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Mirrors > Home > ILE Home > Th. List > ovshftex | GIF version |
Description: Existence of the result of applying shift. (Contributed by Jim Kingdon, 15-Aug-2021.) |
Ref | Expression |
---|---|
ovshftex | ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐴 ∈ ℂ) → (𝐹 shift 𝐴) ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | shftfvalg 10622 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐹 ∈ 𝑉) → (𝐹 shift 𝐴) = {〈𝑧, 𝑤〉 ∣ (𝑧 ∈ ℂ ∧ (𝑧 − 𝐴)𝐹𝑤)}) | |
2 | 1 | ancoms 266 | . 2 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐴 ∈ ℂ) → (𝐹 shift 𝐴) = {〈𝑧, 𝑤〉 ∣ (𝑧 ∈ ℂ ∧ (𝑧 − 𝐴)𝐹𝑤)}) |
3 | cnex 7768 | . . . 4 ⊢ ℂ ∈ V | |
4 | 3 | a1i 9 | . . 3 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐴 ∈ ℂ) → ℂ ∈ V) |
5 | rnexg 4812 | . . . . 5 ⊢ (𝐹 ∈ 𝑉 → ran 𝐹 ∈ V) | |
6 | 5 | ad2antrr 480 | . . . 4 ⊢ (((𝐹 ∈ 𝑉 ∧ 𝐴 ∈ ℂ) ∧ 𝑧 ∈ ℂ) → ran 𝐹 ∈ V) |
7 | vex 2692 | . . . . . . . 8 ⊢ 𝑢 ∈ V | |
8 | breq2 3941 | . . . . . . . 8 ⊢ (𝑤 = 𝑢 → ((𝑧 − 𝐴)𝐹𝑤 ↔ (𝑧 − 𝐴)𝐹𝑢)) | |
9 | 7, 8 | elab 2832 | . . . . . . 7 ⊢ (𝑢 ∈ {𝑤 ∣ (𝑧 − 𝐴)𝐹𝑤} ↔ (𝑧 − 𝐴)𝐹𝑢) |
10 | simpr 109 | . . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝑧 ∈ ℂ) → 𝑧 ∈ ℂ) | |
11 | simpl 108 | . . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝑧 ∈ ℂ) → 𝐴 ∈ ℂ) | |
12 | 10, 11 | subcld 8097 | . . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (𝑧 − 𝐴) ∈ ℂ) |
13 | brelrng 4778 | . . . . . . . . . 10 ⊢ (((𝑧 − 𝐴) ∈ ℂ ∧ 𝑢 ∈ V ∧ (𝑧 − 𝐴)𝐹𝑢) → 𝑢 ∈ ran 𝐹) | |
14 | 7, 13 | mp3an2 1304 | . . . . . . . . 9 ⊢ (((𝑧 − 𝐴) ∈ ℂ ∧ (𝑧 − 𝐴)𝐹𝑢) → 𝑢 ∈ ran 𝐹) |
15 | 12, 14 | sylan 281 | . . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝑧 ∈ ℂ) ∧ (𝑧 − 𝐴)𝐹𝑢) → 𝑢 ∈ ran 𝐹) |
16 | 15 | ex 114 | . . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑧 ∈ ℂ) → ((𝑧 − 𝐴)𝐹𝑢 → 𝑢 ∈ ran 𝐹)) |
17 | 9, 16 | syl5bi 151 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (𝑢 ∈ {𝑤 ∣ (𝑧 − 𝐴)𝐹𝑤} → 𝑢 ∈ ran 𝐹)) |
18 | 17 | ssrdv 3108 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑧 ∈ ℂ) → {𝑤 ∣ (𝑧 − 𝐴)𝐹𝑤} ⊆ ran 𝐹) |
19 | 18 | adantll 468 | . . . 4 ⊢ (((𝐹 ∈ 𝑉 ∧ 𝐴 ∈ ℂ) ∧ 𝑧 ∈ ℂ) → {𝑤 ∣ (𝑧 − 𝐴)𝐹𝑤} ⊆ ran 𝐹) |
20 | 6, 19 | ssexd 4076 | . . 3 ⊢ (((𝐹 ∈ 𝑉 ∧ 𝐴 ∈ ℂ) ∧ 𝑧 ∈ ℂ) → {𝑤 ∣ (𝑧 − 𝐴)𝐹𝑤} ∈ V) |
21 | 4, 20 | opabex3d 6027 | . 2 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐴 ∈ ℂ) → {〈𝑧, 𝑤〉 ∣ (𝑧 ∈ ℂ ∧ (𝑧 − 𝐴)𝐹𝑤)} ∈ V) |
22 | 2, 21 | eqeltrd 2217 | 1 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐴 ∈ ℂ) → (𝐹 shift 𝐴) ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1332 ∈ wcel 1481 {cab 2126 Vcvv 2689 ⊆ wss 3076 class class class wbr 3937 {copab 3996 ran crn 4548 (class class class)co 5782 ℂcc 7642 − cmin 7957 shift cshi 10618 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-coll 4051 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-cnex 7735 ax-resscn 7736 ax-1cn 7737 ax-icn 7739 ax-addcl 7740 ax-addrcl 7741 ax-mulcl 7742 ax-addcom 7744 ax-addass 7746 ax-distr 7748 ax-i2m1 7749 ax-0id 7752 ax-rnegex 7753 ax-cnre 7755 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-ral 2422 df-rex 2423 df-reu 2424 df-rab 2426 df-v 2691 df-sbc 2914 df-csb 3008 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-iun 3823 df-br 3938 df-opab 3998 df-mpt 3999 df-id 4223 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 df-fv 5139 df-riota 5738 df-ov 5785 df-oprab 5786 df-mpo 5787 df-sub 7959 df-shft 10619 |
This theorem is referenced by: 2shfti 10635 climshftlemg 11103 climshft 11105 climshft2 11107 eftlub 11433 |
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