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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 9338 | . . . . 5 ⊢ (𝜑 → (𝑀 + 𝑁) ∈ ℤ) |
5 | 2 | zcnd 9335 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ ℂ) |
6 | 3 | zcnd 9335 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℂ) |
7 | 5, 6 | pncand 8231 | . . . . . . . . 9 ⊢ (𝜑 → ((𝑀 + 𝑁) − 𝑁) = 𝑀) |
8 | 7 | fveq2d 5500 | . . . . . . . 8 ⊢ (𝜑 → (ℤ≥‘((𝑀 + 𝑁) − 𝑁)) = (ℤ≥‘𝑀)) |
9 | 8 | eleq2d 2240 | . . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ (ℤ≥‘((𝑀 + 𝑁) − 𝑁)) ↔ 𝑥 ∈ (ℤ≥‘𝑀))) |
10 | 9 | pm5.32i 451 | . . . . . 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 10802 | . . . 4 ⊢ (𝜑 → seq(𝑀 + 𝑁)( + , (𝐹 shift 𝑁)) = (seq((𝑀 + 𝑁) − 𝑁)( + , 𝐹) shift 𝑁)) |
15 | 7 | seqeq1d 10407 | . . . . 5 ⊢ (𝜑 → seq((𝑀 + 𝑁) − 𝑁)( + , 𝐹) = seq𝑀( + , 𝐹)) |
16 | 15 | oveq1d 5868 | . . . 4 ⊢ (𝜑 → (seq((𝑀 + 𝑁) − 𝑁)( + , 𝐹) shift 𝑁) = (seq𝑀( + , 𝐹) shift 𝑁)) |
17 | 14, 16 | eqtrd 2203 | . . 3 ⊢ (𝜑 → seq(𝑀 + 𝑁)( + , (𝐹 shift 𝑁)) = (seq𝑀( + , 𝐹) shift 𝑁)) |
18 | 17 | breq1d 3999 | . 2 ⊢ (𝜑 → (seq(𝑀 + 𝑁)( + , (𝐹 shift 𝑁)) ⇝ 𝐴 ↔ (seq𝑀( + , 𝐹) shift 𝑁) ⇝ 𝐴)) |
19 | seqex 10403 | . . 3 ⊢ seq𝑀( + , 𝐹) ∈ V | |
20 | climshft 11267 | . . 3 ⊢ ((𝑁 ∈ ℤ ∧ seq𝑀( + , 𝐹) ∈ V) → ((seq𝑀( + , 𝐹) shift 𝑁) ⇝ 𝐴 ↔ seq𝑀( + , 𝐹) ⇝ 𝐴)) | |
21 | 3, 19, 20 | sylancl 411 | . 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 2141 Vcvv 2730 class class class wbr 3989 ‘cfv 5198 (class class class)co 5853 + caddc 7777 − cmin 8090 ℤcz 9212 ℤ≥cuz 9487 seqcseq 10401 shift cshi 10778 ⇝ cli 11241 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4104 ax-sep 4107 ax-nul 4115 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-iinf 4572 ax-cnex 7865 ax-resscn 7866 ax-1cn 7867 ax-1re 7868 ax-icn 7869 ax-addcl 7870 ax-addrcl 7871 ax-mulcl 7872 ax-addcom 7874 ax-addass 7876 ax-distr 7878 ax-i2m1 7879 ax-0lt1 7880 ax-0id 7882 ax-rnegex 7883 ax-cnre 7885 ax-pre-ltirr 7886 ax-pre-ltwlin 7887 ax-pre-lttrn 7888 ax-pre-apti 7889 ax-pre-ltadd 7890 |
This theorem depends on definitions: df-bi 116 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-if 3527 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-iun 3875 df-br 3990 df-opab 4051 df-mpt 4052 df-tr 4088 df-id 4278 df-iord 4351 df-on 4353 df-ilim 4354 df-suc 4356 df-iom 4575 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-f1 5203 df-fo 5204 df-f1o 5205 df-fv 5206 df-riota 5809 df-ov 5856 df-oprab 5857 df-mpo 5858 df-1st 6119 df-2nd 6120 df-recs 6284 df-frec 6370 df-pnf 7956 df-mnf 7957 df-xr 7958 df-ltxr 7959 df-le 7960 df-sub 8092 df-neg 8093 df-inn 8879 df-n0 9136 df-z 9213 df-uz 9488 df-fz 9966 df-seqfrec 10402 df-shft 10779 df-clim 11242 |
This theorem is referenced by: isumshft 11453 |
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