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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 5062 | . . . 4 ⊢ (𝑥 = 𝐶 → ((abs‘(𝐵 − 𝐴)) < 𝑥 ↔ (abs‘(𝐵 − 𝐴)) < 𝐶)) | |
2 | 1 | ralbidv 3197 | . . 3 ⊢ (𝑥 = 𝐶 → (∀𝑧 ∈ 𝑆 (abs‘(𝐵 − 𝐴)) < 𝑥 ↔ ∀𝑧 ∈ 𝑆 (abs‘(𝐵 − 𝐴)) < 𝐶)) |
3 | 2 | rexralbidv 3301 | . 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 24963 | . . . . 5 ⊢ (𝐹(⇝𝑢‘𝑆)𝐺 → 𝐺:𝑆⟶ℂ) | |
11 | 4, 10 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐺:𝑆⟶ℂ) |
12 | ulmscl 24961 | . . . . 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 24967 | . . 3 ⊢ (𝜑 → (𝐹(⇝𝑢‘𝑆)𝐺 ↔ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(𝐵 − 𝐴)) < 𝑥)) |
15 | 4, 14 | mpbid 234 | . 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 3624 | 1 ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(𝐵 − 𝐴)) < 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∀wral 3138 ∃wrex 3139 Vcvv 3494 class class class wbr 5058 ⟶wf 6345 ‘cfv 6349 (class class class)co 7150 ↑m cmap 8400 ℂcc 10529 < clt 10669 − cmin 10864 ℤcz 11975 ℤ≥cuz 12237 ℝ+crp 12383 abscabs 14587 ⇝𝑢culm 24958 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-pre-lttri 10605 ax-pre-lttrn 10606 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-po 5468 df-so 5469 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-ov 7153 df-oprab 7154 df-mpo 7155 df-er 8283 df-map 8402 df-pm 8403 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-neg 10867 df-z 11976 df-uz 12238 df-ulm 24959 |
This theorem is referenced by: ulmshftlem 24971 ulmcau 24977 ulmbdd 24980 ulmcn 24981 iblulm 24989 itgulm 24990 |
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