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Mirrors > Home > MPE Home > Th. List > Mathboxes > imo72b2lem1 | Structured version Visualization version GIF version |
Description: Lemma for imo72b2 41783. (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 6155 | . . 3 ⊢ ((abs ∘ 𝐹) “ ℝ) = (abs “ (𝐹 “ ℝ)) | |
2 | imassrn 5980 | . . . 4 ⊢ ((abs ∘ 𝐹) “ ℝ) ⊆ ran (abs ∘ 𝐹) | |
3 | imo72b2lem1.1 | . . . . . 6 ⊢ (𝜑 → 𝐹:ℝ⟶ℝ) | |
4 | absf 15049 | . . . . . . . 8 ⊢ abs:ℂ⟶ℝ | |
5 | 4 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → abs:ℂ⟶ℝ) |
6 | ax-resscn 10928 | . . . . . . . 8 ⊢ ℝ ⊆ ℂ | |
7 | 6 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → ℝ ⊆ ℂ) |
8 | 5, 7 | fssresd 6641 | . . . . . 6 ⊢ (𝜑 → (abs ↾ ℝ):ℝ⟶ℝ) |
9 | 3, 8 | fco2d 41773 | . . . . 5 ⊢ (𝜑 → (abs ∘ 𝐹):ℝ⟶ℝ) |
10 | 9 | frnd 6608 | . . . 4 ⊢ (𝜑 → ran (abs ∘ 𝐹) ⊆ ℝ) |
11 | 2, 10 | sstrid 3932 | . . 3 ⊢ (𝜑 → ((abs ∘ 𝐹) “ ℝ) ⊆ ℝ) |
12 | 1, 11 | eqsstrrid 3970 | . 2 ⊢ (𝜑 → (abs “ (𝐹 “ ℝ)) ⊆ ℝ) |
13 | 0re 10977 | . . . . . 6 ⊢ 0 ∈ ℝ | |
14 | 13 | ne0ii 4271 | . . . . 5 ⊢ ℝ ≠ ∅ |
15 | 14 | a1i 11 | . . . 4 ⊢ (𝜑 → ℝ ≠ ∅) |
16 | 15, 9 | wnefimgd 41772 | . . 3 ⊢ (𝜑 → ((abs ∘ 𝐹) “ ℝ) ≠ ∅) |
17 | 1, 16 | eqnetrrid 3019 | . 2 ⊢ (𝜑 → (abs “ (𝐹 “ ℝ)) ≠ ∅) |
18 | 1red 10976 | . . 3 ⊢ (𝜑 → 1 ∈ ℝ) | |
19 | simpr 485 | . . . . 5 ⊢ ((𝜑 ∧ 𝑐 = 1) → 𝑐 = 1) | |
20 | 19 | breq2d 5086 | . . . 4 ⊢ ((𝜑 ∧ 𝑐 = 1) → (𝑡 ≤ 𝑐 ↔ 𝑡 ≤ 1)) |
21 | 20 | ralbidv 3112 | . . 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 41775 | . . 3 ⊢ (𝜑 → ∀𝑡 ∈ (abs “ (𝐹 “ ℝ))𝑡 ≤ 1) |
24 | 18, 21, 23 | rspcedvd 3563 | . 2 ⊢ (𝜑 → ∃𝑐 ∈ ℝ ∀𝑡 ∈ (abs “ (𝐹 “ ℝ))𝑡 ≤ 𝑐) |
25 | 0red 10978 | . 2 ⊢ (𝜑 → 0 ∈ ℝ) | |
26 | imo72b2lem1.7 | . . 3 ⊢ (𝜑 → ∃𝑥 ∈ ℝ (𝐹‘𝑥) ≠ 0) | |
27 | 3 | adantr 481 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐹‘𝑥) ≠ 0)) → 𝐹:ℝ⟶ℝ) |
28 | simprl 768 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐹‘𝑥) ≠ 0)) → 𝑥 ∈ ℝ) | |
29 | 27, 28 | fvco3d 6868 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐹‘𝑥) ≠ 0)) → ((abs ∘ 𝐹)‘𝑥) = (abs‘(𝐹‘𝑥))) |
30 | 9 | funfvima2d 7108 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ((abs ∘ 𝐹)‘𝑥) ∈ ((abs ∘ 𝐹) “ ℝ)) |
31 | 30 | adantrr 714 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐹‘𝑥) ≠ 0)) → ((abs ∘ 𝐹)‘𝑥) ∈ ((abs ∘ 𝐹) “ ℝ)) |
32 | 31, 1 | eleqtrdi 2849 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐹‘𝑥) ≠ 0)) → ((abs ∘ 𝐹)‘𝑥) ∈ (abs “ (𝐹 “ ℝ))) |
33 | 29, 32 | eqeltrrd 2840 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐹‘𝑥) ≠ 0)) → (abs‘(𝐹‘𝑥)) ∈ (abs “ (𝐹 “ ℝ))) |
34 | simpr 485 | . . . . 5 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐹‘𝑥) ≠ 0)) ∧ 𝑧 = (abs‘(𝐹‘𝑥))) → 𝑧 = (abs‘(𝐹‘𝑥))) | |
35 | 34 | breq2d 5086 | . . . 4 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐹‘𝑥) ≠ 0)) ∧ 𝑧 = (abs‘(𝐹‘𝑥))) → (0 < 𝑧 ↔ 0 < (abs‘(𝐹‘𝑥)))) |
36 | 3 | ffvelrnda 6961 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐹‘𝑥) ∈ ℝ) |
37 | 36 | adantrr 714 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐹‘𝑥) ≠ 0)) → (𝐹‘𝑥) ∈ ℝ) |
38 | 37 | recnd 11003 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐹‘𝑥) ≠ 0)) → (𝐹‘𝑥) ∈ ℂ) |
39 | simprr 770 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐹‘𝑥) ≠ 0)) → (𝐹‘𝑥) ≠ 0) | |
40 | 38, 39 | absrpcld 15160 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐹‘𝑥) ≠ 0)) → (abs‘(𝐹‘𝑥)) ∈ ℝ+) |
41 | 40 | rpgt0d 12775 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐹‘𝑥) ≠ 0)) → 0 < (abs‘(𝐹‘𝑥))) |
42 | 33, 35, 41 | rspcedvd 3563 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐹‘𝑥) ≠ 0)) → ∃𝑧 ∈ (abs “ (𝐹 “ ℝ))0 < 𝑧) |
43 | 26, 42 | rexlimddv 3220 | . 2 ⊢ (𝜑 → ∃𝑧 ∈ (abs “ (𝐹 “ ℝ))0 < 𝑧) |
44 | 12, 17, 24, 25, 43 | suprlubrd 41779 | 1 ⊢ (𝜑 → 0 < sup((abs “ (𝐹 “ ℝ)), ℝ, < )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ≠ wne 2943 ∀wral 3064 ∃wrex 3065 ⊆ wss 3887 ∅c0 4256 class class class wbr 5074 ran crn 5590 “ cima 5592 ∘ ccom 5593 ⟶wf 6429 ‘cfv 6433 supcsup 9199 ℂcc 10869 ℝcr 10870 0cc0 10871 1c1 10872 < clt 11009 ≤ cle 11010 abscabs 14945 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-er 8498 df-en 8734 df-dom 8735 df-sdom 8736 df-sup 9201 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-n0 12234 df-z 12320 df-uz 12583 df-rp 12731 df-seq 13722 df-exp 13783 df-cj 14810 df-re 14811 df-im 14812 df-sqrt 14946 df-abs 14947 |
This theorem is referenced by: imo72b2 41783 |
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