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Mirrors > Home > MPE Home > Th. List > xrre3 | Structured version Visualization version GIF version |
Description: A way of proving that an extended real is real. (Contributed by FL, 29-May-2014.) |
Ref | Expression |
---|---|
xrre3 | ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ) ∧ (𝐵 ≤ 𝐴 ∧ 𝐴 < +∞)) → 𝐴 ∈ ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mnflt 13051 | . . . . . 6 ⊢ (𝐵 ∈ ℝ → -∞ < 𝐵) | |
2 | 1 | adantl 483 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ) → -∞ < 𝐵) |
3 | mnfxr 11219 | . . . . . 6 ⊢ -∞ ∈ ℝ* | |
4 | rexr 11208 | . . . . . . 7 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℝ*) | |
5 | 4 | adantl 483 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ) → 𝐵 ∈ ℝ*) |
6 | simpl 484 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ) → 𝐴 ∈ ℝ*) | |
7 | xrltletr 13083 | . . . . . 6 ⊢ ((-∞ ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ∈ ℝ*) → ((-∞ < 𝐵 ∧ 𝐵 ≤ 𝐴) → -∞ < 𝐴)) | |
8 | 3, 5, 6, 7 | mp3an2i 1467 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ) → ((-∞ < 𝐵 ∧ 𝐵 ≤ 𝐴) → -∞ < 𝐴)) |
9 | 2, 8 | mpand 694 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ) → (𝐵 ≤ 𝐴 → -∞ < 𝐴)) |
10 | 9 | imp 408 | . . 3 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ) ∧ 𝐵 ≤ 𝐴) → -∞ < 𝐴) |
11 | 10 | adantrr 716 | . 2 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ) ∧ (𝐵 ≤ 𝐴 ∧ 𝐴 < +∞)) → -∞ < 𝐴) |
12 | simprr 772 | . 2 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ) ∧ (𝐵 ≤ 𝐴 ∧ 𝐴 < +∞)) → 𝐴 < +∞) | |
13 | xrrebnd 13094 | . . 3 ⊢ (𝐴 ∈ ℝ* → (𝐴 ∈ ℝ ↔ (-∞ < 𝐴 ∧ 𝐴 < +∞))) | |
14 | 13 | ad2antrr 725 | . 2 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ) ∧ (𝐵 ≤ 𝐴 ∧ 𝐴 < +∞)) → (𝐴 ∈ ℝ ↔ (-∞ < 𝐴 ∧ 𝐴 < +∞))) |
15 | 11, 12, 14 | mpbir2and 712 | 1 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ) ∧ (𝐵 ≤ 𝐴 ∧ 𝐴 < +∞)) → 𝐴 ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∈ wcel 2107 class class class wbr 5110 ℝcr 11057 +∞cpnf 11193 -∞cmnf 11194 ℝ*cxr 11195 < clt 11196 ≤ cle 11197 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-cnex 11114 ax-resscn 11115 ax-pre-lttri 11132 ax-pre-lttrn 11133 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-br 5111 df-opab 5173 df-mpt 5194 df-id 5536 df-po 5550 df-so 5551 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-er 8655 df-en 8891 df-dom 8892 df-sdom 8893 df-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 df-le 11202 |
This theorem is referenced by: elicore 13323 sibfinima 32979 orvcgteel 33107 ismblfin 36148 |
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