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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 6516 | . . 3 ⊢ (𝐹:𝐵⟶𝐶 → 𝐹 Fn 𝐵) | |
2 | shftfval.1 | . . . 4 ⊢ 𝐹 ∈ V | |
3 | 2 | shftfn 14434 | . . 3 ⊢ ((𝐹 Fn 𝐵 ∧ 𝐴 ∈ ℂ) → (𝐹 shift 𝐴) Fn {𝑥 ∈ ℂ ∣ (𝑥 − 𝐴) ∈ 𝐵}) |
4 | 1, 3 | sylan 582 | . 2 ⊢ ((𝐹:𝐵⟶𝐶 ∧ 𝐴 ∈ ℂ) → (𝐹 shift 𝐴) Fn {𝑥 ∈ ℂ ∣ (𝑥 − 𝐴) ∈ 𝐵}) |
5 | oveq1 7165 | . . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 − 𝐴) = (𝑦 − 𝐴)) | |
6 | 5 | eleq1d 2899 | . . . . 5 ⊢ (𝑥 = 𝑦 → ((𝑥 − 𝐴) ∈ 𝐵 ↔ (𝑦 − 𝐴) ∈ 𝐵)) |
7 | 6 | elrab 3682 | . . . 4 ⊢ (𝑦 ∈ {𝑥 ∈ ℂ ∣ (𝑥 − 𝐴) ∈ 𝐵} ↔ (𝑦 ∈ ℂ ∧ (𝑦 − 𝐴) ∈ 𝐵)) |
8 | simpr 487 | . . . . . 6 ⊢ ((𝐹:𝐵⟶𝐶 ∧ 𝐴 ∈ ℂ) → 𝐴 ∈ ℂ) | |
9 | simpl 485 | . . . . . 6 ⊢ ((𝑦 ∈ ℂ ∧ (𝑦 − 𝐴) ∈ 𝐵) → 𝑦 ∈ ℂ) | |
10 | 2 | shftval 14435 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ ℂ) → ((𝐹 shift 𝐴)‘𝑦) = (𝐹‘(𝑦 − 𝐴))) |
11 | 8, 9, 10 | syl2an 597 | . . . . 5 ⊢ (((𝐹:𝐵⟶𝐶 ∧ 𝐴 ∈ ℂ) ∧ (𝑦 ∈ ℂ ∧ (𝑦 − 𝐴) ∈ 𝐵)) → ((𝐹 shift 𝐴)‘𝑦) = (𝐹‘(𝑦 − 𝐴))) |
12 | simpl 485 | . . . . . 6 ⊢ ((𝐹:𝐵⟶𝐶 ∧ 𝐴 ∈ ℂ) → 𝐹:𝐵⟶𝐶) | |
13 | simpr 487 | . . . . . 6 ⊢ ((𝑦 ∈ ℂ ∧ (𝑦 − 𝐴) ∈ 𝐵) → (𝑦 − 𝐴) ∈ 𝐵) | |
14 | ffvelrn 6851 | . . . . . 6 ⊢ ((𝐹:𝐵⟶𝐶 ∧ (𝑦 − 𝐴) ∈ 𝐵) → (𝐹‘(𝑦 − 𝐴)) ∈ 𝐶) | |
15 | 12, 13, 14 | syl2an 597 | . . . . 5 ⊢ (((𝐹:𝐵⟶𝐶 ∧ 𝐴 ∈ ℂ) ∧ (𝑦 ∈ ℂ ∧ (𝑦 − 𝐴) ∈ 𝐵)) → (𝐹‘(𝑦 − 𝐴)) ∈ 𝐶) |
16 | 11, 15 | eqeltrd 2915 | . . . 4 ⊢ (((𝐹:𝐵⟶𝐶 ∧ 𝐴 ∈ ℂ) ∧ (𝑦 ∈ ℂ ∧ (𝑦 − 𝐴) ∈ 𝐵)) → ((𝐹 shift 𝐴)‘𝑦) ∈ 𝐶) |
17 | 7, 16 | sylan2b 595 | . . 3 ⊢ (((𝐹:𝐵⟶𝐶 ∧ 𝐴 ∈ ℂ) ∧ 𝑦 ∈ {𝑥 ∈ ℂ ∣ (𝑥 − 𝐴) ∈ 𝐵}) → ((𝐹 shift 𝐴)‘𝑦) ∈ 𝐶) |
18 | 17 | ralrimiva 3184 | . 2 ⊢ ((𝐹:𝐵⟶𝐶 ∧ 𝐴 ∈ ℂ) → ∀𝑦 ∈ {𝑥 ∈ ℂ ∣ (𝑥 − 𝐴) ∈ 𝐵} ((𝐹 shift 𝐴)‘𝑦) ∈ 𝐶) |
19 | ffnfv 6884 | . 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 585 | 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 398 = wceq 1537 ∈ wcel 2114 ∀wral 3140 {crab 3144 Vcvv 3496 Fn wfn 6352 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 ℂcc 10537 − cmin 10872 shift cshi 14427 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-po 5476 df-so 5477 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-ltxr 10682 df-sub 10874 df-shft 14428 |
This theorem is referenced by: (None) |
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