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Mirrors > Home > MPE Home > Th. List > fsump1i | Structured version Visualization version GIF version |
Description: Optimized version of fsump1 15101 for making sums of a concrete number of terms. (Contributed by Mario Carneiro, 23-Apr-2014.) |
Ref | Expression |
---|---|
fsump1i.1 | ⊢ 𝑍 = (ℤ≥‘𝑀) |
fsump1i.2 | ⊢ 𝑁 = (𝐾 + 1) |
fsump1i.3 | ⊢ (𝑘 = 𝑁 → 𝐴 = 𝐵) |
fsump1i.4 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ ℂ) |
fsump1i.5 | ⊢ (𝜑 → (𝐾 ∈ 𝑍 ∧ Σ𝑘 ∈ (𝑀...𝐾)𝐴 = 𝑆)) |
fsump1i.6 | ⊢ (𝜑 → (𝑆 + 𝐵) = 𝑇) |
Ref | Expression |
---|---|
fsump1i | ⊢ (𝜑 → (𝑁 ∈ 𝑍 ∧ Σ𝑘 ∈ (𝑀...𝑁)𝐴 = 𝑇)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fsump1i.2 | . . 3 ⊢ 𝑁 = (𝐾 + 1) | |
2 | fsump1i.5 | . . . . . 6 ⊢ (𝜑 → (𝐾 ∈ 𝑍 ∧ Σ𝑘 ∈ (𝑀...𝐾)𝐴 = 𝑆)) | |
3 | 2 | simpld 495 | . . . . 5 ⊢ (𝜑 → 𝐾 ∈ 𝑍) |
4 | fsump1i.1 | . . . . 5 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
5 | 3, 4 | eleqtrdi 2923 | . . . 4 ⊢ (𝜑 → 𝐾 ∈ (ℤ≥‘𝑀)) |
6 | peano2uz 12290 | . . . . 5 ⊢ (𝐾 ∈ (ℤ≥‘𝑀) → (𝐾 + 1) ∈ (ℤ≥‘𝑀)) | |
7 | 6, 4 | eleqtrrdi 2924 | . . . 4 ⊢ (𝐾 ∈ (ℤ≥‘𝑀) → (𝐾 + 1) ∈ 𝑍) |
8 | 5, 7 | syl 17 | . . 3 ⊢ (𝜑 → (𝐾 + 1) ∈ 𝑍) |
9 | 1, 8 | eqeltrid 2917 | . 2 ⊢ (𝜑 → 𝑁 ∈ 𝑍) |
10 | 1 | oveq2i 7156 | . . . . 5 ⊢ (𝑀...𝑁) = (𝑀...(𝐾 + 1)) |
11 | 10 | sumeq1i 15045 | . . . 4 ⊢ Σ𝑘 ∈ (𝑀...𝑁)𝐴 = Σ𝑘 ∈ (𝑀...(𝐾 + 1))𝐴 |
12 | elfzuz 12894 | . . . . . . 7 ⊢ (𝑘 ∈ (𝑀...(𝐾 + 1)) → 𝑘 ∈ (ℤ≥‘𝑀)) | |
13 | 12, 4 | eleqtrrdi 2924 | . . . . . 6 ⊢ (𝑘 ∈ (𝑀...(𝐾 + 1)) → 𝑘 ∈ 𝑍) |
14 | fsump1i.4 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ ℂ) | |
15 | 13, 14 | sylan2 592 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...(𝐾 + 1))) → 𝐴 ∈ ℂ) |
16 | 1 | eqeq2i 2834 | . . . . . 6 ⊢ (𝑘 = 𝑁 ↔ 𝑘 = (𝐾 + 1)) |
17 | fsump1i.3 | . . . . . 6 ⊢ (𝑘 = 𝑁 → 𝐴 = 𝐵) | |
18 | 16, 17 | sylbir 236 | . . . . 5 ⊢ (𝑘 = (𝐾 + 1) → 𝐴 = 𝐵) |
19 | 5, 15, 18 | fsump1 15101 | . . . 4 ⊢ (𝜑 → Σ𝑘 ∈ (𝑀...(𝐾 + 1))𝐴 = (Σ𝑘 ∈ (𝑀...𝐾)𝐴 + 𝐵)) |
20 | 11, 19 | syl5eq 2868 | . . 3 ⊢ (𝜑 → Σ𝑘 ∈ (𝑀...𝑁)𝐴 = (Σ𝑘 ∈ (𝑀...𝐾)𝐴 + 𝐵)) |
21 | 2 | simprd 496 | . . . 4 ⊢ (𝜑 → Σ𝑘 ∈ (𝑀...𝐾)𝐴 = 𝑆) |
22 | 21 | oveq1d 7160 | . . 3 ⊢ (𝜑 → (Σ𝑘 ∈ (𝑀...𝐾)𝐴 + 𝐵) = (𝑆 + 𝐵)) |
23 | fsump1i.6 | . . 3 ⊢ (𝜑 → (𝑆 + 𝐵) = 𝑇) | |
24 | 20, 22, 23 | 3eqtrd 2860 | . 2 ⊢ (𝜑 → Σ𝑘 ∈ (𝑀...𝑁)𝐴 = 𝑇) |
25 | 9, 24 | jca 512 | 1 ⊢ (𝜑 → (𝑁 ∈ 𝑍 ∧ Σ𝑘 ∈ (𝑀...𝑁)𝐴 = 𝑇)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ‘cfv 6349 (class class class)co 7145 ℂcc 10524 1c1 10527 + caddc 10529 ℤ≥cuz 12232 ...cfz 12882 Σcsu 15032 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7450 ax-inf2 9093 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-fal 1541 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4833 df-int 4870 df-iun 4914 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-se 5509 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-isom 6358 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7569 df-1st 7680 df-2nd 7681 df-wrecs 7938 df-recs 7999 df-rdg 8037 df-1o 8093 df-oadd 8097 df-er 8279 df-en 8499 df-dom 8500 df-sdom 8501 df-fin 8502 df-sup 8895 df-oi 8963 df-card 9357 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11628 df-2 11689 df-3 11690 df-n0 11887 df-z 11971 df-uz 12233 df-rp 12380 df-fz 12883 df-fzo 13024 df-seq 13360 df-exp 13420 df-hash 13681 df-cj 14448 df-re 14449 df-im 14450 df-sqrt 14584 df-abs 14585 df-clim 14835 df-sum 15033 |
This theorem is referenced by: cphipval 23775 itgcnlem 24319 vieta1 24830 ipval2 28412 subfacval2 32332 |
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