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Mirrors > Home > MPE Home > Th. List > eqneg | Structured version Visualization version GIF version |
Description: A number equal to its negative is zero. (Contributed by NM, 12-Jul-2005.) (Revised by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
eqneg | ⊢ (𝐴 ∈ ℂ → (𝐴 = -𝐴 ↔ 𝐴 = 0)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1p1times 11417 | . . 3 ⊢ (𝐴 ∈ ℂ → ((1 + 1) · 𝐴) = (𝐴 + 𝐴)) | |
2 | ax-1cn 11198 | . . . . . 6 ⊢ 1 ∈ ℂ | |
3 | 2, 2 | addcli 11252 | . . . . 5 ⊢ (1 + 1) ∈ ℂ |
4 | 3 | mul01i 11436 | . . . 4 ⊢ ((1 + 1) · 0) = 0 |
5 | negid 11539 | . . . 4 ⊢ (𝐴 ∈ ℂ → (𝐴 + -𝐴) = 0) | |
6 | 4, 5 | eqtr4id 2784 | . . 3 ⊢ (𝐴 ∈ ℂ → ((1 + 1) · 0) = (𝐴 + -𝐴)) |
7 | 1, 6 | eqeq12d 2741 | . 2 ⊢ (𝐴 ∈ ℂ → (((1 + 1) · 𝐴) = ((1 + 1) · 0) ↔ (𝐴 + 𝐴) = (𝐴 + -𝐴))) |
8 | id 22 | . . 3 ⊢ (𝐴 ∈ ℂ → 𝐴 ∈ ℂ) | |
9 | 0cnd 11239 | . . 3 ⊢ (𝐴 ∈ ℂ → 0 ∈ ℂ) | |
10 | 3 | a1i 11 | . . 3 ⊢ (𝐴 ∈ ℂ → (1 + 1) ∈ ℂ) |
11 | 1re 11246 | . . . . . 6 ⊢ 1 ∈ ℝ | |
12 | 11, 11 | readdcli 11261 | . . . . 5 ⊢ (1 + 1) ∈ ℝ |
13 | 0lt1 11768 | . . . . . 6 ⊢ 0 < 1 | |
14 | 11, 11, 13, 13 | addgt0ii 11788 | . . . . 5 ⊢ 0 < (1 + 1) |
15 | 12, 14 | gt0ne0ii 11782 | . . . 4 ⊢ (1 + 1) ≠ 0 |
16 | 15 | a1i 11 | . . 3 ⊢ (𝐴 ∈ ℂ → (1 + 1) ≠ 0) |
17 | 8, 9, 10, 16 | mulcand 11879 | . 2 ⊢ (𝐴 ∈ ℂ → (((1 + 1) · 𝐴) = ((1 + 1) · 0) ↔ 𝐴 = 0)) |
18 | negcl 11492 | . . 3 ⊢ (𝐴 ∈ ℂ → -𝐴 ∈ ℂ) | |
19 | 8, 8, 18 | addcand 11449 | . 2 ⊢ (𝐴 ∈ ℂ → ((𝐴 + 𝐴) = (𝐴 + -𝐴) ↔ 𝐴 = -𝐴)) |
20 | 7, 17, 19 | 3bitr3rd 309 | 1 ⊢ (𝐴 ∈ ℂ → (𝐴 = -𝐴 ↔ 𝐴 = 0)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1533 ∈ wcel 2098 ≠ wne 2929 (class class class)co 7419 ℂcc 11138 0cc0 11140 1c1 11141 + caddc 11143 · cmul 11145 -cneg 11477 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-po 5590 df-so 5591 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 |
This theorem is referenced by: eqnegd 11968 eqnegi 11976 addsubeq0 46811 |
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