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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 7596 | . 2 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆)) → (𝑎 + 𝑏) ∈ 𝑆) |
3 | simpl 482 | . . . . . . 7 ⊢ ((𝜑 ∧ ((𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆) ∧ (𝑐 ∈ 𝑆 ∧ 𝑑 ∈ 𝑆))) → 𝜑) | |
4 | simprrl 778 | . . . . . . 7 ⊢ ((𝜑 ∧ ((𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆) ∧ (𝑐 ∈ 𝑆 ∧ 𝑑 ∈ 𝑆))) → 𝑐 ∈ 𝑆) | |
5 | simprlr 777 | . . . . . . 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 7599 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑐 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆)) → (𝑐 + 𝑏) = (𝑏 + 𝑐)) |
8 | 3, 4, 5, 7 | syl12anc 834 | . . . . . 6 ⊢ ((𝜑 ∧ ((𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆) ∧ (𝑐 ∈ 𝑆 ∧ 𝑑 ∈ 𝑆))) → (𝑐 + 𝑏) = (𝑏 + 𝑐)) |
9 | 8 | oveq1d 7420 | . . . . 5 ⊢ ((𝜑 ∧ ((𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆) ∧ (𝑐 ∈ 𝑆 ∧ 𝑑 ∈ 𝑆))) → ((𝑐 + 𝑏) + 𝑑) = ((𝑏 + 𝑐) + 𝑑)) |
10 | simprrr 779 | . . . . . 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 7602 | . . . . . 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 7602 | . . . . . 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 7421 | . . 3 ⊢ ((𝜑 ∧ ((𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆) ∧ (𝑐 ∈ 𝑆 ∧ 𝑑 ∈ 𝑆))) → (𝑎 + (𝑐 + (𝑏 + 𝑑))) = (𝑎 + (𝑏 + (𝑐 + 𝑑)))) |
18 | simprll 776 | . . . 4 ⊢ ((𝜑 ∧ ((𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆) ∧ (𝑐 ∈ 𝑆 ∧ 𝑑 ∈ 𝑆))) → 𝑎 ∈ 𝑆) | |
19 | 1 | caovclg 7596 | . . . . 5 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝑆 ∧ 𝑑 ∈ 𝑆)) → (𝑏 + 𝑑) ∈ 𝑆) |
20 | 3, 5, 10, 19 | syl12anc 834 | . . . 4 ⊢ ((𝜑 ∧ ((𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆) ∧ (𝑐 ∈ 𝑆 ∧ 𝑑 ∈ 𝑆))) → (𝑏 + 𝑑) ∈ 𝑆) |
21 | 11 | caovassg 7602 | . . . 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 7596 | . . . . 5 ⊢ ((𝜑 ∧ (𝑐 ∈ 𝑆 ∧ 𝑑 ∈ 𝑆)) → (𝑐 + 𝑑) ∈ 𝑆) |
24 | 23 | adantrl 713 | . . . 4 ⊢ ((𝜑 ∧ ((𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆) ∧ (𝑐 ∈ 𝑆 ∧ 𝑑 ∈ 𝑆))) → (𝑐 + 𝑑) ∈ 𝑆) |
25 | 11 | caovassg 7602 | . . . 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 14009 | 1 ⊢ (𝜑 → (seq𝑀( + , 𝐻)‘𝑁) = ((seq𝑀( + , 𝐹)‘𝑁) + (seq𝑀( + , 𝐺)‘𝑁))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ‘cfv 6537 (class class class)co 7405 ℤ≥cuz 12826 ...cfz 13490 seqcseq 13972 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-n0 12477 df-z 12563 df-uz 12827 df-fz 13491 df-fzo 13634 df-seq 13973 |
This theorem is referenced by: seradd 14015 prodfmul 15842 mulgnn0di 19745 |
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