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Mirrors > Home > MPE Home > Th. List > shftf | Structured version Visualization version 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 6291 | . . 3 ⊢ (𝐹:𝐵⟶𝐶 → 𝐹 Fn 𝐵) | |
2 | shftfval.1 | . . . 4 ⊢ 𝐹 ∈ V | |
3 | 2 | shftfn 14220 | . . 3 ⊢ ((𝐹 Fn 𝐵 ∧ 𝐴 ∈ ℂ) → (𝐹 shift 𝐴) Fn {𝑥 ∈ ℂ ∣ (𝑥 − 𝐴) ∈ 𝐵}) |
4 | 1, 3 | sylan 575 | . 2 ⊢ ((𝐹:𝐵⟶𝐶 ∧ 𝐴 ∈ ℂ) → (𝐹 shift 𝐴) Fn {𝑥 ∈ ℂ ∣ (𝑥 − 𝐴) ∈ 𝐵}) |
5 | oveq1 6929 | . . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 − 𝐴) = (𝑦 − 𝐴)) | |
6 | 5 | eleq1d 2843 | . . . . 5 ⊢ (𝑥 = 𝑦 → ((𝑥 − 𝐴) ∈ 𝐵 ↔ (𝑦 − 𝐴) ∈ 𝐵)) |
7 | 6 | elrab 3571 | . . . 4 ⊢ (𝑦 ∈ {𝑥 ∈ ℂ ∣ (𝑥 − 𝐴) ∈ 𝐵} ↔ (𝑦 ∈ ℂ ∧ (𝑦 − 𝐴) ∈ 𝐵)) |
8 | simpr 479 | . . . . . 6 ⊢ ((𝐹:𝐵⟶𝐶 ∧ 𝐴 ∈ ℂ) → 𝐴 ∈ ℂ) | |
9 | simpl 476 | . . . . . 6 ⊢ ((𝑦 ∈ ℂ ∧ (𝑦 − 𝐴) ∈ 𝐵) → 𝑦 ∈ ℂ) | |
10 | 2 | shftval 14221 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ ℂ) → ((𝐹 shift 𝐴)‘𝑦) = (𝐹‘(𝑦 − 𝐴))) |
11 | 8, 9, 10 | syl2an 589 | . . . . 5 ⊢ (((𝐹:𝐵⟶𝐶 ∧ 𝐴 ∈ ℂ) ∧ (𝑦 ∈ ℂ ∧ (𝑦 − 𝐴) ∈ 𝐵)) → ((𝐹 shift 𝐴)‘𝑦) = (𝐹‘(𝑦 − 𝐴))) |
12 | simpl 476 | . . . . . 6 ⊢ ((𝐹:𝐵⟶𝐶 ∧ 𝐴 ∈ ℂ) → 𝐹:𝐵⟶𝐶) | |
13 | simpr 479 | . . . . . 6 ⊢ ((𝑦 ∈ ℂ ∧ (𝑦 − 𝐴) ∈ 𝐵) → (𝑦 − 𝐴) ∈ 𝐵) | |
14 | ffvelrn 6621 | . . . . . 6 ⊢ ((𝐹:𝐵⟶𝐶 ∧ (𝑦 − 𝐴) ∈ 𝐵) → (𝐹‘(𝑦 − 𝐴)) ∈ 𝐶) | |
15 | 12, 13, 14 | syl2an 589 | . . . . 5 ⊢ (((𝐹:𝐵⟶𝐶 ∧ 𝐴 ∈ ℂ) ∧ (𝑦 ∈ ℂ ∧ (𝑦 − 𝐴) ∈ 𝐵)) → (𝐹‘(𝑦 − 𝐴)) ∈ 𝐶) |
16 | 11, 15 | eqeltrd 2858 | . . . 4 ⊢ (((𝐹:𝐵⟶𝐶 ∧ 𝐴 ∈ ℂ) ∧ (𝑦 ∈ ℂ ∧ (𝑦 − 𝐴) ∈ 𝐵)) → ((𝐹 shift 𝐴)‘𝑦) ∈ 𝐶) |
17 | 7, 16 | sylan2b 587 | . . 3 ⊢ (((𝐹:𝐵⟶𝐶 ∧ 𝐴 ∈ ℂ) ∧ 𝑦 ∈ {𝑥 ∈ ℂ ∣ (𝑥 − 𝐴) ∈ 𝐵}) → ((𝐹 shift 𝐴)‘𝑦) ∈ 𝐶) |
18 | 17 | ralrimiva 3147 | . 2 ⊢ ((𝐹:𝐵⟶𝐶 ∧ 𝐴 ∈ ℂ) → ∀𝑦 ∈ {𝑥 ∈ ℂ ∣ (𝑥 − 𝐴) ∈ 𝐵} ((𝐹 shift 𝐴)‘𝑦) ∈ 𝐶) |
19 | ffnfv 6652 | . 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 578 | 1 ⊢ ((𝐹:𝐵⟶𝐶 ∧ 𝐴 ∈ ℂ) → (𝐹 shift 𝐴):{𝑥 ∈ ℂ ∣ (𝑥 − 𝐴) ∈ 𝐵}⟶𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1601 ∈ wcel 2106 ∀wral 3089 {crab 3093 Vcvv 3397 Fn wfn 6130 ⟶wf 6131 ‘cfv 6135 (class class class)co 6922 ℂcc 10270 − cmin 10606 shift cshi 14213 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rab 3098 df-v 3399 df-sbc 3652 df-csb 3751 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4672 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-id 5261 df-po 5274 df-so 5275 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-er 8026 df-en 8242 df-dom 8243 df-sdom 8244 df-pnf 10413 df-mnf 10414 df-ltxr 10416 df-sub 10608 df-shft 14214 |
This theorem is referenced by: (None) |
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