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Mirrors > Home > MPE Home > Th. List > ulmclm | Structured version Visualization version GIF version |
Description: A uniform limit of functions converges pointwise. (Contributed by Mario Carneiro, 27-Feb-2015.) |
Ref | Expression |
---|---|
ulmclm.z | ⊢ 𝑍 = (ℤ≥‘𝑀) |
ulmclm.m | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
ulmclm.f | ⊢ (𝜑 → 𝐹:𝑍⟶(ℂ ↑m 𝑆)) |
ulmclm.a | ⊢ (𝜑 → 𝐴 ∈ 𝑆) |
ulmclm.h | ⊢ (𝜑 → 𝐻 ∈ 𝑊) |
ulmclm.e | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝐹‘𝑘)‘𝐴) = (𝐻‘𝑘)) |
ulmclm.u | ⊢ (𝜑 → 𝐹(⇝𝑢‘𝑆)𝐺) |
Ref | Expression |
---|---|
ulmclm | ⊢ (𝜑 → 𝐻 ⇝ (𝐺‘𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ulmclm.u | . 2 ⊢ (𝜑 → 𝐹(⇝𝑢‘𝑆)𝐺) | |
2 | ulmclm.a | . . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ 𝑆) | |
3 | fveq2 6665 | . . . . . . . . . . 11 ⊢ (𝑧 = 𝐴 → ((𝐹‘𝑘)‘𝑧) = ((𝐹‘𝑘)‘𝐴)) | |
4 | fveq2 6665 | . . . . . . . . . . 11 ⊢ (𝑧 = 𝐴 → (𝐺‘𝑧) = (𝐺‘𝐴)) | |
5 | 3, 4 | oveq12d 7168 | . . . . . . . . . 10 ⊢ (𝑧 = 𝐴 → (((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧)) = (((𝐹‘𝑘)‘𝐴) − (𝐺‘𝐴))) |
6 | 5 | fveq2d 6669 | . . . . . . . . 9 ⊢ (𝑧 = 𝐴 → (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) = (abs‘(((𝐹‘𝑘)‘𝐴) − (𝐺‘𝐴)))) |
7 | 6 | breq1d 5069 | . . . . . . . 8 ⊢ (𝑧 = 𝐴 → ((abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥 ↔ (abs‘(((𝐹‘𝑘)‘𝐴) − (𝐺‘𝐴))) < 𝑥)) |
8 | 7 | rspcv 3618 | . . . . . . 7 ⊢ (𝐴 ∈ 𝑆 → (∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥 → (abs‘(((𝐹‘𝑘)‘𝐴) − (𝐺‘𝐴))) < 𝑥)) |
9 | 2, 8 | syl 17 | . . . . . 6 ⊢ (𝜑 → (∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥 → (abs‘(((𝐹‘𝑘)‘𝐴) − (𝐺‘𝐴))) < 𝑥)) |
10 | 9 | ralimdv 3178 | . . . . 5 ⊢ (𝜑 → (∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥 → ∀𝑘 ∈ (ℤ≥‘𝑗)(abs‘(((𝐹‘𝑘)‘𝐴) − (𝐺‘𝐴))) < 𝑥)) |
11 | 10 | reximdv 3273 | . . . 4 ⊢ (𝜑 → (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(abs‘(((𝐹‘𝑘)‘𝐴) − (𝐺‘𝐴))) < 𝑥)) |
12 | 11 | ralimdv 3178 | . . 3 ⊢ (𝜑 → (∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥 → ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(abs‘(((𝐹‘𝑘)‘𝐴) − (𝐺‘𝐴))) < 𝑥)) |
13 | ulmclm.z | . . . 4 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
14 | ulmclm.m | . . . 4 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
15 | ulmclm.f | . . . 4 ⊢ (𝜑 → 𝐹:𝑍⟶(ℂ ↑m 𝑆)) | |
16 | eqidd 2822 | . . . 4 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑍 ∧ 𝑧 ∈ 𝑆)) → ((𝐹‘𝑘)‘𝑧) = ((𝐹‘𝑘)‘𝑧)) | |
17 | eqidd 2822 | . . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → (𝐺‘𝑧) = (𝐺‘𝑧)) | |
18 | ulmcl 24963 | . . . . 5 ⊢ (𝐹(⇝𝑢‘𝑆)𝐺 → 𝐺:𝑆⟶ℂ) | |
19 | 1, 18 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐺:𝑆⟶ℂ) |
20 | ulmscl 24961 | . . . . 5 ⊢ (𝐹(⇝𝑢‘𝑆)𝐺 → 𝑆 ∈ V) | |
21 | 1, 20 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑆 ∈ V) |
22 | 13, 14, 15, 16, 17, 19, 21 | ulm2 24967 | . . 3 ⊢ (𝜑 → (𝐹(⇝𝑢‘𝑆)𝐺 ↔ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥)) |
23 | ulmclm.h | . . . 4 ⊢ (𝜑 → 𝐻 ∈ 𝑊) | |
24 | ulmclm.e | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝐹‘𝑘)‘𝐴) = (𝐻‘𝑘)) | |
25 | 24 | eqcomd 2827 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐻‘𝑘) = ((𝐹‘𝑘)‘𝐴)) |
26 | 19, 2 | ffvelrnd 6847 | . . . 4 ⊢ (𝜑 → (𝐺‘𝐴) ∈ ℂ) |
27 | 15 | ffvelrnda 6846 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ (ℂ ↑m 𝑆)) |
28 | elmapi 8422 | . . . . . 6 ⊢ ((𝐹‘𝑘) ∈ (ℂ ↑m 𝑆) → (𝐹‘𝑘):𝑆⟶ℂ) | |
29 | 27, 28 | syl 17 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘):𝑆⟶ℂ) |
30 | 2 | adantr 483 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ 𝑆) |
31 | 29, 30 | ffvelrnd 6847 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝐹‘𝑘)‘𝐴) ∈ ℂ) |
32 | 13, 14, 23, 25, 26, 31 | clim2c 14856 | . . 3 ⊢ (𝜑 → (𝐻 ⇝ (𝐺‘𝐴) ↔ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(abs‘(((𝐹‘𝑘)‘𝐴) − (𝐺‘𝐴))) < 𝑥)) |
33 | 12, 22, 32 | 3imtr4d 296 | . 2 ⊢ (𝜑 → (𝐹(⇝𝑢‘𝑆)𝐺 → 𝐻 ⇝ (𝐺‘𝐴))) |
34 | 1, 33 | mpd 15 | 1 ⊢ (𝜑 → 𝐻 ⇝ (𝐺‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∀wral 3138 ∃wrex 3139 Vcvv 3495 class class class wbr 5059 ⟶wf 6346 ‘cfv 6350 (class class class)co 7150 ↑m cmap 8400 ℂcc 10529 < clt 10669 − cmin 10864 ℤcz 11975 ℤ≥cuz 12237 ℝ+crp 12383 abscabs 14587 ⇝ cli 14835 ⇝𝑢culm 24958 |
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 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-pre-lttri 10605 ax-pre-lttrn 10606 |
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 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5455 df-po 5469 df-so 5470 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-ov 7153 df-oprab 7154 df-mpo 7155 df-1st 7683 df-2nd 7684 df-er 8283 df-map 8402 df-pm 8403 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-neg 10867 df-z 11976 df-uz 12238 df-clim 14839 df-ulm 24959 |
This theorem is referenced by: ulmuni 24974 ulmdvlem3 24984 mbfulm 24988 pserulm 25004 lgamgulm2 25607 lgamcvglem 25611 knoppcnlem9 33835 knoppndvlem4 33849 |
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