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Mirrors > Home > MPE Home > Th. List > Mathboxes > imo72b2lem1 | Structured version Visualization version GIF version |
Description: Lemma for imo72b2 43239. (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 6250 | . . 3 ⊢ ((abs ∘ 𝐹) “ ℝ) = (abs “ (𝐹 “ ℝ)) | |
2 | imassrn 6070 | . . . 4 ⊢ ((abs ∘ 𝐹) “ ℝ) ⊆ ran (abs ∘ 𝐹) | |
3 | imo72b2lem1.1 | . . . . . 6 ⊢ (𝜑 → 𝐹:ℝ⟶ℝ) | |
4 | absf 15291 | . . . . . . . 8 ⊢ abs:ℂ⟶ℝ | |
5 | 4 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → abs:ℂ⟶ℝ) |
6 | ax-resscn 11173 | . . . . . . . 8 ⊢ ℝ ⊆ ℂ | |
7 | 6 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → ℝ ⊆ ℂ) |
8 | 5, 7 | fssresd 6758 | . . . . . 6 ⊢ (𝜑 → (abs ↾ ℝ):ℝ⟶ℝ) |
9 | 3, 8 | fco2d 43229 | . . . . 5 ⊢ (𝜑 → (abs ∘ 𝐹):ℝ⟶ℝ) |
10 | 9 | frnd 6725 | . . . 4 ⊢ (𝜑 → ran (abs ∘ 𝐹) ⊆ ℝ) |
11 | 2, 10 | sstrid 3993 | . . 3 ⊢ (𝜑 → ((abs ∘ 𝐹) “ ℝ) ⊆ ℝ) |
12 | 1, 11 | eqsstrrid 4031 | . 2 ⊢ (𝜑 → (abs “ (𝐹 “ ℝ)) ⊆ ℝ) |
13 | 0re 11223 | . . . . . 6 ⊢ 0 ∈ ℝ | |
14 | 13 | ne0ii 4337 | . . . . 5 ⊢ ℝ ≠ ∅ |
15 | 14 | a1i 11 | . . . 4 ⊢ (𝜑 → ℝ ≠ ∅) |
16 | 15, 9 | wnefimgd 43228 | . . 3 ⊢ (𝜑 → ((abs ∘ 𝐹) “ ℝ) ≠ ∅) |
17 | 1, 16 | eqnetrrid 3015 | . 2 ⊢ (𝜑 → (abs “ (𝐹 “ ℝ)) ≠ ∅) |
18 | 1red 11222 | . . 3 ⊢ (𝜑 → 1 ∈ ℝ) | |
19 | simpr 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝑐 = 1) → 𝑐 = 1) | |
20 | 19 | breq2d 5160 | . . . 4 ⊢ ((𝜑 ∧ 𝑐 = 1) → (𝑡 ≤ 𝑐 ↔ 𝑡 ≤ 1)) |
21 | 20 | ralbidv 3176 | . . 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 43231 | . . 3 ⊢ (𝜑 → ∀𝑡 ∈ (abs “ (𝐹 “ ℝ))𝑡 ≤ 1) |
24 | 18, 21, 23 | rspcedvd 3614 | . 2 ⊢ (𝜑 → ∃𝑐 ∈ ℝ ∀𝑡 ∈ (abs “ (𝐹 “ ℝ))𝑡 ≤ 𝑐) |
25 | 0red 11224 | . 2 ⊢ (𝜑 → 0 ∈ ℝ) | |
26 | imo72b2lem1.7 | . . 3 ⊢ (𝜑 → ∃𝑥 ∈ ℝ (𝐹‘𝑥) ≠ 0) | |
27 | 3 | adantr 480 | . . . . . 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 6991 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐹‘𝑥) ≠ 0)) → ((abs ∘ 𝐹)‘𝑥) = (abs‘(𝐹‘𝑥))) |
30 | 9 | funfvima2d 7236 | . . . . . . 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 2842 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐹‘𝑥) ≠ 0)) → ((abs ∘ 𝐹)‘𝑥) ∈ (abs “ (𝐹 “ ℝ))) |
33 | 29, 32 | eqeltrrd 2833 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐹‘𝑥) ≠ 0)) → (abs‘(𝐹‘𝑥)) ∈ (abs “ (𝐹 “ ℝ))) |
34 | simpr 484 | . . . . 5 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐹‘𝑥) ≠ 0)) ∧ 𝑧 = (abs‘(𝐹‘𝑥))) → 𝑧 = (abs‘(𝐹‘𝑥))) | |
35 | 34 | breq2d 5160 | . . . 4 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐹‘𝑥) ≠ 0)) ∧ 𝑧 = (abs‘(𝐹‘𝑥))) → (0 < 𝑧 ↔ 0 < (abs‘(𝐹‘𝑥)))) |
36 | 3 | ffvelcdmda 7086 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐹‘𝑥) ∈ ℝ) |
37 | 36 | adantrr 714 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐹‘𝑥) ≠ 0)) → (𝐹‘𝑥) ∈ ℝ) |
38 | 37 | recnd 11249 | . . . . . 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 15402 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐹‘𝑥) ≠ 0)) → (abs‘(𝐹‘𝑥)) ∈ ℝ+) |
41 | 40 | rpgt0d 13026 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐹‘𝑥) ≠ 0)) → 0 < (abs‘(𝐹‘𝑥))) |
42 | 33, 35, 41 | rspcedvd 3614 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐹‘𝑥) ≠ 0)) → ∃𝑧 ∈ (abs “ (𝐹 “ ℝ))0 < 𝑧) |
43 | 26, 42 | rexlimddv 3160 | . 2 ⊢ (𝜑 → ∃𝑧 ∈ (abs “ (𝐹 “ ℝ))0 < 𝑧) |
44 | 12, 17, 24, 25, 43 | suprlubrd 43235 | 1 ⊢ (𝜑 → 0 < sup((abs “ (𝐹 “ ℝ)), ℝ, < )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2105 ≠ wne 2939 ∀wral 3060 ∃wrex 3069 ⊆ wss 3948 ∅c0 4322 class class class wbr 5148 ran crn 5677 “ cima 5679 ∘ ccom 5680 ⟶wf 6539 ‘cfv 6543 supcsup 9441 ℂcc 11114 ℝcr 11115 0cc0 11116 1c1 11117 < clt 11255 ≤ cle 11256 abscabs 15188 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 ax-pre-sup 11194 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-2nd 7980 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-er 8709 df-en 8946 df-dom 8947 df-sdom 8948 df-sup 9443 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-div 11879 df-nn 12220 df-2 12282 df-3 12283 df-n0 12480 df-z 12566 df-uz 12830 df-rp 12982 df-seq 13974 df-exp 14035 df-cj 15053 df-re 15054 df-im 15055 df-sqrt 15189 df-abs 15190 |
This theorem is referenced by: imo72b2 43239 |
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