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Mirrors > Home > ILE Home > Th. List > seq3shft2 | GIF version |
Description: Shifting the index set of a sequence. (Contributed by Jim Kingdon, 15-Aug-2021.) (Revised by Jim Kingdon, 17-Oct-2022.) |
Ref | Expression |
---|---|
seq3shft2.1 | ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) |
seq3shft2.2 | ⊢ (𝜑 → 𝐾 ∈ ℤ) |
seq3shft2.3 | ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) = (𝐺‘(𝑘 + 𝐾))) |
seq3shft2.f | ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆) |
seq3shft2.g | ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘(𝑀 + 𝐾))) → (𝐺‘𝑥) ∈ 𝑆) |
seq3shft2.pl | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) |
Ref | Expression |
---|---|
seq3shft2 | ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) = (seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑁 + 𝐾))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | seq3shft2.1 | . . 3 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) | |
2 | seq3shft2.2 | . . 3 ⊢ (𝜑 → 𝐾 ∈ ℤ) | |
3 | seq3shft2.3 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) = (𝐺‘(𝑘 + 𝐾))) | |
4 | seq3shft2.f | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆) | |
5 | seq3shft2.g | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘(𝑀 + 𝐾))) → (𝐺‘𝑥) ∈ 𝑆) | |
6 | seq3shft2.pl | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) | |
7 | 1, 2, 3, 4, 5, 6 | iseqshft2 9959 | . 2 ⊢ (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑁) = (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘(𝑁 + 𝐾))) |
8 | eluzel2 9085 | . . . . . 6 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ ℤ) | |
9 | 1, 8 | syl 14 | . . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℤ) |
10 | ssv 3047 | . . . . . 6 ⊢ 𝑆 ⊆ V | |
11 | 10 | a1i 9 | . . . . 5 ⊢ (𝜑 → 𝑆 ⊆ V) |
12 | 9, 11, 4, 6 | iseqsst 9947 | . . . 4 ⊢ (𝜑 → seq𝑀( + , 𝐹, 𝑆) = seq𝑀( + , 𝐹, V)) |
13 | df-seq3 9915 | . . . 4 ⊢ seq𝑀( + , 𝐹) = seq𝑀( + , 𝐹, V) | |
14 | 12, 13 | syl6eqr 2139 | . . 3 ⊢ (𝜑 → seq𝑀( + , 𝐹, 𝑆) = seq𝑀( + , 𝐹)) |
15 | 14 | fveq1d 5320 | . 2 ⊢ (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑁) = (seq𝑀( + , 𝐹)‘𝑁)) |
16 | 9, 2 | zaddcld 8933 | . . . . 5 ⊢ (𝜑 → (𝑀 + 𝐾) ∈ ℤ) |
17 | 16, 11, 5, 6 | iseqsst 9947 | . . . 4 ⊢ (𝜑 → seq(𝑀 + 𝐾)( + , 𝐺, 𝑆) = seq(𝑀 + 𝐾)( + , 𝐺, V)) |
18 | df-seq3 9915 | . . . 4 ⊢ seq(𝑀 + 𝐾)( + , 𝐺) = seq(𝑀 + 𝐾)( + , 𝐺, V) | |
19 | 17, 18 | syl6eqr 2139 | . . 3 ⊢ (𝜑 → seq(𝑀 + 𝐾)( + , 𝐺, 𝑆) = seq(𝑀 + 𝐾)( + , 𝐺)) |
20 | 19 | fveq1d 5320 | . 2 ⊢ (𝜑 → (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘(𝑁 + 𝐾)) = (seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑁 + 𝐾))) |
21 | 7, 15, 20 | 3eqtr3d 2129 | 1 ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) = (seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑁 + 𝐾))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1290 ∈ wcel 1439 Vcvv 2620 ⊆ wss 3000 ‘cfv 5028 (class class class)co 5666 + caddc 7414 ℤcz 8811 ℤ≥cuz 9080 ...cfz 9485 seqcseq4 9912 seqcseq 9913 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 580 ax-in2 581 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-13 1450 ax-14 1451 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-coll 3960 ax-sep 3963 ax-nul 3971 ax-pow 4015 ax-pr 4045 ax-un 4269 ax-setind 4366 ax-iinf 4416 ax-cnex 7497 ax-resscn 7498 ax-1cn 7499 ax-1re 7500 ax-icn 7501 ax-addcl 7502 ax-addrcl 7503 ax-mulcl 7504 ax-addcom 7506 ax-addass 7508 ax-distr 7510 ax-i2m1 7511 ax-0lt1 7512 ax-0id 7514 ax-rnegex 7515 ax-cnre 7517 ax-pre-ltirr 7518 ax-pre-ltwlin 7519 ax-pre-lttrn 7520 ax-pre-ltadd 7522 |
This theorem depends on definitions: df-bi 116 df-3or 926 df-3an 927 df-tru 1293 df-fal 1296 df-nf 1396 df-sb 1694 df-eu 1952 df-mo 1953 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ne 2257 df-nel 2352 df-ral 2365 df-rex 2366 df-reu 2367 df-rab 2369 df-v 2622 df-sbc 2842 df-csb 2935 df-dif 3002 df-un 3004 df-in 3006 df-ss 3013 df-nul 3288 df-pw 3435 df-sn 3456 df-pr 3457 df-op 3459 df-uni 3660 df-int 3695 df-iun 3738 df-br 3852 df-opab 3906 df-mpt 3907 df-tr 3943 df-id 4129 df-iord 4202 df-on 4204 df-ilim 4205 df-suc 4207 df-iom 4419 df-xp 4458 df-rel 4459 df-cnv 4460 df-co 4461 df-dm 4462 df-rn 4463 df-res 4464 df-ima 4465 df-iota 4993 df-fun 5030 df-fn 5031 df-f 5032 df-f1 5033 df-fo 5034 df-f1o 5035 df-fv 5036 df-riota 5622 df-ov 5669 df-oprab 5670 df-mpt2 5671 df-1st 5925 df-2nd 5926 df-recs 6084 df-frec 6170 df-pnf 7585 df-mnf 7586 df-xr 7587 df-ltxr 7588 df-le 7589 df-sub 7716 df-neg 7717 df-inn 8484 df-n0 8735 df-z 8812 df-uz 9081 df-fz 9486 df-iseq 9914 df-seq3 9915 |
This theorem is referenced by: seq3f1olemqsumkj 9988 seq3shft 10333 |
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