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Mirrors > Home > MPE Home > Th. List > ulmi | Structured version Visualization version GIF version |
Description: The uniform limit property. (Contributed by Mario Carneiro, 27-Feb-2015.) |
Ref | Expression |
---|---|
ulm2.z | ⊢ 𝑍 = (ℤ≥‘𝑀) |
ulm2.m | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
ulm2.f | ⊢ (𝜑 → 𝐹:𝑍⟶(ℂ ↑m 𝑆)) |
ulm2.b | ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑍 ∧ 𝑧 ∈ 𝑆)) → ((𝐹‘𝑘)‘𝑧) = 𝐵) |
ulm2.a | ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → (𝐺‘𝑧) = 𝐴) |
ulmi.u | ⊢ (𝜑 → 𝐹(⇝𝑢‘𝑆)𝐺) |
ulmi.c | ⊢ (𝜑 → 𝐶 ∈ ℝ+) |
Ref | Expression |
---|---|
ulmi | ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(𝐵 − 𝐴)) < 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | breq2 5034 | . . . 4 ⊢ (𝑥 = 𝐶 → ((abs‘(𝐵 − 𝐴)) < 𝑥 ↔ (abs‘(𝐵 − 𝐴)) < 𝐶)) | |
2 | 1 | ralbidv 3162 | . . 3 ⊢ (𝑥 = 𝐶 → (∀𝑧 ∈ 𝑆 (abs‘(𝐵 − 𝐴)) < 𝑥 ↔ ∀𝑧 ∈ 𝑆 (abs‘(𝐵 − 𝐴)) < 𝐶)) |
3 | 2 | rexralbidv 3260 | . 2 ⊢ (𝑥 = 𝐶 → (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(𝐵 − 𝐴)) < 𝑥 ↔ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(𝐵 − 𝐴)) < 𝐶)) |
4 | ulmi.u | . . 3 ⊢ (𝜑 → 𝐹(⇝𝑢‘𝑆)𝐺) | |
5 | ulm2.z | . . . 4 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
6 | ulm2.m | . . . 4 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
7 | ulm2.f | . . . 4 ⊢ (𝜑 → 𝐹:𝑍⟶(ℂ ↑m 𝑆)) | |
8 | ulm2.b | . . . 4 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑍 ∧ 𝑧 ∈ 𝑆)) → ((𝐹‘𝑘)‘𝑧) = 𝐵) | |
9 | ulm2.a | . . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → (𝐺‘𝑧) = 𝐴) | |
10 | ulmcl 24976 | . . . . 5 ⊢ (𝐹(⇝𝑢‘𝑆)𝐺 → 𝐺:𝑆⟶ℂ) | |
11 | 4, 10 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐺:𝑆⟶ℂ) |
12 | ulmscl 24974 | . . . . 5 ⊢ (𝐹(⇝𝑢‘𝑆)𝐺 → 𝑆 ∈ V) | |
13 | 4, 12 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑆 ∈ V) |
14 | 5, 6, 7, 8, 9, 11, 13 | ulm2 24980 | . . 3 ⊢ (𝜑 → (𝐹(⇝𝑢‘𝑆)𝐺 ↔ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(𝐵 − 𝐴)) < 𝑥)) |
15 | 4, 14 | mpbid 235 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(𝐵 − 𝐴)) < 𝑥) |
16 | ulmi.c | . 2 ⊢ (𝜑 → 𝐶 ∈ ℝ+) | |
17 | 3, 15, 16 | rspcdva 3573 | 1 ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(𝐵 − 𝐴)) < 𝐶) |
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 ⇝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-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-ulm 24972 |
This theorem is referenced by: ulmshftlem 24984 ulmcau 24990 ulmbdd 24993 ulmcn 24994 iblulm 25002 itgulm 25003 |
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