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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 10431 | . . 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 7616 | . . . 4 ⊢ ℂ ∈ V | |
4 | 3 | a1i 9 | . . 3 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐴 ∈ ℂ) → ℂ ∈ V) |
5 | rnexg 4740 | . . . . 5 ⊢ (𝐹 ∈ 𝑉 → ran 𝐹 ∈ V) | |
6 | 5 | ad2antrr 475 | . . . 4 ⊢ (((𝐹 ∈ 𝑉 ∧ 𝐴 ∈ ℂ) ∧ 𝑧 ∈ ℂ) → ran 𝐹 ∈ V) |
7 | vex 2644 | . . . . . . . 8 ⊢ 𝑢 ∈ V | |
8 | breq2 3879 | . . . . . . . 8 ⊢ (𝑤 = 𝑢 → ((𝑧 − 𝐴)𝐹𝑤 ↔ (𝑧 − 𝐴)𝐹𝑢)) | |
9 | 7, 8 | elab 2782 | . . . . . . 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 7944 | . . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (𝑧 − 𝐴) ∈ ℂ) |
13 | brelrng 4708 | . . . . . . . . . 10 ⊢ (((𝑧 − 𝐴) ∈ ℂ ∧ 𝑢 ∈ V ∧ (𝑧 − 𝐴)𝐹𝑢) → 𝑢 ∈ ran 𝐹) | |
14 | 7, 13 | mp3an2 1271 | . . . . . . . . 9 ⊢ (((𝑧 − 𝐴) ∈ ℂ ∧ (𝑧 − 𝐴)𝐹𝑢) → 𝑢 ∈ ran 𝐹) |
15 | 12, 14 | sylan 279 | . . . . . . . 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 3053 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑧 ∈ ℂ) → {𝑤 ∣ (𝑧 − 𝐴)𝐹𝑤} ⊆ ran 𝐹) |
19 | 18 | adantll 463 | . . . 4 ⊢ (((𝐹 ∈ 𝑉 ∧ 𝐴 ∈ ℂ) ∧ 𝑧 ∈ ℂ) → {𝑤 ∣ (𝑧 − 𝐴)𝐹𝑤} ⊆ ran 𝐹) |
20 | 6, 19 | ssexd 4008 | . . 3 ⊢ (((𝐹 ∈ 𝑉 ∧ 𝐴 ∈ ℂ) ∧ 𝑧 ∈ ℂ) → {𝑤 ∣ (𝑧 − 𝐴)𝐹𝑤} ∈ V) |
21 | 4, 20 | opabex3d 5950 | . 2 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐴 ∈ ℂ) → {〈𝑧, 𝑤〉 ∣ (𝑧 ∈ ℂ ∧ (𝑧 − 𝐴)𝐹𝑤)} ∈ V) |
22 | 2, 21 | eqeltrd 2176 | 1 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐴 ∈ ℂ) → (𝐹 shift 𝐴) ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1299 ∈ wcel 1448 {cab 2086 Vcvv 2641 ⊆ wss 3021 class class class wbr 3875 {copab 3928 ran crn 4478 (class class class)co 5706 ℂcc 7498 − cmin 7804 shift cshi 10427 |
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 584 ax-in2 585 ax-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-13 1459 ax-14 1460 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 ax-coll 3983 ax-sep 3986 ax-pow 4038 ax-pr 4069 ax-un 4293 ax-setind 4390 ax-cnex 7586 ax-resscn 7587 ax-1cn 7588 ax-icn 7590 ax-addcl 7591 ax-addrcl 7592 ax-mulcl 7593 ax-addcom 7595 ax-addass 7597 ax-distr 7599 ax-i2m1 7600 ax-0id 7603 ax-rnegex 7604 ax-cnre 7606 |
This theorem depends on definitions: df-bi 116 df-3an 932 df-tru 1302 df-fal 1305 df-nf 1405 df-sb 1704 df-eu 1963 df-mo 1964 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-ne 2268 df-ral 2380 df-rex 2381 df-reu 2382 df-rab 2384 df-v 2643 df-sbc 2863 df-csb 2956 df-dif 3023 df-un 3025 df-in 3027 df-ss 3034 df-pw 3459 df-sn 3480 df-pr 3481 df-op 3483 df-uni 3684 df-iun 3762 df-br 3876 df-opab 3930 df-mpt 3931 df-id 4153 df-xp 4483 df-rel 4484 df-cnv 4485 df-co 4486 df-dm 4487 df-rn 4488 df-res 4489 df-ima 4490 df-iota 5024 df-fun 5061 df-fn 5062 df-f 5063 df-f1 5064 df-fo 5065 df-f1o 5066 df-fv 5067 df-riota 5662 df-ov 5709 df-oprab 5710 df-mpo 5711 df-sub 7806 df-shft 10428 |
This theorem is referenced by: 2shfti 10444 climshftlemg 10910 climshft 10912 climshft2 10914 eftlub 11194 |
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