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Mirrors > Home > MPE Home > Th. List > fsum2mul | Structured version Visualization version GIF version |
Description: Separate the nested sum of the product 𝐶(𝑗) · 𝐷(𝑘). (Contributed by NM, 13-Nov-2005.) (Revised by Mario Carneiro, 24-Apr-2014.) |
Ref | Expression |
---|---|
fsum2mul.1 | ⊢ (𝜑 → 𝐴 ∈ Fin) |
fsum2mul.2 | ⊢ (𝜑 → 𝐵 ∈ Fin) |
fsum2mul.3 | ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝐶 ∈ ℂ) |
fsum2mul.4 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → 𝐷 ∈ ℂ) |
Ref | Expression |
---|---|
fsum2mul | ⊢ (𝜑 → Σ𝑗 ∈ 𝐴 Σ𝑘 ∈ 𝐵 (𝐶 · 𝐷) = (Σ𝑗 ∈ 𝐴 𝐶 · Σ𝑘 ∈ 𝐵 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fsum2mul.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ Fin) | |
2 | fsum2mul.2 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ Fin) | |
3 | fsum2mul.4 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → 𝐷 ∈ ℂ) | |
4 | 2, 3 | fsumcl 14848 | . . 3 ⊢ (𝜑 → Σ𝑘 ∈ 𝐵 𝐷 ∈ ℂ) |
5 | fsum2mul.3 | . . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝐶 ∈ ℂ) | |
6 | 1, 4, 5 | fsummulc1 14898 | . 2 ⊢ (𝜑 → (Σ𝑗 ∈ 𝐴 𝐶 · Σ𝑘 ∈ 𝐵 𝐷) = Σ𝑗 ∈ 𝐴 (𝐶 · Σ𝑘 ∈ 𝐵 𝐷)) |
7 | 2 | adantr 474 | . . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝐵 ∈ Fin) |
8 | 3 | adantlr 706 | . . . 4 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝐴) ∧ 𝑘 ∈ 𝐵) → 𝐷 ∈ ℂ) |
9 | 7, 5, 8 | fsummulc2 14897 | . . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → (𝐶 · Σ𝑘 ∈ 𝐵 𝐷) = Σ𝑘 ∈ 𝐵 (𝐶 · 𝐷)) |
10 | 9 | sumeq2dv 14817 | . 2 ⊢ (𝜑 → Σ𝑗 ∈ 𝐴 (𝐶 · Σ𝑘 ∈ 𝐵 𝐷) = Σ𝑗 ∈ 𝐴 Σ𝑘 ∈ 𝐵 (𝐶 · 𝐷)) |
11 | 6, 10 | eqtr2d 2862 | 1 ⊢ (𝜑 → Σ𝑗 ∈ 𝐴 Σ𝑘 ∈ 𝐵 (𝐶 · 𝐷) = (Σ𝑗 ∈ 𝐴 𝐶 · Σ𝑘 ∈ 𝐵 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1656 ∈ wcel 2164 (class class class)co 6910 Fincfn 8228 ℂcc 10257 · cmul 10264 Σcsu 14800 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4996 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 ax-inf2 8822 ax-cnex 10315 ax-resscn 10316 ax-1cn 10317 ax-icn 10318 ax-addcl 10319 ax-addrcl 10320 ax-mulcl 10321 ax-mulrcl 10322 ax-mulcom 10323 ax-addass 10324 ax-mulass 10325 ax-distr 10326 ax-i2m1 10327 ax-1ne0 10328 ax-1rid 10329 ax-rnegex 10330 ax-rrecex 10331 ax-cnre 10332 ax-pre-lttri 10333 ax-pre-lttrn 10334 ax-pre-ltadd 10335 ax-pre-mulgt0 10336 ax-pre-sup 10337 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-fal 1670 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-tp 4404 df-op 4406 df-uni 4661 df-int 4700 df-iun 4744 df-br 4876 df-opab 4938 df-mpt 4955 df-tr 4978 df-id 5252 df-eprel 5257 df-po 5265 df-so 5266 df-fr 5305 df-se 5306 df-we 5307 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-pred 5924 df-ord 5970 df-on 5971 df-lim 5972 df-suc 5973 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-isom 6136 df-riota 6871 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-om 7332 df-1st 7433 df-2nd 7434 df-wrecs 7677 df-recs 7739 df-rdg 7777 df-1o 7831 df-oadd 7835 df-er 8014 df-en 8229 df-dom 8230 df-sdom 8231 df-fin 8232 df-sup 8623 df-oi 8691 df-card 9085 df-pnf 10400 df-mnf 10401 df-xr 10402 df-ltxr 10403 df-le 10404 df-sub 10594 df-neg 10595 df-div 11017 df-nn 11358 df-2 11421 df-3 11422 df-n0 11626 df-z 11712 df-uz 11976 df-rp 12120 df-fz 12627 df-fzo 12768 df-seq 13103 df-exp 13162 df-hash 13418 df-cj 14223 df-re 14224 df-im 14225 df-sqrt 14359 df-abs 14360 df-clim 14603 df-sum 14801 |
This theorem is referenced by: plymullem1 24376 breprexplemc 31255 |
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