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Theorem ex-ceil 12740
Description: Example for df-ceil 9984. (Contributed by AV, 4-Sep-2021.)
Assertion
Ref Expression
ex-ceil  |-  ( ( `  ( 3  /  2
) )  =  2  /\  ( `  -u (
3  /  2 ) )  =  -u 1
)

Proof of Theorem ex-ceil
StepHypRef Expression
1 ex-fl 12739 . 2  |-  ( ( |_ `  ( 3  /  2 ) )  =  1  /\  ( |_ `  -u ( 3  / 
2 ) )  = 
-u 2 )
2 3z 9034 . . . . . . 7  |-  3  e.  ZZ
3 2nn 8832 . . . . . . 7  |-  2  e.  NN
4 znq 9365 . . . . . . 7  |-  ( ( 3  e.  ZZ  /\  2  e.  NN )  ->  ( 3  /  2
)  e.  QQ )
52, 3, 4mp2an 420 . . . . . 6  |-  ( 3  /  2 )  e.  QQ
6 qnegcl 9377 . . . . . 6  |-  ( ( 3  /  2 )  e.  QQ  ->  -u (
3  /  2 )  e.  QQ )
75, 6ax-mp 5 . . . . 5  |-  -u (
3  /  2 )  e.  QQ
8 ceilqval 10019 . . . . 5  |-  ( -u ( 3  /  2
)  e.  QQ  ->  ( `  -u ( 3  / 
2 ) )  = 
-u ( |_ `  -u -u ( 3  /  2
) ) )
97, 8ax-mp 5 . . . 4  |-  ( `  -u (
3  /  2 ) )  =  -u ( |_ `  -u -u ( 3  / 
2 ) )
10 qcn 9375 . . . . . . . . . . 11  |-  ( ( 3  /  2 )  e.  QQ  ->  (
3  /  2 )  e.  CC )
115, 10ax-mp 5 . . . . . . . . . 10  |-  ( 3  /  2 )  e.  CC
1211negnegi 7996 . . . . . . . . 9  |-  -u -u (
3  /  2 )  =  ( 3  / 
2 )
1312eqcomi 2119 . . . . . . . 8  |-  ( 3  /  2 )  = 
-u -u ( 3  / 
2 )
1413fveq2i 5390 . . . . . . 7  |-  ( |_
`  ( 3  / 
2 ) )  =  ( |_ `  -u -u (
3  /  2 ) )
1514eqeq1i 2123 . . . . . 6  |-  ( ( |_ `  ( 3  /  2 ) )  =  1  <->  ( |_ `  -u -u ( 3  / 
2 ) )  =  1 )
1615biimpi 119 . . . . 5  |-  ( ( |_ `  ( 3  /  2 ) )  =  1  ->  ( |_ `  -u -u ( 3  / 
2 ) )  =  1 )
1716negeqd 7921 . . . 4  |-  ( ( |_ `  ( 3  /  2 ) )  =  1  ->  -u ( |_ `  -u -u ( 3  / 
2 ) )  = 
-u 1 )
189, 17syl5eq 2160 . . 3  |-  ( ( |_ `  ( 3  /  2 ) )  =  1  ->  ( `  -u ( 3  /  2
) )  =  -u
1 )
19 ceilqval 10019 . . . . 5  |-  ( ( 3  /  2 )  e.  QQ  ->  ( `  ( 3  /  2
) )  =  -u ( |_ `  -u (
3  /  2 ) ) )
205, 19ax-mp 5 . . . 4  |-  ( `  (
3  /  2 ) )  =  -u ( |_ `  -u ( 3  / 
2 ) )
21 negeq 7919 . . . . 5  |-  ( ( |_ `  -u (
3  /  2 ) )  =  -u 2  -> 
-u ( |_ `  -u ( 3  /  2
) )  =  -u -u 2 )
22 2cn 8748 . . . . . 6  |-  2  e.  CC
2322negnegi 7996 . . . . 5  |-  -u -u 2  =  2
2421, 23syl6eq 2164 . . . 4  |-  ( ( |_ `  -u (
3  /  2 ) )  =  -u 2  -> 
-u ( |_ `  -u ( 3  /  2
) )  =  2 )
2520, 24syl5eq 2160 . . 3  |-  ( ( |_ `  -u (
3  /  2 ) )  =  -u 2  ->  ( `  ( 3  /  2 ) )  =  2 )
2618, 25anim12ci 335 . 2  |-  ( ( ( |_ `  (
3  /  2 ) )  =  1  /\  ( |_ `  -u (
3  /  2 ) )  =  -u 2
)  ->  ( ( `  ( 3  /  2
) )  =  2  /\  ( `  -u (
3  /  2 ) )  =  -u 1
) )
271, 26ax-mp 5 1  |-  ( ( `  ( 3  /  2
) )  =  2  /\  ( `  -u (
3  /  2 ) )  =  -u 1
)
Colors of variables: wff set class
Syntax hints:    /\ wa 103    = wceq 1314    e. wcel 1463   ` cfv 5091  (class class class)co 5740   CCcc 7582   1c1 7585   -ucneg 7898    / cdiv 8392   NNcn 8677   2c2 8728   3c3 8729   ZZcz 9005   QQcq 9360   |_cfl 9981  ⌈cceil 9982
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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420  ax-cnex 7675  ax-resscn 7676  ax-1cn 7677  ax-1re 7678  ax-icn 7679  ax-addcl 7680  ax-addrcl 7681  ax-mulcl 7682  ax-mulrcl 7683  ax-addcom 7684  ax-mulcom 7685  ax-addass 7686  ax-mulass 7687  ax-distr 7688  ax-i2m1 7689  ax-0lt1 7690  ax-1rid 7691  ax-0id 7692  ax-rnegex 7693  ax-precex 7694  ax-cnre 7695  ax-pre-ltirr 7696  ax-pre-ltwlin 7697  ax-pre-lttrn 7698  ax-pre-apti 7699  ax-pre-ltadd 7700  ax-pre-mulgt0 7701  ax-pre-mulext 7702  ax-arch 7703
This theorem depends on definitions:  df-bi 116  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-nel 2379  df-ral 2396  df-rex 2397  df-reu 2398  df-rmo 2399  df-rab 2400  df-v 2660  df-sbc 2881  df-csb 2974  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-int 3740  df-iun 3783  df-br 3898  df-opab 3958  df-mpt 3959  df-id 4183  df-po 4186  df-iso 4187  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-fv 5099  df-riota 5696  df-ov 5743  df-oprab 5744  df-mpo 5745  df-1st 6004  df-2nd 6005  df-pnf 7766  df-mnf 7767  df-xr 7768  df-ltxr 7769  df-le 7770  df-sub 7899  df-neg 7900  df-reap 8300  df-ap 8307  df-div 8393  df-inn 8678  df-2 8736  df-3 8737  df-4 8738  df-n0 8929  df-z 9006  df-q 9361  df-rp 9391  df-fl 9983  df-ceil 9984
This theorem is referenced by: (None)
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