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 | . . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
2 | uzid 11702 | . . . . . . . 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 2712 | . . . . . 6 ⊢ (𝜑 → 𝑀 ∈ 𝑍) |
6 | ne0i 3921 | . . . . . 6 ⊢ (𝑀 ∈ 𝑍 → 𝑍 ≠ ∅) | |
7 | 5, 6 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑍 ≠ ∅) |
8 | 7 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑆 = ∅) → 𝑍 ≠ ∅) |
9 | ral0 4076 | . . . . . . 7 ⊢ ∀𝑧 ∈ ∅ (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥 | |
10 | simpr 477 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑆 = ∅) → 𝑆 = ∅) | |
11 | 10 | raleqdv 3144 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑆 = ∅) → (∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥 ↔ ∀𝑧 ∈ ∅ (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥)) |
12 | 9, 11 | mpbiri 248 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑆 = ∅) → ∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥) |
13 | 12 | ralrimivw 2967 | . . . . 5 ⊢ ((𝜑 ∧ 𝑆 = ∅) → ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥) |
14 | 13 | ralrimivw 2967 | . . . 4 ⊢ ((𝜑 ∧ 𝑆 = ∅) → ∀𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥) |
15 | r19.2z 4060 | . . . 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) | |
16 | 8, 14, 15 | syl2anc 693 | . . 3 ⊢ ((𝜑 ∧ 𝑆 = ∅) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥) |
17 | 16 | ralrimivw 2967 | . 2 ⊢ ((𝜑 ∧ 𝑆 = ∅) → ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥) |
18 | 1 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ 𝑆 = ∅) → 𝑀 ∈ ℤ) |
19 | ulm0.f | . . . 4 ⊢ (𝜑 → 𝐹:𝑍⟶(ℂ ↑𝑚 𝑆)) | |
20 | 19 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ 𝑆 = ∅) → 𝐹:𝑍⟶(ℂ ↑𝑚 𝑆)) |
21 | eqidd 2623 | . . 3 ⊢ (((𝜑 ∧ 𝑆 = ∅) ∧ (𝑘 ∈ 𝑍 ∧ 𝑧 ∈ 𝑆)) → ((𝐹‘𝑘)‘𝑧) = ((𝐹‘𝑘)‘𝑧)) | |
22 | eqidd 2623 | . . 3 ⊢ (((𝜑 ∧ 𝑆 = ∅) ∧ 𝑧 ∈ 𝑆) → (𝐺‘𝑧) = (𝐺‘𝑧)) | |
23 | ulm0.g | . . . 4 ⊢ (𝜑 → 𝐺:𝑆⟶ℂ) | |
24 | 23 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ 𝑆 = ∅) → 𝐺:𝑆⟶ℂ) |
25 | 0ex 4790 | . . . 4 ⊢ ∅ ∈ V | |
26 | 10, 25 | syl6eqel 2709 | . . 3 ⊢ ((𝜑 ∧ 𝑆 = ∅) → 𝑆 ∈ V) |
27 | 4, 18, 20, 21, 22, 24, 26 | ulm2 24139 | . 2 ⊢ ((𝜑 ∧ 𝑆 = ∅) → (𝐹(⇝𝑢‘𝑆)𝐺 ↔ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥)) |
28 | 17, 27 | mpbird 247 | 1 ⊢ ((𝜑 ∧ 𝑆 = ∅) → 𝐹(⇝𝑢‘𝑆)𝐺) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∀wral 2912 ∃wrex 2913 Vcvv 3200 ∅c0 3915 class class class wbr 4653 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 ↑𝑚 cmap 7857 ℂcc 9934 < clt 10074 − cmin 10266 ℤcz 11377 ℤ≥cuz 11687 ℝ+crp 11832 abscabs 13974 ⇝𝑢culm 24130 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-pre-lttri 10010 ax-pre-lttrn 10011 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-po 5035 df-so 5036 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-er 7742 df-map 7859 df-pm 7860 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-neg 10269 df-z 11378 df-uz 11688 df-ulm 24131 |
This theorem is referenced by: pserulm 24176 |
Copyright terms: Public domain | W3C validator |