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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > upbdrech2 | Structured version Visualization version GIF version |
Description: Choice of an upper bound for a possibly empty bunded set (image set version). (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Ref | Expression |
---|---|
upbdrech2.b | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) |
upbdrech2.bd | ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦) |
upbdrech2.c | ⊢ 𝐶 = if(𝐴 = ∅, 0, sup({𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵}, ℝ, < )) |
Ref | Expression |
---|---|
upbdrech2 | ⊢ (𝜑 → (𝐶 ∈ ℝ ∧ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | upbdrech2.c | . . 3 ⊢ 𝐶 = if(𝐴 = ∅, 0, sup({𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵}, ℝ, < )) | |
2 | iftrue 4490 | . . . . . 6 ⊢ (𝐴 = ∅ → if(𝐴 = ∅, 0, sup({𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵}, ℝ, < )) = 0) | |
3 | 0red 11116 | . . . . . 6 ⊢ (𝐴 = ∅ → 0 ∈ ℝ) | |
4 | 2, 3 | eqeltrd 2838 | . . . . 5 ⊢ (𝐴 = ∅ → if(𝐴 = ∅, 0, sup({𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵}, ℝ, < )) ∈ ℝ) |
5 | 4 | adantl 482 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 = ∅) → if(𝐴 = ∅, 0, sup({𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵}, ℝ, < )) ∈ ℝ) |
6 | simpr 485 | . . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐴 = ∅) → ¬ 𝐴 = ∅) | |
7 | 6 | iffalsed 4495 | . . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐴 = ∅) → if(𝐴 = ∅, 0, sup({𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵}, ℝ, < )) = sup({𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵}, ℝ, < )) |
8 | 6 | neqned 2948 | . . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝐴 = ∅) → 𝐴 ≠ ∅) |
9 | upbdrech2.b | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) | |
10 | 9 | adantlr 713 | . . . . . . 7 ⊢ (((𝜑 ∧ ¬ 𝐴 = ∅) ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) |
11 | upbdrech2.bd | . . . . . . . 8 ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦) | |
12 | 11 | adantr 481 | . . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝐴 = ∅) → ∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦) |
13 | eqid 2737 | . . . . . . 7 ⊢ sup({𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵}, ℝ, < ) = sup({𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵}, ℝ, < ) | |
14 | 8, 10, 12, 13 | upbdrech 43437 | . . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐴 = ∅) → (sup({𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵}, ℝ, < ) ∈ ℝ ∧ ∀𝑥 ∈ 𝐴 𝐵 ≤ sup({𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵}, ℝ, < ))) |
15 | 14 | simpld 495 | . . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐴 = ∅) → sup({𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵}, ℝ, < ) ∈ ℝ) |
16 | 7, 15 | eqeltrd 2838 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝐴 = ∅) → if(𝐴 = ∅, 0, sup({𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵}, ℝ, < )) ∈ ℝ) |
17 | 5, 16 | pm2.61dan 811 | . . 3 ⊢ (𝜑 → if(𝐴 = ∅, 0, sup({𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵}, ℝ, < )) ∈ ℝ) |
18 | 1, 17 | eqeltrid 2842 | . 2 ⊢ (𝜑 → 𝐶 ∈ ℝ) |
19 | rzal 4464 | . . . 4 ⊢ (𝐴 = ∅ → ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝐶) | |
20 | 19 | adantl 482 | . . 3 ⊢ ((𝜑 ∧ 𝐴 = ∅) → ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝐶) |
21 | 14 | simprd 496 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝐴 = ∅) → ∀𝑥 ∈ 𝐴 𝐵 ≤ sup({𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵}, ℝ, < )) |
22 | iffalse 4493 | . . . . . . . 8 ⊢ (¬ 𝐴 = ∅ → if(𝐴 = ∅, 0, sup({𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵}, ℝ, < )) = sup({𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵}, ℝ, < )) | |
23 | 1, 22 | eqtrid 2789 | . . . . . . 7 ⊢ (¬ 𝐴 = ∅ → 𝐶 = sup({𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵}, ℝ, < )) |
24 | 23 | breq2d 5115 | . . . . . 6 ⊢ (¬ 𝐴 = ∅ → (𝐵 ≤ 𝐶 ↔ 𝐵 ≤ sup({𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵}, ℝ, < ))) |
25 | 24 | ralbidv 3172 | . . . . 5 ⊢ (¬ 𝐴 = ∅ → (∀𝑥 ∈ 𝐴 𝐵 ≤ 𝐶 ↔ ∀𝑥 ∈ 𝐴 𝐵 ≤ sup({𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵}, ℝ, < ))) |
26 | 25 | adantl 482 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝐴 = ∅) → (∀𝑥 ∈ 𝐴 𝐵 ≤ 𝐶 ↔ ∀𝑥 ∈ 𝐴 𝐵 ≤ sup({𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵}, ℝ, < ))) |
27 | 21, 26 | mpbird 256 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝐴 = ∅) → ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝐶) |
28 | 20, 27 | pm2.61dan 811 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝐶) |
29 | 18, 28 | jca 512 | 1 ⊢ (𝜑 → (𝐶 ∈ ℝ ∧ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 {cab 2714 ∀wral 3062 ∃wrex 3071 ∅c0 4280 ifcif 4484 class class class wbr 5103 supcsup 9334 ℝcr 11008 0cc0 11009 < clt 11147 ≤ cle 11148 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 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 2708 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 ax-pre-sup 11087 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-br 5104 df-opab 5166 df-mpt 5187 df-id 5529 df-po 5543 df-so 5544 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7307 df-ov 7354 df-oprab 7355 df-mpo 7356 df-er 8606 df-en 8842 df-dom 8843 df-sdom 8844 df-sup 9336 df-pnf 11149 df-mnf 11150 df-xr 11151 df-ltxr 11152 df-le 11153 df-sub 11345 df-neg 11346 |
This theorem is referenced by: ssfiunibd 43441 |
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