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Mirrors > Home > MPE Home > Th. List > Mathboxes > imo72b2lem1 | Structured version Visualization version GIF version |
Description: Lemma for imo72b2 40531. (Contributed by Stanislas Polu, 9-Mar-2020.) |
Ref | Expression |
---|---|
imo72b2lem1.1 | ⊢ (𝜑 → 𝐹:ℝ⟶ℝ) |
imo72b2lem1.7 | ⊢ (𝜑 → ∃𝑥 ∈ ℝ (𝐹‘𝑥) ≠ 0) |
imo72b2lem1.6 | ⊢ (𝜑 → ∀𝑦 ∈ ℝ (abs‘(𝐹‘𝑦)) ≤ 1) |
Ref | Expression |
---|---|
imo72b2lem1 | ⊢ (𝜑 → 0 < sup((abs “ (𝐹 “ ℝ)), ℝ, < )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | imaco 6107 | . . 3 ⊢ ((abs ∘ 𝐹) “ ℝ) = (abs “ (𝐹 “ ℝ)) | |
2 | imassrn 5943 | . . . 4 ⊢ ((abs ∘ 𝐹) “ ℝ) ⊆ ran (abs ∘ 𝐹) | |
3 | imo72b2lem1.1 | . . . . . 6 ⊢ (𝜑 → 𝐹:ℝ⟶ℝ) | |
4 | absf 14700 | . . . . . . . 8 ⊢ abs:ℂ⟶ℝ | |
5 | 4 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → abs:ℂ⟶ℝ) |
6 | ax-resscn 10597 | . . . . . . . 8 ⊢ ℝ ⊆ ℂ | |
7 | 6 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → ℝ ⊆ ℂ) |
8 | 5, 7 | fssresd 6548 | . . . . . 6 ⊢ (𝜑 → (abs ↾ ℝ):ℝ⟶ℝ) |
9 | 3, 8 | fco2d 40519 | . . . . 5 ⊢ (𝜑 → (abs ∘ 𝐹):ℝ⟶ℝ) |
10 | 9 | frnd 6524 | . . . 4 ⊢ (𝜑 → ran (abs ∘ 𝐹) ⊆ ℝ) |
11 | 2, 10 | sstrid 3981 | . . 3 ⊢ (𝜑 → ((abs ∘ 𝐹) “ ℝ) ⊆ ℝ) |
12 | 1, 11 | eqsstrrid 4019 | . 2 ⊢ (𝜑 → (abs “ (𝐹 “ ℝ)) ⊆ ℝ) |
13 | 0re 10646 | . . . . . 6 ⊢ 0 ∈ ℝ | |
14 | 13 | ne0ii 4306 | . . . . 5 ⊢ ℝ ≠ ∅ |
15 | 14 | a1i 11 | . . . 4 ⊢ (𝜑 → ℝ ≠ ∅) |
16 | 15, 9 | wnefimgd 40518 | . . 3 ⊢ (𝜑 → ((abs ∘ 𝐹) “ ℝ) ≠ ∅) |
17 | 1, 16 | eqnetrrid 3094 | . 2 ⊢ (𝜑 → (abs “ (𝐹 “ ℝ)) ≠ ∅) |
18 | 1red 10645 | . . 3 ⊢ (𝜑 → 1 ∈ ℝ) | |
19 | simpr 487 | . . . . 5 ⊢ ((𝜑 ∧ 𝑐 = 1) → 𝑐 = 1) | |
20 | 19 | breq2d 5081 | . . . 4 ⊢ ((𝜑 ∧ 𝑐 = 1) → (𝑡 ≤ 𝑐 ↔ 𝑡 ≤ 1)) |
21 | 20 | ralbidv 3200 | . . 3 ⊢ ((𝜑 ∧ 𝑐 = 1) → (∀𝑡 ∈ (abs “ (𝐹 “ ℝ))𝑡 ≤ 𝑐 ↔ ∀𝑡 ∈ (abs “ (𝐹 “ ℝ))𝑡 ≤ 1)) |
22 | imo72b2lem1.6 | . . . 4 ⊢ (𝜑 → ∀𝑦 ∈ ℝ (abs‘(𝐹‘𝑦)) ≤ 1) | |
23 | 3, 22 | extoimad 40521 | . . 3 ⊢ (𝜑 → ∀𝑡 ∈ (abs “ (𝐹 “ ℝ))𝑡 ≤ 1) |
24 | 18, 21, 23 | rspcedvd 3629 | . 2 ⊢ (𝜑 → ∃𝑐 ∈ ℝ ∀𝑡 ∈ (abs “ (𝐹 “ ℝ))𝑡 ≤ 𝑐) |
25 | 0red 10647 | . 2 ⊢ (𝜑 → 0 ∈ ℝ) | |
26 | imo72b2lem1.7 | . . 3 ⊢ (𝜑 → ∃𝑥 ∈ ℝ (𝐹‘𝑥) ≠ 0) | |
27 | 3 | adantr 483 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐹‘𝑥) ≠ 0)) → 𝐹:ℝ⟶ℝ) |
28 | simprl 769 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐹‘𝑥) ≠ 0)) → 𝑥 ∈ ℝ) | |
29 | 27, 28 | fvco3d 6764 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐹‘𝑥) ≠ 0)) → ((abs ∘ 𝐹)‘𝑥) = (abs‘(𝐹‘𝑥))) |
30 | 9 | funfvima2d 6997 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ((abs ∘ 𝐹)‘𝑥) ∈ ((abs ∘ 𝐹) “ ℝ)) |
31 | 30 | adantrr 715 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐹‘𝑥) ≠ 0)) → ((abs ∘ 𝐹)‘𝑥) ∈ ((abs ∘ 𝐹) “ ℝ)) |
32 | 31, 1 | eleqtrdi 2926 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐹‘𝑥) ≠ 0)) → ((abs ∘ 𝐹)‘𝑥) ∈ (abs “ (𝐹 “ ℝ))) |
33 | 29, 32 | eqeltrrd 2917 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐹‘𝑥) ≠ 0)) → (abs‘(𝐹‘𝑥)) ∈ (abs “ (𝐹 “ ℝ))) |
34 | simpr 487 | . . . . 5 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐹‘𝑥) ≠ 0)) ∧ 𝑧 = (abs‘(𝐹‘𝑥))) → 𝑧 = (abs‘(𝐹‘𝑥))) | |
35 | 34 | breq2d 5081 | . . . 4 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐹‘𝑥) ≠ 0)) ∧ 𝑧 = (abs‘(𝐹‘𝑥))) → (0 < 𝑧 ↔ 0 < (abs‘(𝐹‘𝑥)))) |
36 | 3 | ffvelrnda 6854 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐹‘𝑥) ∈ ℝ) |
37 | 36 | adantrr 715 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐹‘𝑥) ≠ 0)) → (𝐹‘𝑥) ∈ ℝ) |
38 | 37 | recnd 10672 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐹‘𝑥) ≠ 0)) → (𝐹‘𝑥) ∈ ℂ) |
39 | simprr 771 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐹‘𝑥) ≠ 0)) → (𝐹‘𝑥) ≠ 0) | |
40 | 38, 39 | absrpcld 14811 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐹‘𝑥) ≠ 0)) → (abs‘(𝐹‘𝑥)) ∈ ℝ+) |
41 | 40 | rpgt0d 12437 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐹‘𝑥) ≠ 0)) → 0 < (abs‘(𝐹‘𝑥))) |
42 | 33, 35, 41 | rspcedvd 3629 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐹‘𝑥) ≠ 0)) → ∃𝑧 ∈ (abs “ (𝐹 “ ℝ))0 < 𝑧) |
43 | 26, 42 | rexlimddv 3294 | . 2 ⊢ (𝜑 → ∃𝑧 ∈ (abs “ (𝐹 “ ℝ))0 < 𝑧) |
44 | 12, 17, 24, 25, 43 | suprlubrd 40526 | 1 ⊢ (𝜑 → 0 < sup((abs “ (𝐹 “ ℝ)), ℝ, < )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ≠ wne 3019 ∀wral 3141 ∃wrex 3142 ⊆ wss 3939 ∅c0 4294 class class class wbr 5069 ran crn 5559 “ cima 5561 ∘ ccom 5562 ⟶wf 6354 ‘cfv 6358 supcsup 8907 ℂcc 10538 ℝcr 10539 0cc0 10540 1c1 10541 < clt 10678 ≤ 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: imo72b2 40531 |
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