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Mirrors > Home > ILE Home > Th. List > xnegid | GIF version |
Description: Extended real version of negid 8009. (Contributed by Mario Carneiro, 20-Aug-2015.) |
Ref | Expression |
---|---|
xnegid | ⊢ (𝐴 ∈ ℝ* → (𝐴 +𝑒 -𝑒𝐴) = 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elxr 9563 | . 2 ⊢ (𝐴 ∈ ℝ* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞)) | |
2 | rexneg 9613 | . . . . 5 ⊢ (𝐴 ∈ ℝ → -𝑒𝐴 = -𝐴) | |
3 | 2 | oveq2d 5790 | . . . 4 ⊢ (𝐴 ∈ ℝ → (𝐴 +𝑒 -𝑒𝐴) = (𝐴 +𝑒 -𝐴)) |
4 | renegcl 8023 | . . . . 5 ⊢ (𝐴 ∈ ℝ → -𝐴 ∈ ℝ) | |
5 | rexadd 9635 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ -𝐴 ∈ ℝ) → (𝐴 +𝑒 -𝐴) = (𝐴 + -𝐴)) | |
6 | 4, 5 | mpdan 417 | . . . 4 ⊢ (𝐴 ∈ ℝ → (𝐴 +𝑒 -𝐴) = (𝐴 + -𝐴)) |
7 | recn 7753 | . . . . 5 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ) | |
8 | 7 | negidd 8063 | . . . 4 ⊢ (𝐴 ∈ ℝ → (𝐴 + -𝐴) = 0) |
9 | 3, 6, 8 | 3eqtrd 2176 | . . 3 ⊢ (𝐴 ∈ ℝ → (𝐴 +𝑒 -𝑒𝐴) = 0) |
10 | id 19 | . . . . 5 ⊢ (𝐴 = +∞ → 𝐴 = +∞) | |
11 | xnegeq 9610 | . . . . . 6 ⊢ (𝐴 = +∞ → -𝑒𝐴 = -𝑒+∞) | |
12 | xnegpnf 9611 | . . . . . 6 ⊢ -𝑒+∞ = -∞ | |
13 | 11, 12 | syl6eq 2188 | . . . . 5 ⊢ (𝐴 = +∞ → -𝑒𝐴 = -∞) |
14 | 10, 13 | oveq12d 5792 | . . . 4 ⊢ (𝐴 = +∞ → (𝐴 +𝑒 -𝑒𝐴) = (+∞ +𝑒 -∞)) |
15 | pnfaddmnf 9633 | . . . 4 ⊢ (+∞ +𝑒 -∞) = 0 | |
16 | 14, 15 | syl6eq 2188 | . . 3 ⊢ (𝐴 = +∞ → (𝐴 +𝑒 -𝑒𝐴) = 0) |
17 | id 19 | . . . . 5 ⊢ (𝐴 = -∞ → 𝐴 = -∞) | |
18 | xnegeq 9610 | . . . . . 6 ⊢ (𝐴 = -∞ → -𝑒𝐴 = -𝑒-∞) | |
19 | xnegmnf 9612 | . . . . . 6 ⊢ -𝑒-∞ = +∞ | |
20 | 18, 19 | syl6eq 2188 | . . . . 5 ⊢ (𝐴 = -∞ → -𝑒𝐴 = +∞) |
21 | 17, 20 | oveq12d 5792 | . . . 4 ⊢ (𝐴 = -∞ → (𝐴 +𝑒 -𝑒𝐴) = (-∞ +𝑒 +∞)) |
22 | mnfaddpnf 9634 | . . . 4 ⊢ (-∞ +𝑒 +∞) = 0 | |
23 | 21, 22 | syl6eq 2188 | . . 3 ⊢ (𝐴 = -∞ → (𝐴 +𝑒 -𝑒𝐴) = 0) |
24 | 9, 16, 23 | 3jaoi 1281 | . 2 ⊢ ((𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞) → (𝐴 +𝑒 -𝑒𝐴) = 0) |
25 | 1, 24 | sylbi 120 | 1 ⊢ (𝐴 ∈ ℝ* → (𝐴 +𝑒 -𝑒𝐴) = 0) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∨ w3o 961 = wceq 1331 ∈ wcel 1480 (class class class)co 5774 ℝcr 7619 0cc0 7620 + caddc 7623 +∞cpnf 7797 -∞cmnf 7798 ℝ*cxr 7799 -cneg 7934 -𝑒cxne 9556 +𝑒 cxad 9557 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-addcom 7720 ax-addass 7722 ax-distr 7724 ax-i2m1 7725 ax-0id 7728 ax-rnegex 7729 ax-cnre 7731 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-if 3475 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-iota 5088 df-fun 5125 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-pnf 7802 df-mnf 7803 df-xr 7804 df-sub 7935 df-neg 7936 df-xneg 9559 df-xadd 9560 |
This theorem is referenced by: (None) |
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