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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 6645 | . . . . . . . . . . 11 ⊢ (𝑧 = 𝐴 → ((𝐹‘𝑘)‘𝑧) = ((𝐹‘𝑘)‘𝐴)) | |
4 | fveq2 6645 | . . . . . . . . . . 11 ⊢ (𝑧 = 𝐴 → (𝐺‘𝑧) = (𝐺‘𝐴)) | |
5 | 3, 4 | oveq12d 7153 | . . . . . . . . . 10 ⊢ (𝑧 = 𝐴 → (((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧)) = (((𝐹‘𝑘)‘𝐴) − (𝐺‘𝐴))) |
6 | 5 | fveq2d 6649 | . . . . . . . . 9 ⊢ (𝑧 = 𝐴 → (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) = (abs‘(((𝐹‘𝑘)‘𝐴) − (𝐺‘𝐴)))) |
7 | 6 | breq1d 5040 | . . . . . . . 8 ⊢ (𝑧 = 𝐴 → ((abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥 ↔ (abs‘(((𝐹‘𝑘)‘𝐴) − (𝐺‘𝐴))) < 𝑥)) |
8 | 7 | rspcv 3566 | . . . . . . 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 3145 | . . . . 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 3232 | . . . 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 3145 | . . 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 2799 | . . . 4 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑍 ∧ 𝑧 ∈ 𝑆)) → ((𝐹‘𝑘)‘𝑧) = ((𝐹‘𝑘)‘𝑧)) | |
17 | eqidd 2799 | . . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → (𝐺‘𝑧) = (𝐺‘𝑧)) | |
18 | ulmcl 24976 | . . . . 5 ⊢ (𝐹(⇝𝑢‘𝑆)𝐺 → 𝐺:𝑆⟶ℂ) | |
19 | 1, 18 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐺:𝑆⟶ℂ) |
20 | ulmscl 24974 | . . . . 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 24980 | . . 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 2804 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐻‘𝑘) = ((𝐹‘𝑘)‘𝐴)) |
26 | 19, 2 | ffvelrnd 6829 | . . . 4 ⊢ (𝜑 → (𝐺‘𝐴) ∈ ℂ) |
27 | 15 | ffvelrnda 6828 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ (ℂ ↑m 𝑆)) |
28 | elmapi 8411 | . . . . . 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 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ 𝑆) |
31 | 29, 30 | ffvelrnd 6829 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝐹‘𝑘)‘𝐴) ∈ ℂ) |
32 | 13, 14, 23, 25, 26, 31 | clim2c 14854 | . . 3 ⊢ (𝜑 → (𝐻 ⇝ (𝐺‘𝐴) ↔ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(abs‘(((𝐹‘𝑘)‘𝐴) − (𝐺‘𝐴))) < 𝑥)) |
33 | 12, 22, 32 | 3imtr4d 297 | . 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 399 = wceq 1538 ∈ wcel 2111 ∀wral 3106 ∃wrex 3107 Vcvv 3441 class class class wbr 5030 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 ↑m cmap 8389 ℂcc 10524 < clt 10664 − cmin 10859 ℤcz 11969 ℤ≥cuz 12231 ℝ+crp 12377 abscabs 14585 ⇝ cli 14833 ⇝𝑢culm 24971 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-pre-lttri 10600 ax-pre-lttrn 10601 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-po 5438 df-so 5439 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-1st 7671 df-2nd 7672 df-er 8272 df-map 8391 df-pm 8392 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-neg 10862 df-z 11970 df-uz 12232 df-clim 14837 df-ulm 24972 |
This theorem is referenced by: ulmuni 24987 ulmdvlem3 24997 mbfulm 25001 pserulm 25017 lgamgulm2 25621 lgamcvglem 25625 knoppcnlem9 33953 knoppndvlem4 33967 |
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