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Mirrors > Home > ILE Home > Th. List > xnegneg | GIF version |
Description: Extended real version of negneg 8210. (Contributed by Mario Carneiro, 20-Aug-2015.) |
Ref | Expression |
---|---|
xnegneg | ⊢ (𝐴 ∈ ℝ* → -𝑒-𝑒𝐴 = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elxr 9779 | . 2 ⊢ (𝐴 ∈ ℝ* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞)) | |
2 | rexneg 9833 | . . . . 5 ⊢ (𝐴 ∈ ℝ → -𝑒𝐴 = -𝐴) | |
3 | xnegeq 9830 | . . . . 5 ⊢ (-𝑒𝐴 = -𝐴 → -𝑒-𝑒𝐴 = -𝑒-𝐴) | |
4 | 2, 3 | syl 14 | . . . 4 ⊢ (𝐴 ∈ ℝ → -𝑒-𝑒𝐴 = -𝑒-𝐴) |
5 | renegcl 8221 | . . . . 5 ⊢ (𝐴 ∈ ℝ → -𝐴 ∈ ℝ) | |
6 | rexneg 9833 | . . . . 5 ⊢ (-𝐴 ∈ ℝ → -𝑒-𝐴 = --𝐴) | |
7 | 5, 6 | syl 14 | . . . 4 ⊢ (𝐴 ∈ ℝ → -𝑒-𝐴 = --𝐴) |
8 | recn 7947 | . . . . 5 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ) | |
9 | 8 | negnegd 8262 | . . . 4 ⊢ (𝐴 ∈ ℝ → --𝐴 = 𝐴) |
10 | 4, 7, 9 | 3eqtrd 2214 | . . 3 ⊢ (𝐴 ∈ ℝ → -𝑒-𝑒𝐴 = 𝐴) |
11 | xnegmnf 9832 | . . . 4 ⊢ -𝑒-∞ = +∞ | |
12 | xnegeq 9830 | . . . . . 6 ⊢ (𝐴 = +∞ → -𝑒𝐴 = -𝑒+∞) | |
13 | xnegpnf 9831 | . . . . . 6 ⊢ -𝑒+∞ = -∞ | |
14 | 12, 13 | eqtrdi 2226 | . . . . 5 ⊢ (𝐴 = +∞ → -𝑒𝐴 = -∞) |
15 | xnegeq 9830 | . . . . 5 ⊢ (-𝑒𝐴 = -∞ → -𝑒-𝑒𝐴 = -𝑒-∞) | |
16 | 14, 15 | syl 14 | . . . 4 ⊢ (𝐴 = +∞ → -𝑒-𝑒𝐴 = -𝑒-∞) |
17 | id 19 | . . . 4 ⊢ (𝐴 = +∞ → 𝐴 = +∞) | |
18 | 11, 16, 17 | 3eqtr4a 2236 | . . 3 ⊢ (𝐴 = +∞ → -𝑒-𝑒𝐴 = 𝐴) |
19 | xnegeq 9830 | . . . . . 6 ⊢ (𝐴 = -∞ → -𝑒𝐴 = -𝑒-∞) | |
20 | 19, 11 | eqtrdi 2226 | . . . . 5 ⊢ (𝐴 = -∞ → -𝑒𝐴 = +∞) |
21 | xnegeq 9830 | . . . . 5 ⊢ (-𝑒𝐴 = +∞ → -𝑒-𝑒𝐴 = -𝑒+∞) | |
22 | 20, 21 | syl 14 | . . . 4 ⊢ (𝐴 = -∞ → -𝑒-𝑒𝐴 = -𝑒+∞) |
23 | id 19 | . . . 4 ⊢ (𝐴 = -∞ → 𝐴 = -∞) | |
24 | 13, 22, 23 | 3eqtr4a 2236 | . . 3 ⊢ (𝐴 = -∞ → -𝑒-𝑒𝐴 = 𝐴) |
25 | 10, 18, 24 | 3jaoi 1303 | . 2 ⊢ ((𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞) → -𝑒-𝑒𝐴 = 𝐴) |
26 | 1, 25 | sylbi 121 | 1 ⊢ (𝐴 ∈ ℝ* → -𝑒-𝑒𝐴 = 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∨ w3o 977 = wceq 1353 ∈ wcel 2148 ℝcr 7813 +∞cpnf 7992 -∞cmnf 7993 ℝ*cxr 7994 -cneg 8132 -𝑒cxne 9772 |
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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4123 ax-pow 4176 ax-pr 4211 ax-un 4435 ax-setind 4538 ax-cnex 7905 ax-resscn 7906 ax-1cn 7907 ax-icn 7909 ax-addcl 7910 ax-addrcl 7911 ax-mulcl 7912 ax-addcom 7914 ax-addass 7916 ax-distr 7918 ax-i2m1 7919 ax-0id 7922 ax-rnegex 7923 ax-cnre 7925 |
This theorem depends on definitions: df-bi 117 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2741 df-sbc 2965 df-dif 3133 df-un 3135 df-in 3137 df-ss 3144 df-if 3537 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-br 4006 df-opab 4067 df-id 4295 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-iota 5180 df-fun 5220 df-fv 5226 df-riota 5834 df-ov 5881 df-oprab 5882 df-mpo 5883 df-pnf 7997 df-mnf 7998 df-xr 7999 df-sub 8133 df-neg 8134 df-xneg 9775 |
This theorem is referenced by: xneg11 9837 xltneg 9839 xnegdi 9871 xnpcan 9875 xrnegiso 11273 infxrnegsupex 11274 xrnegcon1d 11275 xrminmax 11276 xrmin1inf 11278 xrmin2inf 11279 xrltmininf 11281 xrlemininf 11282 xrminltinf 11283 xrminadd 11286 |
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