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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 8023 | . . 3 ⊢ (𝐴 ∈ ℂ → ((1 + 1) · 𝐴) = (𝐴 + 𝐴)) | |
2 | ax-1cn 7837 | . . . . . 6 ⊢ 1 ∈ ℂ | |
3 | 2, 2 | addcli 7894 | . . . . 5 ⊢ (1 + 1) ∈ ℂ |
4 | 3 | mul01i 8280 | . . . 4 ⊢ ((1 + 1) · 0) = 0 |
5 | negid 8136 | . . . 4 ⊢ (𝐴 ∈ ℂ → (𝐴 + -𝐴) = 0) | |
6 | 4, 5 | eqtr4id 2216 | . . 3 ⊢ (𝐴 ∈ ℂ → ((1 + 1) · 0) = (𝐴 + -𝐴)) |
7 | 1, 6 | eqeq12d 2179 | . 2 ⊢ (𝐴 ∈ ℂ → (((1 + 1) · 𝐴) = ((1 + 1) · 0) ↔ (𝐴 + 𝐴) = (𝐴 + -𝐴))) |
8 | id 19 | . . 3 ⊢ (𝐴 ∈ ℂ → 𝐴 ∈ ℂ) | |
9 | 0cnd 7883 | . . 3 ⊢ (𝐴 ∈ ℂ → 0 ∈ ℂ) | |
10 | 3 | a1i 9 | . . 3 ⊢ (𝐴 ∈ ℂ → (1 + 1) ∈ ℂ) |
11 | 1re 7889 | . . . . . 6 ⊢ 1 ∈ ℝ | |
12 | 11, 11 | readdcli 7903 | . . . . 5 ⊢ (1 + 1) ∈ ℝ |
13 | 0lt1 8016 | . . . . . 6 ⊢ 0 < 1 | |
14 | 11, 11, 13, 13 | addgt0ii 8380 | . . . . 5 ⊢ 0 < (1 + 1) |
15 | 12, 14 | gt0ap0ii 8517 | . . . 4 ⊢ (1 + 1) # 0 |
16 | 15 | a1i 9 | . . 3 ⊢ (𝐴 ∈ ℂ → (1 + 1) # 0) |
17 | 8, 9, 10, 16 | mulcanapd 8549 | . 2 ⊢ (𝐴 ∈ ℂ → (((1 + 1) · 𝐴) = ((1 + 1) · 0) ↔ 𝐴 = 0)) |
18 | negcl 8089 | . . 3 ⊢ (𝐴 ∈ ℂ → -𝐴 ∈ ℂ) | |
19 | 8, 8, 18 | addcand 8073 | . 2 ⊢ (𝐴 ∈ ℂ → ((𝐴 + 𝐴) = (𝐴 + -𝐴) ↔ 𝐴 = -𝐴)) |
20 | 7, 17, 19 | 3bitr3rd 218 | 1 ⊢ (𝐴 ∈ ℂ → (𝐴 = -𝐴 ↔ 𝐴 = 0)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 = wceq 1342 ∈ wcel 2135 class class class wbr 3976 (class class class)co 5836 ℂcc 7742 0cc0 7744 1c1 7745 + caddc 7747 · cmul 7749 -cneg 8061 # cap 8470 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-cnex 7835 ax-resscn 7836 ax-1cn 7837 ax-1re 7838 ax-icn 7839 ax-addcl 7840 ax-addrcl 7841 ax-mulcl 7842 ax-mulrcl 7843 ax-addcom 7844 ax-mulcom 7845 ax-addass 7846 ax-mulass 7847 ax-distr 7848 ax-i2m1 7849 ax-0lt1 7850 ax-1rid 7851 ax-0id 7852 ax-rnegex 7853 ax-precex 7854 ax-cnre 7855 ax-pre-ltirr 7856 ax-pre-ltwlin 7857 ax-pre-lttrn 7858 ax-pre-apti 7859 ax-pre-ltadd 7860 ax-pre-mulgt0 7861 ax-pre-mulext 7862 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2723 df-sbc 2947 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-br 3977 df-opab 4038 df-id 4265 df-po 4268 df-iso 4269 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-iota 5147 df-fun 5184 df-fv 5190 df-riota 5792 df-ov 5839 df-oprab 5840 df-mpo 5841 df-pnf 7926 df-mnf 7927 df-xr 7928 df-ltxr 7929 df-le 7930 df-sub 8062 df-neg 8063 df-reap 8464 df-ap 8471 |
This theorem is referenced by: eqnegd 8620 eqnegi 8628 |
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