|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| 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 6905 | . . . . . . . . . . 11 ⊢ (𝑧 = 𝐴 → ((𝐹‘𝑘)‘𝑧) = ((𝐹‘𝑘)‘𝐴)) | |
| 4 | fveq2 6905 | . . . . . . . . . . 11 ⊢ (𝑧 = 𝐴 → (𝐺‘𝑧) = (𝐺‘𝐴)) | |
| 5 | 3, 4 | oveq12d 7450 | . . . . . . . . . 10 ⊢ (𝑧 = 𝐴 → (((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧)) = (((𝐹‘𝑘)‘𝐴) − (𝐺‘𝐴))) | 
| 6 | 5 | fveq2d 6909 | . . . . . . . . 9 ⊢ (𝑧 = 𝐴 → (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) = (abs‘(((𝐹‘𝑘)‘𝐴) − (𝐺‘𝐴)))) | 
| 7 | 6 | breq1d 5152 | . . . . . . . 8 ⊢ (𝑧 = 𝐴 → ((abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥 ↔ (abs‘(((𝐹‘𝑘)‘𝐴) − (𝐺‘𝐴))) < 𝑥)) | 
| 8 | 7 | rspcv 3617 | . . . . . . 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 3168 | . . . . 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 3169 | . . . 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 3168 | . . 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 2737 | . . . 4 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑍 ∧ 𝑧 ∈ 𝑆)) → ((𝐹‘𝑘)‘𝑧) = ((𝐹‘𝑘)‘𝑧)) | |
| 17 | eqidd 2737 | . . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → (𝐺‘𝑧) = (𝐺‘𝑧)) | |
| 18 | ulmcl 26425 | . . . . 5 ⊢ (𝐹(⇝𝑢‘𝑆)𝐺 → 𝐺:𝑆⟶ℂ) | |
| 19 | 1, 18 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐺:𝑆⟶ℂ) | 
| 20 | ulmscl 26423 | . . . . 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 26429 | . . 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 2742 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐻‘𝑘) = ((𝐹‘𝑘)‘𝐴)) | 
| 26 | 19, 2 | ffvelcdmd 7104 | . . . 4 ⊢ (𝜑 → (𝐺‘𝐴) ∈ ℂ) | 
| 27 | 15 | ffvelcdmda 7103 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ (ℂ ↑m 𝑆)) | 
| 28 | elmapi 8890 | . . . . . 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 7104 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝐹‘𝑘)‘𝐴) ∈ ℂ) | 
| 32 | 13, 14, 23, 25, 26, 31 | clim2c 15542 | . . 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 1539 ∈ wcel 2107 ∀wral 3060 ∃wrex 3069 Vcvv 3479 class class class wbr 5142 ⟶wf 6556 ‘cfv 6560 (class class class)co 7432 ↑m cmap 8867 ℂcc 11154 < clt 11296 − cmin 11493 ℤcz 12615 ℤ≥cuz 12879 ℝ+crp 13035 abscabs 15274 ⇝ cli 15521 ⇝𝑢culm 26420 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-pre-lttri 11230 ax-pre-lttrn 11231 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-po 5591 df-so 5592 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-ov 7435 df-oprab 7436 df-mpo 7437 df-1st 8015 df-2nd 8016 df-er 8746 df-map 8869 df-pm 8870 df-en 8987 df-dom 8988 df-sdom 8989 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-neg 11496 df-z 12616 df-uz 12880 df-clim 15525 df-ulm 26421 | 
| This theorem is referenced by: ulmuni 26436 ulmdvlem3 26446 mbfulm 26450 pserulm 26466 lgamgulm2 27080 lgamcvglem 27084 knoppcnlem9 36503 knoppndvlem4 36517 | 
| Copyright terms: Public domain | W3C validator |