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Theorem ex-ceil 12865
Description: Example for df-ceil 10012. (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 12864 . 2  |-  ( ( |_ `  ( 3  /  2 ) )  =  1  /\  ( |_ `  -u ( 3  / 
2 ) )  = 
-u 2 )
2 3z 9051 . . . . . . 7  |-  3  e.  ZZ
3 2nn 8849 . . . . . . 7  |-  2  e.  NN
4 znq 9384 . . . . . . 7  |-  ( ( 3  e.  ZZ  /\  2  e.  NN )  ->  ( 3  /  2
)  e.  QQ )
52, 3, 4mp2an 422 . . . . . 6  |-  ( 3  /  2 )  e.  QQ
6 qnegcl 9396 . . . . . 6  |-  ( ( 3  /  2 )  e.  QQ  ->  -u (
3  /  2 )  e.  QQ )
75, 6ax-mp 5 . . . . 5  |-  -u (
3  /  2 )  e.  QQ
8 ceilqval 10047 . . . . 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 9394 . . . . . . . . . . 11  |-  ( ( 3  /  2 )  e.  QQ  ->  (
3  /  2 )  e.  CC )
115, 10ax-mp 5 . . . . . . . . . 10  |-  ( 3  /  2 )  e.  CC
1211negnegi 8000 . . . . . . . . 9  |-  -u -u (
3  /  2 )  =  ( 3  / 
2 )
1312eqcomi 2121 . . . . . . . 8  |-  ( 3  /  2 )  = 
-u -u ( 3  / 
2 )
1413fveq2i 5392 . . . . . . 7  |-  ( |_
`  ( 3  / 
2 ) )  =  ( |_ `  -u -u (
3  /  2 ) )
1514eqeq1i 2125 . . . . . 6  |-  ( ( |_ `  ( 3  /  2 ) )  =  1  <->  ( |_ `  -u -u ( 3  / 
2 ) )  =  1 )
1615biimpi 119 . . . . 5  |-  ( ( |_ `  ( 3  /  2 ) )  =  1  ->  ( |_ `  -u -u ( 3  / 
2 ) )  =  1 )
1716negeqd 7925 . . . 4  |-  ( ( |_ `  ( 3  /  2 ) )  =  1  ->  -u ( |_ `  -u -u ( 3  / 
2 ) )  = 
-u 1 )
189, 17syl5eq 2162 . . 3  |-  ( ( |_ `  ( 3  /  2 ) )  =  1  ->  ( `  -u ( 3  /  2
) )  =  -u
1 )
19 ceilqval 10047 . . . . 5  |-  ( ( 3  /  2 )  e.  QQ  ->  ( `  ( 3  /  2
) )  =  -u ( |_ `  -u (
3  /  2 ) ) )
205, 19ax-mp 5 . . . 4  |-  ( `  (
3  /  2 ) )  =  -u ( |_ `  -u ( 3  / 
2 ) )
21 negeq 7923 . . . . 5  |-  ( ( |_ `  -u (
3  /  2 ) )  =  -u 2  -> 
-u ( |_ `  -u ( 3  /  2
) )  =  -u -u 2 )
22 2cn 8759 . . . . . 6  |-  2  e.  CC
2322negnegi 8000 . . . . 5  |-  -u -u 2  =  2
2421, 23syl6eq 2166 . . . 4  |-  ( ( |_ `  -u (
3  /  2 ) )  =  -u 2  -> 
-u ( |_ `  -u ( 3  /  2
) )  =  2 )
2520, 24syl5eq 2162 . . 3  |-  ( ( |_ `  -u (
3  /  2 ) )  =  -u 2  ->  ( `  ( 3  /  2 ) )  =  2 )
2618, 25anim12ci 337 . 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 1316    e. wcel 1465   ` cfv 5093  (class class class)co 5742   CCcc 7586   1c1 7589   -ucneg 7902    / cdiv 8400   NNcn 8688   2c2 8739   3c3 8740   ZZcz 9022   QQcq 9379   |_cfl 10009  ⌈cceil 10010
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-cnex 7679  ax-resscn 7680  ax-1cn 7681  ax-1re 7682  ax-icn 7683  ax-addcl 7684  ax-addrcl 7685  ax-mulcl 7686  ax-mulrcl 7687  ax-addcom 7688  ax-mulcom 7689  ax-addass 7690  ax-mulass 7691  ax-distr 7692  ax-i2m1 7693  ax-0lt1 7694  ax-1rid 7695  ax-0id 7696  ax-rnegex 7697  ax-precex 7698  ax-cnre 7699  ax-pre-ltirr 7700  ax-pre-ltwlin 7701  ax-pre-lttrn 7702  ax-pre-apti 7703  ax-pre-ltadd 7704  ax-pre-mulgt0 7705  ax-pre-mulext 7706  ax-arch 7707
This theorem depends on definitions:  df-bi 116  df-3or 948  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-nel 2381  df-ral 2398  df-rex 2399  df-reu 2400  df-rmo 2401  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-id 4185  df-po 4188  df-iso 4189  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-fv 5101  df-riota 5698  df-ov 5745  df-oprab 5746  df-mpo 5747  df-1st 6006  df-2nd 6007  df-pnf 7770  df-mnf 7771  df-xr 7772  df-ltxr 7773  df-le 7774  df-sub 7903  df-neg 7904  df-reap 8305  df-ap 8312  df-div 8401  df-inn 8689  df-2 8747  df-3 8748  df-4 8749  df-n0 8946  df-z 9023  df-q 9380  df-rp 9410  df-fl 10011  df-ceil 10012
This theorem is referenced by: (None)
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