![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 4847 | . . . 4 ⊢ (𝑥 = 𝐶 → ((abs‘(𝐵 − 𝐴)) < 𝑥 ↔ (abs‘(𝐵 − 𝐴)) < 𝐶)) | |
2 | 1 | ralbidv 3167 | . . 3 ⊢ (𝑥 = 𝐶 → (∀𝑧 ∈ 𝑆 (abs‘(𝐵 − 𝐴)) < 𝑥 ↔ ∀𝑧 ∈ 𝑆 (abs‘(𝐵 − 𝐴)) < 𝐶)) |
3 | 2 | rexralbidv 3239 | . 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 24476 | . . . . 5 ⊢ (𝐹(⇝𝑢‘𝑆)𝐺 → 𝐺:𝑆⟶ℂ) | |
11 | 4, 10 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐺:𝑆⟶ℂ) |
12 | ulmscl 24474 | . . . . 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 24480 | . . 3 ⊢ (𝜑 → (𝐹(⇝𝑢‘𝑆)𝐺 ↔ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(𝐵 − 𝐴)) < 𝑥)) |
15 | 4, 14 | mpbid 224 | . 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 3503 | 1 ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(𝐵 − 𝐴)) < 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 385 = wceq 1653 ∈ wcel 2157 ∀wral 3089 ∃wrex 3090 Vcvv 3385 class class class wbr 4843 ⟶wf 6097 ‘cfv 6101 (class class class)co 6878 ↑𝑚 cmap 8095 ℂcc 10222 < clt 10363 − cmin 10556 ℤcz 11666 ℤ≥cuz 11930 ℝ+crp 12074 abscabs 14315 ⇝𝑢culm 24471 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-rep 4964 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 ax-cnex 10280 ax-resscn 10281 ax-pre-lttri 10298 ax-pre-lttrn 10299 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rab 3098 df-v 3387 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-op 4375 df-uni 4629 df-iun 4712 df-br 4844 df-opab 4906 df-mpt 4923 df-id 5220 df-po 5233 df-so 5234 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fo 6107 df-f1o 6108 df-fv 6109 df-ov 6881 df-oprab 6882 df-mpt2 6883 df-er 7982 df-map 8097 df-pm 8098 df-en 8196 df-dom 8197 df-sdom 8198 df-pnf 10365 df-mnf 10366 df-xr 10367 df-ltxr 10368 df-le 10369 df-neg 10559 df-z 11667 df-uz 11931 df-ulm 24472 |
This theorem is referenced by: ulmshftlem 24484 ulmcau 24490 ulmbdd 24493 ulmcn 24494 iblulm 24502 itgulm 24503 |
Copyright terms: Public domain | W3C validator |