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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 | . . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
2 | uzid 11945 | . . . . . . . 8 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ≥‘𝑀)) | |
3 | 1, 2 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘𝑀)) |
4 | ulm0.z | . . . . . . 7 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
5 | 3, 4 | syl6eleqr 2889 | . . . . . 6 ⊢ (𝜑 → 𝑀 ∈ 𝑍) |
6 | 5 | ne0d 4122 | . . . . 5 ⊢ (𝜑 → 𝑍 ≠ ∅) |
7 | 6 | adantr 473 | . . . 4 ⊢ ((𝜑 ∧ 𝑆 = ∅) → 𝑍 ≠ ∅) |
8 | ral0 4269 | . . . . . . 7 ⊢ ∀𝑧 ∈ ∅ (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥 | |
9 | simpr 478 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑆 = ∅) → 𝑆 = ∅) | |
10 | 9 | raleqdv 3327 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑆 = ∅) → (∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥 ↔ ∀𝑧 ∈ ∅ (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥)) |
11 | 8, 10 | mpbiri 250 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑆 = ∅) → ∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥) |
12 | 11 | ralrimivw 3148 | . . . . 5 ⊢ ((𝜑 ∧ 𝑆 = ∅) → ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥) |
13 | 12 | ralrimivw 3148 | . . . 4 ⊢ ((𝜑 ∧ 𝑆 = ∅) → ∀𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥) |
14 | r19.2z 4253 | . . . 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) | |
15 | 7, 13, 14 | syl2anc 580 | . . 3 ⊢ ((𝜑 ∧ 𝑆 = ∅) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥) |
16 | 15 | ralrimivw 3148 | . 2 ⊢ ((𝜑 ∧ 𝑆 = ∅) → ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥) |
17 | 1 | adantr 473 | . . 3 ⊢ ((𝜑 ∧ 𝑆 = ∅) → 𝑀 ∈ ℤ) |
18 | ulm0.f | . . . 4 ⊢ (𝜑 → 𝐹:𝑍⟶(ℂ ↑𝑚 𝑆)) | |
19 | 18 | adantr 473 | . . 3 ⊢ ((𝜑 ∧ 𝑆 = ∅) → 𝐹:𝑍⟶(ℂ ↑𝑚 𝑆)) |
20 | eqidd 2800 | . . 3 ⊢ (((𝜑 ∧ 𝑆 = ∅) ∧ (𝑘 ∈ 𝑍 ∧ 𝑧 ∈ 𝑆)) → ((𝐹‘𝑘)‘𝑧) = ((𝐹‘𝑘)‘𝑧)) | |
21 | eqidd 2800 | . . 3 ⊢ (((𝜑 ∧ 𝑆 = ∅) ∧ 𝑧 ∈ 𝑆) → (𝐺‘𝑧) = (𝐺‘𝑧)) | |
22 | ulm0.g | . . . 4 ⊢ (𝜑 → 𝐺:𝑆⟶ℂ) | |
23 | 22 | adantr 473 | . . 3 ⊢ ((𝜑 ∧ 𝑆 = ∅) → 𝐺:𝑆⟶ℂ) |
24 | 0ex 4984 | . . . 4 ⊢ ∅ ∈ V | |
25 | 9, 24 | syl6eqel 2886 | . . 3 ⊢ ((𝜑 ∧ 𝑆 = ∅) → 𝑆 ∈ V) |
26 | 4, 17, 19, 20, 21, 23, 25 | ulm2 24480 | . 2 ⊢ ((𝜑 ∧ 𝑆 = ∅) → (𝐹(⇝𝑢‘𝑆)𝐺 ↔ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥)) |
27 | 16, 26 | mpbird 249 | 1 ⊢ ((𝜑 ∧ 𝑆 = ∅) → 𝐹(⇝𝑢‘𝑆)𝐺) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 385 = wceq 1653 ∈ wcel 2157 ≠ wne 2971 ∀wral 3089 ∃wrex 3090 Vcvv 3385 ∅c0 4115 class class class wbr 4843 ⟶wf 6097 ‘cfv 6101 (class class class)co 6878 ↑𝑚 cmap 8095 ℂcc 10222 < clt 10363 − cmin 10556 ℤcz 11666 ℤ≥cuz 11930 ℝ+crp 12074 abscabs 14315 ⇝𝑢culm 24471 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-rep 4964 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 ax-cnex 10280 ax-resscn 10281 ax-pre-lttri 10298 ax-pre-lttrn 10299 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rab 3098 df-v 3387 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-op 4375 df-uni 4629 df-iun 4712 df-br 4844 df-opab 4906 df-mpt 4923 df-id 5220 df-po 5233 df-so 5234 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fo 6107 df-f1o 6108 df-fv 6109 df-ov 6881 df-oprab 6882 df-mpt2 6883 df-er 7982 df-map 8097 df-pm 8098 df-en 8196 df-dom 8197 df-sdom 8198 df-pnf 10365 df-mnf 10366 df-xr 10367 df-ltxr 10368 df-le 10369 df-neg 10559 df-z 11667 df-uz 11931 df-ulm 24472 |
This theorem is referenced by: pserulm 24517 |
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