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Mirrors > Home > MPE Home > Th. List > Mathboxes > imo72b2lem1 | Structured version Visualization version GIF version |
Description: Lemma for imo72b2 44162. (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 6273 | . . 3 ⊢ ((abs ∘ 𝐹) “ ℝ) = (abs “ (𝐹 “ ℝ)) | |
2 | imassrn 6091 | . . . 4 ⊢ ((abs ∘ 𝐹) “ ℝ) ⊆ ran (abs ∘ 𝐹) | |
3 | imo72b2lem1.1 | . . . . . 6 ⊢ (𝜑 → 𝐹:ℝ⟶ℝ) | |
4 | absf 15373 | . . . . . . . 8 ⊢ abs:ℂ⟶ℝ | |
5 | 4 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → abs:ℂ⟶ℝ) |
6 | ax-resscn 11210 | . . . . . . . 8 ⊢ ℝ ⊆ ℂ | |
7 | 6 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → ℝ ⊆ ℂ) |
8 | 5, 7 | fssresd 6776 | . . . . . 6 ⊢ (𝜑 → (abs ↾ ℝ):ℝ⟶ℝ) |
9 | 3, 8 | fco2d 44152 | . . . . 5 ⊢ (𝜑 → (abs ∘ 𝐹):ℝ⟶ℝ) |
10 | 9 | frnd 6745 | . . . 4 ⊢ (𝜑 → ran (abs ∘ 𝐹) ⊆ ℝ) |
11 | 2, 10 | sstrid 4007 | . . 3 ⊢ (𝜑 → ((abs ∘ 𝐹) “ ℝ) ⊆ ℝ) |
12 | 1, 11 | eqsstrrid 4045 | . 2 ⊢ (𝜑 → (abs “ (𝐹 “ ℝ)) ⊆ ℝ) |
13 | 0re 11261 | . . . . . 6 ⊢ 0 ∈ ℝ | |
14 | 13 | ne0ii 4350 | . . . . 5 ⊢ ℝ ≠ ∅ |
15 | 14 | a1i 11 | . . . 4 ⊢ (𝜑 → ℝ ≠ ∅) |
16 | 15, 9 | wnefimgd 44151 | . . 3 ⊢ (𝜑 → ((abs ∘ 𝐹) “ ℝ) ≠ ∅) |
17 | 1, 16 | eqnetrrid 3014 | . 2 ⊢ (𝜑 → (abs “ (𝐹 “ ℝ)) ≠ ∅) |
18 | 1red 11260 | . . 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 44154 | . . 3 ⊢ (𝜑 → ∀𝑡 ∈ (abs “ (𝐹 “ ℝ))𝑡 ≤ 1) |
24 | 18, 21, 23 | rspcedvd 3624 | . 2 ⊢ (𝜑 → ∃𝑐 ∈ ℝ ∀𝑡 ∈ (abs “ (𝐹 “ ℝ))𝑡 ≤ 𝑐) |
25 | 0red 11262 | . 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 771 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐹‘𝑥) ≠ 0)) → 𝑥 ∈ ℝ) | |
29 | 27, 28 | fvco3d 7009 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐹‘𝑥) ≠ 0)) → ((abs ∘ 𝐹)‘𝑥) = (abs‘(𝐹‘𝑥))) |
30 | 9 | funfvima2d 7252 | . . . . . . 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 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 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 7104 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐹‘𝑥) ∈ ℝ) |
37 | 36 | adantrr 717 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐹‘𝑥) ≠ 0)) → (𝐹‘𝑥) ∈ ℝ) |
38 | 37 | recnd 11287 | . . . . . 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 15484 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐹‘𝑥) ≠ 0)) → (abs‘(𝐹‘𝑥)) ∈ ℝ+) |
41 | 40 | rpgt0d 13078 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐹‘𝑥) ≠ 0)) → 0 < (abs‘(𝐹‘𝑥))) |
42 | 33, 35, 41 | rspcedvd 3624 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ (𝐹‘𝑥) ≠ 0)) → ∃𝑧 ∈ (abs “ (𝐹 “ ℝ))0 < 𝑧) |
43 | 26, 42 | rexlimddv 3159 | . 2 ⊢ (𝜑 → ∃𝑧 ∈ (abs “ (𝐹 “ ℝ))0 < 𝑧) |
44 | 12, 17, 24, 25, 43 | suprlubrd 44158 | 1 ⊢ (𝜑 → 0 < sup((abs “ (𝐹 “ ℝ)), ℝ, < )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 ∀wral 3059 ∃wrex 3068 ⊆ wss 3963 ∅c0 4339 class class class wbr 5148 ran crn 5690 “ cima 5692 ∘ ccom 5693 ⟶wf 6559 ‘cfv 6563 supcsup 9478 ℂcc 11151 ℝcr 11152 0cc0 11153 1c1 11154 < clt 11293 ≤ cle 11294 abscabs 15270 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-sup 9480 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-n0 12525 df-z 12612 df-uz 12877 df-rp 13033 df-seq 14040 df-exp 14100 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 |
This theorem is referenced by: imo72b2 44162 |
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