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Mirrors > Home > ILE Home > Th. List > 0cnd | GIF version |
Description: 0 is a complex number, deductive form. (Contributed by David A. Wheeler, 8-Dec-2018.) |
Ref | Expression |
---|---|
0cnd | ⊢ (𝜑 → 0 ∈ ℂ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0cn 7782 | . 2 ⊢ 0 ∈ ℂ | |
2 | 1 | a1i 9 | 1 ⊢ (𝜑 → 0 ∈ ℂ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1481 ℂcc 7642 0cc0 7644 |
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-5 1424 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-4 1488 ax-17 1507 ax-ial 1515 ax-ext 2122 ax-1cn 7737 ax-icn 7739 ax-addcl 7740 ax-mulcl 7742 ax-i2m1 7749 |
This theorem depends on definitions: df-bi 116 df-cleq 2133 df-clel 2136 |
This theorem is referenced by: mulap0r 8401 mulap0 8439 mul0eqap 8455 diveqap0 8466 eqneg 8516 div2subap 8620 prodgt0 8634 un0addcl 9034 un0mulcl 9035 modsumfzodifsn 10200 ser0 10318 ser0f 10319 abs00ap 10866 abs00 10868 abssubne0 10895 mul0inf 11044 clim0c 11087 sumrbdclem 11178 summodclem2a 11182 zsumdc 11185 fsum3 11188 isumz 11190 isumss 11192 fisumss 11193 fsum3cvg2 11195 fsum3ser 11198 fsumcl2lem 11199 fsumcl 11201 fsumadd 11207 fsumsplit 11208 sumsnf 11210 sumsplitdc 11233 fsummulc2 11249 ef0lem 11403 ef4p 11437 tanvalap 11451 fsumcncntop 12764 limcimolemlt 12841 dvmptcmulcn 12891 dveflem 12895 dvef 12896 ptolemy 12953 apdiff 13416 |
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