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 6854 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (𝐹‘𝑦) ∈ ℝ) |
4 | 3 | recnd 10672 | . . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (𝐹‘𝑦) ∈ ℂ) |
5 | 4 | abscld 14799 | . . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (abs‘(𝐹‘𝑦)) ∈ ℝ) |
6 | imaco 6107 | . . . . . 6 ⊢ ((abs ∘ 𝐹) “ ℝ) = (abs “ (𝐹 “ ℝ)) | |
7 | 6 | a1i 11 | . . . . 5 ⊢ (𝜑 → ((abs ∘ 𝐹) “ ℝ) = (abs “ (𝐹 “ ℝ))) |
8 | 7 | eleq2d 2901 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ ((abs ∘ 𝐹) “ ℝ) ↔ 𝑥 ∈ (abs “ (𝐹 “ ℝ)))) |
9 | absf 14700 | . . . . . . . . . . 11 ⊢ abs:ℂ⟶ℝ | |
10 | 9 | a1i 11 | . . . . . . . . . 10 ⊢ (𝜑 → abs:ℂ⟶ℝ) |
11 | ax-resscn 10597 | . . . . . . . . . . 11 ⊢ ℝ ⊆ ℂ | |
12 | 11 | a1i 11 | . . . . . . . . . 10 ⊢ (𝜑 → ℝ ⊆ ℂ) |
13 | 10, 12 | fssresd 6548 | . . . . . . . . 9 ⊢ (𝜑 → (abs ↾ ℝ):ℝ⟶ℝ) |
14 | 2, 13 | fco2d 40519 | . . . . . . . 8 ⊢ (𝜑 → (abs ∘ 𝐹):ℝ⟶ℝ) |
15 | 14 | ffnd 6518 | . . . . . . 7 ⊢ (𝜑 → (abs ∘ 𝐹) Fn ℝ) |
16 | ssidd 3993 | . . . . . . 7 ⊢ (𝜑 → ℝ ⊆ ℝ) | |
17 | 15, 16 | fvelimabd 6741 | . . . . . 6 ⊢ (𝜑 → (𝑥 ∈ ((abs ∘ 𝐹) “ ℝ) ↔ ∃𝑦 ∈ ℝ ((abs ∘ 𝐹)‘𝑦) = 𝑥)) |
18 | eqcom 2831 | . . . . . . . 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 3300 | . . . . . 6 ⊢ (𝜑 → (∃𝑦 ∈ ℝ ((abs ∘ 𝐹)‘𝑦) = 𝑥 ↔ ∃𝑦 ∈ ℝ 𝑥 = ((abs ∘ 𝐹)‘𝑦))) |
21 | 17, 20 | bitrd 281 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ ((abs ∘ 𝐹) “ ℝ) ↔ ∃𝑦 ∈ ℝ 𝑥 = ((abs ∘ 𝐹)‘𝑦))) |
22 | 2 | adantr 483 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → 𝐹:ℝ⟶ℝ) |
23 | simpr 487 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ ℝ) | |
24 | 22, 23 | fvco3d 6764 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → ((abs ∘ 𝐹)‘𝑦) = (abs‘(𝐹‘𝑦))) |
25 | 24 | eqcomd 2830 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (abs‘(𝐹‘𝑦)) = ((abs ∘ 𝐹)‘𝑦)) |
26 | 25 | eqeq2d 2835 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (𝑥 = (abs‘(𝐹‘𝑦)) ↔ 𝑥 = ((abs ∘ 𝐹)‘𝑦))) |
27 | 26 | rexbidva 3299 | . . . . 5 ⊢ (𝜑 → (∃𝑦 ∈ ℝ 𝑥 = (abs‘(𝐹‘𝑦)) ↔ ∃𝑦 ∈ ℝ 𝑥 = ((abs ∘ 𝐹)‘𝑦))) |
28 | 21, 27 | bitr4d 284 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ ((abs ∘ 𝐹) “ ℝ) ↔ ∃𝑦 ∈ ℝ 𝑥 = (abs‘(𝐹‘𝑦)))) |
29 | 8, 28 | bitr3d 283 | . . 3 ⊢ (𝜑 → (𝑥 ∈ (abs “ (𝐹 “ ℝ)) ↔ ∃𝑦 ∈ ℝ 𝑥 = (abs‘(𝐹‘𝑦)))) |
30 | simpr 487 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 = (abs‘(𝐹‘𝑦))) → 𝑥 = (abs‘(𝐹‘𝑦))) | |
31 | 30 | breq1d 5079 | . . 3 ⊢ ((𝜑 ∧ 𝑥 = (abs‘(𝐹‘𝑦))) → (𝑥 ≤ 𝐶 ↔ (abs‘(𝐹‘𝑦)) ≤ 𝐶)) |
32 | 5, 29, 31 | ralxfr2d 5314 | . 2 ⊢ (𝜑 → (∀𝑥 ∈ (abs “ (𝐹 “ ℝ))𝑥 ≤ 𝐶 ↔ ∀𝑦 ∈ ℝ (abs‘(𝐹‘𝑦)) ≤ 𝐶)) |
33 | 1, 32 | mpbird 259 | 1 ⊢ (𝜑 → ∀𝑥 ∈ (abs “ (𝐹 “ ℝ))𝑥 ≤ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ∀wral 3141 ∃wrex 3142 ⊆ wss 3939 class class class wbr 5069 “ cima 5561 ∘ ccom 5562 ⟶wf 6354 ‘cfv 6358 ℂcc 10538 ℝcr 10539 ≤ cle 10679 abscabs 14596 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 ax-pre-sup 10618 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-2nd 7693 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-er 8292 df-en 8513 df-dom 8514 df-sdom 8515 df-sup 8909 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-div 11301 df-nn 11642 df-2 11703 df-3 11704 df-n0 11901 df-z 11985 df-uz 12247 df-rp 12393 df-seq 13373 df-exp 13433 df-cj 14461 df-re 14462 df-im 14463 df-sqrt 14597 df-abs 14598 |
This theorem is referenced by: imo72b2lem0 40522 imo72b2lem2 40524 imo72b2lem1 40527 imo72b2 40531 |
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