![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > qcn | GIF version |
Description: A rational number is a complex number. (Contributed by NM, 2-Aug-2004.) |
Ref | Expression |
---|---|
qcn | ⊢ (𝐴 ∈ ℚ → 𝐴 ∈ ℂ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | qsscn 9629 | . 2 ⊢ ℚ ⊆ ℂ | |
2 | 1 | sseli 3151 | 1 ⊢ (𝐴 ∈ ℚ → 𝐴 ∈ ℂ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2148 ℂcc 7808 ℚcq 9617 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4121 ax-pow 4174 ax-pr 4209 ax-un 4433 ax-setind 4536 ax-cnex 7901 ax-resscn 7902 ax-1cn 7903 ax-1re 7904 ax-icn 7905 ax-addcl 7906 ax-addrcl 7907 ax-mulcl 7908 ax-mulrcl 7909 ax-addcom 7910 ax-mulcom 7911 ax-addass 7912 ax-mulass 7913 ax-distr 7914 ax-i2m1 7915 ax-0lt1 7916 ax-1rid 7917 ax-0id 7918 ax-rnegex 7919 ax-precex 7920 ax-cnre 7921 ax-pre-ltirr 7922 ax-pre-ltwlin 7923 ax-pre-lttrn 7924 ax-pre-apti 7925 ax-pre-ltadd 7926 ax-pre-mulgt0 7927 ax-pre-mulext 7928 |
This theorem depends on definitions: df-bi 117 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-int 3845 df-iun 3888 df-br 4004 df-opab 4065 df-mpt 4066 df-id 4293 df-po 4296 df-iso 4297 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-rn 4637 df-res 4638 df-ima 4639 df-iota 5178 df-fun 5218 df-fn 5219 df-f 5220 df-fv 5224 df-riota 5830 df-ov 5877 df-oprab 5878 df-mpo 5879 df-1st 6140 df-2nd 6141 df-pnf 7992 df-mnf 7993 df-xr 7994 df-ltxr 7995 df-le 7996 df-sub 8128 df-neg 8129 df-reap 8530 df-ap 8537 df-div 8628 df-inn 8918 df-z 9252 df-q 9618 |
This theorem is referenced by: qsubcl 9636 qapne 9637 qdivcl 9641 qrevaddcl 9642 irradd 9644 irrmul 9645 qavgle 10256 divfl0 10293 flqzadd 10295 intqfrac2 10316 flqdiv 10318 modqvalr 10322 flqpmodeq 10324 modq0 10326 mulqmod0 10327 negqmod0 10328 modqlt 10330 modqdiffl 10332 modqfrac 10334 flqmod 10335 intqfrac 10336 modqmulnn 10339 modqvalp1 10340 modqid 10346 modqcyc 10356 modqcyc2 10357 modqadd1 10358 modqaddabs 10359 modqmuladdnn0 10365 qnegmod 10366 modqadd2mod 10371 modqm1p1mod0 10372 modqmul1 10374 modqnegd 10376 modqadd12d 10377 modqsub12d 10378 q2txmodxeq0 10381 q2submod 10382 modqmulmodr 10387 modqaddmulmod 10388 modqdi 10389 modqsubdir 10390 modqeqmodmin 10391 qsqcl 10588 qsqeqor 10627 eirraplem 11779 bezoutlemnewy 11991 sqrt2irraplemnn 12173 pcqdiv 12301 pcexp 12303 pcadd 12333 qexpz 12344 4sqlem5 12374 4sqlem10 12379 logbgcd1irraplemap 14280 ex-ceil 14360 qdencn 14657 apdifflemf 14676 apdifflemr 14677 apdiff 14678 |
Copyright terms: Public domain | W3C validator |