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Mirrors > Home > ILE Home > Th. List > zq | Unicode version |
Description: An integer is a rational number. (Contributed by NM, 9-Jan-2002.) |
Ref | Expression |
---|---|
zq |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqcom 2179 |
. . . . 5
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2 | zcn 9256 |
. . . . . . 7
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3 | 2 | div1d 8735 |
. . . . . 6
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4 | 3 | eqeq2d 2189 |
. . . . 5
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5 | 1, 4 | bitr4id 199 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
6 | 1nn 8928 |
. . . . 5
![]() ![]() ![]() ![]() | |
7 | oveq2 5882 |
. . . . . . 7
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8 | 7 | eqeq2d 2189 |
. . . . . 6
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9 | 8 | rspcev 2841 |
. . . . 5
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10 | 6, 9 | mpan 424 |
. . . 4
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11 | 5, 10 | syl6bi 163 |
. . 3
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12 | 11 | reximia 2572 |
. 2
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13 | risset 2505 |
. 2
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14 | elq 9620 |
. 2
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15 | 12, 13, 14 | 3imtr4i 201 |
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: zssq 9625 qdivcl 9641 irrmul 9645 qbtwnz 10249 qbtwnxr 10255 flqlt 10280 flid 10281 flqltnz 10284 flqbi2 10288 flqaddz 10294 flqmulnn0 10296 ceilid 10312 flqeqceilz 10315 flqdiv 10318 modqcl 10323 mulqmod0 10327 modqfrac 10334 zmod10 10337 modqmulnn 10339 zmodcl 10341 zmodfz 10343 zmodid2 10349 q0mod 10352 q1mod 10353 modqcyc 10356 mulp1mod1 10362 modqmuladd 10363 modqmuladdim 10364 modqmuladdnn0 10365 m1modnnsub1 10367 addmodid 10369 modqm1p1mod0 10372 modqltm1p1mod 10373 modqmul1 10374 modqmul12d 10375 q2txmodxeq0 10381 modifeq2int 10383 modaddmodup 10384 modaddmodlo 10385 modqaddmulmod 10388 modqdi 10389 modqsubdir 10390 modsumfzodifsn 10393 addmodlteq 10395 qexpcl 10533 qexpclz 10538 iexpcyc 10621 qsqeqor 10627 facavg 10721 bcval 10724 qabsor 11079 modfsummodlemstep 11460 egt2lt3 11782 dvdsval3 11793 p1modz1 11796 moddvds 11801 modm1div 11802 absdvdsb 11811 dvdsabsb 11812 dvdslelemd 11843 dvdsmod 11862 mulmoddvds 11863 divalglemnn 11917 divalgmod 11926 fldivndvdslt 11934 gcdabs 11983 gcdabs1 11984 modgcd 11986 bezoutlemnewy 11991 bezoutlemstep 11992 eucalglt 12051 lcmabs 12070 sqrt2irraplemnn 12173 nn0sqrtelqelz 12200 crth 12218 phimullem 12219 eulerthlema 12224 eulerthlemh 12225 fermltl 12228 prmdiv 12229 prmdiveq 12230 odzdvds 12239 vfermltl 12245 powm2modprm 12246 modprm0 12248 modprmn0modprm0 12250 pceu 12289 pczpre 12291 pcdiv 12296 pc0 12298 pcqdiv 12301 pcrec 12302 pcexp 12303 pcxcl 12305 pcdvdstr 12320 pcgcd1 12321 pc2dvds 12323 pc11 12324 pcaddlem 12332 pcadd 12333 fldivp1 12340 qexpz 12344 4sqlem5 12374 4sqlem6 12375 4sqlem10 12379 mulgmodid 12975 2logb9irrALT 14285 2irrexpq 14287 2irrexpqap 14289 lgslem1 14294 lgsvalmod 14313 lgsneg 14318 lgsmod 14320 lgsdir2lem4 14325 lgsdirprm 14328 lgsdilem2 14330 lgsne0 14332 apdifflemr 14677 apdiff 14678 |
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