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Mirrors > Home > MPE Home > Th. List > Mathboxes > imo72b2lem2 | Structured version Visualization version GIF version |
Description: Lemma for imo72b2 41490. (Contributed by Stanislas Polu, 9-Mar-2020.) |
Ref | Expression |
---|---|
imo72b2lem2.1 | ⊢ (𝜑 → 𝐹:ℝ⟶ℝ) |
imo72b2lem2.2 | ⊢ (𝜑 → 𝐶 ∈ ℝ) |
imo72b2lem2.3 | ⊢ (𝜑 → ∀𝑧 ∈ ℝ (abs‘(𝐹‘𝑧)) ≤ 𝐶) |
Ref | Expression |
---|---|
imo72b2lem2 | ⊢ (𝜑 → sup((abs “ (𝐹 “ ℝ)), ℝ, < ) ≤ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | imaco 6130 | . . . 4 ⊢ ((abs ∘ 𝐹) “ ℝ) = (abs “ (𝐹 “ ℝ)) | |
2 | 1 | eqcomi 2747 | . . 3 ⊢ (abs “ (𝐹 “ ℝ)) = ((abs ∘ 𝐹) “ ℝ) |
3 | imassrn 5955 | . . . . 5 ⊢ ((abs ∘ 𝐹) “ ℝ) ⊆ ran (abs ∘ 𝐹) | |
4 | 3 | a1i 11 | . . . 4 ⊢ (𝜑 → ((abs ∘ 𝐹) “ ℝ) ⊆ ran (abs ∘ 𝐹)) |
5 | imo72b2lem2.1 | . . . . . 6 ⊢ (𝜑 → 𝐹:ℝ⟶ℝ) | |
6 | absf 14926 | . . . . . . . 8 ⊢ abs:ℂ⟶ℝ | |
7 | 6 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → abs:ℂ⟶ℝ) |
8 | ax-resscn 10811 | . . . . . . . 8 ⊢ ℝ ⊆ ℂ | |
9 | 8 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → ℝ ⊆ ℂ) |
10 | 7, 9 | fssresd 6605 | . . . . . 6 ⊢ (𝜑 → (abs ↾ ℝ):ℝ⟶ℝ) |
11 | 5, 10 | fco2d 41479 | . . . . 5 ⊢ (𝜑 → (abs ∘ 𝐹):ℝ⟶ℝ) |
12 | 11 | frnd 6572 | . . . 4 ⊢ (𝜑 → ran (abs ∘ 𝐹) ⊆ ℝ) |
13 | 4, 12 | sstrd 3926 | . . 3 ⊢ (𝜑 → ((abs ∘ 𝐹) “ ℝ) ⊆ ℝ) |
14 | 2, 13 | eqsstrid 3964 | . 2 ⊢ (𝜑 → (abs “ (𝐹 “ ℝ)) ⊆ ℝ) |
15 | 0re 10860 | . . . . . . . 8 ⊢ 0 ∈ ℝ | |
16 | 15 | ne0ii 4267 | . . . . . . 7 ⊢ ℝ ≠ ∅ |
17 | 16 | a1i 11 | . . . . . 6 ⊢ (𝜑 → ℝ ≠ ∅) |
18 | 17, 11 | wnefimgd 41478 | . . . . 5 ⊢ (𝜑 → ((abs ∘ 𝐹) “ ℝ) ≠ ∅) |
19 | 18 | necomd 2997 | . . . 4 ⊢ (𝜑 → ∅ ≠ ((abs ∘ 𝐹) “ ℝ)) |
20 | 2 | a1i 11 | . . . 4 ⊢ (𝜑 → (abs “ (𝐹 “ ℝ)) = ((abs ∘ 𝐹) “ ℝ)) |
21 | 19, 20 | neeqtrrd 3016 | . . 3 ⊢ (𝜑 → ∅ ≠ (abs “ (𝐹 “ ℝ))) |
22 | 21 | necomd 2997 | . 2 ⊢ (𝜑 → (abs “ (𝐹 “ ℝ)) ≠ ∅) |
23 | imo72b2lem2.2 | . . 3 ⊢ (𝜑 → 𝐶 ∈ ℝ) | |
24 | simpr 488 | . . . . 5 ⊢ ((𝜑 ∧ 𝑐 = 𝐶) → 𝑐 = 𝐶) | |
25 | 24 | breq2d 5080 | . . . 4 ⊢ ((𝜑 ∧ 𝑐 = 𝐶) → (𝑣 ≤ 𝑐 ↔ 𝑣 ≤ 𝐶)) |
26 | 25 | ralbidv 3119 | . . 3 ⊢ ((𝜑 ∧ 𝑐 = 𝐶) → (∀𝑣 ∈ (abs “ (𝐹 “ ℝ))𝑣 ≤ 𝑐 ↔ ∀𝑣 ∈ (abs “ (𝐹 “ ℝ))𝑣 ≤ 𝐶)) |
27 | imo72b2lem2.3 | . . . 4 ⊢ (𝜑 → ∀𝑧 ∈ ℝ (abs‘(𝐹‘𝑧)) ≤ 𝐶) | |
28 | 5, 27 | extoimad 41481 | . . 3 ⊢ (𝜑 → ∀𝑣 ∈ (abs “ (𝐹 “ ℝ))𝑣 ≤ 𝐶) |
29 | 23, 26, 28 | rspcedvd 3553 | . 2 ⊢ (𝜑 → ∃𝑐 ∈ ℝ ∀𝑣 ∈ (abs “ (𝐹 “ ℝ))𝑣 ≤ 𝑐) |
30 | 5, 27 | extoimad 41481 | . 2 ⊢ (𝜑 → ∀𝑡 ∈ (abs “ (𝐹 “ ℝ))𝑡 ≤ 𝐶) |
31 | 14, 22, 29, 23, 30 | suprleubrd 41483 | 1 ⊢ (𝜑 → sup((abs “ (𝐹 “ ℝ)), ℝ, < ) ≤ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2111 ≠ wne 2941 ∀wral 3062 ⊆ wss 3881 ∅c0 4252 class class class wbr 5068 ran crn 5567 “ cima 5569 ∘ ccom 5570 ⟶wf 6394 ‘cfv 6398 supcsup 9081 ℂcc 10752 ℝcr 10753 0cc0 10754 < clt 10892 ≤ cle 10893 abscabs 14822 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2159 ax-12 2176 ax-ext 2709 ax-sep 5207 ax-nul 5214 ax-pow 5273 ax-pr 5337 ax-un 7542 ax-cnex 10810 ax-resscn 10811 ax-1cn 10812 ax-icn 10813 ax-addcl 10814 ax-addrcl 10815 ax-mulcl 10816 ax-mulrcl 10817 ax-mulcom 10818 ax-addass 10819 ax-mulass 10820 ax-distr 10821 ax-i2m1 10822 ax-1ne0 10823 ax-1rid 10824 ax-rnegex 10825 ax-rrecex 10826 ax-cnre 10827 ax-pre-lttri 10828 ax-pre-lttrn 10829 ax-pre-ltadd 10830 ax-pre-mulgt0 10831 ax-pre-sup 10832 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2072 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3067 df-rex 3068 df-reu 3069 df-rmo 3070 df-rab 3071 df-v 3423 df-sbc 3710 df-csb 3827 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4253 df-if 4455 df-pw 4530 df-sn 4557 df-pr 4559 df-tp 4561 df-op 4563 df-uni 4835 df-iun 4921 df-br 5069 df-opab 5131 df-mpt 5151 df-tr 5177 df-id 5470 df-eprel 5475 df-po 5483 df-so 5484 df-fr 5524 df-we 5526 df-xp 5572 df-rel 5573 df-cnv 5574 df-co 5575 df-dm 5576 df-rn 5577 df-res 5578 df-ima 5579 df-pred 6176 df-ord 6234 df-on 6235 df-lim 6236 df-suc 6237 df-iota 6356 df-fun 6400 df-fn 6401 df-f 6402 df-f1 6403 df-fo 6404 df-f1o 6405 df-fv 6406 df-riota 7189 df-ov 7235 df-oprab 7236 df-mpo 7237 df-om 7664 df-2nd 7781 df-wrecs 8068 df-recs 8129 df-rdg 8167 df-er 8412 df-en 8648 df-dom 8649 df-sdom 8650 df-sup 9083 df-pnf 10894 df-mnf 10895 df-xr 10896 df-ltxr 10897 df-le 10898 df-sub 11089 df-neg 11090 df-div 11515 df-nn 11856 df-2 11918 df-3 11919 df-n0 12116 df-z 12202 df-uz 12464 df-rp 12612 df-seq 13600 df-exp 13661 df-cj 14687 df-re 14688 df-im 14689 df-sqrt 14823 df-abs 14824 |
This theorem is referenced by: imo72b2 41490 |
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