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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 11432 | . . 3 ⊢ (𝐴 ∈ ℂ → ((1 + 1) · 𝐴) = (𝐴 + 𝐴)) | |
| 2 | ax-1cn 11213 | . . . . . 6 ⊢ 1 ∈ ℂ | |
| 3 | 2, 2 | addcli 11267 | . . . . 5 ⊢ (1 + 1) ∈ ℂ |
| 4 | 3 | mul01i 11451 | . . . 4 ⊢ ((1 + 1) · 0) = 0 |
| 5 | negid 11556 | . . . 4 ⊢ (𝐴 ∈ ℂ → (𝐴 + -𝐴) = 0) | |
| 6 | 4, 5 | eqtr4id 2796 | . . 3 ⊢ (𝐴 ∈ ℂ → ((1 + 1) · 0) = (𝐴 + -𝐴)) |
| 7 | 1, 6 | eqeq12d 2753 | . 2 ⊢ (𝐴 ∈ ℂ → (((1 + 1) · 𝐴) = ((1 + 1) · 0) ↔ (𝐴 + 𝐴) = (𝐴 + -𝐴))) |
| 8 | id 22 | . . 3 ⊢ (𝐴 ∈ ℂ → 𝐴 ∈ ℂ) | |
| 9 | 0cnd 11254 | . . 3 ⊢ (𝐴 ∈ ℂ → 0 ∈ ℂ) | |
| 10 | 3 | a1i 11 | . . 3 ⊢ (𝐴 ∈ ℂ → (1 + 1) ∈ ℂ) |
| 11 | 1re 11261 | . . . . . 6 ⊢ 1 ∈ ℝ | |
| 12 | 11, 11 | readdcli 11276 | . . . . 5 ⊢ (1 + 1) ∈ ℝ |
| 13 | 0lt1 11785 | . . . . . 6 ⊢ 0 < 1 | |
| 14 | 11, 11, 13, 13 | addgt0ii 11805 | . . . . 5 ⊢ 0 < (1 + 1) |
| 15 | 12, 14 | gt0ne0ii 11799 | . . . 4 ⊢ (1 + 1) ≠ 0 |
| 16 | 15 | a1i 11 | . . 3 ⊢ (𝐴 ∈ ℂ → (1 + 1) ≠ 0) |
| 17 | 8, 9, 10, 16 | mulcand 11896 | . 2 ⊢ (𝐴 ∈ ℂ → (((1 + 1) · 𝐴) = ((1 + 1) · 0) ↔ 𝐴 = 0)) |
| 18 | negcl 11508 | . . 3 ⊢ (𝐴 ∈ ℂ → -𝐴 ∈ ℂ) | |
| 19 | 8, 8, 18 | addcand 11464 | . 2 ⊢ (𝐴 ∈ ℂ → ((𝐴 + 𝐴) = (𝐴 + -𝐴) ↔ 𝐴 = -𝐴)) |
| 20 | 7, 17, 19 | 3bitr3rd 310 | 1 ⊢ (𝐴 ∈ ℂ → (𝐴 = -𝐴 ↔ 𝐴 = 0)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 (class class class)co 7431 ℂcc 11153 0cc0 11155 1c1 11156 + caddc 11158 · cmul 11160 -cneg 11493 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-po 5592 df-so 5593 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 |
| This theorem is referenced by: eqnegd 11988 eqnegi 11996 addsubeq0 47308 |
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