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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 8466 | . 2 ⊢ (A ∈ ℝ* ↔ (A ∈ ℝ ∨ A = +∞ ∨ A = -∞)) | |
2 | rexneg 8513 | . . . . 5 ⊢ (A ∈ ℝ → -𝑒A = -A) | |
3 | renegcl 7068 | . . . . 5 ⊢ (A ∈ ℝ → -A ∈ ℝ) | |
4 | 2, 3 | eqeltrd 2111 | . . . 4 ⊢ (A ∈ ℝ → -𝑒A ∈ ℝ) |
5 | 4 | rexrd 6872 | . . 3 ⊢ (A ∈ ℝ → -𝑒A ∈ ℝ*) |
6 | xnegeq 8510 | . . . 4 ⊢ (A = +∞ → -𝑒A = -𝑒+∞) | |
7 | xnegpnf 8511 | . . . . 5 ⊢ -𝑒+∞ = -∞ | |
8 | mnfxr 8464 | . . . . 5 ⊢ -∞ ∈ ℝ* | |
9 | 7, 8 | eqeltri 2107 | . . . 4 ⊢ -𝑒+∞ ∈ ℝ* |
10 | 6, 9 | syl6eqel 2125 | . . 3 ⊢ (A = +∞ → -𝑒A ∈ ℝ*) |
11 | xnegeq 8510 | . . . 4 ⊢ (A = -∞ → -𝑒A = -𝑒-∞) | |
12 | xnegmnf 8512 | . . . . 5 ⊢ -𝑒-∞ = +∞ | |
13 | pnfxr 8462 | . . . . 5 ⊢ +∞ ∈ ℝ* | |
14 | 12, 13 | eqeltri 2107 | . . . 4 ⊢ -𝑒-∞ ∈ ℝ* |
15 | 11, 14 | syl6eqel 2125 | . . 3 ⊢ (A = -∞ → -𝑒A ∈ ℝ*) |
16 | 5, 10, 15 | 3jaoi 1197 | . 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 883 = wceq 1242 ∈ wcel 1390 ℝcr 6710 +∞cpnf 6854 -∞cmnf 6855 ℝ*cxr 6856 -cneg 6980 -𝑒cxne 8456 |
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 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-13 1401 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-pow 3918 ax-pr 3935 ax-un 4136 ax-setind 4220 ax-cnex 6774 ax-resscn 6775 ax-1cn 6776 ax-icn 6778 ax-addcl 6779 ax-addrcl 6780 ax-mulcl 6781 ax-addcom 6783 ax-addass 6785 ax-distr 6787 ax-i2m1 6788 ax-0id 6791 ax-rnegex 6792 ax-cnre 6794 |
This theorem depends on definitions: df-bi 110 df-3or 885 df-3an 886 df-tru 1245 df-fal 1248 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ne 2203 df-nel 2204 df-ral 2305 df-rex 2306 df-reu 2307 df-rab 2309 df-v 2553 df-sbc 2759 df-dif 2914 df-un 2916 df-in 2918 df-ss 2925 df-if 3326 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-uni 3572 df-br 3756 df-opab 3810 df-id 4021 df-xp 4294 df-rel 4295 df-cnv 4296 df-co 4297 df-dm 4298 df-iota 4810 df-fun 4847 df-fv 4853 df-riota 5411 df-ov 5458 df-oprab 5459 df-mpt2 5460 df-pnf 6859 df-mnf 6860 df-xr 6861 df-sub 6981 df-neg 6982 df-xneg 8459 |
This theorem is referenced by: xltneg 8519 xleneg 8520 xnegcld 8525 |
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