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Theorem ex-ceil 11299
Description: Example for df-ceil 9643. (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 11298 . 2  |-  ( ( |_ `  ( 3  /  2 ) )  =  1  /\  ( |_ `  -u ( 3  / 
2 ) )  = 
-u 2 )
2 3z 8749 . . . . . . 7  |-  3  e.  ZZ
3 2nn 8547 . . . . . . 7  |-  2  e.  NN
4 znq 9078 . . . . . . 7  |-  ( ( 3  e.  ZZ  /\  2  e.  NN )  ->  ( 3  /  2
)  e.  QQ )
52, 3, 4mp2an 417 . . . . . 6  |-  ( 3  /  2 )  e.  QQ
6 qnegcl 9090 . . . . . 6  |-  ( ( 3  /  2 )  e.  QQ  ->  -u (
3  /  2 )  e.  QQ )
75, 6ax-mp 7 . . . . 5  |-  -u (
3  /  2 )  e.  QQ
8 ceilqval 9678 . . . . 5  |-  ( -u ( 3  /  2
)  e.  QQ  ->  ( `  -u ( 3  / 
2 ) )  = 
-u ( |_ `  -u -u ( 3  /  2
) ) )
97, 8ax-mp 7 . . . 4  |-  ( `  -u (
3  /  2 ) )  =  -u ( |_ `  -u -u ( 3  / 
2 ) )
10 qcn 9088 . . . . . . . . . . 11  |-  ( ( 3  /  2 )  e.  QQ  ->  (
3  /  2 )  e.  CC )
115, 10ax-mp 7 . . . . . . . . . 10  |-  ( 3  /  2 )  e.  CC
1211negnegi 7731 . . . . . . . . 9  |-  -u -u (
3  /  2 )  =  ( 3  / 
2 )
1312eqcomi 2092 . . . . . . . 8  |-  ( 3  /  2 )  = 
-u -u ( 3  / 
2 )
1413fveq2i 5292 . . . . . . 7  |-  ( |_
`  ( 3  / 
2 ) )  =  ( |_ `  -u -u (
3  /  2 ) )
1514eqeq1i 2095 . . . . . 6  |-  ( ( |_ `  ( 3  /  2 ) )  =  1  <->  ( |_ `  -u -u ( 3  / 
2 ) )  =  1 )
1615biimpi 118 . . . . 5  |-  ( ( |_ `  ( 3  /  2 ) )  =  1  ->  ( |_ `  -u -u ( 3  / 
2 ) )  =  1 )
1716negeqd 7656 . . . 4  |-  ( ( |_ `  ( 3  /  2 ) )  =  1  ->  -u ( |_ `  -u -u ( 3  / 
2 ) )  = 
-u 1 )
189, 17syl5eq 2132 . . 3  |-  ( ( |_ `  ( 3  /  2 ) )  =  1  ->  ( `  -u ( 3  /  2
) )  =  -u
1 )
19 ceilqval 9678 . . . . 5  |-  ( ( 3  /  2 )  e.  QQ  ->  ( `  ( 3  /  2
) )  =  -u ( |_ `  -u (
3  /  2 ) ) )
205, 19ax-mp 7 . . . 4  |-  ( `  (
3  /  2 ) )  =  -u ( |_ `  -u ( 3  / 
2 ) )
21 negeq 7654 . . . . 5  |-  ( ( |_ `  -u (
3  /  2 ) )  =  -u 2  -> 
-u ( |_ `  -u ( 3  /  2
) )  =  -u -u 2 )
22 2cn 8464 . . . . . 6  |-  2  e.  CC
2322negnegi 7731 . . . . 5  |-  -u -u 2  =  2
2421, 23syl6eq 2136 . . . 4  |-  ( ( |_ `  -u (
3  /  2 ) )  =  -u 2  -> 
-u ( |_ `  -u ( 3  /  2
) )  =  2 )
2520, 24syl5eq 2132 . . 3  |-  ( ( |_ `  -u (
3  /  2 ) )  =  -u 2  ->  ( `  ( 3  /  2 ) )  =  2 )
2618, 25anim12ci 332 . 2  |-  ( ( ( |_ `  (
3  /  2 ) )  =  1  /\  ( |_ `  -u (
3  /  2 ) )  =  -u 2
)  ->  ( ( `  ( 3  /  2
) )  =  2  /\  ( `  -u (
3  /  2 ) )  =  -u 1
) )
271, 26ax-mp 7 1  |-  ( ( `  ( 3  /  2
) )  =  2  /\  ( `  -u (
3  /  2 ) )  =  -u 1
)
Colors of variables: wff set class
Syntax hints:    /\ wa 102    = wceq 1289    e. wcel 1438   ` cfv 5002  (class class class)co 5634   CCcc 7327   1c1 7330   -ucneg 7633    / cdiv 8113   NNcn 8394   2c2 8444   3c3 8445   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-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-2 8452  df-3 8453  df-4 8454  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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