![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > eqneg | 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 7309 | . . 3 ⊢ (𝐴 ∈ ℂ → ((1 + 1) · 𝐴) = (𝐴 + 𝐴)) | |
2 | negid 7422 | . . . 4 ⊢ (𝐴 ∈ ℂ → (𝐴 + -𝐴) = 0) | |
3 | ax-1cn 7131 | . . . . . 6 ⊢ 1 ∈ ℂ | |
4 | 3, 3 | addcli 7185 | . . . . 5 ⊢ (1 + 1) ∈ ℂ |
5 | 4 | mul01i 7562 | . . . 4 ⊢ ((1 + 1) · 0) = 0 |
6 | 2, 5 | syl6reqr 2133 | . . 3 ⊢ (𝐴 ∈ ℂ → ((1 + 1) · 0) = (𝐴 + -𝐴)) |
7 | 1, 6 | eqeq12d 2096 | . 2 ⊢ (𝐴 ∈ ℂ → (((1 + 1) · 𝐴) = ((1 + 1) · 0) ↔ (𝐴 + 𝐴) = (𝐴 + -𝐴))) |
8 | id 19 | . . 3 ⊢ (𝐴 ∈ ℂ → 𝐴 ∈ ℂ) | |
9 | 0cnd 7174 | . . 3 ⊢ (𝐴 ∈ ℂ → 0 ∈ ℂ) | |
10 | 4 | a1i 9 | . . 3 ⊢ (𝐴 ∈ ℂ → (1 + 1) ∈ ℂ) |
11 | 1re 7180 | . . . . . 6 ⊢ 1 ∈ ℝ | |
12 | 11, 11 | readdcli 7194 | . . . . 5 ⊢ (1 + 1) ∈ ℝ |
13 | 0lt1 7303 | . . . . . 6 ⊢ 0 < 1 | |
14 | 11, 11, 13, 13 | addgt0ii 7659 | . . . . 5 ⊢ 0 < (1 + 1) |
15 | 12, 14 | gt0ap0ii 7794 | . . . 4 ⊢ (1 + 1) # 0 |
16 | 15 | a1i 9 | . . 3 ⊢ (𝐴 ∈ ℂ → (1 + 1) # 0) |
17 | 8, 9, 10, 16 | mulcanapd 7818 | . 2 ⊢ (𝐴 ∈ ℂ → (((1 + 1) · 𝐴) = ((1 + 1) · 0) ↔ 𝐴 = 0)) |
18 | negcl 7375 | . . 3 ⊢ (𝐴 ∈ ℂ → -𝐴 ∈ ℂ) | |
19 | 8, 8, 18 | addcand 7359 | . 2 ⊢ (𝐴 ∈ ℂ → ((𝐴 + 𝐴) = (𝐴 + -𝐴) ↔ 𝐴 = -𝐴)) |
20 | 7, 17, 19 | 3bitr3rd 217 | 1 ⊢ (𝐴 ∈ ℂ → (𝐴 = -𝐴 ↔ 𝐴 = 0)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 103 = wceq 1285 ∈ wcel 1434 class class class wbr 3793 (class class class)co 5543 ℂcc 7041 0cc0 7043 1c1 7044 + caddc 7046 · cmul 7048 -cneg 7347 # cap 7748 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 577 ax-in2 578 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-13 1445 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2064 ax-sep 3904 ax-pow 3956 ax-pr 3972 ax-un 4196 ax-setind 4288 ax-cnex 7129 ax-resscn 7130 ax-1cn 7131 ax-1re 7132 ax-icn 7133 ax-addcl 7134 ax-addrcl 7135 ax-mulcl 7136 ax-mulrcl 7137 ax-addcom 7138 ax-mulcom 7139 ax-addass 7140 ax-mulass 7141 ax-distr 7142 ax-i2m1 7143 ax-0lt1 7144 ax-1rid 7145 ax-0id 7146 ax-rnegex 7147 ax-precex 7148 ax-cnre 7149 ax-pre-ltirr 7150 ax-pre-ltwlin 7151 ax-pre-lttrn 7152 ax-pre-apti 7153 ax-pre-ltadd 7154 ax-pre-mulgt0 7155 ax-pre-mulext 7156 |
This theorem depends on definitions: df-bi 115 df-3an 922 df-tru 1288 df-fal 1291 df-nf 1391 df-sb 1687 df-eu 1945 df-mo 1946 df-clab 2069 df-cleq 2075 df-clel 2078 df-nfc 2209 df-ne 2247 df-nel 2341 df-ral 2354 df-rex 2355 df-reu 2356 df-rab 2358 df-v 2604 df-sbc 2817 df-dif 2976 df-un 2978 df-in 2980 df-ss 2987 df-pw 3392 df-sn 3412 df-pr 3413 df-op 3415 df-uni 3610 df-br 3794 df-opab 3848 df-id 4056 df-po 4059 df-iso 4060 df-xp 4377 df-rel 4378 df-cnv 4379 df-co 4380 df-dm 4381 df-iota 4897 df-fun 4934 df-fv 4940 df-riota 5499 df-ov 5546 df-oprab 5547 df-mpt2 5548 df-pnf 7217 df-mnf 7218 df-xr 7219 df-ltxr 7220 df-le 7221 df-sub 7348 df-neg 7349 df-reap 7742 df-ap 7749 |
This theorem is referenced by: eqnegd 7888 eqnegi 7896 |
Copyright terms: Public domain | W3C validator |