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Mirrors > Home > MPE Home > Th. List > isumadd | Structured version Visualization version GIF version |
Description: Addition of infinite sums. (Contributed by Mario Carneiro, 18-Aug-2013.) (Revised by Mario Carneiro, 23-Apr-2014.) |
Ref | Expression |
---|---|
isumadd.1 | ⊢ 𝑍 = (ℤ≥‘𝑀) |
isumadd.2 | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
isumadd.3 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐴) |
isumadd.4 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ ℂ) |
isumadd.5 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) = 𝐵) |
isumadd.6 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 ∈ ℂ) |
isumadd.7 | ⊢ (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ ) |
isumadd.8 | ⊢ (𝜑 → seq𝑀( + , 𝐺) ∈ dom ⇝ ) |
Ref | Expression |
---|---|
isumadd | ⊢ (𝜑 → Σ𝑘 ∈ 𝑍 (𝐴 + 𝐵) = (Σ𝑘 ∈ 𝑍 𝐴 + Σ𝑘 ∈ 𝑍 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isumadd.1 | . 2 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
2 | isumadd.2 | . 2 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
3 | fveq2 6842 | . . . . . 6 ⊢ (𝑚 = 𝑘 → (𝐹‘𝑚) = (𝐹‘𝑘)) | |
4 | fveq2 6842 | . . . . . 6 ⊢ (𝑚 = 𝑘 → (𝐺‘𝑚) = (𝐺‘𝑘)) | |
5 | 3, 4 | oveq12d 7375 | . . . . 5 ⊢ (𝑚 = 𝑘 → ((𝐹‘𝑚) + (𝐺‘𝑚)) = ((𝐹‘𝑘) + (𝐺‘𝑘))) |
6 | eqid 2736 | . . . . 5 ⊢ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚) + (𝐺‘𝑚))) = (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚) + (𝐺‘𝑚))) | |
7 | ovex 7390 | . . . . 5 ⊢ ((𝐹‘𝑘) + (𝐺‘𝑘)) ∈ V | |
8 | 5, 6, 7 | fvmpt 6948 | . . . 4 ⊢ (𝑘 ∈ 𝑍 → ((𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚) + (𝐺‘𝑚)))‘𝑘) = ((𝐹‘𝑘) + (𝐺‘𝑘))) |
9 | 8 | adantl 482 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚) + (𝐺‘𝑚)))‘𝑘) = ((𝐹‘𝑘) + (𝐺‘𝑘))) |
10 | isumadd.3 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐴) | |
11 | isumadd.5 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) = 𝐵) | |
12 | 10, 11 | oveq12d 7375 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝐹‘𝑘) + (𝐺‘𝑘)) = (𝐴 + 𝐵)) |
13 | 9, 12 | eqtrd 2776 | . 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚) + (𝐺‘𝑚)))‘𝑘) = (𝐴 + 𝐵)) |
14 | isumadd.4 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ ℂ) | |
15 | isumadd.6 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 ∈ ℂ) | |
16 | 14, 15 | addcld 11174 | . 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐴 + 𝐵) ∈ ℂ) |
17 | isumadd.7 | . . . 4 ⊢ (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ ) | |
18 | 1, 2, 10, 14, 17 | isumclim2 15643 | . . 3 ⊢ (𝜑 → seq𝑀( + , 𝐹) ⇝ Σ𝑘 ∈ 𝑍 𝐴) |
19 | seqex 13908 | . . . 4 ⊢ seq𝑀( + , (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚) + (𝐺‘𝑚)))) ∈ V | |
20 | 19 | a1i 11 | . . 3 ⊢ (𝜑 → seq𝑀( + , (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚) + (𝐺‘𝑚)))) ∈ V) |
21 | isumadd.8 | . . . 4 ⊢ (𝜑 → seq𝑀( + , 𝐺) ∈ dom ⇝ ) | |
22 | 1, 2, 11, 15, 21 | isumclim2 15643 | . . 3 ⊢ (𝜑 → seq𝑀( + , 𝐺) ⇝ Σ𝑘 ∈ 𝑍 𝐵) |
23 | 10, 14 | eqeltrd 2838 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ) |
24 | 1, 2, 23 | serf 13936 | . . . 4 ⊢ (𝜑 → seq𝑀( + , 𝐹):𝑍⟶ℂ) |
25 | 24 | ffvelcdmda 7035 | . . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (seq𝑀( + , 𝐹)‘𝑗) ∈ ℂ) |
26 | 11, 15 | eqeltrd 2838 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) ∈ ℂ) |
27 | 1, 2, 26 | serf 13936 | . . . 4 ⊢ (𝜑 → seq𝑀( + , 𝐺):𝑍⟶ℂ) |
28 | 27 | ffvelcdmda 7035 | . . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (seq𝑀( + , 𝐺)‘𝑗) ∈ ℂ) |
29 | simpr 485 | . . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ 𝑍) | |
30 | 29, 1 | eleqtrdi 2848 | . . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ (ℤ≥‘𝑀)) |
31 | simpll 765 | . . . . 5 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (𝑀...𝑗)) → 𝜑) | |
32 | elfzuz 13437 | . . . . . . 7 ⊢ (𝑘 ∈ (𝑀...𝑗) → 𝑘 ∈ (ℤ≥‘𝑀)) | |
33 | 32, 1 | eleqtrrdi 2849 | . . . . . 6 ⊢ (𝑘 ∈ (𝑀...𝑗) → 𝑘 ∈ 𝑍) |
34 | 33 | adantl 482 | . . . . 5 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (𝑀...𝑗)) → 𝑘 ∈ 𝑍) |
35 | 31, 34, 23 | syl2anc 584 | . . . 4 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (𝑀...𝑗)) → (𝐹‘𝑘) ∈ ℂ) |
36 | 31, 34, 26 | syl2anc 584 | . . . 4 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (𝑀...𝑗)) → (𝐺‘𝑘) ∈ ℂ) |
37 | 34, 8 | syl 17 | . . . 4 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (𝑀...𝑗)) → ((𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚) + (𝐺‘𝑚)))‘𝑘) = ((𝐹‘𝑘) + (𝐺‘𝑘))) |
38 | 30, 35, 36, 37 | seradd 13950 | . . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (seq𝑀( + , (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚) + (𝐺‘𝑚))))‘𝑗) = ((seq𝑀( + , 𝐹)‘𝑗) + (seq𝑀( + , 𝐺)‘𝑗))) |
39 | 1, 2, 18, 20, 22, 25, 28, 38 | climadd 15514 | . 2 ⊢ (𝜑 → seq𝑀( + , (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚) + (𝐺‘𝑚)))) ⇝ (Σ𝑘 ∈ 𝑍 𝐴 + Σ𝑘 ∈ 𝑍 𝐵)) |
40 | 1, 2, 13, 16, 39 | isumclim 15642 | 1 ⊢ (𝜑 → Σ𝑘 ∈ 𝑍 (𝐴 + 𝐵) = (Σ𝑘 ∈ 𝑍 𝐴 + Σ𝑘 ∈ 𝑍 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 Vcvv 3445 ↦ cmpt 5188 dom cdm 5633 ‘cfv 6496 (class class class)co 7357 ℂcc 11049 + caddc 11054 ℤcz 12499 ℤ≥cuz 12763 ...cfz 13424 seqcseq 13906 ⇝ cli 15366 Σcsu 15570 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-inf2 9577 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 ax-pre-sup 11129 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-se 5589 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-isom 6505 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-om 7803 df-1st 7921 df-2nd 7922 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-er 8648 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-sup 9378 df-oi 9446 df-card 9875 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-div 11813 df-nn 12154 df-2 12216 df-3 12217 df-n0 12414 df-z 12500 df-uz 12764 df-rp 12916 df-fz 13425 df-fzo 13568 df-seq 13907 df-exp 13968 df-hash 14231 df-cj 14984 df-re 14985 df-im 14986 df-sqrt 15120 df-abs 15121 df-clim 15370 df-sum 15571 |
This theorem is referenced by: sumsplit 15653 binomcxplemnotnn0 42626 |
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