Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > infxrunb3rnmpt | Structured version Visualization version GIF version |
Description: The infimum of an unbounded-below set of extended reals is minus infinity. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
Ref | Expression |
---|---|
infxrunb3rnmpt.1 | ⊢ Ⅎ𝑥𝜑 |
infxrunb3rnmpt.2 | ⊢ Ⅎ𝑦𝜑 |
infxrunb3rnmpt.3 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ*) |
Ref | Expression |
---|---|
infxrunb3rnmpt | ⊢ (𝜑 → (∀𝑦 ∈ ℝ ∃𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 ↔ inf(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ*, < ) = -∞)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | infxrunb3rnmpt.2 | . . 3 ⊢ Ⅎ𝑦𝜑 | |
2 | infxrunb3rnmpt.1 | . . . . 5 ⊢ Ⅎ𝑥𝜑 | |
3 | nfmpt1 5128 | . . . . . . 7 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐵) | |
4 | 3 | nfrn 5788 | . . . . . 6 ⊢ Ⅎ𝑥ran (𝑥 ∈ 𝐴 ↦ 𝐵) |
5 | nfv 1915 | . . . . . 6 ⊢ Ⅎ𝑥 𝑧 ≤ 𝑦 | |
6 | 4, 5 | nfrex 3268 | . . . . 5 ⊢ Ⅎ𝑥∃𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝑦 |
7 | simpr 488 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴) | |
8 | infxrunb3rnmpt.3 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ*) | |
9 | eqid 2798 | . . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
10 | 9 | elrnmpt1 5794 | . . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ ℝ*) → 𝐵 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)) |
11 | 7, 8, 10 | syl2anc 587 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)) |
12 | 11 | 3adant3 1129 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝐵 ≤ 𝑦) → 𝐵 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)) |
13 | simp3 1135 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝐵 ≤ 𝑦) → 𝐵 ≤ 𝑦) | |
14 | breq1 5033 | . . . . . . . 8 ⊢ (𝑧 = 𝐵 → (𝑧 ≤ 𝑦 ↔ 𝐵 ≤ 𝑦)) | |
15 | 14 | rspcev 3571 | . . . . . . 7 ⊢ ((𝐵 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ∧ 𝐵 ≤ 𝑦) → ∃𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝑦) |
16 | 12, 13, 15 | syl2anc 587 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝐵 ≤ 𝑦) → ∃𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝑦) |
17 | 16 | 3exp 1116 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → (𝐵 ≤ 𝑦 → ∃𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝑦))) |
18 | 2, 6, 17 | rexlimd 3276 | . . . 4 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 → ∃𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝑦)) |
19 | nfv 1915 | . . . . . 6 ⊢ Ⅎ𝑧∃𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 | |
20 | vex 3444 | . . . . . . . . 9 ⊢ 𝑧 ∈ V | |
21 | 9 | elrnmpt 5792 | . . . . . . . . 9 ⊢ (𝑧 ∈ V → (𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ↔ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵)) |
22 | 20, 21 | ax-mp 5 | . . . . . . . 8 ⊢ (𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ↔ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵) |
23 | 22 | biimpi 219 | . . . . . . 7 ⊢ (𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) → ∃𝑥 ∈ 𝐴 𝑧 = 𝐵) |
24 | 14 | biimpcd 252 | . . . . . . . . . 10 ⊢ (𝑧 ≤ 𝑦 → (𝑧 = 𝐵 → 𝐵 ≤ 𝑦)) |
25 | 24 | a1d 25 | . . . . . . . . 9 ⊢ (𝑧 ≤ 𝑦 → (𝑥 ∈ 𝐴 → (𝑧 = 𝐵 → 𝐵 ≤ 𝑦))) |
26 | 5, 25 | reximdai 3270 | . . . . . . . 8 ⊢ (𝑧 ≤ 𝑦 → (∃𝑥 ∈ 𝐴 𝑧 = 𝐵 → ∃𝑥 ∈ 𝐴 𝐵 ≤ 𝑦)) |
27 | 26 | com12 32 | . . . . . . 7 ⊢ (∃𝑥 ∈ 𝐴 𝑧 = 𝐵 → (𝑧 ≤ 𝑦 → ∃𝑥 ∈ 𝐴 𝐵 ≤ 𝑦)) |
28 | 23, 27 | syl 17 | . . . . . 6 ⊢ (𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) → (𝑧 ≤ 𝑦 → ∃𝑥 ∈ 𝐴 𝐵 ≤ 𝑦)) |
29 | 19, 28 | rexlimi 3274 | . . . . 5 ⊢ (∃𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝑦 → ∃𝑥 ∈ 𝐴 𝐵 ≤ 𝑦) |
30 | 29 | a1i 11 | . . . 4 ⊢ (𝜑 → (∃𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝑦 → ∃𝑥 ∈ 𝐴 𝐵 ≤ 𝑦)) |
31 | 18, 30 | impbid 215 | . . 3 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 ↔ ∃𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝑦)) |
32 | 1, 31 | ralbid 3195 | . 2 ⊢ (𝜑 → (∀𝑦 ∈ ℝ ∃𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 ↔ ∀𝑦 ∈ ℝ ∃𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝑦)) |
33 | 2, 9, 8 | rnmptssd 41824 | . . 3 ⊢ (𝜑 → ran (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ ℝ*) |
34 | infxrunb3 42061 | . . 3 ⊢ (ran (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ ℝ* → (∀𝑦 ∈ ℝ ∃𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝑦 ↔ inf(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ*, < ) = -∞)) | |
35 | 33, 34 | syl 17 | . 2 ⊢ (𝜑 → (∀𝑦 ∈ ℝ ∃𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝑦 ↔ inf(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ*, < ) = -∞)) |
36 | 32, 35 | bitrd 282 | 1 ⊢ (𝜑 → (∀𝑦 ∈ ℝ ∃𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 ↔ inf(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ*, < ) = -∞)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1084 = wceq 1538 Ⅎwnf 1785 ∈ wcel 2111 ∀wral 3106 ∃wrex 3107 Vcvv 3441 ⊆ wss 3881 class class class wbr 5030 ↦ cmpt 5110 ran crn 5520 infcinf 8889 ℝcr 10525 -∞cmnf 10662 ℝ*cxr 10663 < clt 10664 ≤ cle 10665 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-po 5438 df-so 5439 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-sup 8890 df-inf 8891 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 |
This theorem is referenced by: limsupmnflem 42362 |
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