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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 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zcn 9052 | . . . . . . 7 | |
2 | 1 | div1d 8533 | . . . . . 6 |
3 | 2 | eqeq2d 2149 | . . . . 5 |
4 | eqcom 2139 | . . . . 5 | |
5 | 3, 4 | syl6rbbr 198 | . . . 4 |
6 | 1nn 8724 | . . . . 5 | |
7 | oveq2 5775 | . . . . . . 7 | |
8 | 7 | eqeq2d 2149 | . . . . . 6 |
9 | 8 | rspcev 2784 | . . . . 5 |
10 | 6, 9 | mpan 420 | . . . 4 |
11 | 5, 10 | syl6bi 162 | . . 3 |
12 | 11 | reximia 2525 | . 2 |
13 | risset 2461 | . 2 | |
14 | elq 9407 | . 2 | |
15 | 12, 13, 14 | 3imtr4i 200 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wceq 1331 wcel 1480 wrex 2415 (class class class)co 5767 c1 7614 cdiv 8425 cn 8713 cz 9047 cq 9404 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-cnex 7704 ax-resscn 7705 ax-1cn 7706 ax-1re 7707 ax-icn 7708 ax-addcl 7709 ax-addrcl 7710 ax-mulcl 7711 ax-mulrcl 7712 ax-addcom 7713 ax-mulcom 7714 ax-addass 7715 ax-mulass 7716 ax-distr 7717 ax-i2m1 7718 ax-0lt1 7719 ax-1rid 7720 ax-0id 7721 ax-rnegex 7722 ax-precex 7723 ax-cnre 7724 ax-pre-ltirr 7725 ax-pre-ltwlin 7726 ax-pre-lttrn 7727 ax-pre-apti 7728 ax-pre-ltadd 7729 ax-pre-mulgt0 7730 ax-pre-mulext 7731 |
This theorem depends on definitions: df-bi 116 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-nel 2402 df-ral 2419 df-rex 2420 df-reu 2421 df-rmo 2422 df-rab 2423 df-v 2683 df-sbc 2905 df-csb 2999 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-int 3767 df-iun 3810 df-br 3925 df-opab 3985 df-mpt 3986 df-id 4210 df-po 4213 df-iso 4214 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-fv 5126 df-riota 5723 df-ov 5770 df-oprab 5771 df-mpo 5772 df-1st 6031 df-2nd 6032 df-pnf 7795 df-mnf 7796 df-xr 7797 df-ltxr 7798 df-le 7799 df-sub 7928 df-neg 7929 df-reap 8330 df-ap 8337 df-div 8426 df-inn 8714 df-z 9048 df-q 9405 |
This theorem is referenced by: zssq 9412 qdivcl 9428 irrmul 9432 qbtwnz 10022 qbtwnxr 10028 flqlt 10049 flid 10050 flqltnz 10053 flqbi2 10057 flqaddz 10063 flqmulnn0 10065 ceilid 10081 flqeqceilz 10084 flqdiv 10087 modqcl 10092 mulqmod0 10096 modqfrac 10103 zmod10 10106 modqmulnn 10108 zmodcl 10110 zmodfz 10112 zmodid2 10118 q0mod 10121 q1mod 10122 modqcyc 10125 mulp1mod1 10131 modqmuladd 10132 modqmuladdim 10133 modqmuladdnn0 10134 m1modnnsub1 10136 addmodid 10138 modqm1p1mod0 10141 modqltm1p1mod 10142 modqmul1 10143 modqmul12d 10144 q2txmodxeq0 10150 modifeq2int 10152 modaddmodup 10153 modaddmodlo 10154 modqaddmulmod 10157 modqdi 10158 modqsubdir 10159 modsumfzodifsn 10162 addmodlteq 10164 qexpcl 10302 qexpclz 10307 iexpcyc 10390 facavg 10485 bcval 10488 qabsor 10840 modfsummodlemstep 11219 egt2lt3 11475 dvdsval3 11486 moddvds 11491 absdvdsb 11500 dvdsabsb 11501 dvdslelemd 11530 dvdsmod 11549 mulmoddvds 11550 divalglemnn 11604 divalgmod 11613 fldivndvdslt 11621 gcdabs 11665 gcdabs1 11666 modgcd 11668 bezoutlemnewy 11673 bezoutlemstep 11674 eucalglt 11727 lcmabs 11746 sqrt2irraplemnn 11846 nn0sqrtelqelz 11873 crth 11889 phimullem 11890 |
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