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Mirrors > Home > MPE Home > Th. List > fsumsplitf | Structured version Visualization version GIF version |
Description: Split a sum into two parts. A version of fsumsplit 15684 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 5-Apr-2020.) |
Ref | Expression |
---|---|
fsumsplitf.ph | ⊢ Ⅎ𝑘𝜑 |
fsumsplitf.ab | ⊢ (𝜑 → (𝐴 ∩ 𝐵) = ∅) |
fsumsplitf.u | ⊢ (𝜑 → 𝑈 = (𝐴 ∪ 𝐵)) |
fsumsplitf.fi | ⊢ (𝜑 → 𝑈 ∈ Fin) |
fsumsplitf.c | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → 𝐶 ∈ ℂ) |
Ref | Expression |
---|---|
fsumsplitf | ⊢ (𝜑 → Σ𝑘 ∈ 𝑈 𝐶 = (Σ𝑘 ∈ 𝐴 𝐶 + Σ𝑘 ∈ 𝐵 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfcv 2895 | . . . 4 ⊢ Ⅎ𝑗𝐶 | |
2 | nfcsb1v 3910 | . . . 4 ⊢ Ⅎ𝑘⦋𝑗 / 𝑘⦌𝐶 | |
3 | csbeq1a 3899 | . . . 4 ⊢ (𝑘 = 𝑗 → 𝐶 = ⦋𝑗 / 𝑘⦌𝐶) | |
4 | 1, 2, 3 | cbvsumi 15640 | . . 3 ⊢ Σ𝑘 ∈ 𝑈 𝐶 = Σ𝑗 ∈ 𝑈 ⦋𝑗 / 𝑘⦌𝐶 |
5 | 4 | a1i 11 | . 2 ⊢ (𝜑 → Σ𝑘 ∈ 𝑈 𝐶 = Σ𝑗 ∈ 𝑈 ⦋𝑗 / 𝑘⦌𝐶) |
6 | fsumsplitf.ab | . . 3 ⊢ (𝜑 → (𝐴 ∩ 𝐵) = ∅) | |
7 | fsumsplitf.u | . . 3 ⊢ (𝜑 → 𝑈 = (𝐴 ∪ 𝐵)) | |
8 | fsumsplitf.fi | . . 3 ⊢ (𝜑 → 𝑈 ∈ Fin) | |
9 | fsumsplitf.ph | . . . . . 6 ⊢ Ⅎ𝑘𝜑 | |
10 | nfv 1909 | . . . . . 6 ⊢ Ⅎ𝑘 𝑗 ∈ 𝑈 | |
11 | 9, 10 | nfan 1894 | . . . . 5 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑗 ∈ 𝑈) |
12 | 2 | nfel1 2911 | . . . . 5 ⊢ Ⅎ𝑘⦋𝑗 / 𝑘⦌𝐶 ∈ ℂ |
13 | 11, 12 | nfim 1891 | . . . 4 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑗 ∈ 𝑈) → ⦋𝑗 / 𝑘⦌𝐶 ∈ ℂ) |
14 | eleq1w 2808 | . . . . . 6 ⊢ (𝑘 = 𝑗 → (𝑘 ∈ 𝑈 ↔ 𝑗 ∈ 𝑈)) | |
15 | 14 | anbi2d 628 | . . . . 5 ⊢ (𝑘 = 𝑗 → ((𝜑 ∧ 𝑘 ∈ 𝑈) ↔ (𝜑 ∧ 𝑗 ∈ 𝑈))) |
16 | 3 | eleq1d 2810 | . . . . 5 ⊢ (𝑘 = 𝑗 → (𝐶 ∈ ℂ ↔ ⦋𝑗 / 𝑘⦌𝐶 ∈ ℂ)) |
17 | 15, 16 | imbi12d 344 | . . . 4 ⊢ (𝑘 = 𝑗 → (((𝜑 ∧ 𝑘 ∈ 𝑈) → 𝐶 ∈ ℂ) ↔ ((𝜑 ∧ 𝑗 ∈ 𝑈) → ⦋𝑗 / 𝑘⦌𝐶 ∈ ℂ))) |
18 | fsumsplitf.c | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → 𝐶 ∈ ℂ) | |
19 | 13, 17, 18 | chvarfv 2225 | . . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑈) → ⦋𝑗 / 𝑘⦌𝐶 ∈ ℂ) |
20 | 6, 7, 8, 19 | fsumsplit 15684 | . 2 ⊢ (𝜑 → Σ𝑗 ∈ 𝑈 ⦋𝑗 / 𝑘⦌𝐶 = (Σ𝑗 ∈ 𝐴 ⦋𝑗 / 𝑘⦌𝐶 + Σ𝑗 ∈ 𝐵 ⦋𝑗 / 𝑘⦌𝐶)) |
21 | csbeq1a 3899 | . . . . . 6 ⊢ (𝑗 = 𝑘 → ⦋𝑗 / 𝑘⦌𝐶 = ⦋𝑘 / 𝑗⦌⦋𝑗 / 𝑘⦌𝐶) | |
22 | csbcow 3900 | . . . . . . 7 ⊢ ⦋𝑘 / 𝑗⦌⦋𝑗 / 𝑘⦌𝐶 = ⦋𝑘 / 𝑘⦌𝐶 | |
23 | csbid 3898 | . . . . . . 7 ⊢ ⦋𝑘 / 𝑘⦌𝐶 = 𝐶 | |
24 | 22, 23 | eqtri 2752 | . . . . . 6 ⊢ ⦋𝑘 / 𝑗⦌⦋𝑗 / 𝑘⦌𝐶 = 𝐶 |
25 | 21, 24 | eqtrdi 2780 | . . . . 5 ⊢ (𝑗 = 𝑘 → ⦋𝑗 / 𝑘⦌𝐶 = 𝐶) |
26 | 2, 1, 25 | cbvsumi 15640 | . . . 4 ⊢ Σ𝑗 ∈ 𝐴 ⦋𝑗 / 𝑘⦌𝐶 = Σ𝑘 ∈ 𝐴 𝐶 |
27 | 2, 1, 25 | cbvsumi 15640 | . . . 4 ⊢ Σ𝑗 ∈ 𝐵 ⦋𝑗 / 𝑘⦌𝐶 = Σ𝑘 ∈ 𝐵 𝐶 |
28 | 26, 27 | oveq12i 7413 | . . 3 ⊢ (Σ𝑗 ∈ 𝐴 ⦋𝑗 / 𝑘⦌𝐶 + Σ𝑗 ∈ 𝐵 ⦋𝑗 / 𝑘⦌𝐶) = (Σ𝑘 ∈ 𝐴 𝐶 + Σ𝑘 ∈ 𝐵 𝐶) |
29 | 28 | a1i 11 | . 2 ⊢ (𝜑 → (Σ𝑗 ∈ 𝐴 ⦋𝑗 / 𝑘⦌𝐶 + Σ𝑗 ∈ 𝐵 ⦋𝑗 / 𝑘⦌𝐶) = (Σ𝑘 ∈ 𝐴 𝐶 + Σ𝑘 ∈ 𝐵 𝐶)) |
30 | 5, 20, 29 | 3eqtrd 2768 | 1 ⊢ (𝜑 → Σ𝑘 ∈ 𝑈 𝐶 = (Σ𝑘 ∈ 𝐴 𝐶 + Σ𝑘 ∈ 𝐵 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 Ⅎwnf 1777 ∈ wcel 2098 ⦋csb 3885 ∪ cun 3938 ∩ cin 3939 ∅c0 4314 (class class class)co 7401 Fincfn 8935 ℂcc 11104 + caddc 11109 Σcsu 15629 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-inf2 9632 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-int 4941 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-se 5622 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-isom 6542 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-om 7849 df-1st 7968 df-2nd 7969 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-sup 9433 df-oi 9501 df-card 9930 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-div 11869 df-nn 12210 df-2 12272 df-3 12273 df-n0 12470 df-z 12556 df-uz 12820 df-rp 12972 df-fz 13482 df-fzo 13625 df-seq 13964 df-exp 14025 df-hash 14288 df-cj 15043 df-re 15044 df-im 15045 df-sqrt 15179 df-abs 15180 df-clim 15429 df-sum 15630 |
This theorem is referenced by: fsumsplitsn 15687 |
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