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Mirrors > Home > MPE Home > Th. List > Mathboxes > fz1sump1 | Structured version Visualization version GIF version |
Description: Add one more term to a sum. Special case of fsump1 15709 generalized to 𝑁 ∈ ℕ0. (Contributed by SN, 22-Mar-2025.) |
Ref | Expression |
---|---|
fz1sump1.n | ⊢ (𝜑 → 𝑁 ∈ ℕ0) |
fz1sump1.a | ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝑁 + 1))) → 𝐴 ∈ ℂ) |
fz1sump1.s | ⊢ (𝑘 = (𝑁 + 1) → 𝐴 = 𝐵) |
Ref | Expression |
---|---|
fz1sump1 | ⊢ (𝜑 → Σ𝑘 ∈ (1...(𝑁 + 1))𝐴 = (Σ𝑘 ∈ (1...𝑁)𝐴 + 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fz1sump1.n | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℕ0) | |
2 | nn0p1nn 12518 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ) | |
3 | 1, 2 | syl 17 | . . . 4 ⊢ (𝜑 → (𝑁 + 1) ∈ ℕ) |
4 | nnuz 12872 | . . . 4 ⊢ ℕ = (ℤ≥‘1) | |
5 | 3, 4 | eleqtrdi 2842 | . . 3 ⊢ (𝜑 → (𝑁 + 1) ∈ (ℤ≥‘1)) |
6 | fz1sump1.a | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝑁 + 1))) → 𝐴 ∈ ℂ) | |
7 | fz1sump1.s | . . 3 ⊢ (𝑘 = (𝑁 + 1) → 𝐴 = 𝐵) | |
8 | 5, 6, 7 | fsumm1 15704 | . 2 ⊢ (𝜑 → Σ𝑘 ∈ (1...(𝑁 + 1))𝐴 = (Σ𝑘 ∈ (1...((𝑁 + 1) − 1))𝐴 + 𝐵)) |
9 | 1 | nn0cnd 12541 | . . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℂ) |
10 | 1cnd 11216 | . . . . . 6 ⊢ (𝜑 → 1 ∈ ℂ) | |
11 | 9, 10 | pncand 11579 | . . . . 5 ⊢ (𝜑 → ((𝑁 + 1) − 1) = 𝑁) |
12 | 11 | oveq2d 7428 | . . . 4 ⊢ (𝜑 → (1...((𝑁 + 1) − 1)) = (1...𝑁)) |
13 | 12 | sumeq1d 15654 | . . 3 ⊢ (𝜑 → Σ𝑘 ∈ (1...((𝑁 + 1) − 1))𝐴 = Σ𝑘 ∈ (1...𝑁)𝐴) |
14 | 13 | oveq1d 7427 | . 2 ⊢ (𝜑 → (Σ𝑘 ∈ (1...((𝑁 + 1) − 1))𝐴 + 𝐵) = (Σ𝑘 ∈ (1...𝑁)𝐴 + 𝐵)) |
15 | 8, 14 | eqtrd 2771 | 1 ⊢ (𝜑 → Σ𝑘 ∈ (1...(𝑁 + 1))𝐴 = (Σ𝑘 ∈ (1...𝑁)𝐴 + 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2105 ‘cfv 6543 (class class class)co 7412 ℂcc 11114 1c1 11117 + caddc 11119 − cmin 11451 ℕcn 12219 ℕ0cn0 12479 ℤ≥cuz 12829 ...cfz 13491 Σcsu 15639 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-inf2 9642 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 ax-pre-sup 11194 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-1st 7979 df-2nd 7980 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-1o 8472 df-er 8709 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-sup 9443 df-oi 9511 df-card 9940 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-div 11879 df-nn 12220 df-2 12282 df-3 12283 df-n0 12480 df-z 12566 df-uz 12830 df-rp 12982 df-fz 13492 df-fzo 13635 df-seq 13974 df-exp 14035 df-hash 14298 df-cj 15053 df-re 15054 df-im 15055 df-sqrt 15189 df-abs 15190 df-clim 15439 df-sum 15640 |
This theorem is referenced by: sumcubes 41526 |
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