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Mirrors > Home > ILE Home > Th. List > xnegneg | GIF version |
Description: Extended real version of negneg 8169. (Contributed by Mario Carneiro, 20-Aug-2015.) |
Ref | Expression |
---|---|
xnegneg | ⊢ (𝐴 ∈ ℝ* → -𝑒-𝑒𝐴 = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elxr 9733 | . 2 ⊢ (𝐴 ∈ ℝ* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞)) | |
2 | rexneg 9787 | . . . . 5 ⊢ (𝐴 ∈ ℝ → -𝑒𝐴 = -𝐴) | |
3 | xnegeq 9784 | . . . . 5 ⊢ (-𝑒𝐴 = -𝐴 → -𝑒-𝑒𝐴 = -𝑒-𝐴) | |
4 | 2, 3 | syl 14 | . . . 4 ⊢ (𝐴 ∈ ℝ → -𝑒-𝑒𝐴 = -𝑒-𝐴) |
5 | renegcl 8180 | . . . . 5 ⊢ (𝐴 ∈ ℝ → -𝐴 ∈ ℝ) | |
6 | rexneg 9787 | . . . . 5 ⊢ (-𝐴 ∈ ℝ → -𝑒-𝐴 = --𝐴) | |
7 | 5, 6 | syl 14 | . . . 4 ⊢ (𝐴 ∈ ℝ → -𝑒-𝐴 = --𝐴) |
8 | recn 7907 | . . . . 5 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ) | |
9 | 8 | negnegd 8221 | . . . 4 ⊢ (𝐴 ∈ ℝ → --𝐴 = 𝐴) |
10 | 4, 7, 9 | 3eqtrd 2207 | . . 3 ⊢ (𝐴 ∈ ℝ → -𝑒-𝑒𝐴 = 𝐴) |
11 | xnegmnf 9786 | . . . 4 ⊢ -𝑒-∞ = +∞ | |
12 | xnegeq 9784 | . . . . . 6 ⊢ (𝐴 = +∞ → -𝑒𝐴 = -𝑒+∞) | |
13 | xnegpnf 9785 | . . . . . 6 ⊢ -𝑒+∞ = -∞ | |
14 | 12, 13 | eqtrdi 2219 | . . . . 5 ⊢ (𝐴 = +∞ → -𝑒𝐴 = -∞) |
15 | xnegeq 9784 | . . . . 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 2229 | . . 3 ⊢ (𝐴 = +∞ → -𝑒-𝑒𝐴 = 𝐴) |
19 | xnegeq 9784 | . . . . . 6 ⊢ (𝐴 = -∞ → -𝑒𝐴 = -𝑒-∞) | |
20 | 19, 11 | eqtrdi 2219 | . . . . 5 ⊢ (𝐴 = -∞ → -𝑒𝐴 = +∞) |
21 | xnegeq 9784 | . . . . 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 2229 | . . 3 ⊢ (𝐴 = -∞ → -𝑒-𝑒𝐴 = 𝐴) |
25 | 10, 18, 24 | 3jaoi 1298 | . 2 ⊢ ((𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞) → -𝑒-𝑒𝐴 = 𝐴) |
26 | 1, 25 | sylbi 120 | 1 ⊢ (𝐴 ∈ ℝ* → -𝑒-𝑒𝐴 = 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∨ w3o 972 = wceq 1348 ∈ wcel 2141 ℝcr 7773 +∞cpnf 7951 -∞cmnf 7952 ℝ*cxr 7953 -cneg 8091 -𝑒cxne 9726 |
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 4107 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-cnex 7865 ax-resscn 7866 ax-1cn 7867 ax-icn 7869 ax-addcl 7870 ax-addrcl 7871 ax-mulcl 7872 ax-addcom 7874 ax-addass 7876 ax-distr 7878 ax-i2m1 7879 ax-0id 7882 ax-rnegex 7883 ax-cnre 7885 |
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 3527 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-br 3990 df-opab 4051 df-id 4278 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-iota 5160 df-fun 5200 df-fv 5206 df-riota 5809 df-ov 5856 df-oprab 5857 df-mpo 5858 df-pnf 7956 df-mnf 7957 df-xr 7958 df-sub 8092 df-neg 8093 df-xneg 9729 |
This theorem is referenced by: xneg11 9791 xltneg 9793 xnegdi 9825 xnpcan 9829 xrnegiso 11225 infxrnegsupex 11226 xrnegcon1d 11227 xrminmax 11228 xrmin1inf 11230 xrmin2inf 11231 xrltmininf 11233 xrlemininf 11234 xrminltinf 11235 xrminadd 11238 |
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