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Mirrors > Home > MPE Home > Th. List > o1compt | Structured version Visualization version GIF version |
Description: Sufficient condition for transforming the index set of an eventually bounded function. (Contributed by Mario Carneiro, 12-May-2016.) |
Ref | Expression |
---|---|
o1compt.1 | ⊢ (𝜑 → 𝐹:𝐴⟶ℂ) |
o1compt.2 | ⊢ (𝜑 → 𝐹 ∈ 𝑂(1)) |
o1compt.3 | ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝐶 ∈ 𝐴) |
o1compt.4 | ⊢ (𝜑 → 𝐵 ⊆ ℝ) |
o1compt.5 | ⊢ ((𝜑 ∧ 𝑚 ∈ ℝ) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 → 𝑚 ≤ 𝐶)) |
Ref | Expression |
---|---|
o1compt | ⊢ (𝜑 → (𝐹 ∘ (𝑦 ∈ 𝐵 ↦ 𝐶)) ∈ 𝑂(1)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | o1compt.1 | . 2 ⊢ (𝜑 → 𝐹:𝐴⟶ℂ) | |
2 | o1compt.2 | . 2 ⊢ (𝜑 → 𝐹 ∈ 𝑂(1)) | |
3 | o1compt.3 | . . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝐶 ∈ 𝐴) | |
4 | 3 | fmpttd 7135 | . 2 ⊢ (𝜑 → (𝑦 ∈ 𝐵 ↦ 𝐶):𝐵⟶𝐴) |
5 | o1compt.4 | . 2 ⊢ (𝜑 → 𝐵 ⊆ ℝ) | |
6 | o1compt.5 | . . 3 ⊢ ((𝜑 ∧ 𝑚 ∈ ℝ) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 → 𝑚 ≤ 𝐶)) | |
7 | nfv 1912 | . . . . . . . 8 ⊢ Ⅎ𝑦 𝑥 ≤ 𝑧 | |
8 | nfcv 2903 | . . . . . . . . 9 ⊢ Ⅎ𝑦𝑚 | |
9 | nfcv 2903 | . . . . . . . . 9 ⊢ Ⅎ𝑦 ≤ | |
10 | nffvmpt1 6918 | . . . . . . . . 9 ⊢ Ⅎ𝑦((𝑦 ∈ 𝐵 ↦ 𝐶)‘𝑧) | |
11 | 8, 9, 10 | nfbr 5195 | . . . . . . . 8 ⊢ Ⅎ𝑦 𝑚 ≤ ((𝑦 ∈ 𝐵 ↦ 𝐶)‘𝑧) |
12 | 7, 11 | nfim 1894 | . . . . . . 7 ⊢ Ⅎ𝑦(𝑥 ≤ 𝑧 → 𝑚 ≤ ((𝑦 ∈ 𝐵 ↦ 𝐶)‘𝑧)) |
13 | nfv 1912 | . . . . . . 7 ⊢ Ⅎ𝑧(𝑥 ≤ 𝑦 → 𝑚 ≤ ((𝑦 ∈ 𝐵 ↦ 𝐶)‘𝑦)) | |
14 | breq2 5152 | . . . . . . . 8 ⊢ (𝑧 = 𝑦 → (𝑥 ≤ 𝑧 ↔ 𝑥 ≤ 𝑦)) | |
15 | fveq2 6907 | . . . . . . . . 9 ⊢ (𝑧 = 𝑦 → ((𝑦 ∈ 𝐵 ↦ 𝐶)‘𝑧) = ((𝑦 ∈ 𝐵 ↦ 𝐶)‘𝑦)) | |
16 | 15 | breq2d 5160 | . . . . . . . 8 ⊢ (𝑧 = 𝑦 → (𝑚 ≤ ((𝑦 ∈ 𝐵 ↦ 𝐶)‘𝑧) ↔ 𝑚 ≤ ((𝑦 ∈ 𝐵 ↦ 𝐶)‘𝑦))) |
17 | 14, 16 | imbi12d 344 | . . . . . . 7 ⊢ (𝑧 = 𝑦 → ((𝑥 ≤ 𝑧 → 𝑚 ≤ ((𝑦 ∈ 𝐵 ↦ 𝐶)‘𝑧)) ↔ (𝑥 ≤ 𝑦 → 𝑚 ≤ ((𝑦 ∈ 𝐵 ↦ 𝐶)‘𝑦)))) |
18 | 12, 13, 17 | cbvralw 3304 | . . . . . 6 ⊢ (∀𝑧 ∈ 𝐵 (𝑥 ≤ 𝑧 → 𝑚 ≤ ((𝑦 ∈ 𝐵 ↦ 𝐶)‘𝑧)) ↔ ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 → 𝑚 ≤ ((𝑦 ∈ 𝐵 ↦ 𝐶)‘𝑦))) |
19 | simpr 484 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ 𝐵) | |
20 | eqid 2735 | . . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑦 ∈ 𝐵 ↦ 𝐶) | |
21 | 20 | fvmpt2 7027 | . . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝐵 ∧ 𝐶 ∈ 𝐴) → ((𝑦 ∈ 𝐵 ↦ 𝐶)‘𝑦) = 𝐶) |
22 | 19, 3, 21 | syl2anc 584 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → ((𝑦 ∈ 𝐵 ↦ 𝐶)‘𝑦) = 𝐶) |
23 | 22 | breq2d 5160 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (𝑚 ≤ ((𝑦 ∈ 𝐵 ↦ 𝐶)‘𝑦) ↔ 𝑚 ≤ 𝐶)) |
24 | 23 | imbi2d 340 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → ((𝑥 ≤ 𝑦 → 𝑚 ≤ ((𝑦 ∈ 𝐵 ↦ 𝐶)‘𝑦)) ↔ (𝑥 ≤ 𝑦 → 𝑚 ≤ 𝐶))) |
25 | 24 | ralbidva 3174 | . . . . . 6 ⊢ (𝜑 → (∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 → 𝑚 ≤ ((𝑦 ∈ 𝐵 ↦ 𝐶)‘𝑦)) ↔ ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 → 𝑚 ≤ 𝐶))) |
26 | 18, 25 | bitrid 283 | . . . . 5 ⊢ (𝜑 → (∀𝑧 ∈ 𝐵 (𝑥 ≤ 𝑧 → 𝑚 ≤ ((𝑦 ∈ 𝐵 ↦ 𝐶)‘𝑧)) ↔ ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 → 𝑚 ≤ 𝐶))) |
27 | 26 | rexbidv 3177 | . . . 4 ⊢ (𝜑 → (∃𝑥 ∈ ℝ ∀𝑧 ∈ 𝐵 (𝑥 ≤ 𝑧 → 𝑚 ≤ ((𝑦 ∈ 𝐵 ↦ 𝐶)‘𝑧)) ↔ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 → 𝑚 ≤ 𝐶))) |
28 | 27 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝑚 ∈ ℝ) → (∃𝑥 ∈ ℝ ∀𝑧 ∈ 𝐵 (𝑥 ≤ 𝑧 → 𝑚 ≤ ((𝑦 ∈ 𝐵 ↦ 𝐶)‘𝑧)) ↔ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 → 𝑚 ≤ 𝐶))) |
29 | 6, 28 | mpbird 257 | . 2 ⊢ ((𝜑 ∧ 𝑚 ∈ ℝ) → ∃𝑥 ∈ ℝ ∀𝑧 ∈ 𝐵 (𝑥 ≤ 𝑧 → 𝑚 ≤ ((𝑦 ∈ 𝐵 ↦ 𝐶)‘𝑧))) |
30 | 1, 2, 4, 5, 29 | o1co 15619 | 1 ⊢ (𝜑 → (𝐹 ∘ (𝑦 ∈ 𝐵 ↦ 𝐶)) ∈ 𝑂(1)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∀wral 3059 ∃wrex 3068 ⊆ wss 3963 class class class wbr 5148 ↦ cmpt 5231 ∘ ccom 5693 ⟶wf 6559 ‘cfv 6563 ℂcc 11151 ℝcr 11152 ≤ cle 11294 𝑂(1)co1 15519 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-pre-lttri 11227 ax-pre-lttrn 11228 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-po 5597 df-so 5598 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-er 8744 df-pm 8868 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-ico 13390 df-o1 15523 |
This theorem is referenced by: dchrisum0 27579 |
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