043 df-xr 8044 df-ltxr 8045 df-le 8046 df-sub 8178 df-neg 8179 df-reap 8580 df-ap 8587 df-div 8678 df-inn 8969 df-2 9027 df-3 9028 df-4 9029 df-n0 9227 df-z 9304 df-uz 9579 df-q 9671 df-rp 9706 df-fz 10061 df-fzo 10195 df-seqfrec 10505 df-exp 10584 df-ihash 10821 df-cj 10960 df-re 10961 df-im 10962 df-rsqrt 11116 df-abs 11117 df-clim 11396 df-sumdc 11471 |
This theorem is referenced by: isummulc2 11543 isumadd 11548 isumsplit 11608 trirecip 11618 geolim2 11629 geoisum 11634 geoisumr 11635 geoisum1 11636 eftlub 11807 eflegeo 11818 |
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