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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > addeq0 | Structured version Visualization version GIF version |
Description: Two complex which add up to zero are each other's negatives. (Contributed by Thierry Arnoux, 2-May-2017.) |
Ref | Expression |
---|---|
addeq0 | ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 𝐵) = 0 ↔ 𝐴 = -𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-neg 10457 | . . . 4 ⊢ -𝐵 = (0 − 𝐵) | |
2 | 1 | eqeq1i 2761 | . . 3 ⊢ (-𝐵 = 𝐴 ↔ (0 − 𝐵) = 𝐴) |
3 | eqcom 2763 | . . 3 ⊢ (-𝐵 = 𝐴 ↔ 𝐴 = -𝐵) | |
4 | 2, 3 | bitr3i 266 | . 2 ⊢ ((0 − 𝐵) = 𝐴 ↔ 𝐴 = -𝐵) |
5 | 0cnd 10221 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → 0 ∈ ℂ) | |
6 | simpr 479 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → 𝐵 ∈ ℂ) | |
7 | simpl 474 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → 𝐴 ∈ ℂ) | |
8 | 5, 6, 7 | subadd2d 10599 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((0 − 𝐵) = 𝐴 ↔ (𝐴 + 𝐵) = 0)) |
9 | 4, 8 | syl5rbbr 275 | 1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 𝐵) = 0 ↔ 𝐴 = -𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 383 = wceq 1628 ∈ wcel 2135 (class class class)co 6809 ℂcc 10122 0cc0 10124 + caddc 10127 − cmin 10454 -cneg 10455 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1867 ax-4 1882 ax-5 1984 ax-6 2050 ax-7 2086 ax-8 2137 ax-9 2144 ax-10 2164 ax-11 2179 ax-12 2192 ax-13 2387 ax-ext 2736 ax-sep 4929 ax-nul 4937 ax-pow 4988 ax-pr 5051 ax-un 7110 ax-resscn 10181 ax-1cn 10182 ax-icn 10183 ax-addcl 10184 ax-addrcl 10185 ax-mulcl 10186 ax-mulrcl 10187 ax-mulcom 10188 ax-addass 10189 ax-mulass 10190 ax-distr 10191 ax-i2m1 10192 ax-1ne0 10193 ax-1rid 10194 ax-rnegex 10195 ax-rrecex 10196 ax-cnre 10197 ax-pre-lttri 10198 ax-pre-lttrn 10199 ax-pre-ltadd 10200 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1631 df-ex 1850 df-nf 1855 df-sb 2043 df-eu 2607 df-mo 2608 df-clab 2743 df-cleq 2749 df-clel 2752 df-nfc 2887 df-ne 2929 df-nel 3032 df-ral 3051 df-rex 3052 df-reu 3053 df-rab 3055 df-v 3338 df-sbc 3573 df-csb 3671 df-dif 3714 df-un 3716 df-in 3718 df-ss 3725 df-nul 4055 df-if 4227 df-pw 4300 df-sn 4318 df-pr 4320 df-op 4324 df-uni 4585 df-br 4801 df-opab 4861 df-mpt 4878 df-id 5170 df-po 5183 df-so 5184 df-xp 5268 df-rel 5269 df-cnv 5270 df-co 5271 df-dm 5272 df-rn 5273 df-res 5274 df-ima 5275 df-iota 6008 df-fun 6047 df-fn 6048 df-f 6049 df-f1 6050 df-fo 6051 df-f1o 6052 df-fv 6053 df-riota 6770 df-ov 6812 df-oprab 6813 df-mpt2 6814 df-er 7907 df-en 8118 df-dom 8119 df-sdom 8120 df-pnf 10264 df-mnf 10265 df-ltxr 10267 df-sub 10456 df-neg 10457 |
This theorem is referenced by: ballotlemfrceq 30895 |
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