![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > xnegid | Structured version Visualization version GIF version |
Description: Extended real version of negid 11554. (Contributed by Mario Carneiro, 20-Aug-2015.) |
Ref | Expression |
---|---|
xnegid | ⊢ (𝐴 ∈ ℝ* → (𝐴 +𝑒 -𝑒𝐴) = 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elxr 13156 | . 2 ⊢ (𝐴 ∈ ℝ* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞)) | |
2 | rexneg 13250 | . . . . 5 ⊢ (𝐴 ∈ ℝ → -𝑒𝐴 = -𝐴) | |
3 | 2 | oveq2d 7447 | . . . 4 ⊢ (𝐴 ∈ ℝ → (𝐴 +𝑒 -𝑒𝐴) = (𝐴 +𝑒 -𝐴)) |
4 | renegcl 11570 | . . . . 5 ⊢ (𝐴 ∈ ℝ → -𝐴 ∈ ℝ) | |
5 | rexadd 13271 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ -𝐴 ∈ ℝ) → (𝐴 +𝑒 -𝐴) = (𝐴 + -𝐴)) | |
6 | 4, 5 | mpdan 687 | . . . 4 ⊢ (𝐴 ∈ ℝ → (𝐴 +𝑒 -𝐴) = (𝐴 + -𝐴)) |
7 | recn 11243 | . . . . 5 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ) | |
8 | 7 | negidd 11608 | . . . 4 ⊢ (𝐴 ∈ ℝ → (𝐴 + -𝐴) = 0) |
9 | 3, 6, 8 | 3eqtrd 2779 | . . 3 ⊢ (𝐴 ∈ ℝ → (𝐴 +𝑒 -𝑒𝐴) = 0) |
10 | id 22 | . . . . 5 ⊢ (𝐴 = +∞ → 𝐴 = +∞) | |
11 | xnegeq 13246 | . . . . . 6 ⊢ (𝐴 = +∞ → -𝑒𝐴 = -𝑒+∞) | |
12 | xnegpnf 13248 | . . . . . 6 ⊢ -𝑒+∞ = -∞ | |
13 | 11, 12 | eqtrdi 2791 | . . . . 5 ⊢ (𝐴 = +∞ → -𝑒𝐴 = -∞) |
14 | 10, 13 | oveq12d 7449 | . . . 4 ⊢ (𝐴 = +∞ → (𝐴 +𝑒 -𝑒𝐴) = (+∞ +𝑒 -∞)) |
15 | pnfaddmnf 13269 | . . . 4 ⊢ (+∞ +𝑒 -∞) = 0 | |
16 | 14, 15 | eqtrdi 2791 | . . 3 ⊢ (𝐴 = +∞ → (𝐴 +𝑒 -𝑒𝐴) = 0) |
17 | id 22 | . . . . 5 ⊢ (𝐴 = -∞ → 𝐴 = -∞) | |
18 | xnegeq 13246 | . . . . . 6 ⊢ (𝐴 = -∞ → -𝑒𝐴 = -𝑒-∞) | |
19 | xnegmnf 13249 | . . . . . 6 ⊢ -𝑒-∞ = +∞ | |
20 | 18, 19 | eqtrdi 2791 | . . . . 5 ⊢ (𝐴 = -∞ → -𝑒𝐴 = +∞) |
21 | 17, 20 | oveq12d 7449 | . . . 4 ⊢ (𝐴 = -∞ → (𝐴 +𝑒 -𝑒𝐴) = (-∞ +𝑒 +∞)) |
22 | mnfaddpnf 13270 | . . . 4 ⊢ (-∞ +𝑒 +∞) = 0 | |
23 | 21, 22 | eqtrdi 2791 | . . 3 ⊢ (𝐴 = -∞ → (𝐴 +𝑒 -𝑒𝐴) = 0) |
24 | 9, 16, 23 | 3jaoi 1427 | . 2 ⊢ ((𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞) → (𝐴 +𝑒 -𝑒𝐴) = 0) |
25 | 1, 24 | sylbi 217 | 1 ⊢ (𝐴 ∈ ℝ* → (𝐴 +𝑒 -𝑒𝐴) = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ w3o 1085 = wceq 1537 ∈ wcel 2106 (class class class)co 7431 ℝcr 11152 0cc0 11153 + caddc 11156 +∞cpnf 11290 -∞cmnf 11291 ℝ*cxr 11292 -cneg 11491 -𝑒cxne 13149 +𝑒 cxad 13150 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-po 5597 df-so 5598 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-sub 11492 df-neg 11493 df-xneg 13152 df-xadd 13153 |
This theorem is referenced by: xrsxmet 24845 xaddeq0 32764 xlt2addrd 32769 xrge0npcan 33008 carsgclctunlem2 34301 |
Copyright terms: Public domain | W3C validator |