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Mirrors > Home > MPE Home > Th. List > cjne0 | Structured version Visualization version GIF version |
Description: A number is nonzero iff its complex conjugate is nonzero. (Contributed by NM, 29-Apr-2005.) |
Ref | Expression |
---|---|
cjne0 | ⊢ (𝐴 ∈ ℂ → (𝐴 ≠ 0 ↔ (∗‘𝐴) ≠ 0)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0cn 11205 | . . . 4 ⊢ 0 ∈ ℂ | |
2 | cj11 15111 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 0 ∈ ℂ) → ((∗‘𝐴) = (∗‘0) ↔ 𝐴 = 0)) | |
3 | 1, 2 | mpan2 688 | . . 3 ⊢ (𝐴 ∈ ℂ → ((∗‘𝐴) = (∗‘0) ↔ 𝐴 = 0)) |
4 | cj0 15107 | . . . 4 ⊢ (∗‘0) = 0 | |
5 | 4 | eqeq2i 2737 | . . 3 ⊢ ((∗‘𝐴) = (∗‘0) ↔ (∗‘𝐴) = 0) |
6 | 3, 5 | bitr3di 286 | . 2 ⊢ (𝐴 ∈ ℂ → (𝐴 = 0 ↔ (∗‘𝐴) = 0)) |
7 | 6 | necon3bid 2977 | 1 ⊢ (𝐴 ∈ ℂ → (𝐴 ≠ 0 ↔ (∗‘𝐴) ≠ 0)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1533 ∈ wcel 2098 ≠ wne 2932 ‘cfv 6534 ℂcc 11105 0cc0 11107 ∗ccj 15045 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-br 5140 df-opab 5202 df-mpt 5223 df-id 5565 df-po 5579 df-so 5580 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-er 8700 df-en 8937 df-dom 8938 df-sdom 8939 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-div 11871 df-2 12274 df-cj 15048 df-re 15049 df-im 15050 |
This theorem is referenced by: cjdiv 15113 cjne0d 15152 recval 15271 logcj 26480 |
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