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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 13095 | . 2 ⊢ (𝐴 ∈ ℝ* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞)) | |
2 | rexneg 13189 | . . . . 5 ⊢ (𝐴 ∈ ℝ → -𝑒𝐴 = -𝐴) | |
3 | renegcl 11522 | . . . . 5 ⊢ (𝐴 ∈ ℝ → -𝐴 ∈ ℝ) | |
4 | 2, 3 | eqeltrd 2833 | . . . 4 ⊢ (𝐴 ∈ ℝ → -𝑒𝐴 ∈ ℝ) |
5 | 4 | rexrd 11263 | . . 3 ⊢ (𝐴 ∈ ℝ → -𝑒𝐴 ∈ ℝ*) |
6 | xnegeq 13185 | . . . 4 ⊢ (𝐴 = +∞ → -𝑒𝐴 = -𝑒+∞) | |
7 | xnegpnf 13187 | . . . . 5 ⊢ -𝑒+∞ = -∞ | |
8 | mnfxr 11270 | . . . . 5 ⊢ -∞ ∈ ℝ* | |
9 | 7, 8 | eqeltri 2829 | . . . 4 ⊢ -𝑒+∞ ∈ ℝ* |
10 | 6, 9 | eqeltrdi 2841 | . . 3 ⊢ (𝐴 = +∞ → -𝑒𝐴 ∈ ℝ*) |
11 | xnegeq 13185 | . . . 4 ⊢ (𝐴 = -∞ → -𝑒𝐴 = -𝑒-∞) | |
12 | xnegmnf 13188 | . . . . 5 ⊢ -𝑒-∞ = +∞ | |
13 | pnfxr 11267 | . . . . 5 ⊢ +∞ ∈ ℝ* | |
14 | 12, 13 | eqeltri 2829 | . . . 4 ⊢ -𝑒-∞ ∈ ℝ* |
15 | 11, 14 | eqeltrdi 2841 | . . 3 ⊢ (𝐴 = -∞ → -𝑒𝐴 ∈ ℝ*) |
16 | 5, 10, 15 | 3jaoi 1427 | . 2 ⊢ ((𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞) → -𝑒𝐴 ∈ ℝ*) |
17 | 1, 16 | sylbi 216 | 1 ⊢ (𝐴 ∈ ℝ* → -𝑒𝐴 ∈ ℝ*) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ w3o 1086 = wceq 1541 ∈ wcel 2106 ℝcr 11108 +∞cpnf 11244 -∞cmnf 11245 ℝ*cxr 11246 -cneg 11444 -𝑒cxne 13088 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-po 5588 df-so 5589 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7364 df-ov 7411 df-oprab 7412 df-mpo 7413 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-sub 11445 df-neg 11446 df-xneg 13091 |
This theorem is referenced by: xltneg 13195 xleneg 13196 xnegdi 13226 xaddass2 13228 xleadd1 13233 xsubge0 13239 xposdif 13240 xlesubadd 13241 xmulneg1 13247 xmulneg2 13248 xmulpnf1n 13256 xmulasslem 13263 xnegcld 13278 xrsds 20987 xrsxmet 24324 xrhmeo 24461 xaddeq0 31961 xrsinvgval 32173 xrge0npcan 32190 xnegcli 44144 xlenegcon1 44187 xlenegcon2 44188 |
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