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Mirrors > Home > MPE Home > Th. List > isershft | Structured version Visualization version GIF version |
Description: Index shift of the limit of an infinite series. (Contributed by Mario Carneiro, 6-Sep-2013.) (Revised by Mario Carneiro, 5-Nov-2013.) |
Ref | Expression |
---|---|
isershft.1 | ⊢ 𝐹 ∈ V |
Ref | Expression |
---|---|
isershft | ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (seq𝑀( + , 𝐹) ⇝ 𝐴 ↔ seq(𝑀 + 𝑁)( + , (𝐹 shift 𝑁)) ⇝ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zaddcl 12606 | . . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 + 𝑁) ∈ ℤ) | |
2 | isershft.1 | . . . . . 6 ⊢ 𝐹 ∈ V | |
3 | 2 | seqshft 15038 | . . . . 5 ⊢ (((𝑀 + 𝑁) ∈ ℤ ∧ 𝑁 ∈ ℤ) → seq(𝑀 + 𝑁)( + , (𝐹 shift 𝑁)) = (seq((𝑀 + 𝑁) − 𝑁)( + , 𝐹) shift 𝑁)) |
4 | 1, 3 | sylancom 587 | . . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → seq(𝑀 + 𝑁)( + , (𝐹 shift 𝑁)) = (seq((𝑀 + 𝑁) − 𝑁)( + , 𝐹) shift 𝑁)) |
5 | zcn 12567 | . . . . . . 7 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℂ) | |
6 | zcn 12567 | . . . . . . 7 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ) | |
7 | pncan 11470 | . . . . . . 7 ⊢ ((𝑀 ∈ ℂ ∧ 𝑁 ∈ ℂ) → ((𝑀 + 𝑁) − 𝑁) = 𝑀) | |
8 | 5, 6, 7 | syl2an 595 | . . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀 + 𝑁) − 𝑁) = 𝑀) |
9 | 8 | seqeq1d 13978 | . . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → seq((𝑀 + 𝑁) − 𝑁)( + , 𝐹) = seq𝑀( + , 𝐹)) |
10 | 9 | oveq1d 7420 | . . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (seq((𝑀 + 𝑁) − 𝑁)( + , 𝐹) shift 𝑁) = (seq𝑀( + , 𝐹) shift 𝑁)) |
11 | 4, 10 | eqtrd 2766 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → seq(𝑀 + 𝑁)( + , (𝐹 shift 𝑁)) = (seq𝑀( + , 𝐹) shift 𝑁)) |
12 | 11 | breq1d 5151 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (seq(𝑀 + 𝑁)( + , (𝐹 shift 𝑁)) ⇝ 𝐴 ↔ (seq𝑀( + , 𝐹) shift 𝑁) ⇝ 𝐴)) |
13 | seqex 13974 | . . . 4 ⊢ seq𝑀( + , 𝐹) ∈ V | |
14 | climshft 15526 | . . . 4 ⊢ ((𝑁 ∈ ℤ ∧ seq𝑀( + , 𝐹) ∈ V) → ((seq𝑀( + , 𝐹) shift 𝑁) ⇝ 𝐴 ↔ seq𝑀( + , 𝐹) ⇝ 𝐴)) | |
15 | 13, 14 | mpan2 688 | . . 3 ⊢ (𝑁 ∈ ℤ → ((seq𝑀( + , 𝐹) shift 𝑁) ⇝ 𝐴 ↔ seq𝑀( + , 𝐹) ⇝ 𝐴)) |
16 | 15 | adantl 481 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((seq𝑀( + , 𝐹) shift 𝑁) ⇝ 𝐴 ↔ seq𝑀( + , 𝐹) ⇝ 𝐴)) |
17 | 12, 16 | bitr2d 280 | 1 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (seq𝑀( + , 𝐹) ⇝ 𝐴 ↔ seq(𝑀 + 𝑁)( + , (𝐹 shift 𝑁)) ⇝ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1533 ∈ wcel 2098 Vcvv 3468 class class class wbr 5141 (class class class)co 7405 ℂcc 11110 + caddc 11115 − cmin 11448 ℤcz 12562 seqcseq 13972 shift cshi 15019 ⇝ cli 15434 |
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-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-inf2 9638 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-seq 13973 df-shft 15020 df-clim 15438 |
This theorem is referenced by: isumshft 15791 geolim3 26229 dvradcnv 26312 abelthlem6 26328 |
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