Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > ulmcau2 | Structured version Visualization version GIF version |
Description: A sequence of functions converges uniformly iff it is uniformly Cauchy, which is to say that for every 0 < 𝑥 there is a 𝑗 such that for all 𝑗 ≤ 𝑘, 𝑚 the functions 𝐹(𝑘) and 𝐹(𝑚) are uniformly within 𝑥 of each other on 𝑆. (Contributed by Mario Carneiro, 1-Mar-2015.) |
Ref | Expression |
---|---|
ulmcau.z | ⊢ 𝑍 = (ℤ≥‘𝑀) |
ulmcau.m | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
ulmcau.s | ⊢ (𝜑 → 𝑆 ∈ 𝑉) |
ulmcau.f | ⊢ (𝜑 → 𝐹:𝑍⟶(ℂ ↑m 𝑆)) |
Ref | Expression |
---|---|
ulmcau2 | ⊢ (𝜑 → (𝐹 ∈ dom (⇝𝑢‘𝑆) ↔ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑚 ∈ (ℤ≥‘𝑘)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑚)‘𝑧))) < 𝑥)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ulmcau.z | . . 3 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
2 | ulmcau.m | . . 3 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
3 | ulmcau.s | . . 3 ⊢ (𝜑 → 𝑆 ∈ 𝑉) | |
4 | ulmcau.f | . . 3 ⊢ (𝜑 → 𝐹:𝑍⟶(ℂ ↑m 𝑆)) | |
5 | 1, 2, 3, 4 | ulmcau 25543 | . 2 ⊢ (𝜑 → (𝐹 ∈ dom (⇝𝑢‘𝑆) ↔ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑥)) |
6 | 1, 2, 3, 4 | ulmcaulem 25542 | . 2 ⊢ (𝜑 → (∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑥 ↔ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑚 ∈ (ℤ≥‘𝑘)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑚)‘𝑧))) < 𝑥)) |
7 | 5, 6 | bitrd 278 | 1 ⊢ (𝜑 → (𝐹 ∈ dom (⇝𝑢‘𝑆) ↔ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑚 ∈ (ℤ≥‘𝑘)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑚)‘𝑧))) < 𝑥)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1539 ∈ wcel 2106 ∀wral 3064 ∃wrex 3065 class class class wbr 5075 dom cdm 5586 ⟶wf 6424 ‘cfv 6428 (class class class)co 7269 ↑m cmap 8604 ℂcc 10858 < clt 10998 − cmin 11194 ℤcz 12308 ℤ≥cuz 12571 ℝ+crp 12719 abscabs 14934 ⇝𝑢culm 25524 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5210 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7580 ax-cnex 10916 ax-resscn 10917 ax-1cn 10918 ax-icn 10919 ax-addcl 10920 ax-addrcl 10921 ax-mulcl 10922 ax-mulrcl 10923 ax-mulcom 10924 ax-addass 10925 ax-mulass 10926 ax-distr 10927 ax-i2m1 10928 ax-1ne0 10929 ax-1rid 10930 ax-rnegex 10931 ax-rrecex 10932 ax-cnre 10933 ax-pre-lttri 10934 ax-pre-lttrn 10935 ax-pre-ltadd 10936 ax-pre-mulgt0 10937 ax-pre-sup 10938 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3433 df-sbc 3718 df-csb 3834 df-dif 3891 df-un 3893 df-in 3895 df-ss 3905 df-pss 3907 df-nul 4259 df-if 4462 df-pw 4537 df-sn 4564 df-pr 4566 df-op 4570 df-uni 4842 df-iun 4928 df-br 5076 df-opab 5138 df-mpt 5159 df-tr 5193 df-id 5486 df-eprel 5492 df-po 5500 df-so 5501 df-fr 5541 df-we 5543 df-xp 5592 df-rel 5593 df-cnv 5594 df-co 5595 df-dm 5596 df-rn 5597 df-res 5598 df-ima 5599 df-pred 6197 df-ord 6264 df-on 6265 df-lim 6266 df-suc 6267 df-iota 6386 df-fun 6430 df-fn 6431 df-f 6432 df-f1 6433 df-fo 6434 df-f1o 6435 df-fv 6436 df-riota 7226 df-ov 7272 df-oprab 7273 df-mpo 7274 df-om 7705 df-1st 7822 df-2nd 7823 df-frecs 8086 df-wrecs 8117 df-recs 8191 df-rdg 8230 df-er 8487 df-map 8606 df-pm 8607 df-en 8723 df-dom 8724 df-sdom 8725 df-sup 9190 df-inf 9191 df-pnf 11000 df-mnf 11001 df-xr 11002 df-ltxr 11003 df-le 11004 df-sub 11196 df-neg 11197 df-div 11622 df-nn 11963 df-2 12025 df-3 12026 df-n0 12223 df-z 12309 df-uz 12572 df-rp 12720 df-ico 13074 df-fl 13501 df-seq 13711 df-exp 13772 df-cj 14799 df-re 14800 df-im 14801 df-sqrt 14935 df-abs 14936 df-limsup 15169 df-clim 15186 df-rlim 15187 df-ulm 25525 |
This theorem is referenced by: ulmdvlem3 25550 |
Copyright terms: Public domain | W3C validator |