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Mirrors > Home > ILE Home > Th. List > iser3shft | GIF version |
Description: Index shift of the limit of an infinite series. (Contributed by Mario Carneiro, 6-Sep-2013.) (Revised by Jim Kingdon, 17-Oct-2022.) |
Ref | Expression |
---|---|
iser3shft.ex | ⊢ (𝜑 → 𝐹 ∈ 𝑉) |
iser3shft.m | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
iser3shft.n | ⊢ (𝜑 → 𝑁 ∈ ℤ) |
iser3shft.fm | ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆) |
iser3shft.pl | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) |
Ref | Expression |
---|---|
iser3shft | ⊢ (𝜑 → (seq𝑀( + , 𝐹) ⇝ 𝐴 ↔ seq(𝑀 + 𝑁)( + , (𝐹 shift 𝑁)) ⇝ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iser3shft.ex | . . . . 5 ⊢ (𝜑 → 𝐹 ∈ 𝑉) | |
2 | iser3shft.m | . . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
3 | iser3shft.n | . . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℤ) | |
4 | 2, 3 | zaddcld 9201 | . . . . 5 ⊢ (𝜑 → (𝑀 + 𝑁) ∈ ℤ) |
5 | 2 | zcnd 9198 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ ℂ) |
6 | 3 | zcnd 9198 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℂ) |
7 | 5, 6 | pncand 8098 | . . . . . . . . 9 ⊢ (𝜑 → ((𝑀 + 𝑁) − 𝑁) = 𝑀) |
8 | 7 | fveq2d 5433 | . . . . . . . 8 ⊢ (𝜑 → (ℤ≥‘((𝑀 + 𝑁) − 𝑁)) = (ℤ≥‘𝑀)) |
9 | 8 | eleq2d 2210 | . . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ (ℤ≥‘((𝑀 + 𝑁) − 𝑁)) ↔ 𝑥 ∈ (ℤ≥‘𝑀))) |
10 | 9 | pm5.32i 450 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘((𝑀 + 𝑁) − 𝑁))) ↔ (𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀))) |
11 | iser3shft.fm | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆) | |
12 | 10, 11 | sylbi 120 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘((𝑀 + 𝑁) − 𝑁))) → (𝐹‘𝑥) ∈ 𝑆) |
13 | iser3shft.pl | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) | |
14 | 1, 4, 3, 12, 13 | seq3shft 10642 | . . . 4 ⊢ (𝜑 → seq(𝑀 + 𝑁)( + , (𝐹 shift 𝑁)) = (seq((𝑀 + 𝑁) − 𝑁)( + , 𝐹) shift 𝑁)) |
15 | 7 | seqeq1d 10255 | . . . . 5 ⊢ (𝜑 → seq((𝑀 + 𝑁) − 𝑁)( + , 𝐹) = seq𝑀( + , 𝐹)) |
16 | 15 | oveq1d 5797 | . . . 4 ⊢ (𝜑 → (seq((𝑀 + 𝑁) − 𝑁)( + , 𝐹) shift 𝑁) = (seq𝑀( + , 𝐹) shift 𝑁)) |
17 | 14, 16 | eqtrd 2173 | . . 3 ⊢ (𝜑 → seq(𝑀 + 𝑁)( + , (𝐹 shift 𝑁)) = (seq𝑀( + , 𝐹) shift 𝑁)) |
18 | 17 | breq1d 3947 | . 2 ⊢ (𝜑 → (seq(𝑀 + 𝑁)( + , (𝐹 shift 𝑁)) ⇝ 𝐴 ↔ (seq𝑀( + , 𝐹) shift 𝑁) ⇝ 𝐴)) |
19 | seqex 10251 | . . 3 ⊢ seq𝑀( + , 𝐹) ∈ V | |
20 | climshft 11105 | . . 3 ⊢ ((𝑁 ∈ ℤ ∧ seq𝑀( + , 𝐹) ∈ V) → ((seq𝑀( + , 𝐹) shift 𝑁) ⇝ 𝐴 ↔ seq𝑀( + , 𝐹) ⇝ 𝐴)) | |
21 | 3, 19, 20 | sylancl 410 | . 2 ⊢ (𝜑 → ((seq𝑀( + , 𝐹) shift 𝑁) ⇝ 𝐴 ↔ seq𝑀( + , 𝐹) ⇝ 𝐴)) |
22 | 18, 21 | bitr2d 188 | 1 ⊢ (𝜑 → (seq𝑀( + , 𝐹) ⇝ 𝐴 ↔ seq(𝑀 + 𝑁)( + , (𝐹 shift 𝑁)) ⇝ 𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∈ wcel 1481 Vcvv 2689 class class class wbr 3937 ‘cfv 5131 (class class class)co 5782 + caddc 7647 − cmin 7957 ℤcz 9078 ℤ≥cuz 9350 seqcseq 10249 shift cshi 10618 ⇝ cli 11079 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-coll 4051 ax-sep 4054 ax-nul 4062 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-iinf 4510 ax-cnex 7735 ax-resscn 7736 ax-1cn 7737 ax-1re 7738 ax-icn 7739 ax-addcl 7740 ax-addrcl 7741 ax-mulcl 7742 ax-addcom 7744 ax-addass 7746 ax-distr 7748 ax-i2m1 7749 ax-0lt1 7750 ax-0id 7752 ax-rnegex 7753 ax-cnre 7755 ax-pre-ltirr 7756 ax-pre-ltwlin 7757 ax-pre-lttrn 7758 ax-pre-apti 7759 ax-pre-ltadd 7760 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-nel 2405 df-ral 2422 df-rex 2423 df-reu 2424 df-rab 2426 df-v 2691 df-sbc 2914 df-csb 3008 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-if 3480 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-iun 3823 df-br 3938 df-opab 3998 df-mpt 3999 df-tr 4035 df-id 4223 df-iord 4296 df-on 4298 df-ilim 4299 df-suc 4301 df-iom 4513 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 df-fv 5139 df-riota 5738 df-ov 5785 df-oprab 5786 df-mpo 5787 df-1st 6046 df-2nd 6047 df-recs 6210 df-frec 6296 df-pnf 7826 df-mnf 7827 df-xr 7828 df-ltxr 7829 df-le 7830 df-sub 7959 df-neg 7960 df-inn 8745 df-n0 9002 df-z 9079 df-uz 9351 df-fz 9822 df-seqfrec 10250 df-shft 10619 df-clim 11080 |
This theorem is referenced by: isumshft 11291 |
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