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Mirrors > Home > ILE Home > Th. List > qre | Unicode version |
Description: A rational number is a real number. (Contributed by NM, 14-Nov-2002.) |
Ref | Expression |
---|---|
qre |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elq 9441 | . 2 | |
2 | zre 9082 | . . . . 5 | |
3 | nnre 8751 | . . . . . 6 | |
4 | nnap0 8773 | . . . . . 6 # | |
5 | 3, 4 | jca 304 | . . . . 5 # |
6 | redivclap 8515 | . . . . . 6 # | |
7 | 6 | 3expb 1183 | . . . . 5 # |
8 | 2, 5, 7 | syl2an 287 | . . . 4 |
9 | eleq1 2203 | . . . 4 | |
10 | 8, 9 | syl5ibrcom 156 | . . 3 |
11 | 10 | rexlimivv 2558 | . 2 |
12 | 1, 11 | sylbi 120 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1332 wcel 1481 wrex 2418 class class class wbr 3937 (class class class)co 5782 cr 7643 cc0 7644 # cap 8367 cdiv 8456 cn 8744 cz 9078 cq 9438 |
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-in1 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-cnex 7735 ax-resscn 7736 ax-1cn 7737 ax-1re 7738 ax-icn 7739 ax-addcl 7740 ax-addrcl 7741 ax-mulcl 7742 ax-mulrcl 7743 ax-addcom 7744 ax-mulcom 7745 ax-addass 7746 ax-mulass 7747 ax-distr 7748 ax-i2m1 7749 ax-0lt1 7750 ax-1rid 7751 ax-0id 7752 ax-rnegex 7753 ax-precex 7754 ax-cnre 7755 ax-pre-ltirr 7756 ax-pre-ltwlin 7757 ax-pre-lttrn 7758 ax-pre-apti 7759 ax-pre-ltadd 7760 ax-pre-mulgt0 7761 ax-pre-mulext 7762 |
This theorem depends on definitions: df-bi 116 df-3or 964 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-nel 2405 df-ral 2422 df-rex 2423 df-reu 2424 df-rmo 2425 df-rab 2426 df-v 2691 df-sbc 2914 df-csb 3008 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-iun 3823 df-br 3938 df-opab 3998 df-mpt 3999 df-id 4223 df-po 4226 df-iso 4227 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-fv 5139 df-riota 5738 df-ov 5785 df-oprab 5786 df-mpo 5787 df-1st 6046 df-2nd 6047 df-pnf 7826 df-mnf 7827 df-xr 7828 df-ltxr 7829 df-le 7830 df-sub 7959 df-neg 7960 df-reap 8361 df-ap 8368 df-div 8457 df-inn 8745 df-z 9079 df-q 9439 |
This theorem is referenced by: qssre 9449 qltlen 9459 qlttri2 9460 irradd 9465 irrmul 9466 qletric 10052 qlelttric 10053 qltnle 10054 qdceq 10055 qbtwnz 10060 qbtwnxr 10066 qavgle 10067 ioo0 10068 ioom 10069 ico0 10070 ioc0 10071 flqcl 10077 flqlelt 10080 qfraclt1 10084 qfracge0 10085 flqge 10086 flqltnz 10091 flqwordi 10092 flqbi 10094 flqbi2 10095 flqaddz 10101 flqmulnn0 10103 flltdivnn0lt 10108 ceilqval 10110 ceiqge 10113 ceiqm1l 10115 ceiqle 10117 flqleceil 10121 flqeqceilz 10122 intfracq 10124 flqdiv 10125 modqval 10128 modq0 10133 mulqmod0 10134 negqmod0 10135 modqge0 10136 modqlt 10137 modqelico 10138 modqdiffl 10139 modqmulnn 10146 modqid 10153 modqid0 10154 modqabs 10161 modqabs2 10162 modqcyc 10163 mulqaddmodid 10168 modqmuladdim 10171 modqmuladdnn0 10172 modqltm1p1mod 10180 q2txmodxeq0 10188 q2submod 10189 modqdi 10196 modqsubdir 10197 fimaxq 10605 qabsor 10879 qdenre 11006 expcnvre 11304 flodddiv4t2lthalf 11670 sqrt2irraplemnn 11893 sqrt2irrap 11894 qnumgt0 11912 blssps 12635 blss 12636 qtopbas 12730 logbgcd1irraplemap 13094 qdencn 13397 apdifflemf 13414 |
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