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Mirrors > Home > MPE Home > Th. List > xnegid | Structured version Visualization version GIF version |
Description: Extended real version of negid 10328. (Contributed by Mario Carneiro, 20-Aug-2015.) |
Ref | Expression |
---|---|
xnegid | ⊢ (𝐴 ∈ ℝ* → (𝐴 +𝑒 -𝑒𝐴) = 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elxr 11950 | . 2 ⊢ (𝐴 ∈ ℝ* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞)) | |
2 | rexneg 12042 | . . . . 5 ⊢ (𝐴 ∈ ℝ → -𝑒𝐴 = -𝐴) | |
3 | 2 | oveq2d 6666 | . . . 4 ⊢ (𝐴 ∈ ℝ → (𝐴 +𝑒 -𝑒𝐴) = (𝐴 +𝑒 -𝐴)) |
4 | renegcl 10344 | . . . . 5 ⊢ (𝐴 ∈ ℝ → -𝐴 ∈ ℝ) | |
5 | rexadd 12063 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ -𝐴 ∈ ℝ) → (𝐴 +𝑒 -𝐴) = (𝐴 + -𝐴)) | |
6 | 4, 5 | mpdan 702 | . . . 4 ⊢ (𝐴 ∈ ℝ → (𝐴 +𝑒 -𝐴) = (𝐴 + -𝐴)) |
7 | recn 10026 | . . . . 5 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ) | |
8 | 7 | negidd 10382 | . . . 4 ⊢ (𝐴 ∈ ℝ → (𝐴 + -𝐴) = 0) |
9 | 3, 6, 8 | 3eqtrd 2660 | . . 3 ⊢ (𝐴 ∈ ℝ → (𝐴 +𝑒 -𝑒𝐴) = 0) |
10 | id 22 | . . . . 5 ⊢ (𝐴 = +∞ → 𝐴 = +∞) | |
11 | xnegeq 12038 | . . . . . 6 ⊢ (𝐴 = +∞ → -𝑒𝐴 = -𝑒+∞) | |
12 | xnegpnf 12040 | . . . . . 6 ⊢ -𝑒+∞ = -∞ | |
13 | 11, 12 | syl6eq 2672 | . . . . 5 ⊢ (𝐴 = +∞ → -𝑒𝐴 = -∞) |
14 | 10, 13 | oveq12d 6668 | . . . 4 ⊢ (𝐴 = +∞ → (𝐴 +𝑒 -𝑒𝐴) = (+∞ +𝑒 -∞)) |
15 | pnfaddmnf 12061 | . . . 4 ⊢ (+∞ +𝑒 -∞) = 0 | |
16 | 14, 15 | syl6eq 2672 | . . 3 ⊢ (𝐴 = +∞ → (𝐴 +𝑒 -𝑒𝐴) = 0) |
17 | id 22 | . . . . 5 ⊢ (𝐴 = -∞ → 𝐴 = -∞) | |
18 | xnegeq 12038 | . . . . . 6 ⊢ (𝐴 = -∞ → -𝑒𝐴 = -𝑒-∞) | |
19 | xnegmnf 12041 | . . . . . 6 ⊢ -𝑒-∞ = +∞ | |
20 | 18, 19 | syl6eq 2672 | . . . . 5 ⊢ (𝐴 = -∞ → -𝑒𝐴 = +∞) |
21 | 17, 20 | oveq12d 6668 | . . . 4 ⊢ (𝐴 = -∞ → (𝐴 +𝑒 -𝑒𝐴) = (-∞ +𝑒 +∞)) |
22 | mnfaddpnf 12062 | . . . 4 ⊢ (-∞ +𝑒 +∞) = 0 | |
23 | 21, 22 | syl6eq 2672 | . . 3 ⊢ (𝐴 = -∞ → (𝐴 +𝑒 -𝑒𝐴) = 0) |
24 | 9, 16, 23 | 3jaoi 1391 | . 2 ⊢ ((𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞) → (𝐴 +𝑒 -𝑒𝐴) = 0) |
25 | 1, 24 | sylbi 207 | 1 ⊢ (𝐴 ∈ ℝ* → (𝐴 +𝑒 -𝑒𝐴) = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ w3o 1036 = wceq 1483 ∈ wcel 1990 (class class class)co 6650 ℝcr 9935 0cc0 9936 + caddc 9939 +∞cpnf 10071 -∞cmnf 10072 ℝ*cxr 10073 -cneg 10267 -𝑒cxne 11943 +𝑒 cxad 11944 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-po 5035 df-so 5036 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-sub 10268 df-neg 10269 df-xneg 11946 df-xadd 11947 |
This theorem is referenced by: xrsxmet 22612 xaddeq0 29518 xlt2addrd 29523 xrge0npcan 29694 carsgclctunlem2 30381 |
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