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Mirrors > Home > MPE Home > Th. List > xnegid | Structured version Visualization version GIF version |
Description: Extended real version of negid 10207. (Contributed by Mario Carneiro, 20-Aug-2015.) |
Ref | Expression |
---|---|
xnegid | ⊢ (𝐴 ∈ ℝ* → (𝐴 +𝑒 -𝑒𝐴) = 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elxr 11826 | . 2 ⊢ (𝐴 ∈ ℝ* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞)) | |
2 | rexneg 11916 | . . . . 5 ⊢ (𝐴 ∈ ℝ → -𝑒𝐴 = -𝐴) | |
3 | 2 | oveq2d 6565 | . . . 4 ⊢ (𝐴 ∈ ℝ → (𝐴 +𝑒 -𝑒𝐴) = (𝐴 +𝑒 -𝐴)) |
4 | renegcl 10223 | . . . . 5 ⊢ (𝐴 ∈ ℝ → -𝐴 ∈ ℝ) | |
5 | rexadd 11937 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ -𝐴 ∈ ℝ) → (𝐴 +𝑒 -𝐴) = (𝐴 + -𝐴)) | |
6 | 4, 5 | mpdan 699 | . . . 4 ⊢ (𝐴 ∈ ℝ → (𝐴 +𝑒 -𝐴) = (𝐴 + -𝐴)) |
7 | recn 9905 | . . . . 5 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ) | |
8 | 7 | negidd 10261 | . . . 4 ⊢ (𝐴 ∈ ℝ → (𝐴 + -𝐴) = 0) |
9 | 3, 6, 8 | 3eqtrd 2648 | . . 3 ⊢ (𝐴 ∈ ℝ → (𝐴 +𝑒 -𝑒𝐴) = 0) |
10 | id 22 | . . . . 5 ⊢ (𝐴 = +∞ → 𝐴 = +∞) | |
11 | xnegeq 11912 | . . . . . 6 ⊢ (𝐴 = +∞ → -𝑒𝐴 = -𝑒+∞) | |
12 | xnegpnf 11914 | . . . . . 6 ⊢ -𝑒+∞ = -∞ | |
13 | 11, 12 | syl6eq 2660 | . . . . 5 ⊢ (𝐴 = +∞ → -𝑒𝐴 = -∞) |
14 | 10, 13 | oveq12d 6567 | . . . 4 ⊢ (𝐴 = +∞ → (𝐴 +𝑒 -𝑒𝐴) = (+∞ +𝑒 -∞)) |
15 | pnfaddmnf 11935 | . . . 4 ⊢ (+∞ +𝑒 -∞) = 0 | |
16 | 14, 15 | syl6eq 2660 | . . 3 ⊢ (𝐴 = +∞ → (𝐴 +𝑒 -𝑒𝐴) = 0) |
17 | id 22 | . . . . 5 ⊢ (𝐴 = -∞ → 𝐴 = -∞) | |
18 | xnegeq 11912 | . . . . . 6 ⊢ (𝐴 = -∞ → -𝑒𝐴 = -𝑒-∞) | |
19 | xnegmnf 11915 | . . . . . 6 ⊢ -𝑒-∞ = +∞ | |
20 | 18, 19 | syl6eq 2660 | . . . . 5 ⊢ (𝐴 = -∞ → -𝑒𝐴 = +∞) |
21 | 17, 20 | oveq12d 6567 | . . . 4 ⊢ (𝐴 = -∞ → (𝐴 +𝑒 -𝑒𝐴) = (-∞ +𝑒 +∞)) |
22 | mnfaddpnf 11936 | . . . 4 ⊢ (-∞ +𝑒 +∞) = 0 | |
23 | 21, 22 | syl6eq 2660 | . . 3 ⊢ (𝐴 = -∞ → (𝐴 +𝑒 -𝑒𝐴) = 0) |
24 | 9, 16, 23 | 3jaoi 1383 | . 2 ⊢ ((𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞) → (𝐴 +𝑒 -𝑒𝐴) = 0) |
25 | 1, 24 | sylbi 206 | 1 ⊢ (𝐴 ∈ ℝ* → (𝐴 +𝑒 -𝑒𝐴) = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ w3o 1030 = wceq 1475 ∈ wcel 1977 (class class class)co 6549 ℝcr 9814 0cc0 9815 + caddc 9818 +∞cpnf 9950 -∞cmnf 9951 ℝ*cxr 9952 -cneg 10146 -𝑒cxne 11819 +𝑒 cxad 11820 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-po 4959 df-so 4960 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-sub 10147 df-neg 10148 df-xneg 11822 df-xadd 11823 |
This theorem is referenced by: xrsxmet 22420 xaddeq0 28907 xlt2addrd 28913 xrge0npcan 29025 carsgclctunlem2 29708 |
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