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Mirrors > Home > MPE Home > Th. List > Mathboxes > imo72b2lem2 | Structured version Visualization version GIF version |
Description: Lemma for imo72b2 40878. (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 6071 | . . . 4 ⊢ ((abs ∘ 𝐹) “ ℝ) = (abs “ (𝐹 “ ℝ)) | |
2 | 1 | eqcomi 2807 | . . 3 ⊢ (abs “ (𝐹 “ ℝ)) = ((abs ∘ 𝐹) “ ℝ) |
3 | imassrn 5907 | . . . . 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 14689 | . . . . . . . 8 ⊢ abs:ℂ⟶ℝ | |
7 | 6 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → abs:ℂ⟶ℝ) |
8 | ax-resscn 10583 | . . . . . . . 8 ⊢ ℝ ⊆ ℂ | |
9 | 8 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → ℝ ⊆ ℂ) |
10 | 7, 9 | fssresd 6519 | . . . . . 6 ⊢ (𝜑 → (abs ↾ ℝ):ℝ⟶ℝ) |
11 | 5, 10 | fco2d 40866 | . . . . 5 ⊢ (𝜑 → (abs ∘ 𝐹):ℝ⟶ℝ) |
12 | 11 | frnd 6494 | . . . 4 ⊢ (𝜑 → ran (abs ∘ 𝐹) ⊆ ℝ) |
13 | 4, 12 | sstrd 3925 | . . 3 ⊢ (𝜑 → ((abs ∘ 𝐹) “ ℝ) ⊆ ℝ) |
14 | 2, 13 | eqsstrid 3963 | . 2 ⊢ (𝜑 → (abs “ (𝐹 “ ℝ)) ⊆ ℝ) |
15 | 0re 10632 | . . . . . . . 8 ⊢ 0 ∈ ℝ | |
16 | 15 | ne0ii 4253 | . . . . . . 7 ⊢ ℝ ≠ ∅ |
17 | 16 | a1i 11 | . . . . . 6 ⊢ (𝜑 → ℝ ≠ ∅) |
18 | 17, 11 | wnefimgd 40865 | . . . . 5 ⊢ (𝜑 → ((abs ∘ 𝐹) “ ℝ) ≠ ∅) |
19 | 18 | necomd 3042 | . . . 4 ⊢ (𝜑 → ∅ ≠ ((abs ∘ 𝐹) “ ℝ)) |
20 | 2 | a1i 11 | . . . 4 ⊢ (𝜑 → (abs “ (𝐹 “ ℝ)) = ((abs ∘ 𝐹) “ ℝ)) |
21 | 19, 20 | neeqtrrd 3061 | . . 3 ⊢ (𝜑 → ∅ ≠ (abs “ (𝐹 “ ℝ))) |
22 | 21 | necomd 3042 | . 2 ⊢ (𝜑 → (abs “ (𝐹 “ ℝ)) ≠ ∅) |
23 | imo72b2lem2.2 | . . 3 ⊢ (𝜑 → 𝐶 ∈ ℝ) | |
24 | simpr 488 | . . . . 5 ⊢ ((𝜑 ∧ 𝑐 = 𝐶) → 𝑐 = 𝐶) | |
25 | 24 | breq2d 5042 | . . . 4 ⊢ ((𝜑 ∧ 𝑐 = 𝐶) → (𝑣 ≤ 𝑐 ↔ 𝑣 ≤ 𝐶)) |
26 | 25 | ralbidv 3162 | . . 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 40868 | . . 3 ⊢ (𝜑 → ∀𝑣 ∈ (abs “ (𝐹 “ ℝ))𝑣 ≤ 𝐶) |
29 | 23, 26, 28 | rspcedvd 3574 | . 2 ⊢ (𝜑 → ∃𝑐 ∈ ℝ ∀𝑣 ∈ (abs “ (𝐹 “ ℝ))𝑣 ≤ 𝑐) |
30 | 5, 27 | extoimad 40868 | . 2 ⊢ (𝜑 → ∀𝑡 ∈ (abs “ (𝐹 “ ℝ))𝑡 ≤ 𝐶) |
31 | 14, 22, 29, 23, 30 | suprleubrd 40870 | 1 ⊢ (𝜑 → sup((abs “ (𝐹 “ ℝ)), ℝ, < ) ≤ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 ∀wral 3106 ⊆ wss 3881 ∅c0 4243 class class class wbr 5030 ran crn 5520 “ cima 5522 ∘ ccom 5523 ⟶wf 6320 ‘cfv 6324 supcsup 8888 ℂcc 10524 ℝcr 10525 0cc0 10526 < clt 10664 ≤ cle 10665 abscabs 14585 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-sup 8890 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-n0 11886 df-z 11970 df-uz 12232 df-rp 12378 df-seq 13365 df-exp 13426 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 |
This theorem is referenced by: imo72b2 40878 |
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