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Mirrors > Home > MPE Home > Th. List > Mathboxes > imo72b2lem1 | Structured version Visualization version GIF version |
Description: Lemma for imo72b2 41330. (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 6084 | . . 3 ⊢ ((abs ∘ 𝐹) “ ℝ) = (abs “ (𝐹 “ ℝ)) | |
2 | imassrn 5914 | . . . 4 ⊢ ((abs ∘ 𝐹) “ ℝ) ⊆ ran (abs ∘ 𝐹) | |
3 | imo72b2lem1.1 | . . . . . 6 ⊢ (𝜑 → 𝐹:ℝ⟶ℝ) | |
4 | absf 14787 | . . . . . . . 8 ⊢ abs:ℂ⟶ℝ | |
5 | 4 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → abs:ℂ⟶ℝ) |
6 | ax-resscn 10672 | . . . . . . . 8 ⊢ ℝ ⊆ ℂ | |
7 | 6 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → ℝ ⊆ ℂ) |
8 | 5, 7 | fssresd 6545 | . . . . . 6 ⊢ (𝜑 → (abs ↾ ℝ):ℝ⟶ℝ) |
9 | 3, 8 | fco2d 41319 | . . . . 5 ⊢ (𝜑 → (abs ∘ 𝐹):ℝ⟶ℝ) |
10 | 9 | frnd 6512 | . . . 4 ⊢ (𝜑 → ran (abs ∘ 𝐹) ⊆ ℝ) |
11 | 2, 10 | sstrid 3888 | . . 3 ⊢ (𝜑 → ((abs ∘ 𝐹) “ ℝ) ⊆ ℝ) |
12 | 1, 11 | eqsstrrid 3926 | . 2 ⊢ (𝜑 → (abs “ (𝐹 “ ℝ)) ⊆ ℝ) |
13 | 0re 10721 | . . . . . 6 ⊢ 0 ∈ ℝ | |
14 | 13 | ne0ii 4226 | . . . . 5 ⊢ ℝ ≠ ∅ |
15 | 14 | a1i 11 | . . . 4 ⊢ (𝜑 → ℝ ≠ ∅) |
16 | 15, 9 | wnefimgd 41318 | . . 3 ⊢ (𝜑 → ((abs ∘ 𝐹) “ ℝ) ≠ ∅) |
17 | 1, 16 | eqnetrrid 3009 | . 2 ⊢ (𝜑 → (abs “ (𝐹 “ ℝ)) ≠ ∅) |
18 | 1red 10720 | . . 3 ⊢ (𝜑 → 1 ∈ ℝ) | |
19 | simpr 488 | . . . . 5 ⊢ ((𝜑 ∧ 𝑐 = 1) → 𝑐 = 1) | |
20 | 19 | breq2d 5042 | . . . 4 ⊢ ((𝜑 ∧ 𝑐 = 1) → (𝑡 ≤ 𝑐 ↔ 𝑡 ≤ 1)) |
21 | 20 | ralbidv 3109 | . . 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 41321 | . . 3 ⊢ (𝜑 → ∀𝑡 ∈ (abs “ (𝐹 “ ℝ))𝑡 ≤ 1) |
24 | 18, 21, 23 | rspcedvd 3529 | . 2 ⊢ (𝜑 → ∃𝑐 ∈ ℝ ∀𝑡 ∈ (abs “ (𝐹 “ ℝ))𝑡 ≤ 𝑐) |
25 | 0red 10722 | . 2 ⊢ (𝜑 → 0 ∈ ℝ) | |
26 | imo72b2lem1.7 | . . 3 ⊢ (𝜑 → ∃𝑥 ∈ ℝ (𝐹‘𝑥) ≠ 0) | |
27 | 3 | adantr 484 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐹‘𝑥) ≠ 0)) → 𝐹:ℝ⟶ℝ) |
28 | simprl 771 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐹‘𝑥) ≠ 0)) → 𝑥 ∈ ℝ) | |
29 | 27, 28 | fvco3d 6768 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐹‘𝑥) ≠ 0)) → ((abs ∘ 𝐹)‘𝑥) = (abs‘(𝐹‘𝑥))) |
30 | 9 | funfvima2d 7005 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ((abs ∘ 𝐹)‘𝑥) ∈ ((abs ∘ 𝐹) “ ℝ)) |
31 | 30 | adantrr 717 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐹‘𝑥) ≠ 0)) → ((abs ∘ 𝐹)‘𝑥) ∈ ((abs ∘ 𝐹) “ ℝ)) |
32 | 31, 1 | eleqtrdi 2843 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐹‘𝑥) ≠ 0)) → ((abs ∘ 𝐹)‘𝑥) ∈ (abs “ (𝐹 “ ℝ))) |
33 | 29, 32 | eqeltrrd 2834 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐹‘𝑥) ≠ 0)) → (abs‘(𝐹‘𝑥)) ∈ (abs “ (𝐹 “ ℝ))) |
34 | simpr 488 | . . . . 5 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐹‘𝑥) ≠ 0)) ∧ 𝑧 = (abs‘(𝐹‘𝑥))) → 𝑧 = (abs‘(𝐹‘𝑥))) | |
35 | 34 | breq2d 5042 | . . . 4 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐹‘𝑥) ≠ 0)) ∧ 𝑧 = (abs‘(𝐹‘𝑥))) → (0 < 𝑧 ↔ 0 < (abs‘(𝐹‘𝑥)))) |
36 | 3 | ffvelrnda 6861 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐹‘𝑥) ∈ ℝ) |
37 | 36 | adantrr 717 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐹‘𝑥) ≠ 0)) → (𝐹‘𝑥) ∈ ℝ) |
38 | 37 | recnd 10747 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐹‘𝑥) ≠ 0)) → (𝐹‘𝑥) ∈ ℂ) |
39 | simprr 773 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐹‘𝑥) ≠ 0)) → (𝐹‘𝑥) ≠ 0) | |
40 | 38, 39 | absrpcld 14898 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐹‘𝑥) ≠ 0)) → (abs‘(𝐹‘𝑥)) ∈ ℝ+) |
41 | 40 | rpgt0d 12517 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐹‘𝑥) ≠ 0)) → 0 < (abs‘(𝐹‘𝑥))) |
42 | 33, 35, 41 | rspcedvd 3529 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐹‘𝑥) ≠ 0)) → ∃𝑧 ∈ (abs “ (𝐹 “ ℝ))0 < 𝑧) |
43 | 26, 42 | rexlimddv 3201 | . 2 ⊢ (𝜑 → ∃𝑧 ∈ (abs “ (𝐹 “ ℝ))0 < 𝑧) |
44 | 12, 17, 24, 25, 43 | suprlubrd 41325 | 1 ⊢ (𝜑 → 0 < sup((abs “ (𝐹 “ ℝ)), ℝ, < )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1542 ∈ wcel 2114 ≠ wne 2934 ∀wral 3053 ∃wrex 3054 ⊆ wss 3843 ∅c0 4211 class class class wbr 5030 ran crn 5526 “ cima 5528 ∘ ccom 5529 ⟶wf 6335 ‘cfv 6339 supcsup 8977 ℂcc 10613 ℝcr 10614 0cc0 10615 1c1 10616 < clt 10753 ≤ cle 10754 abscabs 14683 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 ax-cnex 10671 ax-resscn 10672 ax-1cn 10673 ax-icn 10674 ax-addcl 10675 ax-addrcl 10676 ax-mulcl 10677 ax-mulrcl 10678 ax-mulcom 10679 ax-addass 10680 ax-mulass 10681 ax-distr 10682 ax-i2m1 10683 ax-1ne0 10684 ax-1rid 10685 ax-rnegex 10686 ax-rrecex 10687 ax-cnre 10688 ax-pre-lttri 10689 ax-pre-lttrn 10690 ax-pre-ltadd 10691 ax-pre-mulgt0 10692 ax-pre-sup 10693 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-uni 4797 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-riota 7127 df-ov 7173 df-oprab 7174 df-mpo 7175 df-om 7600 df-2nd 7715 df-wrecs 7976 df-recs 8037 df-rdg 8075 df-er 8320 df-en 8556 df-dom 8557 df-sdom 8558 df-sup 8979 df-pnf 10755 df-mnf 10756 df-xr 10757 df-ltxr 10758 df-le 10759 df-sub 10950 df-neg 10951 df-div 11376 df-nn 11717 df-2 11779 df-3 11780 df-n0 11977 df-z 12063 df-uz 12325 df-rp 12473 df-seq 13461 df-exp 13522 df-cj 14548 df-re 14549 df-im 14550 df-sqrt 14684 df-abs 14685 |
This theorem is referenced by: imo72b2 41330 |
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