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Mirrors > Home > MPE Home > Th. List > Mathboxes > fzosumm1 | Structured version Visualization version GIF version |
Description: Separate out the last term in a finite sum. (Contributed by Steven Nguyen, 22-Aug-2023.) |
Ref | Expression |
---|---|
fzosumm1.1 | ⊢ (𝜑 → (𝑁 − 1) ∈ (ℤ≥‘𝑀)) |
fzosumm1.2 | ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝐴 ∈ ℂ) |
fzosumm1.3 | ⊢ (𝑘 = (𝑁 − 1) → 𝐴 = 𝐵) |
fzosumm1.n | ⊢ (𝜑 → 𝑁 ∈ ℤ) |
Ref | Expression |
---|---|
fzosumm1 | ⊢ (𝜑 → Σ𝑘 ∈ (𝑀..^𝑁)𝐴 = (Σ𝑘 ∈ (𝑀..^(𝑁 − 1))𝐴 + 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fzosumm1.1 | . . 3 ⊢ (𝜑 → (𝑁 − 1) ∈ (ℤ≥‘𝑀)) | |
2 | fzosumm1.n | . . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℤ) | |
3 | fzoval 13036 | . . . . . . . 8 ⊢ (𝑁 ∈ ℤ → (𝑀..^𝑁) = (𝑀...(𝑁 − 1))) | |
4 | 2, 3 | syl 17 | . . . . . . 7 ⊢ (𝜑 → (𝑀..^𝑁) = (𝑀...(𝑁 − 1))) |
5 | 4 | eqcomd 2826 | . . . . . 6 ⊢ (𝜑 → (𝑀...(𝑁 − 1)) = (𝑀..^𝑁)) |
6 | 5 | eleq2d 2897 | . . . . 5 ⊢ (𝜑 → (𝑘 ∈ (𝑀...(𝑁 − 1)) ↔ 𝑘 ∈ (𝑀..^𝑁))) |
7 | 6 | biimpa 479 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...(𝑁 − 1))) → 𝑘 ∈ (𝑀..^𝑁)) |
8 | fzosumm1.2 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝐴 ∈ ℂ) | |
9 | 7, 8 | syldan 593 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...(𝑁 − 1))) → 𝐴 ∈ ℂ) |
10 | fzosumm1.3 | . . 3 ⊢ (𝑘 = (𝑁 − 1) → 𝐴 = 𝐵) | |
11 | 1, 9, 10 | fsumm1 15101 | . 2 ⊢ (𝜑 → Σ𝑘 ∈ (𝑀...(𝑁 − 1))𝐴 = (Σ𝑘 ∈ (𝑀...((𝑁 − 1) − 1))𝐴 + 𝐵)) |
12 | 4 | sumeq1d 15053 | . 2 ⊢ (𝜑 → Σ𝑘 ∈ (𝑀..^𝑁)𝐴 = Σ𝑘 ∈ (𝑀...(𝑁 − 1))𝐴) |
13 | eluzelz 12247 | . . . . 5 ⊢ ((𝑁 − 1) ∈ (ℤ≥‘𝑀) → (𝑁 − 1) ∈ ℤ) | |
14 | fzoval 13036 | . . . . 5 ⊢ ((𝑁 − 1) ∈ ℤ → (𝑀..^(𝑁 − 1)) = (𝑀...((𝑁 − 1) − 1))) | |
15 | 1, 13, 14 | 3syl 18 | . . . 4 ⊢ (𝜑 → (𝑀..^(𝑁 − 1)) = (𝑀...((𝑁 − 1) − 1))) |
16 | 15 | sumeq1d 15053 | . . 3 ⊢ (𝜑 → Σ𝑘 ∈ (𝑀..^(𝑁 − 1))𝐴 = Σ𝑘 ∈ (𝑀...((𝑁 − 1) − 1))𝐴) |
17 | 16 | oveq1d 7164 | . 2 ⊢ (𝜑 → (Σ𝑘 ∈ (𝑀..^(𝑁 − 1))𝐴 + 𝐵) = (Σ𝑘 ∈ (𝑀...((𝑁 − 1) − 1))𝐴 + 𝐵)) |
18 | 11, 12, 17 | 3eqtr4d 2865 | 1 ⊢ (𝜑 → Σ𝑘 ∈ (𝑀..^𝑁)𝐴 = (Σ𝑘 ∈ (𝑀..^(𝑁 − 1))𝐴 + 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ‘cfv 6348 (class class class)co 7149 ℂcc 10528 1c1 10531 + caddc 10533 − cmin 10863 ℤcz 11975 ℤ≥cuz 12237 ...cfz 12889 ..^cfzo 13030 Σcsu 15037 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 ax-inf2 9097 ax-cnex 10586 ax-resscn 10587 ax-1cn 10588 ax-icn 10589 ax-addcl 10590 ax-addrcl 10591 ax-mulcl 10592 ax-mulrcl 10593 ax-mulcom 10594 ax-addass 10595 ax-mulass 10596 ax-distr 10597 ax-i2m1 10598 ax-1ne0 10599 ax-1rid 10600 ax-rnegex 10601 ax-rrecex 10602 ax-cnre 10603 ax-pre-lttri 10604 ax-pre-lttrn 10605 ax-pre-ltadd 10606 ax-pre-mulgt0 10607 ax-pre-sup 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-fal 1549 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-nel 3123 df-ral 3142 df-rex 3143 df-reu 3144 df-rmo 3145 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-pss 3947 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4870 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7107 df-ov 7152 df-oprab 7153 df-mpo 7154 df-om 7574 df-1st 7682 df-2nd 7683 df-wrecs 7940 df-recs 8001 df-rdg 8039 df-1o 8095 df-oadd 8099 df-er 8282 df-en 8503 df-dom 8504 df-sdom 8505 df-fin 8506 df-sup 8899 df-oi 8967 df-card 9361 df-pnf 10670 df-mnf 10671 df-xr 10672 df-ltxr 10673 df-le 10674 df-sub 10865 df-neg 10866 df-div 11291 df-nn 11632 df-2 11694 df-3 11695 df-n0 11892 df-z 11976 df-uz 12238 df-rp 12384 df-fz 12890 df-fzo 13031 df-seq 13367 df-exp 13427 df-hash 13688 df-cj 14453 df-re 14454 df-im 14455 df-sqrt 14589 df-abs 14590 df-clim 14840 df-sum 15038 |
This theorem is referenced by: fltnltalem 39350 |
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