Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 12257 | . . . . . . 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 2924 | . . . . 5 ⊢ (𝜑 → 𝑀 ∈ 𝑍) |
6 | 5 | ne0d 4300 | . . . 4 ⊢ (𝜑 → 𝑍 ≠ ∅) |
7 | ral0 4455 | . . . . . . 7 ⊢ ∀𝑧 ∈ ∅ (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥 | |
8 | simpr 487 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑆 = ∅) → 𝑆 = ∅) | |
9 | 8 | raleqdv 3415 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑆 = ∅) → (∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥 ↔ ∀𝑧 ∈ ∅ (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥)) |
10 | 7, 9 | mpbiri 260 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑆 = ∅) → ∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥) |
11 | 10 | ralrimivw 3183 | . . . . 5 ⊢ ((𝜑 ∧ 𝑆 = ∅) → ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥) |
12 | 11 | ralrimivw 3183 | . . . 4 ⊢ ((𝜑 ∧ 𝑆 = ∅) → ∀𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥) |
13 | r19.2z 4439 | . . . 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 683 | . . 3 ⊢ ((𝜑 ∧ 𝑆 = ∅) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥) |
15 | 14 | ralrimivw 3183 | . 2 ⊢ ((𝜑 ∧ 𝑆 = ∅) → ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥) |
16 | 1 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ 𝑆 = ∅) → 𝑀 ∈ ℤ) |
17 | ulm0.f | . . . 4 ⊢ (𝜑 → 𝐹:𝑍⟶(ℂ ↑m 𝑆)) | |
18 | 17 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ 𝑆 = ∅) → 𝐹:𝑍⟶(ℂ ↑m 𝑆)) |
19 | eqidd 2822 | . . 3 ⊢ (((𝜑 ∧ 𝑆 = ∅) ∧ (𝑘 ∈ 𝑍 ∧ 𝑧 ∈ 𝑆)) → ((𝐹‘𝑘)‘𝑧) = ((𝐹‘𝑘)‘𝑧)) | |
20 | eqidd 2822 | . . 3 ⊢ (((𝜑 ∧ 𝑆 = ∅) ∧ 𝑧 ∈ 𝑆) → (𝐺‘𝑧) = (𝐺‘𝑧)) | |
21 | ulm0.g | . . . 4 ⊢ (𝜑 → 𝐺:𝑆⟶ℂ) | |
22 | 21 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ 𝑆 = ∅) → 𝐺:𝑆⟶ℂ) |
23 | 0ex 5210 | . . . 4 ⊢ ∅ ∈ V | |
24 | 8, 23 | eqeltrdi 2921 | . . 3 ⊢ ((𝜑 ∧ 𝑆 = ∅) → 𝑆 ∈ V) |
25 | 4, 16, 18, 19, 20, 22, 24 | ulm2 24972 | . 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 259 | 1 ⊢ ((𝜑 ∧ 𝑆 = ∅) → 𝐹(⇝𝑢‘𝑆)𝐺) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 ∀wral 3138 ∃wrex 3139 Vcvv 3494 ∅c0 4290 class class class wbr 5065 ⟶wf 6350 ‘cfv 6354 (class class class)co 7155 ↑m cmap 8405 ℂcc 10534 < clt 10674 − cmin 10869 ℤcz 11980 ℤ≥cuz 12242 ℝ+crp 12388 abscabs 14592 ⇝𝑢culm 24963 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 ax-pre-lttri 10610 ax-pre-lttrn 10611 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-id 5459 df-po 5473 df-so 5474 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-ov 7158 df-oprab 7159 df-mpo 7160 df-er 8288 df-map 8407 df-pm 8408 df-en 8509 df-dom 8510 df-sdom 8511 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-neg 10872 df-z 11981 df-uz 12243 df-ulm 24964 |
This theorem is referenced by: pserulm 25009 |
Copyright terms: Public domain | W3C validator |