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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 9506 | . . . . 5 ⊢ (𝜑 → (𝑀 + 𝑁) ∈ ℤ) |
| 5 | 2 | zcnd 9503 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ ℂ) |
| 6 | 3 | zcnd 9503 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℂ) |
| 7 | 5, 6 | pncand 8391 | . . . . . . . . 9 ⊢ (𝜑 → ((𝑀 + 𝑁) − 𝑁) = 𝑀) |
| 8 | 7 | fveq2d 5587 | . . . . . . . 8 ⊢ (𝜑 → (ℤ≥‘((𝑀 + 𝑁) − 𝑁)) = (ℤ≥‘𝑀)) |
| 9 | 8 | eleq2d 2276 | . . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ (ℤ≥‘((𝑀 + 𝑁) − 𝑁)) ↔ 𝑥 ∈ (ℤ≥‘𝑀))) |
| 10 | 9 | pm5.32i 454 | . . . . . 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 121 | . . . . 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 11193 | . . . 4 ⊢ (𝜑 → seq(𝑀 + 𝑁)( + , (𝐹 shift 𝑁)) = (seq((𝑀 + 𝑁) − 𝑁)( + , 𝐹) shift 𝑁)) |
| 15 | 7 | seqeq1d 10605 | . . . . 5 ⊢ (𝜑 → seq((𝑀 + 𝑁) − 𝑁)( + , 𝐹) = seq𝑀( + , 𝐹)) |
| 16 | 15 | oveq1d 5966 | . . . 4 ⊢ (𝜑 → (seq((𝑀 + 𝑁) − 𝑁)( + , 𝐹) shift 𝑁) = (seq𝑀( + , 𝐹) shift 𝑁)) |
| 17 | 14, 16 | eqtrd 2239 | . . 3 ⊢ (𝜑 → seq(𝑀 + 𝑁)( + , (𝐹 shift 𝑁)) = (seq𝑀( + , 𝐹) shift 𝑁)) |
| 18 | 17 | breq1d 4057 | . 2 ⊢ (𝜑 → (seq(𝑀 + 𝑁)( + , (𝐹 shift 𝑁)) ⇝ 𝐴 ↔ (seq𝑀( + , 𝐹) shift 𝑁) ⇝ 𝐴)) |
| 19 | seqex 10601 | . . 3 ⊢ seq𝑀( + , 𝐹) ∈ V | |
| 20 | climshft 11659 | . . 3 ⊢ ((𝑁 ∈ ℤ ∧ seq𝑀( + , 𝐹) ∈ V) → ((seq𝑀( + , 𝐹) shift 𝑁) ⇝ 𝐴 ↔ seq𝑀( + , 𝐹) ⇝ 𝐴)) | |
| 21 | 3, 19, 20 | sylancl 413 | . 2 ⊢ (𝜑 → ((seq𝑀( + , 𝐹) shift 𝑁) ⇝ 𝐴 ↔ seq𝑀( + , 𝐹) ⇝ 𝐴)) |
| 22 | 18, 21 | bitr2d 189 | 1 ⊢ (𝜑 → (seq𝑀( + , 𝐹) ⇝ 𝐴 ↔ seq(𝑀 + 𝑁)( + , (𝐹 shift 𝑁)) ⇝ 𝐴)) |
| Colors of variables: wff set class |
| Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∈ wcel 2177 Vcvv 2773 class class class wbr 4047 ‘cfv 5276 (class class class)co 5951 + caddc 7935 − cmin 8250 ℤcz 9379 ℤ≥cuz 9655 seqcseq 10599 shift cshi 11169 ⇝ cli 11633 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 711 ax-5 1471 ax-7 1472 ax-gen 1473 ax-ie1 1517 ax-ie2 1518 ax-8 1528 ax-10 1529 ax-11 1530 ax-i12 1531 ax-bndl 1533 ax-4 1534 ax-17 1550 ax-i9 1554 ax-ial 1558 ax-i5r 1559 ax-13 2179 ax-14 2180 ax-ext 2188 ax-coll 4163 ax-sep 4166 ax-nul 4174 ax-pow 4222 ax-pr 4257 ax-un 4484 ax-setind 4589 ax-iinf 4640 ax-cnex 8023 ax-resscn 8024 ax-1cn 8025 ax-1re 8026 ax-icn 8027 ax-addcl 8028 ax-addrcl 8029 ax-mulcl 8030 ax-addcom 8032 ax-addass 8034 ax-distr 8036 ax-i2m1 8037 ax-0lt1 8038 ax-0id 8040 ax-rnegex 8041 ax-cnre 8043 ax-pre-ltirr 8044 ax-pre-ltwlin 8045 ax-pre-lttrn 8046 ax-pre-apti 8047 ax-pre-ltadd 8048 |
| This theorem depends on definitions: df-bi 117 df-dc 837 df-3or 982 df-3an 983 df-tru 1376 df-fal 1379 df-nf 1485 df-sb 1787 df-eu 2058 df-mo 2059 df-clab 2193 df-cleq 2199 df-clel 2202 df-nfc 2338 df-ne 2378 df-nel 2473 df-ral 2490 df-rex 2491 df-reu 2492 df-rab 2494 df-v 2775 df-sbc 3000 df-csb 3095 df-dif 3169 df-un 3171 df-in 3173 df-ss 3180 df-nul 3462 df-if 3573 df-pw 3619 df-sn 3640 df-pr 3641 df-op 3643 df-uni 3853 df-int 3888 df-iun 3931 df-br 4048 df-opab 4110 df-mpt 4111 df-tr 4147 df-id 4344 df-iord 4417 df-on 4419 df-ilim 4420 df-suc 4422 df-iom 4643 df-xp 4685 df-rel 4686 df-cnv 4687 df-co 4688 df-dm 4689 df-rn 4690 df-res 4691 df-ima 4692 df-iota 5237 df-fun 5278 df-fn 5279 df-f 5280 df-f1 5281 df-fo 5282 df-f1o 5283 df-fv 5284 df-riota 5906 df-ov 5954 df-oprab 5955 df-mpo 5956 df-1st 6233 df-2nd 6234 df-recs 6398 df-frec 6484 df-pnf 8116 df-mnf 8117 df-xr 8118 df-ltxr 8119 df-le 8120 df-sub 8252 df-neg 8253 df-inn 9044 df-n0 9303 df-z 9380 df-uz 9656 df-fz 10138 df-seqfrec 10600 df-shft 11170 df-clim 11634 |
| This theorem is referenced by: isumshft 11845 |
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