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Mirrors > Home > ILE Home > Th. List > fsumsplitf | GIF version |
Description: Split a sum into two parts. A version of fsumsplit 11418 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 | csbeq1a 3068 | . . . 4 ⊢ (𝑘 = 𝑗 → 𝐶 = ⦋𝑗 / 𝑘⦌𝐶) | |
2 | nfcv 2319 | . . . 4 ⊢ Ⅎ𝑗𝑈 | |
3 | nfcv 2319 | . . . 4 ⊢ Ⅎ𝑘𝑈 | |
4 | nfcv 2319 | . . . 4 ⊢ Ⅎ𝑗𝐶 | |
5 | nfcsb1v 3092 | . . . 4 ⊢ Ⅎ𝑘⦋𝑗 / 𝑘⦌𝐶 | |
6 | 1, 2, 3, 4, 5 | cbvsum 11371 | . . 3 ⊢ Σ𝑘 ∈ 𝑈 𝐶 = Σ𝑗 ∈ 𝑈 ⦋𝑗 / 𝑘⦌𝐶 |
7 | 6 | a1i 9 | . 2 ⊢ (𝜑 → Σ𝑘 ∈ 𝑈 𝐶 = Σ𝑗 ∈ 𝑈 ⦋𝑗 / 𝑘⦌𝐶) |
8 | fsumsplitf.ab | . . 3 ⊢ (𝜑 → (𝐴 ∩ 𝐵) = ∅) | |
9 | fsumsplitf.u | . . 3 ⊢ (𝜑 → 𝑈 = (𝐴 ∪ 𝐵)) | |
10 | fsumsplitf.fi | . . 3 ⊢ (𝜑 → 𝑈 ∈ Fin) | |
11 | fsumsplitf.ph | . . . . . 6 ⊢ Ⅎ𝑘𝜑 | |
12 | nfv 1528 | . . . . . 6 ⊢ Ⅎ𝑘 𝑗 ∈ 𝑈 | |
13 | 11, 12 | nfan 1565 | . . . . 5 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑗 ∈ 𝑈) |
14 | 5 | nfel1 2330 | . . . . 5 ⊢ Ⅎ𝑘⦋𝑗 / 𝑘⦌𝐶 ∈ ℂ |
15 | 13, 14 | nfim 1572 | . . . 4 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑗 ∈ 𝑈) → ⦋𝑗 / 𝑘⦌𝐶 ∈ ℂ) |
16 | eleq1w 2238 | . . . . . 6 ⊢ (𝑘 = 𝑗 → (𝑘 ∈ 𝑈 ↔ 𝑗 ∈ 𝑈)) | |
17 | 16 | anbi2d 464 | . . . . 5 ⊢ (𝑘 = 𝑗 → ((𝜑 ∧ 𝑘 ∈ 𝑈) ↔ (𝜑 ∧ 𝑗 ∈ 𝑈))) |
18 | 1 | eleq1d 2246 | . . . . 5 ⊢ (𝑘 = 𝑗 → (𝐶 ∈ ℂ ↔ ⦋𝑗 / 𝑘⦌𝐶 ∈ ℂ)) |
19 | 17, 18 | imbi12d 234 | . . . 4 ⊢ (𝑘 = 𝑗 → (((𝜑 ∧ 𝑘 ∈ 𝑈) → 𝐶 ∈ ℂ) ↔ ((𝜑 ∧ 𝑗 ∈ 𝑈) → ⦋𝑗 / 𝑘⦌𝐶 ∈ ℂ))) |
20 | fsumsplitf.c | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → 𝐶 ∈ ℂ) | |
21 | 15, 19, 20 | chvar 1757 | . . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑈) → ⦋𝑗 / 𝑘⦌𝐶 ∈ ℂ) |
22 | 8, 9, 10, 21 | fsumsplit 11418 | . 2 ⊢ (𝜑 → Σ𝑗 ∈ 𝑈 ⦋𝑗 / 𝑘⦌𝐶 = (Σ𝑗 ∈ 𝐴 ⦋𝑗 / 𝑘⦌𝐶 + Σ𝑗 ∈ 𝐵 ⦋𝑗 / 𝑘⦌𝐶)) |
23 | csbeq1a 3068 | . . . . . . 7 ⊢ (𝑗 = 𝑘 → ⦋𝑗 / 𝑘⦌𝐶 = ⦋𝑘 / 𝑗⦌⦋𝑗 / 𝑘⦌𝐶) | |
24 | csbco 3069 | . . . . . . . . 9 ⊢ ⦋𝑘 / 𝑗⦌⦋𝑗 / 𝑘⦌𝐶 = ⦋𝑘 / 𝑘⦌𝐶 | |
25 | csbid 3067 | . . . . . . . . 9 ⊢ ⦋𝑘 / 𝑘⦌𝐶 = 𝐶 | |
26 | 24, 25 | eqtri 2198 | . . . . . . . 8 ⊢ ⦋𝑘 / 𝑗⦌⦋𝑗 / 𝑘⦌𝐶 = 𝐶 |
27 | 26 | a1i 9 | . . . . . . 7 ⊢ (𝑗 = 𝑘 → ⦋𝑘 / 𝑗⦌⦋𝑗 / 𝑘⦌𝐶 = 𝐶) |
28 | 23, 27 | eqtrd 2210 | . . . . . 6 ⊢ (𝑗 = 𝑘 → ⦋𝑗 / 𝑘⦌𝐶 = 𝐶) |
29 | nfcv 2319 | . . . . . 6 ⊢ Ⅎ𝑘𝐴 | |
30 | nfcv 2319 | . . . . . 6 ⊢ Ⅎ𝑗𝐴 | |
31 | 28, 29, 30, 5, 4 | cbvsum 11371 | . . . . 5 ⊢ Σ𝑗 ∈ 𝐴 ⦋𝑗 / 𝑘⦌𝐶 = Σ𝑘 ∈ 𝐴 𝐶 |
32 | eqid 2177 | . . . . 5 ⊢ Σ𝑘 ∈ 𝐴 𝐶 = Σ𝑘 ∈ 𝐴 𝐶 | |
33 | 31, 32 | eqtri 2198 | . . . 4 ⊢ Σ𝑗 ∈ 𝐴 ⦋𝑗 / 𝑘⦌𝐶 = Σ𝑘 ∈ 𝐴 𝐶 |
34 | 5, 4, 28 | cbvsumi 11373 | . . . 4 ⊢ Σ𝑗 ∈ 𝐵 ⦋𝑗 / 𝑘⦌𝐶 = Σ𝑘 ∈ 𝐵 𝐶 |
35 | 33, 34 | oveq12i 5890 | . . 3 ⊢ (Σ𝑗 ∈ 𝐴 ⦋𝑗 / 𝑘⦌𝐶 + Σ𝑗 ∈ 𝐵 ⦋𝑗 / 𝑘⦌𝐶) = (Σ𝑘 ∈ 𝐴 𝐶 + Σ𝑘 ∈ 𝐵 𝐶) |
36 | 35 | a1i 9 | . 2 ⊢ (𝜑 → (Σ𝑗 ∈ 𝐴 ⦋𝑗 / 𝑘⦌𝐶 + Σ𝑗 ∈ 𝐵 ⦋𝑗 / 𝑘⦌𝐶) = (Σ𝑘 ∈ 𝐴 𝐶 + Σ𝑘 ∈ 𝐵 𝐶)) |
37 | 7, 22, 36 | 3eqtrd 2214 | 1 ⊢ (𝜑 → Σ𝑘 ∈ 𝑈 𝐶 = (Σ𝑘 ∈ 𝐴 𝐶 + Σ𝑘 ∈ 𝐵 𝐶)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1353 Ⅎwnf 1460 ∈ wcel 2148 ⦋csb 3059 ∪ cun 3129 ∩ cin 3130 ∅c0 3424 (class class class)co 5878 Fincfn 6743 ℂcc 7812 + caddc 7817 Σcsu 11364 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4120 ax-sep 4123 ax-nul 4131 ax-pow 4176 ax-pr 4211 ax-un 4435 ax-setind 4538 ax-iinf 4589 ax-cnex 7905 ax-resscn 7906 ax-1cn 7907 ax-1re 7908 ax-icn 7909 ax-addcl 7910 ax-addrcl 7911 ax-mulcl 7912 ax-mulrcl 7913 ax-addcom 7914 ax-mulcom 7915 ax-addass 7916 ax-mulass 7917 ax-distr 7918 ax-i2m1 7919 ax-0lt1 7920 ax-1rid 7921 ax-0id 7922 ax-rnegex 7923 ax-precex 7924 ax-cnre 7925 ax-pre-ltirr 7926 ax-pre-ltwlin 7927 ax-pre-lttrn 7928 ax-pre-apti 7929 ax-pre-ltadd 7930 ax-pre-mulgt0 7931 ax-pre-mulext 7932 ax-arch 7933 ax-caucvg 7934 |
This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2741 df-sbc 2965 df-csb 3060 df-dif 3133 df-un 3135 df-in 3137 df-ss 3144 df-nul 3425 df-if 3537 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-int 3847 df-iun 3890 df-br 4006 df-opab 4067 df-mpt 4068 df-tr 4104 df-id 4295 df-po 4298 df-iso 4299 df-iord 4368 df-on 4370 df-ilim 4371 df-suc 4373 df-iom 4592 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-rn 4639 df-res 4640 df-ima 4641 df-iota 5180 df-fun 5220 df-fn 5221 df-f 5222 df-f1 5223 df-fo 5224 df-f1o 5225 df-fv 5226 df-isom 5227 df-riota 5834 df-ov 5881 df-oprab 5882 df-mpo 5883 df-1st 6144 df-2nd 6145 df-recs 6309 df-irdg 6374 df-frec 6395 df-1o 6420 df-oadd 6424 df-er 6538 df-en 6744 df-dom 6745 df-fin 6746 df-pnf 7997 df-mnf 7998 df-xr 7999 df-ltxr 8000 df-le 8001 df-sub 8133 df-neg 8134 df-reap 8535 df-ap 8542 df-div 8633 df-inn 8923 df-2 8981 df-3 8982 df-4 8983 df-n0 9180 df-z 9257 df-uz 9532 df-q 9623 df-rp 9657 df-fz 10012 df-fzo 10146 df-seqfrec 10449 df-exp 10523 df-ihash 10759 df-cj 10854 df-re 10855 df-im 10856 df-rsqrt 11010 df-abs 11011 df-clim 11290 df-sumdc 11365 |
This theorem is referenced by: fsumsplitsn 11421 |
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