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Mirrors > Home > ILE Home > Th. List > xnegcl | GIF version |
Description: Closure of extended real negative. (Contributed by Mario Carneiro, 20-Aug-2015.) |
Ref | Expression |
---|---|
xnegcl | ⊢ (𝐴 ∈ ℝ* → -𝑒𝐴 ∈ ℝ*) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elxr 8696 | . 2 ⊢ (𝐴 ∈ ℝ* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞)) | |
2 | rexneg 8743 | . . . . 5 ⊢ (𝐴 ∈ ℝ → -𝑒𝐴 = -𝐴) | |
3 | renegcl 7272 | . . . . 5 ⊢ (𝐴 ∈ ℝ → -𝐴 ∈ ℝ) | |
4 | 2, 3 | eqeltrd 2114 | . . . 4 ⊢ (𝐴 ∈ ℝ → -𝑒𝐴 ∈ ℝ) |
5 | 4 | rexrd 7075 | . . 3 ⊢ (𝐴 ∈ ℝ → -𝑒𝐴 ∈ ℝ*) |
6 | xnegeq 8740 | . . . 4 ⊢ (𝐴 = +∞ → -𝑒𝐴 = -𝑒+∞) | |
7 | xnegpnf 8741 | . . . . 5 ⊢ -𝑒+∞ = -∞ | |
8 | mnfxr 8694 | . . . . 5 ⊢ -∞ ∈ ℝ* | |
9 | 7, 8 | eqeltri 2110 | . . . 4 ⊢ -𝑒+∞ ∈ ℝ* |
10 | 6, 9 | syl6eqel 2128 | . . 3 ⊢ (𝐴 = +∞ → -𝑒𝐴 ∈ ℝ*) |
11 | xnegeq 8740 | . . . 4 ⊢ (𝐴 = -∞ → -𝑒𝐴 = -𝑒-∞) | |
12 | xnegmnf 8742 | . . . . 5 ⊢ -𝑒-∞ = +∞ | |
13 | pnfxr 8692 | . . . . 5 ⊢ +∞ ∈ ℝ* | |
14 | 12, 13 | eqeltri 2110 | . . . 4 ⊢ -𝑒-∞ ∈ ℝ* |
15 | 11, 14 | syl6eqel 2128 | . . 3 ⊢ (𝐴 = -∞ → -𝑒𝐴 ∈ ℝ*) |
16 | 5, 10, 15 | 3jaoi 1198 | . 2 ⊢ ((𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞) → -𝑒𝐴 ∈ ℝ*) |
17 | 1, 16 | sylbi 114 | 1 ⊢ (𝐴 ∈ ℝ* → -𝑒𝐴 ∈ ℝ*) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∨ w3o 884 = wceq 1243 ∈ wcel 1393 ℝcr 6888 +∞cpnf 7057 -∞cmnf 7058 ℝ*cxr 7059 -cneg 7183 -𝑒cxne 8686 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-setind 4262 ax-cnex 6975 ax-resscn 6976 ax-1cn 6977 ax-icn 6979 ax-addcl 6980 ax-addrcl 6981 ax-mulcl 6982 ax-addcom 6984 ax-addass 6986 ax-distr 6988 ax-i2m1 6989 ax-0id 6992 ax-rnegex 6993 ax-cnre 6995 |
This theorem depends on definitions: df-bi 110 df-3or 886 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-nel 2207 df-ral 2311 df-rex 2312 df-reu 2313 df-rab 2315 df-v 2559 df-sbc 2765 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-if 3332 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-br 3765 df-opab 3819 df-id 4030 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-iota 4867 df-fun 4904 df-fv 4910 df-riota 5468 df-ov 5515 df-oprab 5516 df-mpt2 5517 df-pnf 7062 df-mnf 7063 df-xr 7064 df-sub 7184 df-neg 7185 df-xneg 8689 |
This theorem is referenced by: xltneg 8749 xleneg 8750 xnegcld 8755 |
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