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Mirrors > Home > MPE Home > Th. List > Mathboxes > elbigo2r | Structured version Visualization version GIF version |
Description: Sufficient condition for a function to be of order G(x). (Contributed by AV, 18-May-2020.) |
Ref | Expression |
---|---|
elbigo2r | ⊢ (((𝐺:𝐴⟶ℝ ∧ 𝐴 ⊆ ℝ) ∧ (𝐹:𝐵⟶ℝ ∧ 𝐵 ⊆ 𝐴) ∧ (𝐶 ∈ ℝ ∧ 𝑀 ∈ ℝ ∧ ∀𝑥 ∈ 𝐵 (𝐶 ≤ 𝑥 → (𝐹‘𝑥) ≤ (𝑀 · (𝐺‘𝑥))))) → 𝐹 ∈ (Ο‘𝐺)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | breq1 5055 | . . . . . 6 ⊢ (𝑦 = 𝐶 → (𝑦 ≤ 𝑥 ↔ 𝐶 ≤ 𝑥)) | |
2 | 1 | imbi1d 344 | . . . . 5 ⊢ (𝑦 = 𝐶 → ((𝑦 ≤ 𝑥 → (𝐹‘𝑥) ≤ (𝑚 · (𝐺‘𝑥))) ↔ (𝐶 ≤ 𝑥 → (𝐹‘𝑥) ≤ (𝑚 · (𝐺‘𝑥))))) |
3 | 2 | ralbidv 3197 | . . . 4 ⊢ (𝑦 = 𝐶 → (∀𝑥 ∈ 𝐵 (𝑦 ≤ 𝑥 → (𝐹‘𝑥) ≤ (𝑚 · (𝐺‘𝑥))) ↔ ∀𝑥 ∈ 𝐵 (𝐶 ≤ 𝑥 → (𝐹‘𝑥) ≤ (𝑚 · (𝐺‘𝑥))))) |
4 | oveq1 7149 | . . . . . . 7 ⊢ (𝑚 = 𝑀 → (𝑚 · (𝐺‘𝑥)) = (𝑀 · (𝐺‘𝑥))) | |
5 | 4 | breq2d 5064 | . . . . . 6 ⊢ (𝑚 = 𝑀 → ((𝐹‘𝑥) ≤ (𝑚 · (𝐺‘𝑥)) ↔ (𝐹‘𝑥) ≤ (𝑀 · (𝐺‘𝑥)))) |
6 | 5 | imbi2d 343 | . . . . 5 ⊢ (𝑚 = 𝑀 → ((𝐶 ≤ 𝑥 → (𝐹‘𝑥) ≤ (𝑚 · (𝐺‘𝑥))) ↔ (𝐶 ≤ 𝑥 → (𝐹‘𝑥) ≤ (𝑀 · (𝐺‘𝑥))))) |
7 | 6 | ralbidv 3197 | . . . 4 ⊢ (𝑚 = 𝑀 → (∀𝑥 ∈ 𝐵 (𝐶 ≤ 𝑥 → (𝐹‘𝑥) ≤ (𝑚 · (𝐺‘𝑥))) ↔ ∀𝑥 ∈ 𝐵 (𝐶 ≤ 𝑥 → (𝐹‘𝑥) ≤ (𝑀 · (𝐺‘𝑥))))) |
8 | 3, 7 | rspc2ev 3627 | . . 3 ⊢ ((𝐶 ∈ ℝ ∧ 𝑀 ∈ ℝ ∧ ∀𝑥 ∈ 𝐵 (𝐶 ≤ 𝑥 → (𝐹‘𝑥) ≤ (𝑀 · (𝐺‘𝑥)))) → ∃𝑦 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑥 ∈ 𝐵 (𝑦 ≤ 𝑥 → (𝐹‘𝑥) ≤ (𝑚 · (𝐺‘𝑥)))) |
9 | 8 | 3ad2ant3 1131 | . 2 ⊢ (((𝐺:𝐴⟶ℝ ∧ 𝐴 ⊆ ℝ) ∧ (𝐹:𝐵⟶ℝ ∧ 𝐵 ⊆ 𝐴) ∧ (𝐶 ∈ ℝ ∧ 𝑀 ∈ ℝ ∧ ∀𝑥 ∈ 𝐵 (𝐶 ≤ 𝑥 → (𝐹‘𝑥) ≤ (𝑀 · (𝐺‘𝑥))))) → ∃𝑦 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑥 ∈ 𝐵 (𝑦 ≤ 𝑥 → (𝐹‘𝑥) ≤ (𝑚 · (𝐺‘𝑥)))) |
10 | elbigo2 44697 | . . 3 ⊢ (((𝐺:𝐴⟶ℝ ∧ 𝐴 ⊆ ℝ) ∧ (𝐹:𝐵⟶ℝ ∧ 𝐵 ⊆ 𝐴)) → (𝐹 ∈ (Ο‘𝐺) ↔ ∃𝑦 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑥 ∈ 𝐵 (𝑦 ≤ 𝑥 → (𝐹‘𝑥) ≤ (𝑚 · (𝐺‘𝑥))))) | |
11 | 10 | 3adant3 1128 | . 2 ⊢ (((𝐺:𝐴⟶ℝ ∧ 𝐴 ⊆ ℝ) ∧ (𝐹:𝐵⟶ℝ ∧ 𝐵 ⊆ 𝐴) ∧ (𝐶 ∈ ℝ ∧ 𝑀 ∈ ℝ ∧ ∀𝑥 ∈ 𝐵 (𝐶 ≤ 𝑥 → (𝐹‘𝑥) ≤ (𝑀 · (𝐺‘𝑥))))) → (𝐹 ∈ (Ο‘𝐺) ↔ ∃𝑦 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑥 ∈ 𝐵 (𝑦 ≤ 𝑥 → (𝐹‘𝑥) ≤ (𝑚 · (𝐺‘𝑥))))) |
12 | 9, 11 | mpbird 259 | 1 ⊢ (((𝐺:𝐴⟶ℝ ∧ 𝐴 ⊆ ℝ) ∧ (𝐹:𝐵⟶ℝ ∧ 𝐵 ⊆ 𝐴) ∧ (𝐶 ∈ ℝ ∧ 𝑀 ∈ ℝ ∧ ∀𝑥 ∈ 𝐵 (𝐶 ≤ 𝑥 → (𝐹‘𝑥) ≤ (𝑀 · (𝐺‘𝑥))))) → 𝐹 ∈ (Ο‘𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ∀wral 3138 ∃wrex 3139 ⊆ wss 3924 class class class wbr 5052 ⟶wf 6337 ‘cfv 6341 (class class class)co 7142 ℝcr 10522 · cmul 10528 ≤ cle 10662 Οcbigo 44692 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5189 ax-nul 5196 ax-pow 5252 ax-pr 5316 ax-un 7447 ax-cnex 10579 ax-resscn 10580 ax-pre-lttri 10597 ax-pre-lttrn 10598 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 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-rab 3147 df-v 3488 df-sbc 3764 df-csb 3872 df-dif 3927 df-un 3929 df-in 3931 df-ss 3940 df-nul 4280 df-if 4454 df-pw 4527 df-sn 4554 df-pr 4556 df-op 4560 df-uni 4825 df-br 5053 df-opab 5115 df-mpt 5133 df-id 5446 df-po 5460 df-so 5461 df-xp 5547 df-rel 5548 df-cnv 5549 df-co 5550 df-dm 5551 df-rn 5552 df-res 5553 df-ima 5554 df-iota 6300 df-fun 6343 df-fn 6344 df-f 6345 df-f1 6346 df-fo 6347 df-f1o 6348 df-fv 6349 df-ov 7145 df-oprab 7146 df-mpo 7147 df-er 8275 df-pm 8395 df-en 8496 df-dom 8497 df-sdom 8498 df-pnf 10663 df-mnf 10664 df-xr 10665 df-ltxr 10666 df-le 10667 df-ico 12731 df-bigo 44693 |
This theorem is referenced by: (None) |
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