Mathbox for Stanislas Polu |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > extoimad | Structured version Visualization version GIF version |
Description: If |f(x)| <= C for all x then it applies to all x in the image of |f(x)| (Contributed by Stanislas Polu, 9-Mar-2020.) |
Ref | Expression |
---|---|
extoimad.1 | ⊢ (𝜑 → 𝐹:ℝ⟶ℝ) |
extoimad.2 | ⊢ (𝜑 → ∀𝑦 ∈ ℝ (abs‘(𝐹‘𝑦)) ≤ 𝐶) |
Ref | Expression |
---|---|
extoimad | ⊢ (𝜑 → ∀𝑥 ∈ (abs “ (𝐹 “ ℝ))𝑥 ≤ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | extoimad.2 | . 2 ⊢ (𝜑 → ∀𝑦 ∈ ℝ (abs‘(𝐹‘𝑦)) ≤ 𝐶) | |
2 | extoimad.1 | . . . . . 6 ⊢ (𝜑 → 𝐹:ℝ⟶ℝ) | |
3 | 2 | ffvelrnda 6843 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (𝐹‘𝑦) ∈ ℝ) |
4 | 3 | recnd 10657 | . . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (𝐹‘𝑦) ∈ ℂ) |
5 | 4 | abscld 14784 | . . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (abs‘(𝐹‘𝑦)) ∈ ℝ) |
6 | imaco 6097 | . . . . . 6 ⊢ ((abs ∘ 𝐹) “ ℝ) = (abs “ (𝐹 “ ℝ)) | |
7 | 6 | a1i 11 | . . . . 5 ⊢ (𝜑 → ((abs ∘ 𝐹) “ ℝ) = (abs “ (𝐹 “ ℝ))) |
8 | 7 | eleq2d 2895 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ ((abs ∘ 𝐹) “ ℝ) ↔ 𝑥 ∈ (abs “ (𝐹 “ ℝ)))) |
9 | absf 14685 | . . . . . . . . . . 11 ⊢ abs:ℂ⟶ℝ | |
10 | 9 | a1i 11 | . . . . . . . . . 10 ⊢ (𝜑 → abs:ℂ⟶ℝ) |
11 | ax-resscn 10582 | . . . . . . . . . . 11 ⊢ ℝ ⊆ ℂ | |
12 | 11 | a1i 11 | . . . . . . . . . 10 ⊢ (𝜑 → ℝ ⊆ ℂ) |
13 | 10, 12 | fssresd 6538 | . . . . . . . . 9 ⊢ (𝜑 → (abs ↾ ℝ):ℝ⟶ℝ) |
14 | 2, 13 | fco2d 40391 | . . . . . . . 8 ⊢ (𝜑 → (abs ∘ 𝐹):ℝ⟶ℝ) |
15 | 14 | ffnd 6508 | . . . . . . 7 ⊢ (𝜑 → (abs ∘ 𝐹) Fn ℝ) |
16 | ssidd 3987 | . . . . . . 7 ⊢ (𝜑 → ℝ ⊆ ℝ) | |
17 | 15, 16 | fvelimabd 6731 | . . . . . 6 ⊢ (𝜑 → (𝑥 ∈ ((abs ∘ 𝐹) “ ℝ) ↔ ∃𝑦 ∈ ℝ ((abs ∘ 𝐹)‘𝑦) = 𝑥)) |
18 | eqcom 2825 | . . . . . . . 8 ⊢ (((abs ∘ 𝐹)‘𝑦) = 𝑥 ↔ 𝑥 = ((abs ∘ 𝐹)‘𝑦)) | |
19 | 18 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → (((abs ∘ 𝐹)‘𝑦) = 𝑥 ↔ 𝑥 = ((abs ∘ 𝐹)‘𝑦))) |
20 | 19 | rexbidv 3294 | . . . . . 6 ⊢ (𝜑 → (∃𝑦 ∈ ℝ ((abs ∘ 𝐹)‘𝑦) = 𝑥 ↔ ∃𝑦 ∈ ℝ 𝑥 = ((abs ∘ 𝐹)‘𝑦))) |
21 | 17, 20 | bitrd 280 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ ((abs ∘ 𝐹) “ ℝ) ↔ ∃𝑦 ∈ ℝ 𝑥 = ((abs ∘ 𝐹)‘𝑦))) |
22 | 2 | adantr 481 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → 𝐹:ℝ⟶ℝ) |
23 | simpr 485 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ ℝ) | |
24 | 22, 23 | fvco3d 6754 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → ((abs ∘ 𝐹)‘𝑦) = (abs‘(𝐹‘𝑦))) |
25 | 24 | eqcomd 2824 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (abs‘(𝐹‘𝑦)) = ((abs ∘ 𝐹)‘𝑦)) |
26 | 25 | eqeq2d 2829 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (𝑥 = (abs‘(𝐹‘𝑦)) ↔ 𝑥 = ((abs ∘ 𝐹)‘𝑦))) |
27 | 26 | rexbidva 3293 | . . . . 5 ⊢ (𝜑 → (∃𝑦 ∈ ℝ 𝑥 = (abs‘(𝐹‘𝑦)) ↔ ∃𝑦 ∈ ℝ 𝑥 = ((abs ∘ 𝐹)‘𝑦))) |
28 | 21, 27 | bitr4d 283 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ ((abs ∘ 𝐹) “ ℝ) ↔ ∃𝑦 ∈ ℝ 𝑥 = (abs‘(𝐹‘𝑦)))) |
29 | 8, 28 | bitr3d 282 | . . 3 ⊢ (𝜑 → (𝑥 ∈ (abs “ (𝐹 “ ℝ)) ↔ ∃𝑦 ∈ ℝ 𝑥 = (abs‘(𝐹‘𝑦)))) |
30 | simpr 485 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 = (abs‘(𝐹‘𝑦))) → 𝑥 = (abs‘(𝐹‘𝑦))) | |
31 | 30 | breq1d 5067 | . . 3 ⊢ ((𝜑 ∧ 𝑥 = (abs‘(𝐹‘𝑦))) → (𝑥 ≤ 𝐶 ↔ (abs‘(𝐹‘𝑦)) ≤ 𝐶)) |
32 | 5, 29, 31 | ralxfr2d 5301 | . 2 ⊢ (𝜑 → (∀𝑥 ∈ (abs “ (𝐹 “ ℝ))𝑥 ≤ 𝐶 ↔ ∀𝑦 ∈ ℝ (abs‘(𝐹‘𝑦)) ≤ 𝐶)) |
33 | 1, 32 | mpbird 258 | 1 ⊢ (𝜑 → ∀𝑥 ∈ (abs “ (𝐹 “ ℝ))𝑥 ≤ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ∀wral 3135 ∃wrex 3136 ⊆ wss 3933 class class class wbr 5057 “ cima 5551 ∘ ccom 5552 ⟶wf 6344 ‘cfv 6348 ℂcc 10523 ℝcr 10524 ≤ cle 10664 abscabs 14581 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-sup 8894 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-2 11688 df-3 11689 df-n0 11886 df-z 11970 df-uz 12232 df-rp 12378 df-seq 13358 df-exp 13418 df-cj 14446 df-re 14447 df-im 14448 df-sqrt 14582 df-abs 14583 |
This theorem is referenced by: imo72b2lem0 40394 imo72b2lem2 40396 imo72b2lem1 40399 imo72b2 40403 |
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