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Mirrors > Home > MPE Home > Th. List > fsumsplitsn | Structured version Visualization version GIF version |
Description: Separate out a term in a finite sum. (Contributed by Glauco Siliprandi, 5-Apr-2020.) |
Ref | Expression |
---|---|
fsumsplitsn.ph | ⊢ Ⅎ𝑘𝜑 |
fsumsplitsn.kd | ⊢ Ⅎ𝑘𝐷 |
fsumsplitsn.a | ⊢ (𝜑 → 𝐴 ∈ Fin) |
fsumsplitsn.b | ⊢ (𝜑 → 𝐵 ∈ 𝑉) |
fsumsplitsn.ba | ⊢ (𝜑 → ¬ 𝐵 ∈ 𝐴) |
fsumsplitsn.c | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ ℂ) |
fsumsplitsn.d | ⊢ (𝑘 = 𝐵 → 𝐶 = 𝐷) |
fsumsplitsn.dcn | ⊢ (𝜑 → 𝐷 ∈ ℂ) |
Ref | Expression |
---|---|
fsumsplitsn | ⊢ (𝜑 → Σ𝑘 ∈ (𝐴 ∪ {𝐵})𝐶 = (Σ𝑘 ∈ 𝐴 𝐶 + 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fsumsplitsn.ph | . . 3 ⊢ Ⅎ𝑘𝜑 | |
2 | fsumsplitsn.ba | . . . 4 ⊢ (𝜑 → ¬ 𝐵 ∈ 𝐴) | |
3 | disjsn 4647 | . . . 4 ⊢ ((𝐴 ∩ {𝐵}) = ∅ ↔ ¬ 𝐵 ∈ 𝐴) | |
4 | 2, 3 | sylibr 236 | . . 3 ⊢ (𝜑 → (𝐴 ∩ {𝐵}) = ∅) |
5 | eqidd 2822 | . . 3 ⊢ (𝜑 → (𝐴 ∪ {𝐵}) = (𝐴 ∪ {𝐵})) | |
6 | fsumsplitsn.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ Fin) | |
7 | snfi 8594 | . . . 4 ⊢ {𝐵} ∈ Fin | |
8 | unfi 8785 | . . . 4 ⊢ ((𝐴 ∈ Fin ∧ {𝐵} ∈ Fin) → (𝐴 ∪ {𝐵}) ∈ Fin) | |
9 | 6, 7, 8 | sylancl 588 | . . 3 ⊢ (𝜑 → (𝐴 ∪ {𝐵}) ∈ Fin) |
10 | fsumsplitsn.c | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ ℂ) | |
11 | 10 | adantlr 713 | . . . 4 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝐴 ∪ {𝐵})) ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ ℂ) |
12 | simpll 765 | . . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝐴 ∪ {𝐵})) ∧ ¬ 𝑘 ∈ 𝐴) → 𝜑) | |
13 | elunnel1 4126 | . . . . . . 7 ⊢ ((𝑘 ∈ (𝐴 ∪ {𝐵}) ∧ ¬ 𝑘 ∈ 𝐴) → 𝑘 ∈ {𝐵}) | |
14 | elsni 4584 | . . . . . . 7 ⊢ (𝑘 ∈ {𝐵} → 𝑘 = 𝐵) | |
15 | 13, 14 | syl 17 | . . . . . 6 ⊢ ((𝑘 ∈ (𝐴 ∪ {𝐵}) ∧ ¬ 𝑘 ∈ 𝐴) → 𝑘 = 𝐵) |
16 | 15 | adantll 712 | . . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝐴 ∪ {𝐵})) ∧ ¬ 𝑘 ∈ 𝐴) → 𝑘 = 𝐵) |
17 | fsumsplitsn.d | . . . . . . 7 ⊢ (𝑘 = 𝐵 → 𝐶 = 𝐷) | |
18 | 17 | adantl 484 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 = 𝐵) → 𝐶 = 𝐷) |
19 | fsumsplitsn.dcn | . . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ ℂ) | |
20 | 19 | adantr 483 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 = 𝐵) → 𝐷 ∈ ℂ) |
21 | 18, 20 | eqeltrd 2913 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 = 𝐵) → 𝐶 ∈ ℂ) |
22 | 12, 16, 21 | syl2anc 586 | . . . 4 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝐴 ∪ {𝐵})) ∧ ¬ 𝑘 ∈ 𝐴) → 𝐶 ∈ ℂ) |
23 | 11, 22 | pm2.61dan 811 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∪ {𝐵})) → 𝐶 ∈ ℂ) |
24 | 1, 4, 5, 9, 23 | fsumsplitf 15098 | . 2 ⊢ (𝜑 → Σ𝑘 ∈ (𝐴 ∪ {𝐵})𝐶 = (Σ𝑘 ∈ 𝐴 𝐶 + Σ𝑘 ∈ {𝐵}𝐶)) |
25 | fsumsplitsn.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑉) | |
26 | fsumsplitsn.kd | . . . . 5 ⊢ Ⅎ𝑘𝐷 | |
27 | 26, 17 | sumsnf 15099 | . . . 4 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐷 ∈ ℂ) → Σ𝑘 ∈ {𝐵}𝐶 = 𝐷) |
28 | 25, 19, 27 | syl2anc 586 | . . 3 ⊢ (𝜑 → Σ𝑘 ∈ {𝐵}𝐶 = 𝐷) |
29 | 28 | oveq2d 7172 | . 2 ⊢ (𝜑 → (Σ𝑘 ∈ 𝐴 𝐶 + Σ𝑘 ∈ {𝐵}𝐶) = (Σ𝑘 ∈ 𝐴 𝐶 + 𝐷)) |
30 | 24, 29 | eqtrd 2856 | 1 ⊢ (𝜑 → Σ𝑘 ∈ (𝐴 ∪ {𝐵})𝐶 = (Σ𝑘 ∈ 𝐴 𝐶 + 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 = wceq 1537 Ⅎwnf 1784 ∈ wcel 2114 Ⅎwnfc 2961 ∪ cun 3934 ∩ cin 3935 ∅c0 4291 {csn 4567 (class class class)co 7156 Fincfn 8509 ℂcc 10535 + caddc 10540 Σcsu 15042 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-inf2 9104 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-sup 8906 df-oi 8974 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-n0 11899 df-z 11983 df-uz 12245 df-rp 12391 df-fz 12894 df-fzo 13035 df-seq 13371 df-exp 13431 df-hash 13692 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-clim 14845 df-sum 15043 |
This theorem is referenced by: reprsuc 31886 hgt750lemd 31919 fsumnncl 41872 fsumsplit1 41873 mccllem 41898 dvmptfprodlem 42249 dvnprodlem1 42251 sge0iunmptlemfi 42715 |
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