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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 9382 | . . . . 5 ⊢ (𝜑 → (𝑀 + 𝑁) ∈ ℤ) |
5 | 2 | zcnd 9379 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ ℂ) |
6 | 3 | zcnd 9379 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℂ) |
7 | 5, 6 | pncand 8272 | . . . . . . . . 9 ⊢ (𝜑 → ((𝑀 + 𝑁) − 𝑁) = 𝑀) |
8 | 7 | fveq2d 5521 | . . . . . . . 8 ⊢ (𝜑 → (ℤ≥‘((𝑀 + 𝑁) − 𝑁)) = (ℤ≥‘𝑀)) |
9 | 8 | eleq2d 2247 | . . . . . . 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 10850 | . . . 4 ⊢ (𝜑 → seq(𝑀 + 𝑁)( + , (𝐹 shift 𝑁)) = (seq((𝑀 + 𝑁) − 𝑁)( + , 𝐹) shift 𝑁)) |
15 | 7 | seqeq1d 10454 | . . . . 5 ⊢ (𝜑 → seq((𝑀 + 𝑁) − 𝑁)( + , 𝐹) = seq𝑀( + , 𝐹)) |
16 | 15 | oveq1d 5893 | . . . 4 ⊢ (𝜑 → (seq((𝑀 + 𝑁) − 𝑁)( + , 𝐹) shift 𝑁) = (seq𝑀( + , 𝐹) shift 𝑁)) |
17 | 14, 16 | eqtrd 2210 | . . 3 ⊢ (𝜑 → seq(𝑀 + 𝑁)( + , (𝐹 shift 𝑁)) = (seq𝑀( + , 𝐹) shift 𝑁)) |
18 | 17 | breq1d 4015 | . 2 ⊢ (𝜑 → (seq(𝑀 + 𝑁)( + , (𝐹 shift 𝑁)) ⇝ 𝐴 ↔ (seq𝑀( + , 𝐹) shift 𝑁) ⇝ 𝐴)) |
19 | seqex 10450 | . . 3 ⊢ seq𝑀( + , 𝐹) ∈ V | |
20 | climshft 11315 | . . 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 2148 Vcvv 2739 class class class wbr 4005 ‘cfv 5218 (class class class)co 5878 + caddc 7817 − cmin 8131 ℤcz 9256 ℤ≥cuz 9531 seqcseq 10448 shift cshi 10826 ⇝ cli 11289 |
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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4120 ax-sep 4123 ax-nul 4131 ax-pow 4176 ax-pr 4211 ax-un 4435 ax-setind 4538 ax-iinf 4589 ax-cnex 7905 ax-resscn 7906 ax-1cn 7907 ax-1re 7908 ax-icn 7909 ax-addcl 7910 ax-addrcl 7911 ax-mulcl 7912 ax-addcom 7914 ax-addass 7916 ax-distr 7918 ax-i2m1 7919 ax-0lt1 7920 ax-0id 7922 ax-rnegex 7923 ax-cnre 7925 ax-pre-ltirr 7926 ax-pre-ltwlin 7927 ax-pre-lttrn 7928 ax-pre-apti 7929 ax-pre-ltadd 7930 |
This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2741 df-sbc 2965 df-csb 3060 df-dif 3133 df-un 3135 df-in 3137 df-ss 3144 df-nul 3425 df-if 3537 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-int 3847 df-iun 3890 df-br 4006 df-opab 4067 df-mpt 4068 df-tr 4104 df-id 4295 df-iord 4368 df-on 4370 df-ilim 4371 df-suc 4373 df-iom 4592 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-rn 4639 df-res 4640 df-ima 4641 df-iota 5180 df-fun 5220 df-fn 5221 df-f 5222 df-f1 5223 df-fo 5224 df-f1o 5225 df-fv 5226 df-riota 5834 df-ov 5881 df-oprab 5882 df-mpo 5883 df-1st 6144 df-2nd 6145 df-recs 6309 df-frec 6395 df-pnf 7997 df-mnf 7998 df-xr 7999 df-ltxr 8000 df-le 8001 df-sub 8133 df-neg 8134 df-inn 8923 df-n0 9180 df-z 9257 df-uz 9532 df-fz 10012 df-seqfrec 10449 df-shft 10827 df-clim 11290 |
This theorem is referenced by: isumshft 11501 |
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