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Mirrors > Home > MPE Home > Th. List > addeq0 | Structured version Visualization version GIF version |
Description: Two complex numbers add up to zero iff they are each other's opposites. (Contributed by Thierry Arnoux, 2-May-2017.) |
Ref | Expression |
---|---|
addeq0 | ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 𝐵) = 0 ↔ 𝐴 = -𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0cnd 11186 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → 0 ∈ ℂ) | |
2 | simpr 485 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → 𝐵 ∈ ℂ) | |
3 | simpl 483 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → 𝐴 ∈ ℂ) | |
4 | 1, 2, 3 | subadd2d 11569 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((0 − 𝐵) = 𝐴 ↔ (𝐴 + 𝐵) = 0)) |
5 | df-neg 11426 | . . . 4 ⊢ -𝐵 = (0 − 𝐵) | |
6 | 5 | eqeq1i 2736 | . . 3 ⊢ (-𝐵 = 𝐴 ↔ (0 − 𝐵) = 𝐴) |
7 | eqcom 2738 | . . 3 ⊢ (-𝐵 = 𝐴 ↔ 𝐴 = -𝐵) | |
8 | 6, 7 | bitr3i 276 | . 2 ⊢ ((0 − 𝐵) = 𝐴 ↔ 𝐴 = -𝐵) |
9 | 4, 8 | bitr3di 285 | 1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 𝐵) = 0 ↔ 𝐴 = -𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 (class class class)co 7390 ℂcc 11087 0cc0 11089 + caddc 11092 − cmin 11423 -cneg 11424 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7705 ax-resscn 11146 ax-1cn 11147 ax-icn 11148 ax-addcl 11149 ax-addrcl 11150 ax-mulcl 11151 ax-mulrcl 11152 ax-mulcom 11153 ax-addass 11154 ax-mulass 11155 ax-distr 11156 ax-i2m1 11157 ax-1ne0 11158 ax-1rid 11159 ax-rnegex 11160 ax-rrecex 11161 ax-cnre 11162 ax-pre-lttri 11163 ax-pre-lttrn 11164 ax-pre-ltadd 11165 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3430 df-v 3472 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4520 df-pw 4595 df-sn 4620 df-pr 4622 df-op 4626 df-uni 4899 df-br 5139 df-opab 5201 df-mpt 5222 df-id 5564 df-po 5578 df-so 5579 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-iota 6481 df-fun 6531 df-fn 6532 df-f 6533 df-f1 6534 df-fo 6535 df-f1o 6536 df-fv 6537 df-riota 7346 df-ov 7393 df-oprab 7394 df-mpo 7395 df-er 8683 df-en 8920 df-dom 8921 df-sdom 8922 df-pnf 11229 df-mnf 11230 df-ltxr 11232 df-sub 11425 df-neg 11426 |
This theorem is referenced by: ballotlemfrceq 33342 sqrtcval 42149 rrx2linest 47062 itsclquadb 47096 |
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