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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 9726 | . 2 ⊢ (𝐴 ∈ ℝ* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞)) | |
2 | rexneg 9780 | . . . . 5 ⊢ (𝐴 ∈ ℝ → -𝑒𝐴 = -𝐴) | |
3 | renegcl 8173 | . . . . 5 ⊢ (𝐴 ∈ ℝ → -𝐴 ∈ ℝ) | |
4 | 2, 3 | eqeltrd 2247 | . . . 4 ⊢ (𝐴 ∈ ℝ → -𝑒𝐴 ∈ ℝ) |
5 | 4 | rexrd 7962 | . . 3 ⊢ (𝐴 ∈ ℝ → -𝑒𝐴 ∈ ℝ*) |
6 | xnegeq 9777 | . . . 4 ⊢ (𝐴 = +∞ → -𝑒𝐴 = -𝑒+∞) | |
7 | xnegpnf 9778 | . . . . 5 ⊢ -𝑒+∞ = -∞ | |
8 | mnfxr 7969 | . . . . 5 ⊢ -∞ ∈ ℝ* | |
9 | 7, 8 | eqeltri 2243 | . . . 4 ⊢ -𝑒+∞ ∈ ℝ* |
10 | 6, 9 | eqeltrdi 2261 | . . 3 ⊢ (𝐴 = +∞ → -𝑒𝐴 ∈ ℝ*) |
11 | xnegeq 9777 | . . . 4 ⊢ (𝐴 = -∞ → -𝑒𝐴 = -𝑒-∞) | |
12 | xnegmnf 9779 | . . . . 5 ⊢ -𝑒-∞ = +∞ | |
13 | pnfxr 7965 | . . . . 5 ⊢ +∞ ∈ ℝ* | |
14 | 12, 13 | eqeltri 2243 | . . . 4 ⊢ -𝑒-∞ ∈ ℝ* |
15 | 11, 14 | eqeltrdi 2261 | . . 3 ⊢ (𝐴 = -∞ → -𝑒𝐴 ∈ ℝ*) |
16 | 5, 10, 15 | 3jaoi 1298 | . 2 ⊢ ((𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞) → -𝑒𝐴 ∈ ℝ*) |
17 | 1, 16 | sylbi 120 | 1 ⊢ (𝐴 ∈ ℝ* → -𝑒𝐴 ∈ ℝ*) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∨ w3o 972 = wceq 1348 ∈ wcel 2141 ℝcr 7766 +∞cpnf 7944 -∞cmnf 7945 ℝ*cxr 7946 -cneg 8084 -𝑒cxne 9719 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4105 ax-pow 4158 ax-pr 4192 ax-un 4416 ax-setind 4519 ax-cnex 7858 ax-resscn 7859 ax-1cn 7860 ax-icn 7862 ax-addcl 7863 ax-addrcl 7864 ax-mulcl 7865 ax-addcom 7867 ax-addass 7869 ax-distr 7871 ax-i2m1 7872 ax-0id 7875 ax-rnegex 7876 ax-cnre 7878 |
This theorem depends on definitions: df-bi 116 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-if 3526 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-br 3988 df-opab 4049 df-id 4276 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-iota 5158 df-fun 5198 df-fv 5204 df-riota 5807 df-ov 5854 df-oprab 5855 df-mpo 5856 df-pnf 7949 df-mnf 7950 df-xr 7951 df-sub 8085 df-neg 8086 df-xneg 9722 |
This theorem is referenced by: xltneg 9786 xleneg 9787 xnegcld 9805 xnegdi 9818 xaddass2 9820 xleadd1 9825 xsubge0 9831 xrnegiso 11218 xrminmax 11221 xrmincl 11222 xrmin1inf 11223 xrmin2inf 11224 xrlemininf 11227 xrminltinf 11228 |
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