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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 9794 | . 2 ⊢ (𝐴 ∈ ℝ* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞)) | |
2 | rexneg 9848 | . . . . 5 ⊢ (𝐴 ∈ ℝ → -𝑒𝐴 = -𝐴) | |
3 | renegcl 8236 | . . . . 5 ⊢ (𝐴 ∈ ℝ → -𝐴 ∈ ℝ) | |
4 | 2, 3 | eqeltrd 2266 | . . . 4 ⊢ (𝐴 ∈ ℝ → -𝑒𝐴 ∈ ℝ) |
5 | 4 | rexrd 8025 | . . 3 ⊢ (𝐴 ∈ ℝ → -𝑒𝐴 ∈ ℝ*) |
6 | xnegeq 9845 | . . . 4 ⊢ (𝐴 = +∞ → -𝑒𝐴 = -𝑒+∞) | |
7 | xnegpnf 9846 | . . . . 5 ⊢ -𝑒+∞ = -∞ | |
8 | mnfxr 8032 | . . . . 5 ⊢ -∞ ∈ ℝ* | |
9 | 7, 8 | eqeltri 2262 | . . . 4 ⊢ -𝑒+∞ ∈ ℝ* |
10 | 6, 9 | eqeltrdi 2280 | . . 3 ⊢ (𝐴 = +∞ → -𝑒𝐴 ∈ ℝ*) |
11 | xnegeq 9845 | . . . 4 ⊢ (𝐴 = -∞ → -𝑒𝐴 = -𝑒-∞) | |
12 | xnegmnf 9847 | . . . . 5 ⊢ -𝑒-∞ = +∞ | |
13 | pnfxr 8028 | . . . . 5 ⊢ +∞ ∈ ℝ* | |
14 | 12, 13 | eqeltri 2262 | . . . 4 ⊢ -𝑒-∞ ∈ ℝ* |
15 | 11, 14 | eqeltrdi 2280 | . . 3 ⊢ (𝐴 = -∞ → -𝑒𝐴 ∈ ℝ*) |
16 | 5, 10, 15 | 3jaoi 1314 | . 2 ⊢ ((𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞) → -𝑒𝐴 ∈ ℝ*) |
17 | 1, 16 | sylbi 121 | 1 ⊢ (𝐴 ∈ ℝ* → -𝑒𝐴 ∈ ℝ*) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∨ w3o 979 = wceq 1364 ∈ wcel 2160 ℝcr 7828 +∞cpnf 8007 -∞cmnf 8008 ℝ*cxr 8009 -cneg 8147 -𝑒cxne 9787 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4189 ax-pr 4224 ax-un 4448 ax-setind 4551 ax-cnex 7920 ax-resscn 7921 ax-1cn 7922 ax-icn 7924 ax-addcl 7925 ax-addrcl 7926 ax-mulcl 7927 ax-addcom 7929 ax-addass 7931 ax-distr 7933 ax-i2m1 7934 ax-0id 7937 ax-rnegex 7938 ax-cnre 7940 |
This theorem depends on definitions: df-bi 117 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rab 2477 df-v 2754 df-sbc 2978 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-if 3550 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-br 4019 df-opab 4080 df-id 4308 df-xp 4647 df-rel 4648 df-cnv 4649 df-co 4650 df-dm 4651 df-iota 5193 df-fun 5233 df-fv 5239 df-riota 5847 df-ov 5894 df-oprab 5895 df-mpo 5896 df-pnf 8012 df-mnf 8013 df-xr 8014 df-sub 8148 df-neg 8149 df-xneg 9790 |
This theorem is referenced by: xltneg 9854 xleneg 9855 xnegcld 9873 xnegdi 9886 xaddass2 9888 xleadd1 9893 xsubge0 9899 xrnegiso 11288 xrminmax 11291 xrmincl 11292 xrmin1inf 11293 xrmin2inf 11294 xrlemininf 11297 xrminltinf 11298 |
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