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Mirrors > Home > MPE Home > Th. List > ulm0 | Structured version Visualization version GIF version |
Description: Every function converges uniformly on the empty set. (Contributed by Mario Carneiro, 3-Mar-2015.) |
Ref | Expression |
---|---|
ulm0.z | ⊢ 𝑍 = (ℤ≥‘𝑀) |
ulm0.m | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
ulm0.f | ⊢ (𝜑 → 𝐹:𝑍⟶(ℂ ↑m 𝑆)) |
ulm0.g | ⊢ (𝜑 → 𝐺:𝑆⟶ℂ) |
Ref | Expression |
---|---|
ulm0 | ⊢ ((𝜑 ∧ 𝑆 = ∅) → 𝐹(⇝𝑢‘𝑆)𝐺) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ulm0.m | . . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
2 | uzid 12246 | . . . . . . 7 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ≥‘𝑀)) | |
3 | 1, 2 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘𝑀)) |
4 | ulm0.z | . . . . . 6 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
5 | 3, 4 | eleqtrrdi 2901 | . . . . 5 ⊢ (𝜑 → 𝑀 ∈ 𝑍) |
6 | 5 | ne0d 4251 | . . . 4 ⊢ (𝜑 → 𝑍 ≠ ∅) |
7 | ral0 4414 | . . . . . . 7 ⊢ ∀𝑧 ∈ ∅ (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥 | |
8 | simpr 488 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑆 = ∅) → 𝑆 = ∅) | |
9 | 8 | raleqdv 3364 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑆 = ∅) → (∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥 ↔ ∀𝑧 ∈ ∅ (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥)) |
10 | 7, 9 | mpbiri 261 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑆 = ∅) → ∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥) |
11 | 10 | ralrimivw 3150 | . . . . 5 ⊢ ((𝜑 ∧ 𝑆 = ∅) → ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥) |
12 | 11 | ralrimivw 3150 | . . . 4 ⊢ ((𝜑 ∧ 𝑆 = ∅) → ∀𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥) |
13 | r19.2z 4398 | . . . 4 ⊢ ((𝑍 ≠ ∅ ∧ ∀𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥) | |
14 | 6, 12, 13 | syl2an2r 684 | . . 3 ⊢ ((𝜑 ∧ 𝑆 = ∅) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥) |
15 | 14 | ralrimivw 3150 | . 2 ⊢ ((𝜑 ∧ 𝑆 = ∅) → ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥) |
16 | 1 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ 𝑆 = ∅) → 𝑀 ∈ ℤ) |
17 | ulm0.f | . . . 4 ⊢ (𝜑 → 𝐹:𝑍⟶(ℂ ↑m 𝑆)) | |
18 | 17 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ 𝑆 = ∅) → 𝐹:𝑍⟶(ℂ ↑m 𝑆)) |
19 | eqidd 2799 | . . 3 ⊢ (((𝜑 ∧ 𝑆 = ∅) ∧ (𝑘 ∈ 𝑍 ∧ 𝑧 ∈ 𝑆)) → ((𝐹‘𝑘)‘𝑧) = ((𝐹‘𝑘)‘𝑧)) | |
20 | eqidd 2799 | . . 3 ⊢ (((𝜑 ∧ 𝑆 = ∅) ∧ 𝑧 ∈ 𝑆) → (𝐺‘𝑧) = (𝐺‘𝑧)) | |
21 | ulm0.g | . . . 4 ⊢ (𝜑 → 𝐺:𝑆⟶ℂ) | |
22 | 21 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ 𝑆 = ∅) → 𝐺:𝑆⟶ℂ) |
23 | 0ex 5175 | . . . 4 ⊢ ∅ ∈ V | |
24 | 8, 23 | eqeltrdi 2898 | . . 3 ⊢ ((𝜑 ∧ 𝑆 = ∅) → 𝑆 ∈ V) |
25 | 4, 16, 18, 19, 20, 22, 24 | ulm2 24980 | . 2 ⊢ ((𝜑 ∧ 𝑆 = ∅) → (𝐹(⇝𝑢‘𝑆)𝐺 ↔ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥)) |
26 | 15, 25 | mpbird 260 | 1 ⊢ ((𝜑 ∧ 𝑆 = ∅) → 𝐹(⇝𝑢‘𝑆)𝐺) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 ∀wral 3106 ∃wrex 3107 Vcvv 3441 ∅c0 4243 class class class wbr 5030 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 ↑m cmap 8389 ℂcc 10524 < clt 10664 − cmin 10859 ℤcz 11969 ℤ≥cuz 12231 ℝ+crp 12377 abscabs 14585 ⇝𝑢culm 24971 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-pre-lttri 10600 ax-pre-lttrn 10601 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-po 5438 df-so 5439 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-er 8272 df-map 8391 df-pm 8392 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-neg 10862 df-z 11970 df-uz 12232 df-ulm 24972 |
This theorem is referenced by: pserulm 25017 |
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