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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > fsumsplit1 | Structured version Visualization version GIF version |
Description: Separate out a term in a finite sum. (Contributed by Glauco Siliprandi, 5-Apr-2020.) |
Ref | Expression |
---|---|
fsumsplit1.kph | ⊢ Ⅎ𝑘𝜑 |
fsumsplit1.kd | ⊢ Ⅎ𝑘𝐷 |
fsumsplit1.a | ⊢ (𝜑 → 𝐴 ∈ Fin) |
fsumsplit1.b | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) |
fsumsplit1.c | ⊢ (𝜑 → 𝐶 ∈ 𝐴) |
fsumsplit1.bd | ⊢ (𝑘 = 𝐶 → 𝐵 = 𝐷) |
Ref | Expression |
---|---|
fsumsplit1 | ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 = (𝐷 + Σ𝑘 ∈ (𝐴 ∖ {𝐶})𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uncom 4054 | . . . . 5 ⊢ ((𝐴 ∖ {𝐶}) ∪ {𝐶}) = ({𝐶} ∪ (𝐴 ∖ {𝐶})) | |
2 | 1 | a1i 11 | . . . 4 ⊢ (𝜑 → ((𝐴 ∖ {𝐶}) ∪ {𝐶}) = ({𝐶} ∪ (𝐴 ∖ {𝐶}))) |
3 | fsumsplit1.c | . . . . . 6 ⊢ (𝜑 → 𝐶 ∈ 𝐴) | |
4 | 3 | snssd 4692 | . . . . 5 ⊢ (𝜑 → {𝐶} ⊆ 𝐴) |
5 | undif 4371 | . . . . 5 ⊢ ({𝐶} ⊆ 𝐴 ↔ ({𝐶} ∪ (𝐴 ∖ {𝐶})) = 𝐴) | |
6 | 4, 5 | sylib 221 | . . . 4 ⊢ (𝜑 → ({𝐶} ∪ (𝐴 ∖ {𝐶})) = 𝐴) |
7 | eqidd 2760 | . . . 4 ⊢ (𝜑 → 𝐴 = 𝐴) | |
8 | 2, 6, 7 | 3eqtrrd 2799 | . . 3 ⊢ (𝜑 → 𝐴 = ((𝐴 ∖ {𝐶}) ∪ {𝐶})) |
9 | 8 | sumeq1d 15091 | . 2 ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 = Σ𝑘 ∈ ((𝐴 ∖ {𝐶}) ∪ {𝐶})𝐵) |
10 | fsumsplit1.kph | . . 3 ⊢ Ⅎ𝑘𝜑 | |
11 | fsumsplit1.kd | . . 3 ⊢ Ⅎ𝑘𝐷 | |
12 | fsumsplit1.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ Fin) | |
13 | diffi 8764 | . . . 4 ⊢ (𝐴 ∈ Fin → (𝐴 ∖ {𝐶}) ∈ Fin) | |
14 | 12, 13 | syl 17 | . . 3 ⊢ (𝜑 → (𝐴 ∖ {𝐶}) ∈ Fin) |
15 | neldifsnd 4676 | . . 3 ⊢ (𝜑 → ¬ 𝐶 ∈ (𝐴 ∖ {𝐶})) | |
16 | simpl 487 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ {𝐶})) → 𝜑) | |
17 | eldifi 4028 | . . . . 5 ⊢ (𝑘 ∈ (𝐴 ∖ {𝐶}) → 𝑘 ∈ 𝐴) | |
18 | 17 | adantl 486 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ {𝐶})) → 𝑘 ∈ 𝐴) |
19 | fsumsplit1.b | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) | |
20 | 16, 18, 19 | syl2anc 588 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ {𝐶})) → 𝐵 ∈ ℂ) |
21 | fsumsplit1.bd | . . 3 ⊢ (𝑘 = 𝐶 → 𝐵 = 𝐷) | |
22 | 11 | a1i 11 | . . . . . 6 ⊢ (𝜑 → Ⅎ𝑘𝐷) |
23 | simpr 489 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 = 𝐶) → 𝑘 = 𝐶) | |
24 | 23, 21 | syl 17 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 = 𝐶) → 𝐵 = 𝐷) |
25 | 10, 22, 3, 24 | csbiedf 3831 | . . . . 5 ⊢ (𝜑 → ⦋𝐶 / 𝑘⦌𝐵 = 𝐷) |
26 | 25 | eqcomd 2765 | . . . 4 ⊢ (𝜑 → 𝐷 = ⦋𝐶 / 𝑘⦌𝐵) |
27 | 3 | ancli 553 | . . . . 5 ⊢ (𝜑 → (𝜑 ∧ 𝐶 ∈ 𝐴)) |
28 | nfcv 2917 | . . . . . 6 ⊢ Ⅎ𝑘𝐶 | |
29 | nfv 1916 | . . . . . . . 8 ⊢ Ⅎ𝑘 𝐶 ∈ 𝐴 | |
30 | 10, 29 | nfan 1901 | . . . . . . 7 ⊢ Ⅎ𝑘(𝜑 ∧ 𝐶 ∈ 𝐴) |
31 | 28 | nfcsb1 3824 | . . . . . . . 8 ⊢ Ⅎ𝑘⦋𝐶 / 𝑘⦌𝐵 |
32 | nfcv 2917 | . . . . . . . 8 ⊢ Ⅎ𝑘ℂ | |
33 | 31, 32 | nfel 2931 | . . . . . . 7 ⊢ Ⅎ𝑘⦋𝐶 / 𝑘⦌𝐵 ∈ ℂ |
34 | 30, 33 | nfim 1898 | . . . . . 6 ⊢ Ⅎ𝑘((𝜑 ∧ 𝐶 ∈ 𝐴) → ⦋𝐶 / 𝑘⦌𝐵 ∈ ℂ) |
35 | eleq1 2838 | . . . . . . . 8 ⊢ (𝑘 = 𝐶 → (𝑘 ∈ 𝐴 ↔ 𝐶 ∈ 𝐴)) | |
36 | 35 | anbi2d 632 | . . . . . . 7 ⊢ (𝑘 = 𝐶 → ((𝜑 ∧ 𝑘 ∈ 𝐴) ↔ (𝜑 ∧ 𝐶 ∈ 𝐴))) |
37 | csbeq1a 3815 | . . . . . . . 8 ⊢ (𝑘 = 𝐶 → 𝐵 = ⦋𝐶 / 𝑘⦌𝐵) | |
38 | 37 | eleq1d 2835 | . . . . . . 7 ⊢ (𝑘 = 𝐶 → (𝐵 ∈ ℂ ↔ ⦋𝐶 / 𝑘⦌𝐵 ∈ ℂ)) |
39 | 36, 38 | imbi12d 349 | . . . . . 6 ⊢ (𝑘 = 𝐶 → (((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) ↔ ((𝜑 ∧ 𝐶 ∈ 𝐴) → ⦋𝐶 / 𝑘⦌𝐵 ∈ ℂ))) |
40 | 28, 34, 39, 19 | vtoclgf 3481 | . . . . 5 ⊢ (𝐶 ∈ 𝐴 → ((𝜑 ∧ 𝐶 ∈ 𝐴) → ⦋𝐶 / 𝑘⦌𝐵 ∈ ℂ)) |
41 | 3, 27, 40 | sylc 65 | . . . 4 ⊢ (𝜑 → ⦋𝐶 / 𝑘⦌𝐵 ∈ ℂ) |
42 | 26, 41 | eqeltrd 2851 | . . 3 ⊢ (𝜑 → 𝐷 ∈ ℂ) |
43 | 10, 11, 14, 3, 15, 20, 21, 42 | fsumsplitsn 15133 | . 2 ⊢ (𝜑 → Σ𝑘 ∈ ((𝐴 ∖ {𝐶}) ∪ {𝐶})𝐵 = (Σ𝑘 ∈ (𝐴 ∖ {𝐶})𝐵 + 𝐷)) |
44 | 10, 14, 20 | fsumclf 42562 | . . 3 ⊢ (𝜑 → Σ𝑘 ∈ (𝐴 ∖ {𝐶})𝐵 ∈ ℂ) |
45 | 44, 42 | addcomd 10865 | . 2 ⊢ (𝜑 → (Σ𝑘 ∈ (𝐴 ∖ {𝐶})𝐵 + 𝐷) = (𝐷 + Σ𝑘 ∈ (𝐴 ∖ {𝐶})𝐵)) |
46 | 9, 43, 45 | 3eqtrd 2798 | 1 ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 = (𝐷 + Σ𝑘 ∈ (𝐴 ∖ {𝐶})𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 400 = wceq 1539 Ⅎwnf 1786 ∈ wcel 2112 Ⅎwnfc 2897 ⦋csb 3801 ∖ cdif 3851 ∪ cun 3852 ⊆ wss 3854 {csn 4515 (class class class)co 7143 Fincfn 8520 ℂcc 10558 + caddc 10563 Σcsu 15075 |
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 1912 ax-6 1971 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2159 ax-12 2176 ax-ext 2730 ax-rep 5149 ax-sep 5162 ax-nul 5169 ax-pow 5227 ax-pr 5291 ax-un 7452 ax-inf2 9122 ax-cnex 10616 ax-resscn 10617 ax-1cn 10618 ax-icn 10619 ax-addcl 10620 ax-addrcl 10621 ax-mulcl 10622 ax-mulrcl 10623 ax-mulcom 10624 ax-addass 10625 ax-mulass 10626 ax-distr 10627 ax-i2m1 10628 ax-1ne0 10629 ax-1rid 10630 ax-rnegex 10631 ax-rrecex 10632 ax-cnre 10633 ax-pre-lttri 10634 ax-pre-lttrn 10635 ax-pre-ltadd 10636 ax-pre-mulgt0 10637 ax-pre-sup 10638 |
This theorem depends on definitions: df-bi 210 df-an 401 df-or 846 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2071 df-mo 2558 df-eu 2589 df-clab 2737 df-cleq 2751 df-clel 2831 df-nfc 2899 df-ne 2950 df-nel 3054 df-ral 3073 df-rex 3074 df-reu 3075 df-rmo 3076 df-rab 3077 df-v 3409 df-sbc 3694 df-csb 3802 df-dif 3857 df-un 3859 df-in 3861 df-ss 3871 df-pss 3873 df-nul 4222 df-if 4414 df-pw 4489 df-sn 4516 df-pr 4518 df-tp 4520 df-op 4522 df-uni 4792 df-int 4832 df-iun 4878 df-br 5026 df-opab 5088 df-mpt 5106 df-tr 5132 df-id 5423 df-eprel 5428 df-po 5436 df-so 5437 df-fr 5476 df-se 5477 df-we 5478 df-xp 5523 df-rel 5524 df-cnv 5525 df-co 5526 df-dm 5527 df-rn 5528 df-res 5529 df-ima 5530 df-pred 6119 df-ord 6165 df-on 6166 df-lim 6167 df-suc 6168 df-iota 6287 df-fun 6330 df-fn 6331 df-f 6332 df-f1 6333 df-fo 6334 df-f1o 6335 df-fv 6336 df-isom 6337 df-riota 7101 df-ov 7146 df-oprab 7147 df-mpo 7148 df-om 7573 df-1st 7686 df-2nd 7687 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-1o 8105 df-oadd 8109 df-er 8292 df-en 8521 df-dom 8522 df-sdom 8523 df-fin 8524 df-sup 8924 df-oi 8992 df-card 9386 df-pnf 10700 df-mnf 10701 df-xr 10702 df-ltxr 10703 df-le 10704 df-sub 10895 df-neg 10896 df-div 11321 df-nn 11660 df-2 11722 df-3 11723 df-n0 11920 df-z 12006 df-uz 12268 df-rp 12416 df-fz 12925 df-fzo 13068 df-seq 13404 df-exp 13465 df-hash 13726 df-cj 14491 df-re 14492 df-im 14493 df-sqrt 14627 df-abs 14628 df-clim 14878 df-sum 15076 |
This theorem is referenced by: dvnmul 42936 etransclem35 43262 etransclem44 43271 |
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