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Mirrors > Home > MPE Home > Th. List > fsump1i | Structured version Visualization version GIF version |
Description: Optimized version of fsump1 15742 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 493 | . . . . 5 ⊢ (𝜑 → 𝐾 ∈ 𝑍) |
4 | fsump1i.1 | . . . . 5 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
5 | 3, 4 | eleqtrdi 2839 | . . . 4 ⊢ (𝜑 → 𝐾 ∈ (ℤ≥‘𝑀)) |
6 | peano2uz 12923 | . . . . 5 ⊢ (𝐾 ∈ (ℤ≥‘𝑀) → (𝐾 + 1) ∈ (ℤ≥‘𝑀)) | |
7 | 6, 4 | eleqtrrdi 2840 | . . . 4 ⊢ (𝐾 ∈ (ℤ≥‘𝑀) → (𝐾 + 1) ∈ 𝑍) |
8 | 5, 7 | syl 17 | . . 3 ⊢ (𝜑 → (𝐾 + 1) ∈ 𝑍) |
9 | 1, 8 | eqeltrid 2833 | . 2 ⊢ (𝜑 → 𝑁 ∈ 𝑍) |
10 | 1 | oveq2i 7437 | . . . . 5 ⊢ (𝑀...𝑁) = (𝑀...(𝐾 + 1)) |
11 | 10 | sumeq1i 15684 | . . . 4 ⊢ Σ𝑘 ∈ (𝑀...𝑁)𝐴 = Σ𝑘 ∈ (𝑀...(𝐾 + 1))𝐴 |
12 | elfzuz 13537 | . . . . . . 7 ⊢ (𝑘 ∈ (𝑀...(𝐾 + 1)) → 𝑘 ∈ (ℤ≥‘𝑀)) | |
13 | 12, 4 | eleqtrrdi 2840 | . . . . . 6 ⊢ (𝑘 ∈ (𝑀...(𝐾 + 1)) → 𝑘 ∈ 𝑍) |
14 | fsump1i.4 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ ℂ) | |
15 | 13, 14 | sylan2 591 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...(𝐾 + 1))) → 𝐴 ∈ ℂ) |
16 | 1 | eqeq2i 2741 | . . . . . 6 ⊢ (𝑘 = 𝑁 ↔ 𝑘 = (𝐾 + 1)) |
17 | fsump1i.3 | . . . . . 6 ⊢ (𝑘 = 𝑁 → 𝐴 = 𝐵) | |
18 | 16, 17 | sylbir 234 | . . . . 5 ⊢ (𝑘 = (𝐾 + 1) → 𝐴 = 𝐵) |
19 | 5, 15, 18 | fsump1 15742 | . . . 4 ⊢ (𝜑 → Σ𝑘 ∈ (𝑀...(𝐾 + 1))𝐴 = (Σ𝑘 ∈ (𝑀...𝐾)𝐴 + 𝐵)) |
20 | 11, 19 | eqtrid 2780 | . . 3 ⊢ (𝜑 → Σ𝑘 ∈ (𝑀...𝑁)𝐴 = (Σ𝑘 ∈ (𝑀...𝐾)𝐴 + 𝐵)) |
21 | 2 | simprd 494 | . . . 4 ⊢ (𝜑 → Σ𝑘 ∈ (𝑀...𝐾)𝐴 = 𝑆) |
22 | 21 | oveq1d 7441 | . . 3 ⊢ (𝜑 → (Σ𝑘 ∈ (𝑀...𝐾)𝐴 + 𝐵) = (𝑆 + 𝐵)) |
23 | fsump1i.6 | . . 3 ⊢ (𝜑 → (𝑆 + 𝐵) = 𝑇) | |
24 | 20, 22, 23 | 3eqtrd 2772 | . 2 ⊢ (𝜑 → Σ𝑘 ∈ (𝑀...𝑁)𝐴 = 𝑇) |
25 | 9, 24 | jca 510 | 1 ⊢ (𝜑 → (𝑁 ∈ 𝑍 ∧ Σ𝑘 ∈ (𝑀...𝑁)𝐴 = 𝑇)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ‘cfv 6553 (class class class)co 7426 ℂcc 11144 1c1 11147 + caddc 11149 ℤ≥cuz 12860 ...cfz 13524 Σcsu 15672 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-inf2 9672 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 ax-pre-sup 11224 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-isom 6562 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-1st 7999 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-er 8731 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-sup 9473 df-oi 9541 df-card 9970 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-div 11910 df-nn 12251 df-2 12313 df-3 12314 df-n0 12511 df-z 12597 df-uz 12861 df-rp 13015 df-fz 13525 df-fzo 13668 df-seq 14007 df-exp 14067 df-hash 14330 df-cj 15086 df-re 15087 df-im 15088 df-sqrt 15222 df-abs 15223 df-clim 15472 df-sum 15673 |
This theorem is referenced by: cphipval 25191 itgcnlem 25739 vieta1 26267 ipval2 30537 subfacval2 34830 |
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