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Mirrors > Home > MPE Home > Th. List > Mathboxes > imo72b2lem2 | Structured version Visualization version GIF version |
Description: Lemma for imo72b2 39315. (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 5881 | . . . 4 ⊢ ((abs ∘ 𝐹) “ ℝ) = (abs “ (𝐹 “ ℝ)) | |
2 | 1 | eqcomi 2834 | . . 3 ⊢ (abs “ (𝐹 “ ℝ)) = ((abs ∘ 𝐹) “ ℝ) |
3 | imassrn 5718 | . . . . 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 14454 | . . . . . . . 8 ⊢ abs:ℂ⟶ℝ | |
7 | 6 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → abs:ℂ⟶ℝ) |
8 | ax-resscn 10309 | . . . . . . . 8 ⊢ ℝ ⊆ ℂ | |
9 | 8 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → ℝ ⊆ ℂ) |
10 | 7, 9 | fssresd 6308 | . . . . . 6 ⊢ (𝜑 → (abs ↾ ℝ):ℝ⟶ℝ) |
11 | 5, 10 | fco2d 39301 | . . . . 5 ⊢ (𝜑 → (abs ∘ 𝐹):ℝ⟶ℝ) |
12 | 11 | frnd 6285 | . . . 4 ⊢ (𝜑 → ran (abs ∘ 𝐹) ⊆ ℝ) |
13 | 4, 12 | sstrd 3837 | . . 3 ⊢ (𝜑 → ((abs ∘ 𝐹) “ ℝ) ⊆ ℝ) |
14 | 2, 13 | syl5eqss 3874 | . 2 ⊢ (𝜑 → (abs “ (𝐹 “ ℝ)) ⊆ ℝ) |
15 | 0re 10358 | . . . . . . . 8 ⊢ 0 ∈ ℝ | |
16 | 15 | ne0ii 4153 | . . . . . . 7 ⊢ ℝ ≠ ∅ |
17 | 16 | a1i 11 | . . . . . 6 ⊢ (𝜑 → ℝ ≠ ∅) |
18 | 17, 11 | wnefimgd 39300 | . . . . 5 ⊢ (𝜑 → ((abs ∘ 𝐹) “ ℝ) ≠ ∅) |
19 | 18 | necomd 3054 | . . . 4 ⊢ (𝜑 → ∅ ≠ ((abs ∘ 𝐹) “ ℝ)) |
20 | 2 | a1i 11 | . . . 4 ⊢ (𝜑 → (abs “ (𝐹 “ ℝ)) = ((abs ∘ 𝐹) “ ℝ)) |
21 | 19, 20 | neeqtrrd 3073 | . . 3 ⊢ (𝜑 → ∅ ≠ (abs “ (𝐹 “ ℝ))) |
22 | 21 | necomd 3054 | . 2 ⊢ (𝜑 → (abs “ (𝐹 “ ℝ)) ≠ ∅) |
23 | imo72b2lem2.2 | . . 3 ⊢ (𝜑 → 𝐶 ∈ ℝ) | |
24 | simpr 479 | . . . . 5 ⊢ ((𝜑 ∧ 𝑐 = 𝐶) → 𝑐 = 𝐶) | |
25 | 24 | breq2d 4885 | . . . 4 ⊢ ((𝜑 ∧ 𝑐 = 𝐶) → (𝑣 ≤ 𝑐 ↔ 𝑣 ≤ 𝐶)) |
26 | 25 | ralbidv 3195 | . . 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 39304 | . . 3 ⊢ (𝜑 → ∀𝑣 ∈ (abs “ (𝐹 “ ℝ))𝑣 ≤ 𝐶) |
29 | 23, 26, 28 | rspcedvd 3533 | . 2 ⊢ (𝜑 → ∃𝑐 ∈ ℝ ∀𝑣 ∈ (abs “ (𝐹 “ ℝ))𝑣 ≤ 𝑐) |
30 | 5, 27 | extoimad 39304 | . 2 ⊢ (𝜑 → ∀𝑡 ∈ (abs “ (𝐹 “ ℝ))𝑡 ≤ 𝐶) |
31 | 14, 22, 29, 23, 30 | suprleubrd 39306 | 1 ⊢ (𝜑 → sup((abs “ (𝐹 “ ℝ)), ℝ, < ) ≤ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1658 ∈ wcel 2166 ≠ wne 2999 ∀wral 3117 ⊆ wss 3798 ∅c0 4144 class class class wbr 4873 ran crn 5343 “ cima 5345 ∘ ccom 5346 ⟶wf 6119 ‘cfv 6123 supcsup 8615 ℂcc 10250 ℝcr 10251 0cc0 10252 < clt 10391 ≤ cle 10392 abscabs 14351 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-cnex 10308 ax-resscn 10309 ax-1cn 10310 ax-icn 10311 ax-addcl 10312 ax-addrcl 10313 ax-mulcl 10314 ax-mulrcl 10315 ax-mulcom 10316 ax-addass 10317 ax-mulass 10318 ax-distr 10319 ax-i2m1 10320 ax-1ne0 10321 ax-1rid 10322 ax-rnegex 10323 ax-rrecex 10324 ax-cnre 10325 ax-pre-lttri 10326 ax-pre-lttrn 10327 ax-pre-ltadd 10328 ax-pre-mulgt0 10329 ax-pre-sup 10330 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4659 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-pred 5920 df-ord 5966 df-on 5967 df-lim 5968 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-riota 6866 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-om 7327 df-2nd 7429 df-wrecs 7672 df-recs 7734 df-rdg 7772 df-er 8009 df-en 8223 df-dom 8224 df-sdom 8225 df-sup 8617 df-pnf 10393 df-mnf 10394 df-xr 10395 df-ltxr 10396 df-le 10397 df-sub 10587 df-neg 10588 df-div 11010 df-nn 11351 df-2 11414 df-3 11415 df-n0 11619 df-z 11705 df-uz 11969 df-rp 12113 df-seq 13096 df-exp 13155 df-cj 14216 df-re 14217 df-im 14218 df-sqrt 14352 df-abs 14353 |
This theorem is referenced by: imo72b2 39315 |
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