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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 12800 | . . . . . . 7 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (𝐵 ∈ (𝑀[,]𝑁) ↔ (𝐵 ∈ ℝ ∧ 𝑀 ≤ 𝐵 ∧ 𝐵 ≤ 𝑁))) | |
8 | 6, 4, 7 | syl2anc 586 | . . . . . 6 ⊢ (𝜑 → (𝐵 ∈ (𝑀[,]𝑁) ↔ (𝐵 ∈ ℝ ∧ 𝑀 ≤ 𝐵 ∧ 𝐵 ≤ 𝑁))) |
9 | 8 | adantr 483 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝐶 ≤ 𝑥)) → (𝐵 ∈ (𝑀[,]𝑁) ↔ (𝐵 ∈ ℝ ∧ 𝑀 ≤ 𝐵 ∧ 𝐵 ≤ 𝑁))) |
10 | 5, 9 | mpbid 234 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝐶 ≤ 𝑥)) → (𝐵 ∈ ℝ ∧ 𝑀 ≤ 𝐵 ∧ 𝐵 ≤ 𝑁)) |
11 | 10 | simp3d 1140 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝐶 ≤ 𝑥)) → 𝐵 ≤ 𝑁) |
12 | 1, 2, 3, 4, 11 | ello1d 14879 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ ≤𝑂(1)) |
13 | 2 | renegcld 11066 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → -𝐵 ∈ ℝ) |
14 | 6 | renegcld 11066 | . . 3 ⊢ (𝜑 → -𝑀 ∈ ℝ) |
15 | 10 | simp2d 1139 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝐶 ≤ 𝑥)) → 𝑀 ≤ 𝐵) |
16 | 6 | adantr 483 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝐶 ≤ 𝑥)) → 𝑀 ∈ ℝ) |
17 | 2 | adantrr 715 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝐶 ≤ 𝑥)) → 𝐵 ∈ ℝ) |
18 | 16, 17 | lenegd 11218 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝐶 ≤ 𝑥)) → (𝑀 ≤ 𝐵 ↔ -𝐵 ≤ -𝑀)) |
19 | 15, 18 | mpbid 234 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝐶 ≤ 𝑥)) → -𝐵 ≤ -𝑀) |
20 | 1, 13, 3, 14, 19 | ello1d 14879 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ -𝐵) ∈ ≤𝑂(1)) |
21 | 2 | o1lo1 14893 | . 2 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝑂(1) ↔ ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ ≤𝑂(1) ∧ (𝑥 ∈ 𝐴 ↦ -𝐵) ∈ ≤𝑂(1)))) |
22 | 12, 20, 21 | mpbir2and 711 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝑂(1)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 ∈ wcel 2110 ⊆ wss 3935 class class class wbr 5065 ↦ cmpt 5145 (class class class)co 7155 ℝcr 10535 ≤ cle 10675 -cneg 10870 [,]cicc 12740 𝑂(1)co1 14842 ≤𝑂(1)clo1 14843 |
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-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 ax-pre-sup 10614 |
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-rmo 3146 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-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-om 7580 df-2nd 7689 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-er 8288 df-pm 8408 df-en 8509 df-dom 8510 df-sdom 8511 df-sup 8905 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-div 11297 df-nn 11638 df-2 11699 df-3 11700 df-n0 11897 df-z 11981 df-uz 12243 df-rp 12389 df-ico 12743 df-icc 12744 df-seq 13369 df-exp 13429 df-cj 14457 df-re 14458 df-im 14459 df-sqrt 14593 df-abs 14594 df-o1 14846 df-lo1 14847 |
This theorem is referenced by: (None) |
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