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Mirrors > Home > MPE Home > Th. List > Mathboxes > fsummulc1f | Structured version Visualization version GIF version |
Description: Closure of a finite sum of complex numbers 𝐴(𝑘). A version of fsummulc1 15349 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 5-Apr-2020.) |
Ref | Expression |
---|---|
fsummulc1f.ph | ⊢ Ⅎ𝑘𝜑 |
fsummulclf.a | ⊢ (𝜑 → 𝐴 ∈ Fin) |
fsummulclf.c | ⊢ (𝜑 → 𝐶 ∈ ℂ) |
fsummulclf.b | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) |
Ref | Expression |
---|---|
fsummulc1f | ⊢ (𝜑 → (Σ𝑘 ∈ 𝐴 𝐵 · 𝐶) = Σ𝑘 ∈ 𝐴 (𝐵 · 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | csbeq1a 3825 | . . . . 5 ⊢ (𝑘 = 𝑗 → 𝐵 = ⦋𝑗 / 𝑘⦌𝐵) | |
2 | nfcv 2904 | . . . . 5 ⊢ Ⅎ𝑗𝐴 | |
3 | nfcv 2904 | . . . . 5 ⊢ Ⅎ𝑘𝐴 | |
4 | nfcv 2904 | . . . . 5 ⊢ Ⅎ𝑗𝐵 | |
5 | nfcsb1v 3836 | . . . . 5 ⊢ Ⅎ𝑘⦋𝑗 / 𝑘⦌𝐵 | |
6 | 1, 2, 3, 4, 5 | cbvsum 15259 | . . . 4 ⊢ Σ𝑘 ∈ 𝐴 𝐵 = Σ𝑗 ∈ 𝐴 ⦋𝑗 / 𝑘⦌𝐵 |
7 | 6 | oveq1i 7223 | . . 3 ⊢ (Σ𝑘 ∈ 𝐴 𝐵 · 𝐶) = (Σ𝑗 ∈ 𝐴 ⦋𝑗 / 𝑘⦌𝐵 · 𝐶) |
8 | 7 | a1i 11 | . 2 ⊢ (𝜑 → (Σ𝑘 ∈ 𝐴 𝐵 · 𝐶) = (Σ𝑗 ∈ 𝐴 ⦋𝑗 / 𝑘⦌𝐵 · 𝐶)) |
9 | fsummulclf.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ Fin) | |
10 | fsummulclf.c | . . 3 ⊢ (𝜑 → 𝐶 ∈ ℂ) | |
11 | fsummulc1f.ph | . . . . . 6 ⊢ Ⅎ𝑘𝜑 | |
12 | nfv 1922 | . . . . . 6 ⊢ Ⅎ𝑘 𝑗 ∈ 𝐴 | |
13 | 11, 12 | nfan 1907 | . . . . 5 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑗 ∈ 𝐴) |
14 | 5 | nfel1 2920 | . . . . 5 ⊢ Ⅎ𝑘⦋𝑗 / 𝑘⦌𝐵 ∈ ℂ |
15 | 13, 14 | nfim 1904 | . . . 4 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑗 ∈ 𝐴) → ⦋𝑗 / 𝑘⦌𝐵 ∈ ℂ) |
16 | eleq1w 2820 | . . . . . 6 ⊢ (𝑘 = 𝑗 → (𝑘 ∈ 𝐴 ↔ 𝑗 ∈ 𝐴)) | |
17 | 16 | anbi2d 632 | . . . . 5 ⊢ (𝑘 = 𝑗 → ((𝜑 ∧ 𝑘 ∈ 𝐴) ↔ (𝜑 ∧ 𝑗 ∈ 𝐴))) |
18 | 1 | eleq1d 2822 | . . . . 5 ⊢ (𝑘 = 𝑗 → (𝐵 ∈ ℂ ↔ ⦋𝑗 / 𝑘⦌𝐵 ∈ ℂ)) |
19 | 17, 18 | imbi12d 348 | . . . 4 ⊢ (𝑘 = 𝑗 → (((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) ↔ ((𝜑 ∧ 𝑗 ∈ 𝐴) → ⦋𝑗 / 𝑘⦌𝐵 ∈ ℂ))) |
20 | fsummulclf.b | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) | |
21 | 15, 19, 20 | chvarfv 2238 | . . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → ⦋𝑗 / 𝑘⦌𝐵 ∈ ℂ) |
22 | 9, 10, 21 | fsummulc1 15349 | . 2 ⊢ (𝜑 → (Σ𝑗 ∈ 𝐴 ⦋𝑗 / 𝑘⦌𝐵 · 𝐶) = Σ𝑗 ∈ 𝐴 (⦋𝑗 / 𝑘⦌𝐵 · 𝐶)) |
23 | eqcom 2744 | . . . . . . . 8 ⊢ (𝑘 = 𝑗 ↔ 𝑗 = 𝑘) | |
24 | 23 | imbi1i 353 | . . . . . . 7 ⊢ ((𝑘 = 𝑗 → 𝐵 = ⦋𝑗 / 𝑘⦌𝐵) ↔ (𝑗 = 𝑘 → 𝐵 = ⦋𝑗 / 𝑘⦌𝐵)) |
25 | eqcom 2744 | . . . . . . . 8 ⊢ (𝐵 = ⦋𝑗 / 𝑘⦌𝐵 ↔ ⦋𝑗 / 𝑘⦌𝐵 = 𝐵) | |
26 | 25 | imbi2i 339 | . . . . . . 7 ⊢ ((𝑗 = 𝑘 → 𝐵 = ⦋𝑗 / 𝑘⦌𝐵) ↔ (𝑗 = 𝑘 → ⦋𝑗 / 𝑘⦌𝐵 = 𝐵)) |
27 | 24, 26 | bitri 278 | . . . . . 6 ⊢ ((𝑘 = 𝑗 → 𝐵 = ⦋𝑗 / 𝑘⦌𝐵) ↔ (𝑗 = 𝑘 → ⦋𝑗 / 𝑘⦌𝐵 = 𝐵)) |
28 | 1, 27 | mpbi 233 | . . . . 5 ⊢ (𝑗 = 𝑘 → ⦋𝑗 / 𝑘⦌𝐵 = 𝐵) |
29 | 28 | oveq1d 7228 | . . . 4 ⊢ (𝑗 = 𝑘 → (⦋𝑗 / 𝑘⦌𝐵 · 𝐶) = (𝐵 · 𝐶)) |
30 | nfcv 2904 | . . . . 5 ⊢ Ⅎ𝑘 · | |
31 | nfcv 2904 | . . . . 5 ⊢ Ⅎ𝑘𝐶 | |
32 | 5, 30, 31 | nfov 7243 | . . . 4 ⊢ Ⅎ𝑘(⦋𝑗 / 𝑘⦌𝐵 · 𝐶) |
33 | nfcv 2904 | . . . 4 ⊢ Ⅎ𝑗(𝐵 · 𝐶) | |
34 | 29, 3, 2, 32, 33 | cbvsum 15259 | . . 3 ⊢ Σ𝑗 ∈ 𝐴 (⦋𝑗 / 𝑘⦌𝐵 · 𝐶) = Σ𝑘 ∈ 𝐴 (𝐵 · 𝐶) |
35 | 34 | a1i 11 | . 2 ⊢ (𝜑 → Σ𝑗 ∈ 𝐴 (⦋𝑗 / 𝑘⦌𝐵 · 𝐶) = Σ𝑘 ∈ 𝐴 (𝐵 · 𝐶)) |
36 | 8, 22, 35 | 3eqtrd 2781 | 1 ⊢ (𝜑 → (Σ𝑘 ∈ 𝐴 𝐵 · 𝐶) = Σ𝑘 ∈ 𝐴 (𝐵 · 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 Ⅎwnf 1791 ∈ wcel 2110 ⦋csb 3811 (class class class)co 7213 Fincfn 8626 ℂcc 10727 · cmul 10734 Σcsu 15249 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-inf2 9256 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 ax-pre-sup 10807 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-int 4860 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-se 5510 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-isom 6389 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-om 7645 df-1st 7761 df-2nd 7762 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-1o 8202 df-er 8391 df-en 8627 df-dom 8628 df-sdom 8629 df-fin 8630 df-sup 9058 df-oi 9126 df-card 9555 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-div 11490 df-nn 11831 df-2 11893 df-3 11894 df-n0 12091 df-z 12177 df-uz 12439 df-rp 12587 df-fz 13096 df-fzo 13239 df-seq 13575 df-exp 13636 df-hash 13897 df-cj 14662 df-re 14663 df-im 14664 df-sqrt 14798 df-abs 14799 df-clim 15049 df-sum 15250 |
This theorem is referenced by: dvmptfprodlem 43160 |
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