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Theorem flqeqceilz 9690
Description: A rational number is an integer iff its floor equals its ceiling. (Contributed by Jim Kingdon, 11-Oct-2021.)
Assertion
Ref Expression
flqeqceilz  |-  ( A  e.  QQ  ->  ( A  e.  ZZ  <->  ( |_ `  A )  =  ( `  A ) ) )

Proof of Theorem flqeqceilz
StepHypRef Expression
1 flid 9656 . . 3  |-  ( A  e.  ZZ  ->  ( |_ `  A )  =  A )
2 ceilid 9687 . . 3  |-  ( A  e.  ZZ  ->  ( `  A )  =  A )
31, 2eqtr4d 2123 . 2  |-  ( A  e.  ZZ  ->  ( |_ `  A )  =  ( `  A )
)
4 flqcl 9645 . . . . . 6  |-  ( A  e.  QQ  ->  ( |_ `  A )  e.  ZZ )
5 zq 9080 . . . . . 6  |-  ( ( |_ `  A )  e.  ZZ  ->  ( |_ `  A )  e.  QQ )
64, 5syl 14 . . . . 5  |-  ( A  e.  QQ  ->  ( |_ `  A )  e.  QQ )
7 qdceq 9623 . . . . 5  |-  ( ( ( |_ `  A
)  e.  QQ  /\  A  e.  QQ )  -> DECID  ( |_ `  A )  =  A )
86, 7mpancom 413 . . . 4  |-  ( A  e.  QQ  -> DECID  ( |_ `  A
)  =  A )
9 exmiddc 782 . . . 4  |-  (DECID  ( |_
`  A )  =  A  ->  ( ( |_ `  A )  =  A  \/  -.  ( |_ `  A )  =  A ) )
108, 9syl 14 . . 3  |-  ( A  e.  QQ  ->  (
( |_ `  A
)  =  A  \/  -.  ( |_ `  A
)  =  A ) )
11 eqeq1 2094 . . . . . . 7  |-  ( ( |_ `  A )  =  A  ->  (
( |_ `  A
)  =  ( `  A
)  <->  A  =  ( `  A ) ) )
1211adantr 270 . . . . . 6  |-  ( ( ( |_ `  A
)  =  A  /\  A  e.  QQ )  ->  ( ( |_ `  A )  =  ( `  A )  <->  A  =  ( `  A ) ) )
13 ceilqidz 9688 . . . . . . . . 9  |-  ( A  e.  QQ  ->  ( A  e.  ZZ  <->  ( `  A
)  =  A ) )
14 eqcom 2090 . . . . . . . . 9  |-  ( ( `  A )  =  A  <-> 
A  =  ( `  A
) )
1513, 14syl6bb 194 . . . . . . . 8  |-  ( A  e.  QQ  ->  ( A  e.  ZZ  <->  A  =  ( `  A ) ) )
1615biimprd 156 . . . . . . 7  |-  ( A  e.  QQ  ->  ( A  =  ( `  A
)  ->  A  e.  ZZ ) )
1716adantl 271 . . . . . 6  |-  ( ( ( |_ `  A
)  =  A  /\  A  e.  QQ )  ->  ( A  =  ( `  A )  ->  A  e.  ZZ ) )
1812, 17sylbid 148 . . . . 5  |-  ( ( ( |_ `  A
)  =  A  /\  A  e.  QQ )  ->  ( ( |_ `  A )  =  ( `  A )  ->  A  e.  ZZ ) )
1918ex 113 . . . 4  |-  ( ( |_ `  A )  =  A  ->  ( A  e.  QQ  ->  ( ( |_ `  A
)  =  ( `  A
)  ->  A  e.  ZZ ) ) )
20 flqle 9650 . . . . 5  |-  ( A  e.  QQ  ->  ( |_ `  A )  <_  A )
21 df-ne 2256 . . . . . 6  |-  ( ( |_ `  A )  =/=  A  <->  -.  ( |_ `  A )  =  A )
22 necom 2339 . . . . . . 7  |-  ( ( |_ `  A )  =/=  A  <->  A  =/=  ( |_ `  A ) )
23 qltlen 9094 . . . . . . . . . . 11  |-  ( ( ( |_ `  A
)  e.  QQ  /\  A  e.  QQ )  ->  ( ( |_ `  A )  <  A  <->  ( ( |_ `  A
)  <_  A  /\  A  =/=  ( |_ `  A ) ) ) )
246, 23mpancom 413 . . . . . . . . . 10  |-  ( A  e.  QQ  ->  (
( |_ `  A
)  <  A  <->  ( ( |_ `  A )  <_  A  /\  A  =/=  ( |_ `  A ) ) ) )
25 breq1 3840 . . . . . . . . . . . . . 14  |-  ( ( |_ `  A )  =  ( `  A
)  ->  ( ( |_ `  A )  < 
A  <->  ( `  A )  <  A ) )
2625adantl 271 . . . . . . . . . . . . 13  |-  ( ( A  e.  QQ  /\  ( |_ `  A )  =  ( `  A
) )  ->  (
( |_ `  A
)  <  A  <->  ( `  A
)  <  A )
)
27 ceilqge 9682 . . . . . . . . . . . . . . 15  |-  ( A  e.  QQ  ->  A  <_  ( `  A )
)
28 qre 9079 . . . . . . . . . . . . . . . . 17  |-  ( A  e.  QQ  ->  A  e.  RR )
29 ceilqcl 9680 . . . . . . . . . . . . . . . . . 18  |-  ( A  e.  QQ  ->  ( `  A )  e.  ZZ )
3029zred 8838 . . . . . . . . . . . . . . . . 17  |-  ( A  e.  QQ  ->  ( `  A )  e.  RR )
3128, 30lenltd 7580 . . . . . . . . . . . . . . . 16  |-  ( A  e.  QQ  ->  ( A  <_  ( `  A )  <->  -.  ( `  A )  <  A ) )
32 pm2.21 582 . . . . . . . . . . . . . . . 16  |-  ( -.  ( `  A )  <  A  ->  ( ( `  A )  <  A  ->  A  e.  ZZ ) )
3331, 32syl6bi 161 . . . . . . . . . . . . . . 15  |-  ( A  e.  QQ  ->  ( A  <_  ( `  A )  ->  ( ( `  A
)  <  A  ->  A  e.  ZZ ) ) )
3427, 33mpd 13 . . . . . . . . . . . . . 14  |-  ( A  e.  QQ  ->  (
( `  A )  < 
A  ->  A  e.  ZZ ) )
3534adantr 270 . . . . . . . . . . . . 13  |-  ( ( A  e.  QQ  /\  ( |_ `  A )  =  ( `  A
) )  ->  (
( `  A )  < 
A  ->  A  e.  ZZ ) )
3626, 35sylbid 148 . . . . . . . . . . . 12  |-  ( ( A  e.  QQ  /\  ( |_ `  A )  =  ( `  A
) )  ->  (
( |_ `  A
)  <  A  ->  A  e.  ZZ ) )
3736ex 113 . . . . . . . . . . 11  |-  ( A  e.  QQ  ->  (
( |_ `  A
)  =  ( `  A
)  ->  ( ( |_ `  A )  < 
A  ->  A  e.  ZZ ) ) )
3837com23 77 . . . . . . . . . 10  |-  ( A  e.  QQ  ->  (
( |_ `  A
)  <  A  ->  ( ( |_ `  A
)  =  ( `  A
)  ->  A  e.  ZZ ) ) )
3924, 38sylbird 168 . . . . . . . . 9  |-  ( A  e.  QQ  ->  (
( ( |_ `  A )  <_  A  /\  A  =/=  ( |_ `  A ) )  ->  ( ( |_
`  A )  =  ( `  A )  ->  A  e.  ZZ ) ) )
4039expd 254 . . . . . . . 8  |-  ( A  e.  QQ  ->  (
( |_ `  A
)  <_  A  ->  ( A  =/=  ( |_
`  A )  -> 
( ( |_ `  A )  =  ( `  A )  ->  A  e.  ZZ ) ) ) )
4140com3r 78 . . . . . . 7  |-  ( A  =/=  ( |_ `  A )  ->  ( A  e.  QQ  ->  ( ( |_ `  A
)  <_  A  ->  ( ( |_ `  A
)  =  ( `  A
)  ->  A  e.  ZZ ) ) ) )
4222, 41sylbi 119 . . . . . 6  |-  ( ( |_ `  A )  =/=  A  ->  ( A  e.  QQ  ->  ( ( |_ `  A
)  <_  A  ->  ( ( |_ `  A
)  =  ( `  A
)  ->  A  e.  ZZ ) ) ) )
4321, 42sylbir 133 . . . . 5  |-  ( -.  ( |_ `  A
)  =  A  -> 
( A  e.  QQ  ->  ( ( |_ `  A )  <_  A  ->  ( ( |_ `  A )  =  ( `  A )  ->  A  e.  ZZ ) ) ) )
4420, 43mpdi 42 . . . 4  |-  ( -.  ( |_ `  A
)  =  A  -> 
( A  e.  QQ  ->  ( ( |_ `  A )  =  ( `  A )  ->  A  e.  ZZ ) ) )
4519, 44jaoi 671 . . 3  |-  ( ( ( |_ `  A
)  =  A  \/  -.  ( |_ `  A
)  =  A )  ->  ( A  e.  QQ  ->  ( ( |_ `  A )  =  ( `  A )  ->  A  e.  ZZ ) ) )
4610, 45mpcom 36 . 2  |-  ( A  e.  QQ  ->  (
( |_ `  A
)  =  ( `  A
)  ->  A  e.  ZZ ) )
473, 46impbid2 141 1  |-  ( A  e.  QQ  ->  ( A  e.  ZZ  <->  ( |_ `  A )  =  ( `  A ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    <-> wb 103    \/ wo 664  DECID wdc 780    = wceq 1289    e. wcel 1438    =/= wne 2255   class class class wbr 3837   ` cfv 5002    < clt 7501    <_ cle 7502   ZZcz 8720   QQcq 9073   |_cfl 9640  ⌈cceil 9641
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343  ax-cnex 7415  ax-resscn 7416  ax-1cn 7417  ax-1re 7418  ax-icn 7419  ax-addcl 7420  ax-addrcl 7421  ax-mulcl 7422  ax-mulrcl 7423  ax-addcom 7424  ax-mulcom 7425  ax-addass 7426  ax-mulass 7427  ax-distr 7428  ax-i2m1 7429  ax-0lt1 7430  ax-1rid 7431  ax-0id 7432  ax-rnegex 7433  ax-precex 7434  ax-cnre 7435  ax-pre-ltirr 7436  ax-pre-ltwlin 7437  ax-pre-lttrn 7438  ax-pre-apti 7439  ax-pre-ltadd 7440  ax-pre-mulgt0 7441  ax-pre-mulext 7442  ax-arch 7443
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rmo 2367  df-rab 2368  df-v 2621  df-sbc 2839  df-csb 2932  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-int 3684  df-iun 3727  df-br 3838  df-opab 3892  df-mpt 3893  df-id 4111  df-po 4114  df-iso 4115  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-fv 5010  df-riota 5590  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-1st 5893  df-2nd 5894  df-pnf 7503  df-mnf 7504  df-xr 7505  df-ltxr 7506  df-le 7507  df-sub 7634  df-neg 7635  df-reap 8028  df-ap 8035  df-div 8114  df-inn 8395  df-n0 8644  df-z 8721  df-q 9074  df-rp 9104  df-fl 9642  df-ceil 9643
This theorem is referenced by: (None)
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