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Mirrors > Home > MPE Home > Th. List > xnegneg | Structured version Visualization version GIF version |
Description: Extended real version of negneg 11534. (Contributed by Mario Carneiro, 20-Aug-2015.) |
Ref | Expression |
---|---|
xnegneg | ⊢ (𝐴 ∈ ℝ* → -𝑒-𝑒𝐴 = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elxr 13122 | . 2 ⊢ (𝐴 ∈ ℝ* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞)) | |
2 | rexneg 13216 | . . . . 5 ⊢ (𝐴 ∈ ℝ → -𝑒𝐴 = -𝐴) | |
3 | xnegeq 13212 | . . . . 5 ⊢ (-𝑒𝐴 = -𝐴 → -𝑒-𝑒𝐴 = -𝑒-𝐴) | |
4 | 2, 3 | syl 17 | . . . 4 ⊢ (𝐴 ∈ ℝ → -𝑒-𝑒𝐴 = -𝑒-𝐴) |
5 | renegcl 11547 | . . . . 5 ⊢ (𝐴 ∈ ℝ → -𝐴 ∈ ℝ) | |
6 | rexneg 13216 | . . . . 5 ⊢ (-𝐴 ∈ ℝ → -𝑒-𝐴 = --𝐴) | |
7 | 5, 6 | syl 17 | . . . 4 ⊢ (𝐴 ∈ ℝ → -𝑒-𝐴 = --𝐴) |
8 | recn 11222 | . . . . 5 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ) | |
9 | 8 | negnegd 11586 | . . . 4 ⊢ (𝐴 ∈ ℝ → --𝐴 = 𝐴) |
10 | 4, 7, 9 | 3eqtrd 2772 | . . 3 ⊢ (𝐴 ∈ ℝ → -𝑒-𝑒𝐴 = 𝐴) |
11 | xnegmnf 13215 | . . . 4 ⊢ -𝑒-∞ = +∞ | |
12 | xnegeq 13212 | . . . . . 6 ⊢ (𝐴 = +∞ → -𝑒𝐴 = -𝑒+∞) | |
13 | xnegpnf 13214 | . . . . . 6 ⊢ -𝑒+∞ = -∞ | |
14 | 12, 13 | eqtrdi 2784 | . . . . 5 ⊢ (𝐴 = +∞ → -𝑒𝐴 = -∞) |
15 | xnegeq 13212 | . . . . 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 2794 | . . 3 ⊢ (𝐴 = +∞ → -𝑒-𝑒𝐴 = 𝐴) |
19 | xnegeq 13212 | . . . . . 6 ⊢ (𝐴 = -∞ → -𝑒𝐴 = -𝑒-∞) | |
20 | 19, 11 | eqtrdi 2784 | . . . . 5 ⊢ (𝐴 = -∞ → -𝑒𝐴 = +∞) |
21 | xnegeq 13212 | . . . . 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 2794 | . . 3 ⊢ (𝐴 = -∞ → -𝑒-𝑒𝐴 = 𝐴) |
25 | 10, 18, 24 | 3jaoi 1425 | . 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 1084 = wceq 1534 ∈ wcel 2099 ℝcr 11131 +∞cpnf 11269 -∞cmnf 11270 ℝ*cxr 11271 -cneg 11469 -𝑒cxne 13115 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11188 ax-resscn 11189 ax-1cn 11190 ax-icn 11191 ax-addcl 11192 ax-addrcl 11193 ax-mulcl 11194 ax-mulrcl 11195 ax-mulcom 11196 ax-addass 11197 ax-mulass 11198 ax-distr 11199 ax-i2m1 11200 ax-1ne0 11201 ax-1rid 11202 ax-rnegex 11203 ax-rrecex 11204 ax-cnre 11205 ax-pre-lttri 11206 ax-pre-lttrn 11207 ax-pre-ltadd 11208 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-po 5584 df-so 5585 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-er 8718 df-en 8958 df-dom 8959 df-sdom 8960 df-pnf 11274 df-mnf 11275 df-xr 11276 df-ltxr 11277 df-sub 11470 df-neg 11471 df-xneg 13118 |
This theorem is referenced by: xneg11 13220 xltneg 13222 xnegdi 13253 xnpcan 13257 xposdif 13267 xrsxmet 24718 xrhmeo 24864 xaddeq0 32517 xrge0npcan 32744 carsgclctunlem2 33933 xnegnegi 44815 xnegnegd 44818 xnegrecl2 44836 supminfxr2 44845 supminfxrrnmpt 44847 xlenegcon1 44863 xlenegcon2 44864 |
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