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Mirrors > Home > ILE Home > Th. List > nn0sqrtelqelz | Unicode version |
Description: If a nonnegative integer has a rational square root, that root must be an integer. (Contributed by Jim Kingdon, 24-May-2022.) |
Ref | Expression |
---|---|
nn0sqrtelqelz |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | qdencl 12115 | . . . . 5 denom | |
2 | 1 | adantl 275 | . . . 4 denom |
3 | 2 | nnred 8864 | . . 3 denom |
4 | 1red 7908 | . . 3 | |
5 | 2 | nnnn0d 9161 | . . . 4 denom |
6 | 5 | nn0ge0d 9164 | . . 3 denom |
7 | 0le1 8373 | . . . 4 | |
8 | 7 | a1i 9 | . . 3 |
9 | sq1 10542 | . . . . 5 | |
10 | 9 | a1i 9 | . . . 4 |
11 | simpl 108 | . . . . . . . 8 | |
12 | 11 | nn0red 9162 | . . . . . . 7 |
13 | 11 | nn0ge0d 9164 | . . . . . . 7 |
14 | resqrtth 10967 | . . . . . . 7 | |
15 | 12, 13, 14 | syl2anc 409 | . . . . . 6 |
16 | 15 | fveq2d 5487 | . . . . 5 denom denom |
17 | nn0z 9205 | . . . . . . 7 | |
18 | 11, 17 | syl 14 | . . . . . 6 |
19 | zq 9558 | . . . . . . . . 9 | |
20 | 17, 19 | syl 14 | . . . . . . . 8 |
21 | 11, 20 | syl 14 | . . . . . . 7 |
22 | qden1elz 12131 | . . . . . . 7 denom | |
23 | 21, 22 | syl 14 | . . . . . 6 denom |
24 | 18, 23 | mpbird 166 | . . . . 5 denom |
25 | 16, 24 | eqtrd 2197 | . . . 4 denom |
26 | densq 12130 | . . . . 5 denom denom | |
27 | 26 | adantl 275 | . . . 4 denom denom |
28 | 10, 25, 27 | 3eqtr2rd 2204 | . . 3 denom |
29 | 3, 4, 6, 8, 28 | sq11d 10615 | . 2 denom |
30 | qden1elz 12131 | . . 3 denom | |
31 | 30 | adantl 275 | . 2 denom |
32 | 29, 31 | mpbid 146 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1342 wcel 2135 class class class wbr 3979 cfv 5185 (class class class)co 5839 cr 7746 cc0 7747 c1 7748 cle 7928 cn 8851 c2 8902 cn0 9108 cz 9185 cq 9551 cexp 10448 csqrt 10932 denomcdenom 12108 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-coll 4094 ax-sep 4097 ax-nul 4105 ax-pow 4150 ax-pr 4184 ax-un 4408 ax-setind 4511 ax-iinf 4562 ax-cnex 7838 ax-resscn 7839 ax-1cn 7840 ax-1re 7841 ax-icn 7842 ax-addcl 7843 ax-addrcl 7844 ax-mulcl 7845 ax-mulrcl 7846 ax-addcom 7847 ax-mulcom 7848 ax-addass 7849 ax-mulass 7850 ax-distr 7851 ax-i2m1 7852 ax-0lt1 7853 ax-1rid 7854 ax-0id 7855 ax-rnegex 7856 ax-precex 7857 ax-cnre 7858 ax-pre-ltirr 7859 ax-pre-ltwlin 7860 ax-pre-lttrn 7861 ax-pre-apti 7862 ax-pre-ltadd 7863 ax-pre-mulgt0 7864 ax-pre-mulext 7865 ax-arch 7866 ax-caucvg 7867 |
This theorem depends on definitions: df-bi 116 df-stab 821 df-dc 825 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-reu 2449 df-rmo 2450 df-rab 2451 df-v 2726 df-sbc 2950 df-csb 3044 df-dif 3116 df-un 3118 df-in 3120 df-ss 3127 df-nul 3408 df-if 3519 df-pw 3558 df-sn 3579 df-pr 3580 df-op 3582 df-uni 3787 df-int 3822 df-iun 3865 df-br 3980 df-opab 4041 df-mpt 4042 df-tr 4078 df-id 4268 df-po 4271 df-iso 4272 df-iord 4341 df-on 4343 df-ilim 4344 df-suc 4346 df-iom 4565 df-xp 4607 df-rel 4608 df-cnv 4609 df-co 4610 df-dm 4611 df-rn 4612 df-res 4613 df-ima 4614 df-iota 5150 df-fun 5187 df-fn 5188 df-f 5189 df-f1 5190 df-fo 5191 df-f1o 5192 df-fv 5193 df-riota 5795 df-ov 5842 df-oprab 5843 df-mpo 5844 df-1st 6103 df-2nd 6104 df-recs 6267 df-frec 6353 df-sup 6943 df-pnf 7929 df-mnf 7930 df-xr 7931 df-ltxr 7932 df-le 7933 df-sub 8065 df-neg 8066 df-reap 8467 df-ap 8474 df-div 8563 df-inn 8852 df-2 8910 df-3 8911 df-4 8912 df-n0 9109 df-z 9186 df-uz 9461 df-q 9552 df-rp 9584 df-fz 9939 df-fzo 10072 df-fl 10199 df-mod 10252 df-seqfrec 10375 df-exp 10449 df-cj 10778 df-re 10779 df-im 10780 df-rsqrt 10934 df-abs 10935 df-dvds 11722 df-gcd 11870 df-numer 12109 df-denom 12110 |
This theorem is referenced by: nonsq 12133 |
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