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Mirrors > Home > MPE Home > Th. List > Mathboxes > imo72b2lem2 | Structured version Visualization version GIF version |
Description: Lemma for imo72b2 40518. (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 6098 | . . . 4 ⊢ ((abs ∘ 𝐹) “ ℝ) = (abs “ (𝐹 “ ℝ)) | |
2 | 1 | eqcomi 2830 | . . 3 ⊢ (abs “ (𝐹 “ ℝ)) = ((abs ∘ 𝐹) “ ℝ) |
3 | imassrn 5934 | . . . . 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 14691 | . . . . . . . 8 ⊢ abs:ℂ⟶ℝ | |
7 | 6 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → abs:ℂ⟶ℝ) |
8 | ax-resscn 10588 | . . . . . . . 8 ⊢ ℝ ⊆ ℂ | |
9 | 8 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → ℝ ⊆ ℂ) |
10 | 7, 9 | fssresd 6539 | . . . . . 6 ⊢ (𝜑 → (abs ↾ ℝ):ℝ⟶ℝ) |
11 | 5, 10 | fco2d 40506 | . . . . 5 ⊢ (𝜑 → (abs ∘ 𝐹):ℝ⟶ℝ) |
12 | 11 | frnd 6515 | . . . 4 ⊢ (𝜑 → ran (abs ∘ 𝐹) ⊆ ℝ) |
13 | 4, 12 | sstrd 3976 | . . 3 ⊢ (𝜑 → ((abs ∘ 𝐹) “ ℝ) ⊆ ℝ) |
14 | 2, 13 | eqsstrid 4014 | . 2 ⊢ (𝜑 → (abs “ (𝐹 “ ℝ)) ⊆ ℝ) |
15 | 0re 10637 | . . . . . . . 8 ⊢ 0 ∈ ℝ | |
16 | 15 | ne0ii 4302 | . . . . . . 7 ⊢ ℝ ≠ ∅ |
17 | 16 | a1i 11 | . . . . . 6 ⊢ (𝜑 → ℝ ≠ ∅) |
18 | 17, 11 | wnefimgd 40505 | . . . . 5 ⊢ (𝜑 → ((abs ∘ 𝐹) “ ℝ) ≠ ∅) |
19 | 18 | necomd 3071 | . . . 4 ⊢ (𝜑 → ∅ ≠ ((abs ∘ 𝐹) “ ℝ)) |
20 | 2 | a1i 11 | . . . 4 ⊢ (𝜑 → (abs “ (𝐹 “ ℝ)) = ((abs ∘ 𝐹) “ ℝ)) |
21 | 19, 20 | neeqtrrd 3090 | . . 3 ⊢ (𝜑 → ∅ ≠ (abs “ (𝐹 “ ℝ))) |
22 | 21 | necomd 3071 | . 2 ⊢ (𝜑 → (abs “ (𝐹 “ ℝ)) ≠ ∅) |
23 | imo72b2lem2.2 | . . 3 ⊢ (𝜑 → 𝐶 ∈ ℝ) | |
24 | simpr 487 | . . . . 5 ⊢ ((𝜑 ∧ 𝑐 = 𝐶) → 𝑐 = 𝐶) | |
25 | 24 | breq2d 5070 | . . . 4 ⊢ ((𝜑 ∧ 𝑐 = 𝐶) → (𝑣 ≤ 𝑐 ↔ 𝑣 ≤ 𝐶)) |
26 | 25 | ralbidv 3197 | . . 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 40508 | . . 3 ⊢ (𝜑 → ∀𝑣 ∈ (abs “ (𝐹 “ ℝ))𝑣 ≤ 𝐶) |
29 | 23, 26, 28 | rspcedvd 3625 | . 2 ⊢ (𝜑 → ∃𝑐 ∈ ℝ ∀𝑣 ∈ (abs “ (𝐹 “ ℝ))𝑣 ≤ 𝑐) |
30 | 5, 27 | extoimad 40508 | . 2 ⊢ (𝜑 → ∀𝑡 ∈ (abs “ (𝐹 “ ℝ))𝑡 ≤ 𝐶) |
31 | 14, 22, 29, 23, 30 | suprleubrd 40510 | 1 ⊢ (𝜑 → sup((abs “ (𝐹 “ ℝ)), ℝ, < ) ≤ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 ∀wral 3138 ⊆ wss 3935 ∅c0 4290 class class class wbr 5058 ran crn 5550 “ cima 5552 ∘ ccom 5553 ⟶wf 6345 ‘cfv 6349 supcsup 8898 ℂcc 10529 ℝcr 10530 0cc0 10531 < clt 10669 ≤ cle 10670 abscabs 14587 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 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-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-sup 8900 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-2 11694 df-3 11695 df-n0 11892 df-z 11976 df-uz 12238 df-rp 12384 df-seq 13364 df-exp 13424 df-cj 14452 df-re 14453 df-im 14454 df-sqrt 14588 df-abs 14589 |
This theorem is referenced by: imo72b2 40518 |
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