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Mirrors > Home > ILE Home > Th. List > shftf | GIF version |
Description: Functionality of a shifted sequence. (Contributed by NM, 19-Aug-2005.) (Revised by Mario Carneiro, 5-Nov-2013.) |
Ref | Expression |
---|---|
shftfval.1 | ⊢ 𝐹 ∈ V |
Ref | Expression |
---|---|
shftf | ⊢ ((𝐹:𝐵⟶𝐶 ∧ 𝐴 ∈ ℂ) → (𝐹 shift 𝐴):{𝑥 ∈ ℂ ∣ (𝑥 − 𝐴) ∈ 𝐵}⟶𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ffn 5046 | . . 3 ⊢ (𝐹:𝐵⟶𝐶 → 𝐹 Fn 𝐵) | |
2 | shftfval.1 | . . . 4 ⊢ 𝐹 ∈ V | |
3 | 2 | shftfn 9425 | . . 3 ⊢ ((𝐹 Fn 𝐵 ∧ 𝐴 ∈ ℂ) → (𝐹 shift 𝐴) Fn {𝑥 ∈ ℂ ∣ (𝑥 − 𝐴) ∈ 𝐵}) |
4 | 1, 3 | sylan 267 | . 2 ⊢ ((𝐹:𝐵⟶𝐶 ∧ 𝐴 ∈ ℂ) → (𝐹 shift 𝐴) Fn {𝑥 ∈ ℂ ∣ (𝑥 − 𝐴) ∈ 𝐵}) |
5 | oveq1 5519 | . . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 − 𝐴) = (𝑦 − 𝐴)) | |
6 | 5 | eleq1d 2106 | . . . . 5 ⊢ (𝑥 = 𝑦 → ((𝑥 − 𝐴) ∈ 𝐵 ↔ (𝑦 − 𝐴) ∈ 𝐵)) |
7 | 6 | elrab 2698 | . . . 4 ⊢ (𝑦 ∈ {𝑥 ∈ ℂ ∣ (𝑥 − 𝐴) ∈ 𝐵} ↔ (𝑦 ∈ ℂ ∧ (𝑦 − 𝐴) ∈ 𝐵)) |
8 | simpr 103 | . . . . . 6 ⊢ ((𝐹:𝐵⟶𝐶 ∧ 𝐴 ∈ ℂ) → 𝐴 ∈ ℂ) | |
9 | simpl 102 | . . . . . 6 ⊢ ((𝑦 ∈ ℂ ∧ (𝑦 − 𝐴) ∈ 𝐵) → 𝑦 ∈ ℂ) | |
10 | 2 | shftval 9426 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ ℂ) → ((𝐹 shift 𝐴)‘𝑦) = (𝐹‘(𝑦 − 𝐴))) |
11 | 8, 9, 10 | syl2an 273 | . . . . 5 ⊢ (((𝐹:𝐵⟶𝐶 ∧ 𝐴 ∈ ℂ) ∧ (𝑦 ∈ ℂ ∧ (𝑦 − 𝐴) ∈ 𝐵)) → ((𝐹 shift 𝐴)‘𝑦) = (𝐹‘(𝑦 − 𝐴))) |
12 | simpl 102 | . . . . . 6 ⊢ ((𝐹:𝐵⟶𝐶 ∧ 𝐴 ∈ ℂ) → 𝐹:𝐵⟶𝐶) | |
13 | simpr 103 | . . . . . 6 ⊢ ((𝑦 ∈ ℂ ∧ (𝑦 − 𝐴) ∈ 𝐵) → (𝑦 − 𝐴) ∈ 𝐵) | |
14 | ffvelrn 5300 | . . . . . 6 ⊢ ((𝐹:𝐵⟶𝐶 ∧ (𝑦 − 𝐴) ∈ 𝐵) → (𝐹‘(𝑦 − 𝐴)) ∈ 𝐶) | |
15 | 12, 13, 14 | syl2an 273 | . . . . 5 ⊢ (((𝐹:𝐵⟶𝐶 ∧ 𝐴 ∈ ℂ) ∧ (𝑦 ∈ ℂ ∧ (𝑦 − 𝐴) ∈ 𝐵)) → (𝐹‘(𝑦 − 𝐴)) ∈ 𝐶) |
16 | 11, 15 | eqeltrd 2114 | . . . 4 ⊢ (((𝐹:𝐵⟶𝐶 ∧ 𝐴 ∈ ℂ) ∧ (𝑦 ∈ ℂ ∧ (𝑦 − 𝐴) ∈ 𝐵)) → ((𝐹 shift 𝐴)‘𝑦) ∈ 𝐶) |
17 | 7, 16 | sylan2b 271 | . . 3 ⊢ (((𝐹:𝐵⟶𝐶 ∧ 𝐴 ∈ ℂ) ∧ 𝑦 ∈ {𝑥 ∈ ℂ ∣ (𝑥 − 𝐴) ∈ 𝐵}) → ((𝐹 shift 𝐴)‘𝑦) ∈ 𝐶) |
18 | 17 | ralrimiva 2392 | . 2 ⊢ ((𝐹:𝐵⟶𝐶 ∧ 𝐴 ∈ ℂ) → ∀𝑦 ∈ {𝑥 ∈ ℂ ∣ (𝑥 − 𝐴) ∈ 𝐵} ((𝐹 shift 𝐴)‘𝑦) ∈ 𝐶) |
19 | ffnfv 5323 | . 2 ⊢ ((𝐹 shift 𝐴):{𝑥 ∈ ℂ ∣ (𝑥 − 𝐴) ∈ 𝐵}⟶𝐶 ↔ ((𝐹 shift 𝐴) Fn {𝑥 ∈ ℂ ∣ (𝑥 − 𝐴) ∈ 𝐵} ∧ ∀𝑦 ∈ {𝑥 ∈ ℂ ∣ (𝑥 − 𝐴) ∈ 𝐵} ((𝐹 shift 𝐴)‘𝑦) ∈ 𝐶)) | |
20 | 4, 18, 19 | sylanbrc 394 | 1 ⊢ ((𝐹:𝐵⟶𝐶 ∧ 𝐴 ∈ ℂ) → (𝐹 shift 𝐴):{𝑥 ∈ ℂ ∣ (𝑥 − 𝐴) ∈ 𝐵}⟶𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 = wceq 1243 ∈ wcel 1393 ∀wral 2306 {crab 2310 Vcvv 2557 Fn wfn 4897 ⟶wf 4898 ‘cfv 4902 (class class class)co 5512 ℂcc 6887 − cmin 7182 shift cshi 9415 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-coll 3872 ax-sep 3875 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-setind 4262 ax-resscn 6976 ax-1cn 6977 ax-icn 6979 ax-addcl 6980 ax-addrcl 6981 ax-mulcl 6982 ax-addcom 6984 ax-addass 6986 ax-distr 6988 ax-i2m1 6989 ax-0id 6992 ax-rnegex 6993 ax-cnre 6995 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-ral 2311 df-rex 2312 df-reu 2313 df-rab 2315 df-v 2559 df-sbc 2765 df-csb 2853 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-iun 3659 df-br 3765 df-opab 3819 df-mpt 3820 df-id 4030 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-f1 4907 df-fo 4908 df-f1o 4909 df-fv 4910 df-riota 5468 df-ov 5515 df-oprab 5516 df-mpt2 5517 df-sub 7184 df-shft 9416 |
This theorem is referenced by: (None) |
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