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Mirrors > Home > ILE Home > Th. List > xnegid | GIF version |
Description: Extended real version of negid 8206. (Contributed by Mario Carneiro, 20-Aug-2015.) |
Ref | Expression |
---|---|
xnegid | ⊢ (𝐴 ∈ ℝ* → (𝐴 +𝑒 -𝑒𝐴) = 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elxr 9778 | . 2 ⊢ (𝐴 ∈ ℝ* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞)) | |
2 | rexneg 9832 | . . . . 5 ⊢ (𝐴 ∈ ℝ → -𝑒𝐴 = -𝐴) | |
3 | 2 | oveq2d 5893 | . . . 4 ⊢ (𝐴 ∈ ℝ → (𝐴 +𝑒 -𝑒𝐴) = (𝐴 +𝑒 -𝐴)) |
4 | renegcl 8220 | . . . . 5 ⊢ (𝐴 ∈ ℝ → -𝐴 ∈ ℝ) | |
5 | rexadd 9854 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ -𝐴 ∈ ℝ) → (𝐴 +𝑒 -𝐴) = (𝐴 + -𝐴)) | |
6 | 4, 5 | mpdan 421 | . . . 4 ⊢ (𝐴 ∈ ℝ → (𝐴 +𝑒 -𝐴) = (𝐴 + -𝐴)) |
7 | recn 7946 | . . . . 5 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ) | |
8 | 7 | negidd 8260 | . . . 4 ⊢ (𝐴 ∈ ℝ → (𝐴 + -𝐴) = 0) |
9 | 3, 6, 8 | 3eqtrd 2214 | . . 3 ⊢ (𝐴 ∈ ℝ → (𝐴 +𝑒 -𝑒𝐴) = 0) |
10 | id 19 | . . . . 5 ⊢ (𝐴 = +∞ → 𝐴 = +∞) | |
11 | xnegeq 9829 | . . . . . 6 ⊢ (𝐴 = +∞ → -𝑒𝐴 = -𝑒+∞) | |
12 | xnegpnf 9830 | . . . . . 6 ⊢ -𝑒+∞ = -∞ | |
13 | 11, 12 | eqtrdi 2226 | . . . . 5 ⊢ (𝐴 = +∞ → -𝑒𝐴 = -∞) |
14 | 10, 13 | oveq12d 5895 | . . . 4 ⊢ (𝐴 = +∞ → (𝐴 +𝑒 -𝑒𝐴) = (+∞ +𝑒 -∞)) |
15 | pnfaddmnf 9852 | . . . 4 ⊢ (+∞ +𝑒 -∞) = 0 | |
16 | 14, 15 | eqtrdi 2226 | . . 3 ⊢ (𝐴 = +∞ → (𝐴 +𝑒 -𝑒𝐴) = 0) |
17 | id 19 | . . . . 5 ⊢ (𝐴 = -∞ → 𝐴 = -∞) | |
18 | xnegeq 9829 | . . . . . 6 ⊢ (𝐴 = -∞ → -𝑒𝐴 = -𝑒-∞) | |
19 | xnegmnf 9831 | . . . . . 6 ⊢ -𝑒-∞ = +∞ | |
20 | 18, 19 | eqtrdi 2226 | . . . . 5 ⊢ (𝐴 = -∞ → -𝑒𝐴 = +∞) |
21 | 17, 20 | oveq12d 5895 | . . . 4 ⊢ (𝐴 = -∞ → (𝐴 +𝑒 -𝑒𝐴) = (-∞ +𝑒 +∞)) |
22 | mnfaddpnf 9853 | . . . 4 ⊢ (-∞ +𝑒 +∞) = 0 | |
23 | 21, 22 | eqtrdi 2226 | . . 3 ⊢ (𝐴 = -∞ → (𝐴 +𝑒 -𝑒𝐴) = 0) |
24 | 9, 16, 23 | 3jaoi 1303 | . 2 ⊢ ((𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞) → (𝐴 +𝑒 -𝑒𝐴) = 0) |
25 | 1, 24 | sylbi 121 | 1 ⊢ (𝐴 ∈ ℝ* → (𝐴 +𝑒 -𝑒𝐴) = 0) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∨ w3o 977 = wceq 1353 ∈ wcel 2148 (class class class)co 5877 ℝcr 7812 0cc0 7813 + caddc 7816 +∞cpnf 7991 -∞cmnf 7992 ℝ*cxr 7993 -cneg 8131 -𝑒cxne 9771 +𝑒 cxad 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 7904 ax-resscn 7905 ax-1cn 7906 ax-1re 7907 ax-icn 7908 ax-addcl 7909 ax-addrcl 7910 ax-mulcl 7911 ax-addcom 7913 ax-addass 7915 ax-distr 7917 ax-i2m1 7918 ax-0id 7921 ax-rnegex 7922 ax-cnre 7924 |
This theorem depends on definitions: df-bi 117 df-dc 835 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 5833 df-ov 5880 df-oprab 5881 df-mpo 5882 df-pnf 7996 df-mnf 7997 df-xr 7998 df-sub 8132 df-neg 8133 df-xneg 9774 df-xadd 9775 |
This theorem is referenced by: (None) |
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