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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 4715 | . . . 4 ⊢ ((𝐴 ∩ {𝐵}) = ∅ ↔ ¬ 𝐵 ∈ 𝐴) | |
4 | 2, 3 | sylibr 233 | . . 3 ⊢ (𝜑 → (𝐴 ∩ {𝐵}) = ∅) |
5 | eqidd 2732 | . . 3 ⊢ (𝜑 → (𝐴 ∪ {𝐵}) = (𝐴 ∪ {𝐵})) | |
6 | fsumsplitsn.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ Fin) | |
7 | snfi 9048 | . . . 4 ⊢ {𝐵} ∈ Fin | |
8 | unfi 9176 | . . . 4 ⊢ ((𝐴 ∈ Fin ∧ {𝐵} ∈ Fin) → (𝐴 ∪ {𝐵}) ∈ Fin) | |
9 | 6, 7, 8 | sylancl 585 | . . 3 ⊢ (𝜑 → (𝐴 ∪ {𝐵}) ∈ Fin) |
10 | fsumsplitsn.c | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ ℂ) | |
11 | 10 | adantlr 712 | . . . 4 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝐴 ∪ {𝐵})) ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ ℂ) |
12 | simpll 764 | . . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝐴 ∪ {𝐵})) ∧ ¬ 𝑘 ∈ 𝐴) → 𝜑) | |
13 | elunnel1 4149 | . . . . . . 7 ⊢ ((𝑘 ∈ (𝐴 ∪ {𝐵}) ∧ ¬ 𝑘 ∈ 𝐴) → 𝑘 ∈ {𝐵}) | |
14 | elsni 4645 | . . . . . . 7 ⊢ (𝑘 ∈ {𝐵} → 𝑘 = 𝐵) | |
15 | 13, 14 | syl 17 | . . . . . 6 ⊢ ((𝑘 ∈ (𝐴 ∪ {𝐵}) ∧ ¬ 𝑘 ∈ 𝐴) → 𝑘 = 𝐵) |
16 | 15 | adantll 711 | . . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝐴 ∪ {𝐵})) ∧ ¬ 𝑘 ∈ 𝐴) → 𝑘 = 𝐵) |
17 | fsumsplitsn.d | . . . . . . 7 ⊢ (𝑘 = 𝐵 → 𝐶 = 𝐷) | |
18 | 17 | adantl 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 = 𝐵) → 𝐶 = 𝐷) |
19 | fsumsplitsn.dcn | . . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ ℂ) | |
20 | 19 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 = 𝐵) → 𝐷 ∈ ℂ) |
21 | 18, 20 | eqeltrd 2832 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 = 𝐵) → 𝐶 ∈ ℂ) |
22 | 12, 16, 21 | syl2anc 583 | . . . 4 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝐴 ∪ {𝐵})) ∧ ¬ 𝑘 ∈ 𝐴) → 𝐶 ∈ ℂ) |
23 | 11, 22 | pm2.61dan 810 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∪ {𝐵})) → 𝐶 ∈ ℂ) |
24 | 1, 4, 5, 9, 23 | fsumsplitf 15693 | . 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 15694 | . . . 4 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐷 ∈ ℂ) → Σ𝑘 ∈ {𝐵}𝐶 = 𝐷) |
28 | 25, 19, 27 | syl2anc 583 | . . 3 ⊢ (𝜑 → Σ𝑘 ∈ {𝐵}𝐶 = 𝐷) |
29 | 28 | oveq2d 7428 | . 2 ⊢ (𝜑 → (Σ𝑘 ∈ 𝐴 𝐶 + Σ𝑘 ∈ {𝐵}𝐶) = (Σ𝑘 ∈ 𝐴 𝐶 + 𝐷)) |
30 | 24, 29 | eqtrd 2771 | 1 ⊢ (𝜑 → Σ𝑘 ∈ (𝐴 ∪ {𝐵})𝐶 = (Σ𝑘 ∈ 𝐴 𝐶 + 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1540 Ⅎwnf 1784 ∈ wcel 2105 Ⅎwnfc 2882 ∪ cun 3946 ∩ cin 3947 ∅c0 4322 {csn 4628 (class class class)co 7412 Fincfn 8943 ℂcc 11112 + caddc 11117 Σcsu 15637 |
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 9640 ax-cnex 11170 ax-resscn 11171 ax-1cn 11172 ax-icn 11173 ax-addcl 11174 ax-addrcl 11175 ax-mulcl 11176 ax-mulrcl 11177 ax-mulcom 11178 ax-addass 11179 ax-mulass 11180 ax-distr 11181 ax-i2m1 11182 ax-1ne0 11183 ax-1rid 11184 ax-rnegex 11185 ax-rrecex 11186 ax-cnre 11187 ax-pre-lttri 11188 ax-pre-lttrn 11189 ax-pre-ltadd 11190 ax-pre-mulgt0 11191 ax-pre-sup 11192 |
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 8270 df-wrecs 8301 df-recs 8375 df-rdg 8414 df-1o 8470 df-er 8707 df-en 8944 df-dom 8945 df-sdom 8946 df-fin 8947 df-sup 9441 df-oi 9509 df-card 9938 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-sub 11451 df-neg 11452 df-div 11877 df-nn 12218 df-2 12280 df-3 12281 df-n0 12478 df-z 12564 df-uz 12828 df-rp 12980 df-fz 13490 df-fzo 13633 df-seq 13972 df-exp 14033 df-hash 14296 df-cj 15051 df-re 15052 df-im 15053 df-sqrt 15187 df-abs 15188 df-clim 15437 df-sum 15638 |
This theorem is referenced by: fsumsplit1 15696 reprsuc 33926 hgt750lemd 33959 fsumnncl 44587 mccllem 44612 dvmptfprodlem 44959 dvnprodlem1 44961 sge0iunmptlemfi 45428 |
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