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Mirrors > Home > ILE Home > Th. List > qden1elz | Unicode version |
Description: A rational is an integer iff it has denominator 1. (Contributed by Stefan O'Rear, 15-Sep-2014.) |
Ref | Expression |
---|---|
qden1elz |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | qeqnumdivden 12221 |
. . . . 5
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2 | 1 | adantr 276 |
. . . 4
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3 | oveq2 5900 |
. . . . 5
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4 | 3 | adantl 277 |
. . . 4
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5 | qnumcl 12215 |
. . . . . . 7
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6 | 5 | adantr 276 |
. . . . . 6
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7 | 6 | zcnd 9401 |
. . . . 5
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8 | 7 | div1d 8762 |
. . . 4
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9 | 2, 4, 8 | 3eqtrd 2226 |
. . 3
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10 | 9, 6 | eqeltrd 2266 |
. 2
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11 | simpr 110 |
. . . . . . 7
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12 | 11 | zcnd 9401 |
. . . . . 6
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13 | 12 | div1d 8762 |
. . . . 5
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14 | 13 | fveq2d 5535 |
. . . 4
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15 | 1nn 8955 |
. . . . 5
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16 | divdenle 12224 |
. . . . 5
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17 | 11, 15, 16 | sylancl 413 |
. . . 4
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18 | 14, 17 | eqbrtrrd 4042 |
. . 3
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19 | qdencl 12216 |
. . . . 5
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20 | 19 | adantr 276 |
. . . 4
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21 | nnle1eq1 8968 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
22 | 20, 21 | syl 14 |
. . 3
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23 | 18, 22 | mpbid 147 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
24 | 10, 23 | impbida 596 |
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 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-coll 4133 ax-sep 4136 ax-nul 4144 ax-pow 4189 ax-pr 4224 ax-un 4448 ax-setind 4551 ax-iinf 4602 ax-cnex 7927 ax-resscn 7928 ax-1cn 7929 ax-1re 7930 ax-icn 7931 ax-addcl 7932 ax-addrcl 7933 ax-mulcl 7934 ax-mulrcl 7935 ax-addcom 7936 ax-mulcom 7937 ax-addass 7938 ax-mulass 7939 ax-distr 7940 ax-i2m1 7941 ax-0lt1 7942 ax-1rid 7943 ax-0id 7944 ax-rnegex 7945 ax-precex 7946 ax-cnre 7947 ax-pre-ltirr 7948 ax-pre-ltwlin 7949 ax-pre-lttrn 7950 ax-pre-apti 7951 ax-pre-ltadd 7952 ax-pre-mulgt0 7953 ax-pre-mulext 7954 ax-arch 7955 ax-caucvg 7956 |
This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rmo 2476 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-if 3550 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-iun 3903 df-br 4019 df-opab 4080 df-mpt 4081 df-tr 4117 df-id 4308 df-po 4311 df-iso 4312 df-iord 4381 df-on 4383 df-ilim 4384 df-suc 4386 df-iom 4605 df-xp 4647 df-rel 4648 df-cnv 4649 df-co 4650 df-dm 4651 df-rn 4652 df-res 4653 df-ima 4654 df-iota 5193 df-fun 5234 df-fn 5235 df-f 5236 df-f1 5237 df-fo 5238 df-f1o 5239 df-fv 5240 df-riota 5848 df-ov 5895 df-oprab 5896 df-mpo 5897 df-1st 6160 df-2nd 6161 df-recs 6325 df-frec 6411 df-sup 7008 df-pnf 8019 df-mnf 8020 df-xr 8021 df-ltxr 8022 df-le 8023 df-sub 8155 df-neg 8156 df-reap 8557 df-ap 8564 df-div 8655 df-inn 8945 df-2 9003 df-3 9004 df-4 9005 df-n0 9202 df-z 9279 df-uz 9554 df-q 9645 df-rp 9679 df-fz 10034 df-fzo 10168 df-fl 10296 df-mod 10349 df-seqfrec 10472 df-exp 10546 df-cj 10878 df-re 10879 df-im 10880 df-rsqrt 11034 df-abs 11035 df-dvds 11822 df-gcd 11971 df-numer 12210 df-denom 12211 |
This theorem is referenced by: nn0sqrtelqelz 12233 |
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