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Mirrors > Home > MPE Home > Th. List > xnegneg | Structured version Visualization version GIF version |
Description: Extended real version of negneg 11459. (Contributed by Mario Carneiro, 20-Aug-2015.) |
Ref | Expression |
---|---|
xnegneg | ⊢ (𝐴 ∈ ℝ* → -𝑒-𝑒𝐴 = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elxr 13045 | . 2 ⊢ (𝐴 ∈ ℝ* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞)) | |
2 | rexneg 13139 | . . . . 5 ⊢ (𝐴 ∈ ℝ → -𝑒𝐴 = -𝐴) | |
3 | xnegeq 13135 | . . . . 5 ⊢ (-𝑒𝐴 = -𝐴 → -𝑒-𝑒𝐴 = -𝑒-𝐴) | |
4 | 2, 3 | syl 17 | . . . 4 ⊢ (𝐴 ∈ ℝ → -𝑒-𝑒𝐴 = -𝑒-𝐴) |
5 | renegcl 11472 | . . . . 5 ⊢ (𝐴 ∈ ℝ → -𝐴 ∈ ℝ) | |
6 | rexneg 13139 | . . . . 5 ⊢ (-𝐴 ∈ ℝ → -𝑒-𝐴 = --𝐴) | |
7 | 5, 6 | syl 17 | . . . 4 ⊢ (𝐴 ∈ ℝ → -𝑒-𝐴 = --𝐴) |
8 | recn 11149 | . . . . 5 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ) | |
9 | 8 | negnegd 11511 | . . . 4 ⊢ (𝐴 ∈ ℝ → --𝐴 = 𝐴) |
10 | 4, 7, 9 | 3eqtrd 2777 | . . 3 ⊢ (𝐴 ∈ ℝ → -𝑒-𝑒𝐴 = 𝐴) |
11 | xnegmnf 13138 | . . . 4 ⊢ -𝑒-∞ = +∞ | |
12 | xnegeq 13135 | . . . . . 6 ⊢ (𝐴 = +∞ → -𝑒𝐴 = -𝑒+∞) | |
13 | xnegpnf 13137 | . . . . . 6 ⊢ -𝑒+∞ = -∞ | |
14 | 12, 13 | eqtrdi 2789 | . . . . 5 ⊢ (𝐴 = +∞ → -𝑒𝐴 = -∞) |
15 | xnegeq 13135 | . . . . 5 ⊢ (-𝑒𝐴 = -∞ → -𝑒-𝑒𝐴 = -𝑒-∞) | |
16 | 14, 15 | syl 17 | . . . 4 ⊢ (𝐴 = +∞ → -𝑒-𝑒𝐴 = -𝑒-∞) |
17 | id 22 | . . . 4 ⊢ (𝐴 = +∞ → 𝐴 = +∞) | |
18 | 11, 16, 17 | 3eqtr4a 2799 | . . 3 ⊢ (𝐴 = +∞ → -𝑒-𝑒𝐴 = 𝐴) |
19 | xnegeq 13135 | . . . . . 6 ⊢ (𝐴 = -∞ → -𝑒𝐴 = -𝑒-∞) | |
20 | 19, 11 | eqtrdi 2789 | . . . . 5 ⊢ (𝐴 = -∞ → -𝑒𝐴 = +∞) |
21 | xnegeq 13135 | . . . . 5 ⊢ (-𝑒𝐴 = +∞ → -𝑒-𝑒𝐴 = -𝑒+∞) | |
22 | 20, 21 | syl 17 | . . . 4 ⊢ (𝐴 = -∞ → -𝑒-𝑒𝐴 = -𝑒+∞) |
23 | id 22 | . . . 4 ⊢ (𝐴 = -∞ → 𝐴 = -∞) | |
24 | 13, 22, 23 | 3eqtr4a 2799 | . . 3 ⊢ (𝐴 = -∞ → -𝑒-𝑒𝐴 = 𝐴) |
25 | 10, 18, 24 | 3jaoi 1428 | . 2 ⊢ ((𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞) → -𝑒-𝑒𝐴 = 𝐴) |
26 | 1, 25 | sylbi 216 | 1 ⊢ (𝐴 ∈ ℝ* → -𝑒-𝑒𝐴 = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ w3o 1087 = wceq 1542 ∈ wcel 2107 ℝcr 11058 +∞cpnf 11194 -∞cmnf 11195 ℝ*cxr 11196 -cneg 11394 -𝑒cxne 13038 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-br 5110 df-opab 5172 df-mpt 5193 df-id 5535 df-po 5549 df-so 5550 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-er 8654 df-en 8890 df-dom 8891 df-sdom 8892 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-sub 11395 df-neg 11396 df-xneg 13041 |
This theorem is referenced by: xneg11 13143 xltneg 13145 xnegdi 13176 xnpcan 13180 xposdif 13190 xrsxmet 24195 xrhmeo 24332 xaddeq0 31712 xrge0npcan 31941 carsgclctunlem2 32983 xnegnegi 43764 xnegnegd 43767 xnegrecl2 43785 supminfxr2 43794 supminfxrrnmpt 43796 xlenegcon1 43812 xlenegcon2 43813 |
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