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Theorem fleqceilz 12087
Description: A real number is an integer iff its floor equals its ceiling. (Contributed by AV, 30-Nov-2018.)
Assertion
Ref Expression
fleqceilz  |-  ( A  e.  RR  ->  ( A  e.  ZZ  <->  ( |_ `  A )  =  ( `  A ) ) )

Proof of Theorem fleqceilz
StepHypRef Expression
1 flid 12050 . . 3  |-  ( A  e.  ZZ  ->  ( |_ `  A )  =  A )
2 ceilid 12084 . . 3  |-  ( A  e.  ZZ  ->  ( `  A )  =  A )
31, 2eqtr4d 2466 . 2  |-  ( A  e.  ZZ  ->  ( |_ `  A )  =  ( `  A )
)
4 eqeq1 2426 . . . . . 6  |-  ( ( |_ `  A )  =  A  ->  (
( |_ `  A
)  =  ( `  A
)  <->  A  =  ( `  A ) ) )
54adantr 466 . . . . 5  |-  ( ( ( |_ `  A
)  =  A  /\  A  e.  RR )  ->  ( ( |_ `  A )  =  ( `  A )  <->  A  =  ( `  A ) ) )
6 ceilidz 12085 . . . . . . . 8  |-  ( A  e.  RR  ->  ( A  e.  ZZ  <->  ( `  A
)  =  A ) )
7 eqcom 2431 . . . . . . . 8  |-  ( ( `  A )  =  A  <-> 
A  =  ( `  A
) )
86, 7syl6bb 264 . . . . . . 7  |-  ( A  e.  RR  ->  ( A  e.  ZZ  <->  A  =  ( `  A ) ) )
98biimprd 226 . . . . . 6  |-  ( A  e.  RR  ->  ( A  =  ( `  A
)  ->  A  e.  ZZ ) )
109adantl 467 . . . . 5  |-  ( ( ( |_ `  A
)  =  A  /\  A  e.  RR )  ->  ( A  =  ( `  A )  ->  A  e.  ZZ ) )
115, 10sylbid 218 . . . 4  |-  ( ( ( |_ `  A
)  =  A  /\  A  e.  RR )  ->  ( ( |_ `  A )  =  ( `  A )  ->  A  e.  ZZ ) )
1211ex 435 . . 3  |-  ( ( |_ `  A )  =  A  ->  ( A  e.  RR  ->  ( ( |_ `  A
)  =  ( `  A
)  ->  A  e.  ZZ ) ) )
13 flle 12041 . . . 4  |-  ( A  e.  RR  ->  ( |_ `  A )  <_  A )
14 df-ne 2616 . . . . 5  |-  ( ( |_ `  A )  =/=  A  <->  -.  ( |_ `  A )  =  A )
15 necom 2689 . . . . . 6  |-  ( ( |_ `  A )  =/=  A  <->  A  =/=  ( |_ `  A ) )
16 reflcl 12038 . . . . . . . . . 10  |-  ( A  e.  RR  ->  ( |_ `  A )  e.  RR )
17 id 22 . . . . . . . . . 10  |-  ( A  e.  RR  ->  A  e.  RR )
1816, 17ltlend 9787 . . . . . . . . 9  |-  ( A  e.  RR  ->  (
( |_ `  A
)  <  A  <->  ( ( |_ `  A )  <_  A  /\  A  =/=  ( |_ `  A ) ) ) )
19 breq1 4426 . . . . . . . . . . . . 13  |-  ( ( |_ `  A )  =  ( `  A
)  ->  ( ( |_ `  A )  < 
A  <->  ( `  A )  <  A ) )
2019adantl 467 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  ( |_ `  A )  =  ( `  A
) )  ->  (
( |_ `  A
)  <  A  <->  ( `  A
)  <  A )
)
21 ceilge 12079 . . . . . . . . . . . . . 14  |-  ( A  e.  RR  ->  A  <_  ( `  A )
)
22 ceilcl 12077 . . . . . . . . . . . . . . . . 17  |-  ( A  e.  RR  ->  ( `  A )  e.  ZZ )
2322zred 11047 . . . . . . . . . . . . . . . 16  |-  ( A  e.  RR  ->  ( `  A )  e.  RR )
2417, 23lenltd 9788 . . . . . . . . . . . . . . 15  |-  ( A  e.  RR  ->  ( A  <_  ( `  A )  <->  -.  ( `  A )  <  A ) )
25 pm2.21 111 . . . . . . . . . . . . . . 15  |-  ( -.  ( `  A )  <  A  ->  ( ( `  A )  <  A  ->  A  e.  ZZ ) )
2624, 25syl6bi 231 . . . . . . . . . . . . . 14  |-  ( A  e.  RR  ->  ( A  <_  ( `  A )  ->  ( ( `  A
)  <  A  ->  A  e.  ZZ ) ) )
2721, 26mpd 15 . . . . . . . . . . . . 13  |-  ( A  e.  RR  ->  (
( `  A )  < 
A  ->  A  e.  ZZ ) )
2827adantr 466 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  ( |_ `  A )  =  ( `  A
) )  ->  (
( `  A )  < 
A  ->  A  e.  ZZ ) )
2920, 28sylbid 218 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  ( |_ `  A )  =  ( `  A
) )  ->  (
( |_ `  A
)  <  A  ->  A  e.  ZZ ) )
3029ex 435 . . . . . . . . . 10  |-  ( A  e.  RR  ->  (
( |_ `  A
)  =  ( `  A
)  ->  ( ( |_ `  A )  < 
A  ->  A  e.  ZZ ) ) )
3130com23 81 . . . . . . . . 9  |-  ( A  e.  RR  ->  (
( |_ `  A
)  <  A  ->  ( ( |_ `  A
)  =  ( `  A
)  ->  A  e.  ZZ ) ) )
3218, 31sylbird 238 . . . . . . . 8  |-  ( A  e.  RR  ->  (
( ( |_ `  A )  <_  A  /\  A  =/=  ( |_ `  A ) )  ->  ( ( |_
`  A )  =  ( `  A )  ->  A  e.  ZZ ) ) )
3332expd 437 . . . . . . 7  |-  ( A  e.  RR  ->  (
( |_ `  A
)  <_  A  ->  ( A  =/=  ( |_
`  A )  -> 
( ( |_ `  A )  =  ( `  A )  ->  A  e.  ZZ ) ) ) )
3433com3r 82 . . . . . 6  |-  ( A  =/=  ( |_ `  A )  ->  ( A  e.  RR  ->  ( ( |_ `  A
)  <_  A  ->  ( ( |_ `  A
)  =  ( `  A
)  ->  A  e.  ZZ ) ) ) )
3515, 34sylbi 198 . . . . 5  |-  ( ( |_ `  A )  =/=  A  ->  ( A  e.  RR  ->  ( ( |_ `  A
)  <_  A  ->  ( ( |_ `  A
)  =  ( `  A
)  ->  A  e.  ZZ ) ) ) )
3614, 35sylbir 216 . . . 4  |-  ( -.  ( |_ `  A
)  =  A  -> 
( A  e.  RR  ->  ( ( |_ `  A )  <_  A  ->  ( ( |_ `  A )  =  ( `  A )  ->  A  e.  ZZ ) ) ) )
3713, 36mpdi 43 . . 3  |-  ( -.  ( |_ `  A
)  =  A  -> 
( A  e.  RR  ->  ( ( |_ `  A )  =  ( `  A )  ->  A  e.  ZZ ) ) )
3812, 37pm2.61i 167 . 2  |-  ( A  e.  RR  ->  (
( |_ `  A
)  =  ( `  A
)  ->  A  e.  ZZ ) )
393, 38impbid2 207 1  |-  ( A  e.  RR  ->  ( A  e.  ZZ  <->  ( |_ `  A )  =  ( `  A ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1872    =/= wne 2614   class class class wbr 4423   ` cfv 5601   RRcr 9545    < clt 9682    <_ cle 9683   ZZcz 10944   |_cfl 12032  ⌈cceil 12033
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6597  ax-cnex 9602  ax-resscn 9603  ax-1cn 9604  ax-icn 9605  ax-addcl 9606  ax-addrcl 9607  ax-mulcl 9608  ax-mulrcl 9609  ax-mulcom 9610  ax-addass 9611  ax-mulass 9612  ax-distr 9613  ax-i2m1 9614  ax-1ne0 9615  ax-1rid 9616  ax-rnegex 9617  ax-rrecex 9618  ax-cnre 9619  ax-pre-lttri 9620  ax-pre-lttrn 9621  ax-pre-ltadd 9622  ax-pre-mulgt0 9623  ax-pre-sup 9624
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-nel 2617  df-ral 2776  df-rex 2777  df-reu 2778  df-rmo 2779  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-tp 4003  df-op 4005  df-uni 4220  df-iun 4301  df-br 4424  df-opab 4483  df-mpt 4484  df-tr 4519  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-wrecs 7039  df-recs 7101  df-rdg 7139  df-er 7374  df-en 7581  df-dom 7582  df-sdom 7583  df-sup 7965  df-inf 7966  df-pnf 9684  df-mnf 9685  df-xr 9686  df-ltxr 9687  df-le 9688  df-sub 9869  df-neg 9870  df-nn 10617  df-n0 10877  df-z 10945  df-uz 11167  df-fl 12034  df-ceil 12035
This theorem is referenced by: (None)
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