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Mirrors > Home > MPE Home > Th. List > seqcaopr | Structured version Visualization version GIF version |
Description: The sum of two infinite series (generalized to an arbitrary commutative and associative operation). (Contributed by NM, 17-Mar-2005.) (Revised by Mario Carneiro, 30-May-2014.) |
Ref | Expression |
---|---|
seqcaopr.1 | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) |
seqcaopr.2 | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) = (𝑦 + 𝑥)) |
seqcaopr.3 | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))) |
seqcaopr.4 | ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) |
seqcaopr.5 | ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) ∈ 𝑆) |
seqcaopr.6 | ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐺‘𝑘) ∈ 𝑆) |
seqcaopr.7 | ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐻‘𝑘) = ((𝐹‘𝑘) + (𝐺‘𝑘))) |
Ref | Expression |
---|---|
seqcaopr | ⊢ (𝜑 → (seq𝑀( + , 𝐻)‘𝑁) = ((seq𝑀( + , 𝐹)‘𝑁) + (seq𝑀( + , 𝐺)‘𝑁))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | seqcaopr.1 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) | |
2 | 1 | caovclg 7607 | . 2 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆)) → (𝑎 + 𝑏) ∈ 𝑆) |
3 | simpl 481 | . . . . . . 7 ⊢ ((𝜑 ∧ ((𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆) ∧ (𝑐 ∈ 𝑆 ∧ 𝑑 ∈ 𝑆))) → 𝜑) | |
4 | simprrl 779 | . . . . . . 7 ⊢ ((𝜑 ∧ ((𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆) ∧ (𝑐 ∈ 𝑆 ∧ 𝑑 ∈ 𝑆))) → 𝑐 ∈ 𝑆) | |
5 | simprlr 778 | . . . . . . 7 ⊢ ((𝜑 ∧ ((𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆) ∧ (𝑐 ∈ 𝑆 ∧ 𝑑 ∈ 𝑆))) → 𝑏 ∈ 𝑆) | |
6 | seqcaopr.2 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) = (𝑦 + 𝑥)) | |
7 | 6 | caovcomg 7610 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑐 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆)) → (𝑐 + 𝑏) = (𝑏 + 𝑐)) |
8 | 3, 4, 5, 7 | syl12anc 835 | . . . . . 6 ⊢ ((𝜑 ∧ ((𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆) ∧ (𝑐 ∈ 𝑆 ∧ 𝑑 ∈ 𝑆))) → (𝑐 + 𝑏) = (𝑏 + 𝑐)) |
9 | 8 | oveq1d 7428 | . . . . 5 ⊢ ((𝜑 ∧ ((𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆) ∧ (𝑐 ∈ 𝑆 ∧ 𝑑 ∈ 𝑆))) → ((𝑐 + 𝑏) + 𝑑) = ((𝑏 + 𝑐) + 𝑑)) |
10 | simprrr 780 | . . . . . 6 ⊢ ((𝜑 ∧ ((𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆) ∧ (𝑐 ∈ 𝑆 ∧ 𝑑 ∈ 𝑆))) → 𝑑 ∈ 𝑆) | |
11 | seqcaopr.3 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))) | |
12 | 11 | caovassg 7613 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑐 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆 ∧ 𝑑 ∈ 𝑆)) → ((𝑐 + 𝑏) + 𝑑) = (𝑐 + (𝑏 + 𝑑))) |
13 | 3, 4, 5, 10, 12 | syl13anc 1369 | . . . . 5 ⊢ ((𝜑 ∧ ((𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆) ∧ (𝑐 ∈ 𝑆 ∧ 𝑑 ∈ 𝑆))) → ((𝑐 + 𝑏) + 𝑑) = (𝑐 + (𝑏 + 𝑑))) |
14 | 11 | caovassg 7613 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝑆 ∧ 𝑐 ∈ 𝑆 ∧ 𝑑 ∈ 𝑆)) → ((𝑏 + 𝑐) + 𝑑) = (𝑏 + (𝑐 + 𝑑))) |
15 | 3, 5, 4, 10, 14 | syl13anc 1369 | . . . . 5 ⊢ ((𝜑 ∧ ((𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆) ∧ (𝑐 ∈ 𝑆 ∧ 𝑑 ∈ 𝑆))) → ((𝑏 + 𝑐) + 𝑑) = (𝑏 + (𝑐 + 𝑑))) |
16 | 9, 13, 15 | 3eqtr3d 2774 | . . . 4 ⊢ ((𝜑 ∧ ((𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆) ∧ (𝑐 ∈ 𝑆 ∧ 𝑑 ∈ 𝑆))) → (𝑐 + (𝑏 + 𝑑)) = (𝑏 + (𝑐 + 𝑑))) |
17 | 16 | oveq2d 7429 | . . 3 ⊢ ((𝜑 ∧ ((𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆) ∧ (𝑐 ∈ 𝑆 ∧ 𝑑 ∈ 𝑆))) → (𝑎 + (𝑐 + (𝑏 + 𝑑))) = (𝑎 + (𝑏 + (𝑐 + 𝑑)))) |
18 | simprll 777 | . . . 4 ⊢ ((𝜑 ∧ ((𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆) ∧ (𝑐 ∈ 𝑆 ∧ 𝑑 ∈ 𝑆))) → 𝑎 ∈ 𝑆) | |
19 | 1 | caovclg 7607 | . . . . 5 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝑆 ∧ 𝑑 ∈ 𝑆)) → (𝑏 + 𝑑) ∈ 𝑆) |
20 | 3, 5, 10, 19 | syl12anc 835 | . . . 4 ⊢ ((𝜑 ∧ ((𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆) ∧ (𝑐 ∈ 𝑆 ∧ 𝑑 ∈ 𝑆))) → (𝑏 + 𝑑) ∈ 𝑆) |
21 | 11 | caovassg 7613 | . . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑆 ∧ 𝑐 ∈ 𝑆 ∧ (𝑏 + 𝑑) ∈ 𝑆)) → ((𝑎 + 𝑐) + (𝑏 + 𝑑)) = (𝑎 + (𝑐 + (𝑏 + 𝑑)))) |
22 | 3, 18, 4, 20, 21 | syl13anc 1369 | . . 3 ⊢ ((𝜑 ∧ ((𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆) ∧ (𝑐 ∈ 𝑆 ∧ 𝑑 ∈ 𝑆))) → ((𝑎 + 𝑐) + (𝑏 + 𝑑)) = (𝑎 + (𝑐 + (𝑏 + 𝑑)))) |
23 | 1 | caovclg 7607 | . . . . 5 ⊢ ((𝜑 ∧ (𝑐 ∈ 𝑆 ∧ 𝑑 ∈ 𝑆)) → (𝑐 + 𝑑) ∈ 𝑆) |
24 | 23 | adantrl 714 | . . . 4 ⊢ ((𝜑 ∧ ((𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆) ∧ (𝑐 ∈ 𝑆 ∧ 𝑑 ∈ 𝑆))) → (𝑐 + 𝑑) ∈ 𝑆) |
25 | 11 | caovassg 7613 | . . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆 ∧ (𝑐 + 𝑑) ∈ 𝑆)) → ((𝑎 + 𝑏) + (𝑐 + 𝑑)) = (𝑎 + (𝑏 + (𝑐 + 𝑑)))) |
26 | 3, 18, 5, 24, 25 | syl13anc 1369 | . . 3 ⊢ ((𝜑 ∧ ((𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆) ∧ (𝑐 ∈ 𝑆 ∧ 𝑑 ∈ 𝑆))) → ((𝑎 + 𝑏) + (𝑐 + 𝑑)) = (𝑎 + (𝑏 + (𝑐 + 𝑑)))) |
27 | 17, 22, 26 | 3eqtr4d 2776 | . 2 ⊢ ((𝜑 ∧ ((𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆) ∧ (𝑐 ∈ 𝑆 ∧ 𝑑 ∈ 𝑆))) → ((𝑎 + 𝑐) + (𝑏 + 𝑑)) = ((𝑎 + 𝑏) + (𝑐 + 𝑑))) |
28 | seqcaopr.4 | . 2 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) | |
29 | seqcaopr.5 | . 2 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) ∈ 𝑆) | |
30 | seqcaopr.6 | . 2 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐺‘𝑘) ∈ 𝑆) | |
31 | seqcaopr.7 | . 2 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐻‘𝑘) = ((𝐹‘𝑘) + (𝐺‘𝑘))) | |
32 | 2, 2, 27, 28, 29, 30, 31 | seqcaopr2 14049 | 1 ⊢ (𝜑 → (seq𝑀( + , 𝐻)‘𝑁) = ((seq𝑀( + , 𝐹)‘𝑁) + (seq𝑀( + , 𝐺)‘𝑁))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∧ w3a 1084 = wceq 1534 ∈ wcel 2099 ‘cfv 6543 (class class class)co 7413 ℤ≥cuz 12865 ...cfz 13529 seqcseq 14012 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7735 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-iun 4995 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6302 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7866 df-1st 7992 df-2nd 7993 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-er 8723 df-en 8964 df-dom 8965 df-sdom 8966 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-nn 12256 df-n0 12516 df-z 12602 df-uz 12866 df-fz 13530 df-fzo 13673 df-seq 14013 |
This theorem is referenced by: seradd 14055 prodfmul 15886 mulgnn0di 19816 |
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