Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > nncnd | GIF version |
Description: A positive integer is a complex number. (Contributed by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
nnred.1 | ⊢ (𝜑 → 𝐴 ∈ ℕ) |
Ref | Expression |
---|---|
nncnd | ⊢ (𝜑 → 𝐴 ∈ ℂ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnsscn 8693 | . 2 ⊢ ℕ ⊆ ℂ | |
2 | nnred.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℕ) | |
3 | 1, 2 | sseldi 3065 | 1 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1465 ℂcc 7586 ℕcn 8688 |
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-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-cnex 7679 ax-resscn 7680 ax-1re 7682 ax-addrcl 7685 |
This theorem depends on definitions: df-bi 116 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-v 2662 df-in 3047 df-ss 3054 df-int 3742 df-inn 8689 |
This theorem is referenced by: peano5uzti 9127 qapne 9399 qtri3or 9988 exbtwnzlemstep 9993 intfracq 10061 flqdiv 10062 modqmulnn 10083 addmodid 10113 modaddmodup 10128 modsumfzodifsn 10137 addmodlteq 10139 facdiv 10452 facndiv 10453 faclbnd 10455 faclbnd6 10458 facubnd 10459 facavg 10460 bccmpl 10468 bcn0 10469 bcn1 10472 bcm1k 10474 bcp1n 10475 bcp1nk 10476 bcval5 10477 bcpasc 10480 permnn 10485 cvg1nlemcxze 10722 cvg1nlemcau 10724 resqrexlemcalc3 10756 binom11 11223 binom1dif 11224 divcnv 11234 arisum2 11236 trireciplem 11237 trirecip 11238 expcnvap0 11239 geo2sum 11251 geo2lim 11253 cvgratnnlembern 11260 cvgratnnlemnexp 11261 cvgratnnlemmn 11262 cvgratnnlemsumlt 11265 cvgratnnlemfm 11266 cvgratnnlemrate 11267 cvgratz 11269 eftcl 11287 eftabs 11289 efcllemp 11291 ege2le3 11304 efcj 11306 efaddlem 11307 eftlub 11323 eirraplem 11410 oexpneg 11501 divalglemnn 11542 dvdsgcdidd 11609 bezoutlemnewy 11611 mulgcd 11631 rplpwr 11642 sqgcd 11644 lcmgcdlem 11685 3lcm2e6woprm 11694 cncongr1 11711 cncongr2 11712 prmind2 11728 divgcdodd 11748 prmdvdsexpr 11755 sqrt2irrlem 11766 oddpwdclemxy 11774 oddpwdclemodd 11777 oddpwdclemdc 11778 oddpwdc 11779 sqpweven 11780 2sqpwodd 11781 sqrt2irraplemnn 11784 sqrt2irrap 11785 qmuldeneqnum 11800 divnumden 11801 qnumgt0 11803 numdensq 11807 hashdvds 11824 phiprmpw 11825 oddennn 11832 evenennn 11833 trilpolemeq1 13160 trilpolemlt1 13161 |
Copyright terms: Public domain | W3C validator |