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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 6907 | . . . . . . . . . . 11 ⊢ (𝑧 = 𝐴 → ((𝐹‘𝑘)‘𝑧) = ((𝐹‘𝑘)‘𝐴)) | |
4 | fveq2 6907 | . . . . . . . . . . 11 ⊢ (𝑧 = 𝐴 → (𝐺‘𝑧) = (𝐺‘𝐴)) | |
5 | 3, 4 | oveq12d 7449 | . . . . . . . . . 10 ⊢ (𝑧 = 𝐴 → (((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧)) = (((𝐹‘𝑘)‘𝐴) − (𝐺‘𝐴))) |
6 | 5 | fveq2d 6911 | . . . . . . . . 9 ⊢ (𝑧 = 𝐴 → (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) = (abs‘(((𝐹‘𝑘)‘𝐴) − (𝐺‘𝐴)))) |
7 | 6 | breq1d 5158 | . . . . . . . 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 3167 | . . . . 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 3168 | . . . 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 3167 | . . 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 2736 | . . . 4 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑍 ∧ 𝑧 ∈ 𝑆)) → ((𝐹‘𝑘)‘𝑧) = ((𝐹‘𝑘)‘𝑧)) | |
17 | eqidd 2736 | . . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → (𝐺‘𝑧) = (𝐺‘𝑧)) | |
18 | ulmcl 26439 | . . . . 5 ⊢ (𝐹(⇝𝑢‘𝑆)𝐺 → 𝐺:𝑆⟶ℂ) | |
19 | 1, 18 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐺:𝑆⟶ℂ) |
20 | ulmscl 26437 | . . . . 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 26443 | . . 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 2741 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐻‘𝑘) = ((𝐹‘𝑘)‘𝐴)) |
26 | 19, 2 | ffvelcdmd 7105 | . . . 4 ⊢ (𝜑 → (𝐺‘𝐴) ∈ ℂ) |
27 | 15 | ffvelcdmda 7104 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ (ℂ ↑m 𝑆)) |
28 | elmapi 8888 | . . . . . 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 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ 𝑆) |
31 | 29, 30 | ffvelcdmd 7105 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝐹‘𝑘)‘𝐴) ∈ ℂ) |
32 | 13, 14, 23, 25, 26, 31 | clim2c 15538 | . . 3 ⊢ (𝜑 → (𝐻 ⇝ (𝐺‘𝐴) ↔ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(abs‘(((𝐹‘𝑘)‘𝐴) − (𝐺‘𝐴))) < 𝑥)) |
33 | 12, 22, 32 | 3imtr4d 294 | . 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 395 = wceq 1537 ∈ wcel 2106 ∀wral 3059 ∃wrex 3068 Vcvv 3478 class class class wbr 5148 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 ↑m cmap 8865 ℂcc 11151 < clt 11293 − cmin 11490 ℤcz 12611 ℤ≥cuz 12876 ℝ+crp 13032 abscabs 15270 ⇝ cli 15517 ⇝𝑢culm 26434 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-pre-lttri 11227 ax-pre-lttrn 11228 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-po 5597 df-so 5598 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8013 df-2nd 8014 df-er 8744 df-map 8867 df-pm 8868 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-neg 11493 df-z 12612 df-uz 12877 df-clim 15521 df-ulm 26435 |
This theorem is referenced by: ulmuni 26450 ulmdvlem3 26460 mbfulm 26464 pserulm 26480 lgamgulm2 27094 lgamcvglem 27098 knoppcnlem9 36484 knoppndvlem4 36498 |
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