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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 10495 | . . 3 ⊢ (𝐴 ∈ ℂ → ((1 + 1) · 𝐴) = (𝐴 + 𝐴)) | |
2 | negid 10618 | . . . 4 ⊢ (𝐴 ∈ ℂ → (𝐴 + -𝐴) = 0) | |
3 | ax-1cn 10280 | . . . . . 6 ⊢ 1 ∈ ℂ | |
4 | 3, 3 | addcli 10333 | . . . . 5 ⊢ (1 + 1) ∈ ℂ |
5 | 4 | mul01i 10514 | . . . 4 ⊢ ((1 + 1) · 0) = 0 |
6 | 2, 5 | syl6reqr 2850 | . . 3 ⊢ (𝐴 ∈ ℂ → ((1 + 1) · 0) = (𝐴 + -𝐴)) |
7 | 1, 6 | eqeq12d 2812 | . 2 ⊢ (𝐴 ∈ ℂ → (((1 + 1) · 𝐴) = ((1 + 1) · 0) ↔ (𝐴 + 𝐴) = (𝐴 + -𝐴))) |
8 | id 22 | . . 3 ⊢ (𝐴 ∈ ℂ → 𝐴 ∈ ℂ) | |
9 | 0cnd 10319 | . . 3 ⊢ (𝐴 ∈ ℂ → 0 ∈ ℂ) | |
10 | 4 | a1i 11 | . . 3 ⊢ (𝐴 ∈ ℂ → (1 + 1) ∈ ℂ) |
11 | 1re 10326 | . . . . . 6 ⊢ 1 ∈ ℝ | |
12 | 11, 11 | readdcli 10342 | . . . . 5 ⊢ (1 + 1) ∈ ℝ |
13 | 0lt1 10840 | . . . . . 6 ⊢ 0 < 1 | |
14 | 11, 11, 13, 13 | addgt0ii 10860 | . . . . 5 ⊢ 0 < (1 + 1) |
15 | 12, 14 | gt0ne0ii 10854 | . . . 4 ⊢ (1 + 1) ≠ 0 |
16 | 15 | a1i 11 | . . 3 ⊢ (𝐴 ∈ ℂ → (1 + 1) ≠ 0) |
17 | 8, 9, 10, 16 | mulcand 10950 | . 2 ⊢ (𝐴 ∈ ℂ → (((1 + 1) · 𝐴) = ((1 + 1) · 0) ↔ 𝐴 = 0)) |
18 | negcl 10570 | . . 3 ⊢ (𝐴 ∈ ℂ → -𝐴 ∈ ℂ) | |
19 | 8, 8, 18 | addcand 10527 | . 2 ⊢ (𝐴 ∈ ℂ → ((𝐴 + 𝐴) = (𝐴 + -𝐴) ↔ 𝐴 = -𝐴)) |
20 | 7, 17, 19 | 3bitr3rd 302 | 1 ⊢ (𝐴 ∈ ℂ → (𝐴 = -𝐴 ↔ 𝐴 = 0)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 = wceq 1653 ∈ wcel 2157 ≠ wne 2969 (class class class)co 6876 ℂcc 10220 0cc0 10222 1c1 10223 + caddc 10225 · cmul 10227 -cneg 10555 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2375 ax-ext 2775 ax-sep 4973 ax-nul 4981 ax-pow 5033 ax-pr 5095 ax-un 7181 ax-resscn 10279 ax-1cn 10280 ax-icn 10281 ax-addcl 10282 ax-addrcl 10283 ax-mulcl 10284 ax-mulrcl 10285 ax-mulcom 10286 ax-addass 10287 ax-mulass 10288 ax-distr 10289 ax-i2m1 10290 ax-1ne0 10291 ax-1rid 10292 ax-rnegex 10293 ax-rrecex 10294 ax-cnre 10295 ax-pre-lttri 10296 ax-pre-lttrn 10297 ax-pre-ltadd 10298 ax-pre-mulgt0 10299 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2590 df-eu 2607 df-clab 2784 df-cleq 2790 df-clel 2793 df-nfc 2928 df-ne 2970 df-nel 3073 df-ral 3092 df-rex 3093 df-reu 3094 df-rab 3096 df-v 3385 df-sbc 3632 df-csb 3727 df-dif 3770 df-un 3772 df-in 3774 df-ss 3781 df-nul 4114 df-if 4276 df-pw 4349 df-sn 4367 df-pr 4369 df-op 4373 df-uni 4627 df-br 4842 df-opab 4904 df-mpt 4921 df-id 5218 df-po 5231 df-so 5232 df-xp 5316 df-rel 5317 df-cnv 5318 df-co 5319 df-dm 5320 df-rn 5321 df-res 5322 df-ima 5323 df-iota 6062 df-fun 6101 df-fn 6102 df-f 6103 df-f1 6104 df-fo 6105 df-f1o 6106 df-fv 6107 df-riota 6837 df-ov 6879 df-oprab 6880 df-mpt2 6881 df-er 7980 df-en 8194 df-dom 8195 df-sdom 8196 df-pnf 10363 df-mnf 10364 df-xr 10365 df-ltxr 10366 df-le 10367 df-sub 10556 df-neg 10557 |
This theorem is referenced by: eqnegd 11036 eqnegi 11044 |
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