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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 6672 | . . . . . . . . . . 11 ⊢ (𝑧 = 𝐴 → ((𝐹‘𝑘)‘𝑧) = ((𝐹‘𝑘)‘𝐴)) | |
4 | fveq2 6672 | . . . . . . . . . . 11 ⊢ (𝑧 = 𝐴 → (𝐺‘𝑧) = (𝐺‘𝐴)) | |
5 | 3, 4 | oveq12d 7176 | . . . . . . . . . 10 ⊢ (𝑧 = 𝐴 → (((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧)) = (((𝐹‘𝑘)‘𝐴) − (𝐺‘𝐴))) |
6 | 5 | fveq2d 6676 | . . . . . . . . 9 ⊢ (𝑧 = 𝐴 → (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) = (abs‘(((𝐹‘𝑘)‘𝐴) − (𝐺‘𝐴)))) |
7 | 6 | breq1d 5078 | . . . . . . . 8 ⊢ (𝑧 = 𝐴 → ((abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥 ↔ (abs‘(((𝐹‘𝑘)‘𝐴) − (𝐺‘𝐴))) < 𝑥)) |
8 | 7 | rspcv 3620 | . . . . . . 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 3180 | . . . . 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 3275 | . . . 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 3180 | . . 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 2824 | . . . 4 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑍 ∧ 𝑧 ∈ 𝑆)) → ((𝐹‘𝑘)‘𝑧) = ((𝐹‘𝑘)‘𝑧)) | |
17 | eqidd 2824 | . . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → (𝐺‘𝑧) = (𝐺‘𝑧)) | |
18 | ulmcl 24971 | . . . . 5 ⊢ (𝐹(⇝𝑢‘𝑆)𝐺 → 𝐺:𝑆⟶ℂ) | |
19 | 1, 18 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐺:𝑆⟶ℂ) |
20 | ulmscl 24969 | . . . . 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 24975 | . . 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 2829 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐻‘𝑘) = ((𝐹‘𝑘)‘𝐴)) |
26 | 19, 2 | ffvelrnd 6854 | . . . 4 ⊢ (𝜑 → (𝐺‘𝐴) ∈ ℂ) |
27 | 15 | ffvelrnda 6853 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ (ℂ ↑m 𝑆)) |
28 | elmapi 8430 | . . . . . 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 6854 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝐹‘𝑘)‘𝐴) ∈ ℂ) |
32 | 13, 14, 23, 25, 26, 31 | clim2c 14864 | . . 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 1537 ∈ wcel 2114 ∀wral 3140 ∃wrex 3141 Vcvv 3496 class class class wbr 5068 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 ↑m cmap 8408 ℂcc 10537 < clt 10677 − cmin 10872 ℤcz 11984 ℤ≥cuz 12246 ℝ+crp 12392 abscabs 14595 ⇝ cli 14843 ⇝𝑢culm 24966 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-pre-lttri 10613 ax-pre-lttrn 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-po 5476 df-so 5477 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-1st 7691 df-2nd 7692 df-er 8291 df-map 8410 df-pm 8411 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-neg 10875 df-z 11985 df-uz 12247 df-clim 14847 df-ulm 24967 |
This theorem is referenced by: ulmuni 24982 ulmdvlem3 24992 mbfulm 24996 pserulm 25012 lgamgulm2 25615 lgamcvglem 25619 knoppcnlem9 33842 knoppndvlem4 33856 |
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