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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 9512 | . . . . 5 ⊢ (𝜑 → (𝑀 + 𝑁) ∈ ℤ) |
| 5 | 2 | zcnd 9509 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ ℂ) |
| 6 | 3 | zcnd 9509 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℂ) |
| 7 | 5, 6 | pncand 8397 | . . . . . . . . 9 ⊢ (𝜑 → ((𝑀 + 𝑁) − 𝑁) = 𝑀) |
| 8 | 7 | fveq2d 5590 | . . . . . . . 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 11199 | . . . 4 ⊢ (𝜑 → seq(𝑀 + 𝑁)( + , (𝐹 shift 𝑁)) = (seq((𝑀 + 𝑁) − 𝑁)( + , 𝐹) shift 𝑁)) |
| 15 | 7 | seqeq1d 10611 | . . . . 5 ⊢ (𝜑 → seq((𝑀 + 𝑁) − 𝑁)( + , 𝐹) = seq𝑀( + , 𝐹)) |
| 16 | 15 | oveq1d 5969 | . . . 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 4058 | . 2 ⊢ (𝜑 → (seq(𝑀 + 𝑁)( + , (𝐹 shift 𝑁)) ⇝ 𝐴 ↔ (seq𝑀( + , 𝐹) shift 𝑁) ⇝ 𝐴)) |
| 19 | seqex 10607 | . . 3 ⊢ seq𝑀( + , 𝐹) ∈ V | |
| 20 | climshft 11665 | . . 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 4048 ‘cfv 5277 (class class class)co 5954 + caddc 7941 − cmin 8256 ℤcz 9385 ℤ≥cuz 9661 seqcseq 10605 shift cshi 11175 ⇝ cli 11639 |
| 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 4164 ax-sep 4167 ax-nul 4175 ax-pow 4223 ax-pr 4258 ax-un 4485 ax-setind 4590 ax-iinf 4641 ax-cnex 8029 ax-resscn 8030 ax-1cn 8031 ax-1re 8032 ax-icn 8033 ax-addcl 8034 ax-addrcl 8035 ax-mulcl 8036 ax-addcom 8038 ax-addass 8040 ax-distr 8042 ax-i2m1 8043 ax-0lt1 8044 ax-0id 8046 ax-rnegex 8047 ax-cnre 8049 ax-pre-ltirr 8050 ax-pre-ltwlin 8051 ax-pre-lttrn 8052 ax-pre-apti 8053 ax-pre-ltadd 8054 |
| 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 3001 df-csb 3096 df-dif 3170 df-un 3172 df-in 3174 df-ss 3181 df-nul 3463 df-if 3574 df-pw 3620 df-sn 3641 df-pr 3642 df-op 3644 df-uni 3854 df-int 3889 df-iun 3932 df-br 4049 df-opab 4111 df-mpt 4112 df-tr 4148 df-id 4345 df-iord 4418 df-on 4420 df-ilim 4421 df-suc 4423 df-iom 4644 df-xp 4686 df-rel 4687 df-cnv 4688 df-co 4689 df-dm 4690 df-rn 4691 df-res 4692 df-ima 4693 df-iota 5238 df-fun 5279 df-fn 5280 df-f 5281 df-f1 5282 df-fo 5283 df-f1o 5284 df-fv 5285 df-riota 5909 df-ov 5957 df-oprab 5958 df-mpo 5959 df-1st 6236 df-2nd 6237 df-recs 6401 df-frec 6487 df-pnf 8122 df-mnf 8123 df-xr 8124 df-ltxr 8125 df-le 8126 df-sub 8258 df-neg 8259 df-inn 9050 df-n0 9309 df-z 9386 df-uz 9662 df-fz 10144 df-seqfrec 10606 df-shft 11176 df-clim 11640 |
| This theorem is referenced by: isumshft 11851 |
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