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Mirrors > Home > MPE Home > Th. List > fsumm1 | Structured version Visualization version GIF version |
Description: Separate out the last term in a finite sum. (Contributed by Mario Carneiro, 26-Apr-2014.) |
Ref | Expression |
---|---|
fsumm1.1 | ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) |
fsumm1.2 | ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝐴 ∈ ℂ) |
fsumm1.3 | ⊢ (𝑘 = 𝑁 → 𝐴 = 𝐵) |
Ref | Expression |
---|---|
fsumm1 | ⊢ (𝜑 → Σ𝑘 ∈ (𝑀...𝑁)𝐴 = (Σ𝑘 ∈ (𝑀...(𝑁 − 1))𝐴 + 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fsumm1.1 | . . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) | |
2 | eluzelz 12778 | . . . . . . 7 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ ℤ) | |
3 | 1, 2 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℤ) |
4 | fzsn 13489 | . . . . . 6 ⊢ (𝑁 ∈ ℤ → (𝑁...𝑁) = {𝑁}) | |
5 | 3, 4 | syl 17 | . . . . 5 ⊢ (𝜑 → (𝑁...𝑁) = {𝑁}) |
6 | 5 | ineq2d 4173 | . . . 4 ⊢ (𝜑 → ((𝑀...(𝑁 − 1)) ∩ (𝑁...𝑁)) = ((𝑀...(𝑁 − 1)) ∩ {𝑁})) |
7 | 3 | zred 12612 | . . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℝ) |
8 | 7 | ltm1d 12092 | . . . . 5 ⊢ (𝜑 → (𝑁 − 1) < 𝑁) |
9 | fzdisj 13474 | . . . . 5 ⊢ ((𝑁 − 1) < 𝑁 → ((𝑀...(𝑁 − 1)) ∩ (𝑁...𝑁)) = ∅) | |
10 | 8, 9 | syl 17 | . . . 4 ⊢ (𝜑 → ((𝑀...(𝑁 − 1)) ∩ (𝑁...𝑁)) = ∅) |
11 | 6, 10 | eqtr3d 2775 | . . 3 ⊢ (𝜑 → ((𝑀...(𝑁 − 1)) ∩ {𝑁}) = ∅) |
12 | eluzel2 12773 | . . . . . . 7 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ ℤ) | |
13 | 1, 12 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℤ) |
14 | peano2zm 12551 | . . . . . . . 8 ⊢ (𝑀 ∈ ℤ → (𝑀 − 1) ∈ ℤ) | |
15 | 13, 14 | syl 17 | . . . . . . 7 ⊢ (𝜑 → (𝑀 − 1) ∈ ℤ) |
16 | 13 | zcnd 12613 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ ℂ) |
17 | ax-1cn 11114 | . . . . . . . . . 10 ⊢ 1 ∈ ℂ | |
18 | npcan 11415 | . . . . . . . . . 10 ⊢ ((𝑀 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑀 − 1) + 1) = 𝑀) | |
19 | 16, 17, 18 | sylancl 587 | . . . . . . . . 9 ⊢ (𝜑 → ((𝑀 − 1) + 1) = 𝑀) |
20 | 19 | fveq2d 6847 | . . . . . . . 8 ⊢ (𝜑 → (ℤ≥‘((𝑀 − 1) + 1)) = (ℤ≥‘𝑀)) |
21 | 1, 20 | eleqtrrd 2837 | . . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘((𝑀 − 1) + 1))) |
22 | eluzp1m1 12794 | . . . . . . 7 ⊢ (((𝑀 − 1) ∈ ℤ ∧ 𝑁 ∈ (ℤ≥‘((𝑀 − 1) + 1))) → (𝑁 − 1) ∈ (ℤ≥‘(𝑀 − 1))) | |
23 | 15, 21, 22 | syl2anc 585 | . . . . . 6 ⊢ (𝜑 → (𝑁 − 1) ∈ (ℤ≥‘(𝑀 − 1))) |
24 | fzsuc2 13505 | . . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ (𝑁 − 1) ∈ (ℤ≥‘(𝑀 − 1))) → (𝑀...((𝑁 − 1) + 1)) = ((𝑀...(𝑁 − 1)) ∪ {((𝑁 − 1) + 1)})) | |
25 | 13, 23, 24 | syl2anc 585 | . . . . 5 ⊢ (𝜑 → (𝑀...((𝑁 − 1) + 1)) = ((𝑀...(𝑁 − 1)) ∪ {((𝑁 − 1) + 1)})) |
26 | 3 | zcnd 12613 | . . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℂ) |
27 | npcan 11415 | . . . . . . 7 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 − 1) + 1) = 𝑁) | |
28 | 26, 17, 27 | sylancl 587 | . . . . . 6 ⊢ (𝜑 → ((𝑁 − 1) + 1) = 𝑁) |
29 | 28 | oveq2d 7374 | . . . . 5 ⊢ (𝜑 → (𝑀...((𝑁 − 1) + 1)) = (𝑀...𝑁)) |
30 | 25, 29 | eqtr3d 2775 | . . . 4 ⊢ (𝜑 → ((𝑀...(𝑁 − 1)) ∪ {((𝑁 − 1) + 1)}) = (𝑀...𝑁)) |
31 | 28 | sneqd 4599 | . . . . 5 ⊢ (𝜑 → {((𝑁 − 1) + 1)} = {𝑁}) |
32 | 31 | uneq2d 4124 | . . . 4 ⊢ (𝜑 → ((𝑀...(𝑁 − 1)) ∪ {((𝑁 − 1) + 1)}) = ((𝑀...(𝑁 − 1)) ∪ {𝑁})) |
33 | 30, 32 | eqtr3d 2775 | . . 3 ⊢ (𝜑 → (𝑀...𝑁) = ((𝑀...(𝑁 − 1)) ∪ {𝑁})) |
34 | fzfid 13884 | . . 3 ⊢ (𝜑 → (𝑀...𝑁) ∈ Fin) | |
35 | fsumm1.2 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝐴 ∈ ℂ) | |
36 | 11, 33, 34, 35 | fsumsplit 15631 | . 2 ⊢ (𝜑 → Σ𝑘 ∈ (𝑀...𝑁)𝐴 = (Σ𝑘 ∈ (𝑀...(𝑁 − 1))𝐴 + Σ𝑘 ∈ {𝑁}𝐴)) |
37 | fsumm1.3 | . . . . . 6 ⊢ (𝑘 = 𝑁 → 𝐴 = 𝐵) | |
38 | 37 | eleq1d 2819 | . . . . 5 ⊢ (𝑘 = 𝑁 → (𝐴 ∈ ℂ ↔ 𝐵 ∈ ℂ)) |
39 | 35 | ralrimiva 3140 | . . . . 5 ⊢ (𝜑 → ∀𝑘 ∈ (𝑀...𝑁)𝐴 ∈ ℂ) |
40 | eluzfz2 13455 | . . . . . 6 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ (𝑀...𝑁)) | |
41 | 1, 40 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ (𝑀...𝑁)) |
42 | 38, 39, 41 | rspcdva 3581 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
43 | 37 | sumsn 15636 | . . . 4 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ 𝐵 ∈ ℂ) → Σ𝑘 ∈ {𝑁}𝐴 = 𝐵) |
44 | 1, 42, 43 | syl2anc 585 | . . 3 ⊢ (𝜑 → Σ𝑘 ∈ {𝑁}𝐴 = 𝐵) |
45 | 44 | oveq2d 7374 | . 2 ⊢ (𝜑 → (Σ𝑘 ∈ (𝑀...(𝑁 − 1))𝐴 + Σ𝑘 ∈ {𝑁}𝐴) = (Σ𝑘 ∈ (𝑀...(𝑁 − 1))𝐴 + 𝐵)) |
46 | 36, 45 | eqtrd 2773 | 1 ⊢ (𝜑 → Σ𝑘 ∈ (𝑀...𝑁)𝐴 = (Σ𝑘 ∈ (𝑀...(𝑁 − 1))𝐴 + 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∪ cun 3909 ∩ cin 3910 ∅c0 4283 {csn 4587 class class class wbr 5106 ‘cfv 6497 (class class class)co 7358 ℂcc 11054 1c1 11057 + caddc 11059 < clt 11194 − cmin 11390 ℤcz 12504 ℤ≥cuz 12768 ...cfz 13430 Σcsu 15576 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-inf2 9582 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 ax-pre-sup 11134 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-int 4909 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-se 5590 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-isom 6506 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-er 8651 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-sup 9383 df-oi 9451 df-card 9880 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-div 11818 df-nn 12159 df-2 12221 df-3 12222 df-n0 12419 df-z 12505 df-uz 12769 df-rp 12921 df-fz 13431 df-fzo 13574 df-seq 13913 df-exp 13974 df-hash 14237 df-cj 14990 df-re 14991 df-im 14992 df-sqrt 15126 df-abs 15127 df-clim 15376 df-sum 15577 |
This theorem is referenced by: fzosump1 15642 fsump1 15646 telfsumo 15692 fsumparts 15696 binom1dif 15723 pwdif 15758 bpolysum 15941 bpolydiflem 15942 pwp1fsum 16278 prmreclem4 16796 ovolicc2lem4 24900 dvfsumlem1 25406 abelthlem6 25811 log2ublem2 26313 harmonicbnd4 26376 ftalem1 26438 ftalem5 26442 chpp1 26520 1sgmprm 26563 chtublem 26575 logdivbnd 26920 pntrlog2bndlem1 26941 knoppndvlem15 35035 mettrifi 36262 sticksstones12a 40611 sticksstones12 40612 fzosumm1 40712 stoweidlem17 44344 nnsum4primeseven 46078 nnsum4primesevenALTV 46079 |
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