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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 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elq 9620 |
. 2
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2 | zre 9255 |
. . . . 5
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3 | nnre 8924 |
. . . . . 6
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4 | nnap0 8946 |
. . . . . 6
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5 | 3, 4 | jca 306 |
. . . . 5
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6 | redivclap 8686 |
. . . . . 6
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7 | 6 | 3expb 1204 |
. . . . 5
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8 | 2, 5, 7 | syl2an 289 |
. . . 4
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9 | eleq1 2240 |
. . . 4
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10 | 8, 9 | syl5ibrcom 157 |
. . 3
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11 | 10 | rexlimivv 2600 |
. 2
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12 | 1, 11 | sylbi 121 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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: qssre 9628 qltlen 9638 qlttri2 9639 irradd 9644 irrmul 9645 qletric 10241 qlelttric 10242 qltnle 10243 qdceq 10244 qbtwnz 10249 qbtwnxr 10255 qavgle 10256 ioo0 10257 ioom 10258 ico0 10259 ioc0 10260 flqcl 10270 flqlelt 10273 qfraclt1 10277 qfracge0 10278 flqge 10279 flqltnz 10284 flqwordi 10285 flqbi 10287 flqbi2 10288 flqaddz 10294 flqmulnn0 10296 flltdivnn0lt 10301 ceilqval 10303 ceiqge 10306 ceiqm1l 10308 ceiqle 10310 flqleceil 10314 flqeqceilz 10315 intfracq 10317 flqdiv 10318 modqval 10321 modq0 10326 mulqmod0 10327 negqmod0 10328 modqge0 10329 modqlt 10330 modqelico 10331 modqdiffl 10332 modqmulnn 10339 modqid 10346 modqid0 10347 modqabs 10354 modqabs2 10355 modqcyc 10356 mulqaddmodid 10361 modqmuladdim 10364 modqmuladdnn0 10365 modqltm1p1mod 10373 q2txmodxeq0 10381 q2submod 10382 modqdi 10389 modqsubdir 10390 qsqeqor 10627 fimaxq 10802 qabsor 11079 qdenre 11206 expcnvre 11506 flodddiv4t2lthalf 11936 sqrt2irraplemnn 12173 sqrt2irrap 12174 qnumgt0 12192 4sqlem6 12375 blssps 13820 blss 13821 qtopbas 13915 logbgcd1irraplemap 14280 qdencn 14657 apdifflemf 14676 |
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