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Theorem ex-fl 20834
Description: Example for df-fl 10925. Example by David A. Wheeler. (Contributed by Mario Carneiro, 18-Jun-2015.)
Assertion
Ref Expression
ex-fl  |-  ( ( |_ `  ( 3  /  2 ) )  =  1  /\  ( |_ `  -u ( 3  / 
2 ) )  = 
-u 2 )

Proof of Theorem ex-fl
StepHypRef Expression
1 1re 8837 . . . 4  |-  1  e.  RR
2 3re 9817 . . . . 5  |-  3  e.  RR
3 rehalfcl 9938 . . . . 5  |-  ( 3  e.  RR  ->  (
3  /  2 )  e.  RR )
42, 3ax-mp 8 . . . 4  |-  ( 3  /  2 )  e.  RR
5 2cn 9816 . . . . . . 7  |-  2  e.  CC
65mulid2i 8840 . . . . . 6  |-  ( 1  x.  2 )  =  2
7 2lt3 9887 . . . . . 6  |-  2  <  3
86, 7eqbrtri 4042 . . . . 5  |-  ( 1  x.  2 )  <  3
9 2pos 9828 . . . . . 6  |-  0  <  2
10 2re 9815 . . . . . . 7  |-  2  e.  RR
111, 2, 10ltmuldivi 9677 . . . . . 6  |-  ( 0  <  2  ->  (
( 1  x.  2 )  <  3  <->  1  <  ( 3  / 
2 ) ) )
129, 11ax-mp 8 . . . . 5  |-  ( ( 1  x.  2 )  <  3  <->  1  <  ( 3  /  2 ) )
138, 12mpbi 199 . . . 4  |-  1  <  ( 3  /  2
)
141, 4, 13ltleii 8941 . . 3  |-  1  <_  ( 3  /  2
)
15 3lt4 9889 . . . . . 6  |-  3  <  4
16 2t2e4 9871 . . . . . 6  |-  ( 2  x.  2 )  =  4
1715, 16breqtrri 4048 . . . . 5  |-  3  <  ( 2  x.  2 )
1810, 9pm3.2i 441 . . . . . 6  |-  ( 2  e.  RR  /\  0  <  2 )
19 ltdivmul 9628 . . . . . 6  |-  ( ( 3  e.  RR  /\  2  e.  RR  /\  (
2  e.  RR  /\  0  <  2 ) )  ->  ( ( 3  /  2 )  <  2  <->  3  <  (
2  x.  2 ) ) )
202, 10, 18, 19mp3an 1277 . . . . 5  |-  ( ( 3  /  2 )  <  2  <->  3  <  ( 2  x.  2 ) )
2117, 20mpbir 200 . . . 4  |-  ( 3  /  2 )  <  2
22 df-2 9804 . . . 4  |-  2  =  ( 1  +  1 )
2321, 22breqtri 4046 . . 3  |-  ( 3  /  2 )  < 
( 1  +  1 )
24 1z 10053 . . . 4  |-  1  e.  ZZ
25 flbi 10946 . . . 4  |-  ( ( ( 3  /  2
)  e.  RR  /\  1  e.  ZZ )  ->  ( ( |_ `  ( 3  /  2
) )  =  1  <-> 
( 1  <_  (
3  /  2 )  /\  ( 3  / 
2 )  <  (
1  +  1 ) ) ) )
264, 24, 25mp2an 653 . . 3  |-  ( ( |_ `  ( 3  /  2 ) )  =  1  <->  ( 1  <_  ( 3  / 
2 )  /\  (
3  /  2 )  <  ( 1  +  1 ) ) )
2714, 23, 26mpbir2an 886 . 2  |-  ( |_
`  ( 3  / 
2 ) )  =  1
2810renegcli 9108 . . . 4  |-  -u 2  e.  RR
294renegcli 9108 . . . 4  |-  -u (
3  /  2 )  e.  RR
304, 10ltnegi 9317 . . . . 5  |-  ( ( 3  /  2 )  <  2  <->  -u 2  <  -u ( 3  /  2
) )
3121, 30mpbi 199 . . . 4  |-  -u 2  <  -u ( 3  / 
2 )
3228, 29, 31ltleii 8941 . . 3  |-  -u 2  <_ 
-u ( 3  / 
2 )
335negcli 9114 . . . . . . 7  |-  -u 2  e.  CC
34 ax-1cn 8795 . . . . . . 7  |-  1  e.  CC
35 negdi2 9105 . . . . . . 7  |-  ( (
-u 2  e.  CC  /\  1  e.  CC )  ->  -u ( -u 2  +  1 )  =  ( -u -u 2  -  1 ) )
3633, 34, 35mp2an 653 . . . . . 6  |-  -u ( -u 2  +  1 )  =  ( -u -u 2  -  1 )
375negnegi 9116 . . . . . . 7  |-  -u -u 2  =  2
3837oveq1i 5868 . . . . . 6  |-  ( -u -u 2  -  1 )  =  ( 2  -  1 )
3936, 38eqtri 2303 . . . . 5  |-  -u ( -u 2  +  1 )  =  ( 2  -  1 )
40 1p1e2 9840 . . . . . . 7  |-  ( 1  +  1 )  =  2
415, 34, 34, 40subaddrii 9135 . . . . . 6  |-  ( 2  -  1 )  =  1
4241, 13eqbrtri 4042 . . . . 5  |-  ( 2  -  1 )  < 
( 3  /  2
)
4339, 42eqbrtri 4042 . . . 4  |-  -u ( -u 2  +  1 )  <  ( 3  / 
2 )
4428, 1readdcli 8850 . . . . 5  |-  ( -u
2  +  1 )  e.  RR
4544, 4ltnegcon1i 9324 . . . 4  |-  ( -u ( -u 2  +  1 )  <  ( 3  /  2 )  <->  -u ( 3  /  2 )  < 
( -u 2  +  1 ) )
4643, 45mpbi 199 . . 3  |-  -u (
3  /  2 )  <  ( -u 2  +  1 )
47 2z 10054 . . . . 5  |-  2  e.  ZZ
48 znegcl 10055 . . . . 5  |-  ( 2  e.  ZZ  ->  -u 2  e.  ZZ )
4947, 48ax-mp 8 . . . 4  |-  -u 2  e.  ZZ
50 flbi 10946 . . . 4  |-  ( (
-u ( 3  / 
2 )  e.  RR  /\  -u 2  e.  ZZ )  ->  ( ( |_
`  -u ( 3  / 
2 ) )  = 
-u 2  <->  ( -u 2  <_ 
-u ( 3  / 
2 )  /\  -u (
3  /  2 )  <  ( -u 2  +  1 ) ) ) )
5129, 49, 50mp2an 653 . . 3  |-  ( ( |_ `  -u (
3  /  2 ) )  =  -u 2  <->  (
-u 2  <_  -u (
3  /  2 )  /\  -u ( 3  / 
2 )  <  ( -u 2  +  1 ) ) )
5232, 46, 51mpbir2an 886 . 2  |-  ( |_
`  -u ( 3  / 
2 ) )  = 
-u 2
5327, 52pm3.2i 441 1  |-  ( ( |_ `  ( 3  /  2 ) )  =  1  /\  ( |_ `  -u ( 3  / 
2 ) )  = 
-u 2 )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   CCcc 8735   RRcr 8736   0cc0 8737   1c1 8738    + caddc 8740    x. cmul 8742    < clt 8867    <_ cle 8868    - cmin 9037   -ucneg 9038    / cdiv 9423   2c2 9795   3c3 9796   4c4 9797   ZZcz 10024   |_cfl 10924
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-n0 9966  df-z 10025  df-uz 10231  df-fl 10925
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