![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > fsumsplitf | Structured version Visualization version GIF version |
Description: Split a sum into two parts. A version of fsumsplit 15774 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 3922 | . . . 4 ⊢ (𝑘 = 𝑗 → 𝐶 = ⦋𝑗 / 𝑘⦌𝐶) | |
2 | nfcv 2903 | . . . 4 ⊢ Ⅎ𝑗𝐶 | |
3 | nfcsb1v 3933 | . . . 4 ⊢ Ⅎ𝑘⦋𝑗 / 𝑘⦌𝐶 | |
4 | 1, 2, 3 | cbvsum 15728 | . . 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 1912 | . . . . . 6 ⊢ Ⅎ𝑘 𝑗 ∈ 𝑈 | |
11 | 9, 10 | nfan 1897 | . . . . 5 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑗 ∈ 𝑈) |
12 | 3 | nfel1 2920 | . . . . 5 ⊢ Ⅎ𝑘⦋𝑗 / 𝑘⦌𝐶 ∈ ℂ |
13 | 11, 12 | nfim 1894 | . . . 4 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑗 ∈ 𝑈) → ⦋𝑗 / 𝑘⦌𝐶 ∈ ℂ) |
14 | eleq1w 2822 | . . . . . 6 ⊢ (𝑘 = 𝑗 → (𝑘 ∈ 𝑈 ↔ 𝑗 ∈ 𝑈)) | |
15 | 14 | anbi2d 630 | . . . . 5 ⊢ (𝑘 = 𝑗 → ((𝜑 ∧ 𝑘 ∈ 𝑈) ↔ (𝜑 ∧ 𝑗 ∈ 𝑈))) |
16 | 1 | eleq1d 2824 | . . . . 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 2238 | . . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑈) → ⦋𝑗 / 𝑘⦌𝐶 ∈ ℂ) |
20 | 6, 7, 8, 19 | fsumsplit 15774 | . 2 ⊢ (𝜑 → Σ𝑗 ∈ 𝑈 ⦋𝑗 / 𝑘⦌𝐶 = (Σ𝑗 ∈ 𝐴 ⦋𝑗 / 𝑘⦌𝐶 + Σ𝑗 ∈ 𝐵 ⦋𝑗 / 𝑘⦌𝐶)) |
21 | csbeq1a 3922 | . . . . . 6 ⊢ (𝑗 = 𝑘 → ⦋𝑗 / 𝑘⦌𝐶 = ⦋𝑘 / 𝑗⦌⦋𝑗 / 𝑘⦌𝐶) | |
22 | csbcow 3923 | . . . . . . 7 ⊢ ⦋𝑘 / 𝑗⦌⦋𝑗 / 𝑘⦌𝐶 = ⦋𝑘 / 𝑘⦌𝐶 | |
23 | csbid 3921 | . . . . . . 7 ⊢ ⦋𝑘 / 𝑘⦌𝐶 = 𝐶 | |
24 | 22, 23 | eqtri 2763 | . . . . . 6 ⊢ ⦋𝑘 / 𝑗⦌⦋𝑗 / 𝑘⦌𝐶 = 𝐶 |
25 | 21, 24 | eqtrdi 2791 | . . . . 5 ⊢ (𝑗 = 𝑘 → ⦋𝑗 / 𝑘⦌𝐶 = 𝐶) |
26 | 25, 3, 2 | cbvsum 15728 | . . . 4 ⊢ Σ𝑗 ∈ 𝐴 ⦋𝑗 / 𝑘⦌𝐶 = Σ𝑘 ∈ 𝐴 𝐶 |
27 | 25, 3, 2 | cbvsum 15728 | . . . 4 ⊢ Σ𝑗 ∈ 𝐵 ⦋𝑗 / 𝑘⦌𝐶 = Σ𝑘 ∈ 𝐵 𝐶 |
28 | 26, 27 | oveq12i 7443 | . . 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 2779 | 1 ⊢ (𝜑 → Σ𝑘 ∈ 𝑈 𝐶 = (Σ𝑘 ∈ 𝐴 𝐶 + Σ𝑘 ∈ 𝐵 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 Ⅎwnf 1780 ∈ wcel 2106 ⦋csb 3908 ∪ cun 3961 ∩ cin 3962 ∅c0 4339 (class class class)co 7431 Fincfn 8984 ℂcc 11151 + caddc 11156 Σcsu 15719 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-inf2 9679 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-sup 9480 df-oi 9548 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-n0 12525 df-z 12612 df-uz 12877 df-rp 13033 df-fz 13545 df-fzo 13692 df-seq 14040 df-exp 14100 df-hash 14367 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-clim 15521 df-sum 15720 |
This theorem is referenced by: fsumsplitsn 15777 |
Copyright terms: Public domain | W3C validator |