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Mirrors > Home > MPE Home > Th. List > icco1 | Structured version Visualization version GIF version |
Description: Derive eventual boundedness from separate upper and lower eventual bounds. (Contributed by Mario Carneiro, 15-Apr-2016.) |
Ref | Expression |
---|---|
icco1.1 | ⊢ (𝜑 → 𝐴 ⊆ ℝ) |
icco1.2 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) |
icco1.3 | ⊢ (𝜑 → 𝐶 ∈ ℝ) |
icco1.4 | ⊢ (𝜑 → 𝑀 ∈ ℝ) |
icco1.5 | ⊢ (𝜑 → 𝑁 ∈ ℝ) |
icco1.6 | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝐶 ≤ 𝑥)) → 𝐵 ∈ (𝑀[,]𝑁)) |
Ref | Expression |
---|---|
icco1 | ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝑂(1)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | icco1.1 | . . 3 ⊢ (𝜑 → 𝐴 ⊆ ℝ) | |
2 | icco1.2 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) | |
3 | icco1.3 | . . 3 ⊢ (𝜑 → 𝐶 ∈ ℝ) | |
4 | icco1.5 | . . 3 ⊢ (𝜑 → 𝑁 ∈ ℝ) | |
5 | icco1.6 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝐶 ≤ 𝑥)) → 𝐵 ∈ (𝑀[,]𝑁)) | |
6 | icco1.4 | . . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ℝ) | |
7 | elicc2 13390 | . . . . . . 7 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (𝐵 ∈ (𝑀[,]𝑁) ↔ (𝐵 ∈ ℝ ∧ 𝑀 ≤ 𝐵 ∧ 𝐵 ≤ 𝑁))) | |
8 | 6, 4, 7 | syl2anc 583 | . . . . . 6 ⊢ (𝜑 → (𝐵 ∈ (𝑀[,]𝑁) ↔ (𝐵 ∈ ℝ ∧ 𝑀 ≤ 𝐵 ∧ 𝐵 ≤ 𝑁))) |
9 | 8 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝐶 ≤ 𝑥)) → (𝐵 ∈ (𝑀[,]𝑁) ↔ (𝐵 ∈ ℝ ∧ 𝑀 ≤ 𝐵 ∧ 𝐵 ≤ 𝑁))) |
10 | 5, 9 | mpbid 231 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝐶 ≤ 𝑥)) → (𝐵 ∈ ℝ ∧ 𝑀 ≤ 𝐵 ∧ 𝐵 ≤ 𝑁)) |
11 | 10 | simp3d 1141 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝐶 ≤ 𝑥)) → 𝐵 ≤ 𝑁) |
12 | 1, 2, 3, 4, 11 | ello1d 15469 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ ≤𝑂(1)) |
13 | 2 | renegcld 11640 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → -𝐵 ∈ ℝ) |
14 | 6 | renegcld 11640 | . . 3 ⊢ (𝜑 → -𝑀 ∈ ℝ) |
15 | 10 | simp2d 1140 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝐶 ≤ 𝑥)) → 𝑀 ≤ 𝐵) |
16 | 6 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝐶 ≤ 𝑥)) → 𝑀 ∈ ℝ) |
17 | 2 | adantrr 714 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝐶 ≤ 𝑥)) → 𝐵 ∈ ℝ) |
18 | 16, 17 | lenegd 11792 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝐶 ≤ 𝑥)) → (𝑀 ≤ 𝐵 ↔ -𝐵 ≤ -𝑀)) |
19 | 15, 18 | mpbid 231 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝐶 ≤ 𝑥)) → -𝐵 ≤ -𝑀) |
20 | 1, 13, 3, 14, 19 | ello1d 15469 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ -𝐵) ∈ ≤𝑂(1)) |
21 | 2 | o1lo1 15483 | . 2 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝑂(1) ↔ ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ ≤𝑂(1) ∧ (𝑥 ∈ 𝐴 ↦ -𝐵) ∈ ≤𝑂(1)))) |
22 | 12, 20, 21 | mpbir2and 710 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝑂(1)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1084 ∈ wcel 2098 ⊆ wss 3941 class class class wbr 5139 ↦ cmpt 5222 (class class class)co 7402 ℝcr 11106 ≤ cle 11248 -cneg 11444 [,]cicc 13328 𝑂(1)co1 15432 ≤𝑂(1)clo1 15433 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 ax-pre-sup 11185 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-om 7850 df-2nd 7970 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-er 8700 df-pm 8820 df-en 8937 df-dom 8938 df-sdom 8939 df-sup 9434 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-div 11871 df-nn 12212 df-2 12274 df-3 12275 df-n0 12472 df-z 12558 df-uz 12822 df-rp 12976 df-ico 13331 df-icc 13332 df-seq 13968 df-exp 14029 df-cj 15048 df-re 15049 df-im 15050 df-sqrt 15184 df-abs 15185 df-o1 15436 df-lo1 15437 |
This theorem is referenced by: (None) |
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