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Mirrors > Home > MPE Home > Th. List > xnegcl | Structured version Visualization version 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 12156 | . 2 ⊢ (𝐴 ∈ ℝ* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞)) | |
2 | rexneg 12248 | . . . . 5 ⊢ (𝐴 ∈ ℝ → -𝑒𝐴 = -𝐴) | |
3 | renegcl 10547 | . . . . 5 ⊢ (𝐴 ∈ ℝ → -𝐴 ∈ ℝ) | |
4 | 2, 3 | eqeltrd 2850 | . . . 4 ⊢ (𝐴 ∈ ℝ → -𝑒𝐴 ∈ ℝ) |
5 | 4 | rexrd 10292 | . . 3 ⊢ (𝐴 ∈ ℝ → -𝑒𝐴 ∈ ℝ*) |
6 | xnegeq 12244 | . . . 4 ⊢ (𝐴 = +∞ → -𝑒𝐴 = -𝑒+∞) | |
7 | xnegpnf 12246 | . . . . 5 ⊢ -𝑒+∞ = -∞ | |
8 | mnfxr 10299 | . . . . 5 ⊢ -∞ ∈ ℝ* | |
9 | 7, 8 | eqeltri 2846 | . . . 4 ⊢ -𝑒+∞ ∈ ℝ* |
10 | 6, 9 | syl6eqel 2858 | . . 3 ⊢ (𝐴 = +∞ → -𝑒𝐴 ∈ ℝ*) |
11 | xnegeq 12244 | . . . 4 ⊢ (𝐴 = -∞ → -𝑒𝐴 = -𝑒-∞) | |
12 | xnegmnf 12247 | . . . . 5 ⊢ -𝑒-∞ = +∞ | |
13 | pnfxr 10295 | . . . . 5 ⊢ +∞ ∈ ℝ* | |
14 | 12, 13 | eqeltri 2846 | . . . 4 ⊢ -𝑒-∞ ∈ ℝ* |
15 | 11, 14 | syl6eqel 2858 | . . 3 ⊢ (𝐴 = -∞ → -𝑒𝐴 ∈ ℝ*) |
16 | 5, 10, 15 | 3jaoi 1539 | . 2 ⊢ ((𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞) → -𝑒𝐴 ∈ ℝ*) |
17 | 1, 16 | sylbi 207 | 1 ⊢ (𝐴 ∈ ℝ* → -𝑒𝐴 ∈ ℝ*) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ w3o 1070 = wceq 1631 ∈ wcel 2145 ℝcr 10138 +∞cpnf 10274 -∞cmnf 10275 ℝ*cxr 10276 -cneg 10470 -𝑒cxne 12149 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4916 ax-nul 4924 ax-pow 4975 ax-pr 5035 ax-un 7097 ax-cnex 10195 ax-resscn 10196 ax-1cn 10197 ax-icn 10198 ax-addcl 10199 ax-addrcl 10200 ax-mulcl 10201 ax-mulrcl 10202 ax-mulcom 10203 ax-addass 10204 ax-mulass 10205 ax-distr 10206 ax-i2m1 10207 ax-1ne0 10208 ax-1rid 10209 ax-rnegex 10210 ax-rrecex 10211 ax-cnre 10212 ax-pre-lttri 10213 ax-pre-lttrn 10214 ax-pre-ltadd 10215 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 829 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3353 df-sbc 3589 df-csb 3684 df-dif 3727 df-un 3729 df-in 3731 df-ss 3738 df-nul 4065 df-if 4227 df-pw 4300 df-sn 4318 df-pr 4320 df-op 4324 df-uni 4576 df-br 4788 df-opab 4848 df-mpt 4865 df-id 5158 df-po 5171 df-so 5172 df-xp 5256 df-rel 5257 df-cnv 5258 df-co 5259 df-dm 5260 df-rn 5261 df-res 5262 df-ima 5263 df-iota 5995 df-fun 6034 df-fn 6035 df-f 6036 df-f1 6037 df-fo 6038 df-f1o 6039 df-fv 6040 df-riota 6755 df-ov 6797 df-oprab 6798 df-mpt2 6799 df-er 7897 df-en 8111 df-dom 8112 df-sdom 8113 df-pnf 10279 df-mnf 10280 df-xr 10281 df-ltxr 10282 df-sub 10471 df-neg 10472 df-xneg 12152 |
This theorem is referenced by: xltneg 12254 xleneg 12255 xnegdi 12284 xaddass2 12286 xleadd1 12291 xsubge0 12297 xposdif 12298 xlesubadd 12299 xmulneg1 12305 xmulneg2 12306 xmulpnf1n 12314 xmulasslem 12321 xnegcld 12336 xrsds 20005 xrsxmet 22833 xrhmeo 22966 xaddeq0 29859 xrsinvgval 30018 xrge0npcan 30035 xnegcli 40188 |
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