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Mirrors > Home > MPE Home > Th. List > zsqrtelqelz | Structured version Visualization version GIF version |
Description: If an integer has a rational square root, that root is must be an integer. (Contributed by Stefan O'Rear, 15-Sep-2014.) |
Ref | Expression |
---|---|
zsqrtelqelz | β’ ((π΄ β β€ β§ (ββπ΄) β β) β (ββπ΄) β β€) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | qdencl 16707 | . . . . 5 β’ ((ββπ΄) β β β (denomβ(ββπ΄)) β β) | |
2 | 1 | adantl 480 | . . . 4 β’ ((π΄ β β€ β§ (ββπ΄) β β) β (denomβ(ββπ΄)) β β) |
3 | 2 | nnred 12252 | . . 3 β’ ((π΄ β β€ β§ (ββπ΄) β β) β (denomβ(ββπ΄)) β β) |
4 | 1red 11240 | . . 3 β’ ((π΄ β β€ β§ (ββπ΄) β β) β 1 β β) | |
5 | 2 | nnnn0d 12557 | . . . 4 β’ ((π΄ β β€ β§ (ββπ΄) β β) β (denomβ(ββπ΄)) β β0) |
6 | 5 | nn0ge0d 12560 | . . 3 β’ ((π΄ β β€ β§ (ββπ΄) β β) β 0 β€ (denomβ(ββπ΄))) |
7 | 0le1 11762 | . . . 4 β’ 0 β€ 1 | |
8 | 7 | a1i 11 | . . 3 β’ ((π΄ β β€ β§ (ββπ΄) β β) β 0 β€ 1) |
9 | sq1 14185 | . . . . 5 β’ (1β2) = 1 | |
10 | 9 | a1i 11 | . . . 4 β’ ((π΄ β β€ β§ (ββπ΄) β β) β (1β2) = 1) |
11 | zcn 12588 | . . . . . . . 8 β’ (π΄ β β€ β π΄ β β) | |
12 | 11 | sqsqrtd 15413 | . . . . . . 7 β’ (π΄ β β€ β ((ββπ΄)β2) = π΄) |
13 | 12 | adantr 479 | . . . . . 6 β’ ((π΄ β β€ β§ (ββπ΄) β β) β ((ββπ΄)β2) = π΄) |
14 | 13 | fveq2d 6894 | . . . . 5 β’ ((π΄ β β€ β§ (ββπ΄) β β) β (denomβ((ββπ΄)β2)) = (denomβπ΄)) |
15 | simpl 481 | . . . . . 6 β’ ((π΄ β β€ β§ (ββπ΄) β β) β π΄ β β€) | |
16 | zq 12963 | . . . . . . . 8 β’ (π΄ β β€ β π΄ β β) | |
17 | 16 | adantr 479 | . . . . . . 7 β’ ((π΄ β β€ β§ (ββπ΄) β β) β π΄ β β) |
18 | qden1elz 16723 | . . . . . . 7 β’ (π΄ β β β ((denomβπ΄) = 1 β π΄ β β€)) | |
19 | 17, 18 | syl 17 | . . . . . 6 β’ ((π΄ β β€ β§ (ββπ΄) β β) β ((denomβπ΄) = 1 β π΄ β β€)) |
20 | 15, 19 | mpbird 256 | . . . . 5 β’ ((π΄ β β€ β§ (ββπ΄) β β) β (denomβπ΄) = 1) |
21 | 14, 20 | eqtrd 2765 | . . . 4 β’ ((π΄ β β€ β§ (ββπ΄) β β) β (denomβ((ββπ΄)β2)) = 1) |
22 | densq 16722 | . . . . 5 β’ ((ββπ΄) β β β (denomβ((ββπ΄)β2)) = ((denomβ(ββπ΄))β2)) | |
23 | 22 | adantl 480 | . . . 4 β’ ((π΄ β β€ β§ (ββπ΄) β β) β (denomβ((ββπ΄)β2)) = ((denomβ(ββπ΄))β2)) |
24 | 10, 21, 23 | 3eqtr2rd 2772 | . . 3 β’ ((π΄ β β€ β§ (ββπ΄) β β) β ((denomβ(ββπ΄))β2) = (1β2)) |
25 | 3, 4, 6, 8, 24 | sq11d 14247 | . 2 β’ ((π΄ β β€ β§ (ββπ΄) β β) β (denomβ(ββπ΄)) = 1) |
26 | qden1elz 16723 | . . 3 β’ ((ββπ΄) β β β ((denomβ(ββπ΄)) = 1 β (ββπ΄) β β€)) | |
27 | 26 | adantl 480 | . 2 β’ ((π΄ β β€ β§ (ββπ΄) β β) β ((denomβ(ββπ΄)) = 1 β (ββπ΄) β β€)) |
28 | 25, 27 | mpbid 231 | 1 β’ ((π΄ β β€ β§ (ββπ΄) β β) β (ββπ΄) β β€) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 394 = wceq 1533 β wcel 2098 class class class wbr 5144 βcfv 6543 (class class class)co 7413 0cc0 11133 1c1 11134 β€ cle 11274 βcn 12237 2c2 12292 β€cz 12583 βcq 12957 βcexp 14053 βcsqrt 15207 denomcdenom 16700 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-cnex 11189 ax-resscn 11190 ax-1cn 11191 ax-icn 11192 ax-addcl 11193 ax-addrcl 11194 ax-mulcl 11195 ax-mulrcl 11196 ax-mulcom 11197 ax-addass 11198 ax-mulass 11199 ax-distr 11200 ax-i2m1 11201 ax-1ne0 11202 ax-1rid 11203 ax-rnegex 11204 ax-rrecex 11205 ax-cnre 11206 ax-pre-lttri 11207 ax-pre-lttrn 11208 ax-pre-ltadd 11209 ax-pre-mulgt0 11210 ax-pre-sup 11211 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3961 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-iun 4994 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7866 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-er 8718 df-en 8958 df-dom 8959 df-sdom 8960 df-sup 9460 df-inf 9461 df-pnf 11275 df-mnf 11276 df-xr 11277 df-ltxr 11278 df-le 11279 df-sub 11471 df-neg 11472 df-div 11897 df-nn 12238 df-2 12300 df-3 12301 df-n0 12498 df-z 12584 df-uz 12848 df-q 12958 df-rp 13002 df-fl 13784 df-mod 13862 df-seq 13994 df-exp 14054 df-cj 15073 df-re 15074 df-im 15075 df-sqrt 15209 df-abs 15210 df-dvds 16226 df-gcd 16464 df-numer 16701 df-denom 16702 |
This theorem is referenced by: nonsq 16725 dchrisum0flblem2 27455 posqsqznn 41964 |
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