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Theorem ex-fl 20850
Description: Example for df-fl 10941. 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 8853 . . . 4  |-  1  e.  RR
2 3re 9833 . . . . 5  |-  3  e.  RR
3 rehalfcl 9954 . . . . 5  |-  ( 3  e.  RR  ->  (
3  /  2 )  e.  RR )
42, 3ax-mp 8 . . . 4  |-  ( 3  /  2 )  e.  RR
5 2cn 9832 . . . . . . 7  |-  2  e.  CC
65mulid2i 8856 . . . . . 6  |-  ( 1  x.  2 )  =  2
7 2lt3 9903 . . . . . 6  |-  2  <  3
86, 7eqbrtri 4058 . . . . 5  |-  ( 1  x.  2 )  <  3
9 2pos 9844 . . . . . 6  |-  0  <  2
10 2re 9831 . . . . . . 7  |-  2  e.  RR
111, 2, 10ltmuldivi 9693 . . . . . 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 8957 . . 3  |-  1  <_  ( 3  /  2
)
15 3lt4 9905 . . . . . 6  |-  3  <  4
16 2t2e4 9887 . . . . . 6  |-  ( 2  x.  2 )  =  4
1715, 16breqtrri 4064 . . . . 5  |-  3  <  ( 2  x.  2 )
1810, 9pm3.2i 441 . . . . . 6  |-  ( 2  e.  RR  /\  0  <  2 )
19 ltdivmul 9644 . . . . . 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 9820 . . . 4  |-  2  =  ( 1  +  1 )
2321, 22breqtri 4062 . . 3  |-  ( 3  /  2 )  < 
( 1  +  1 )
24 1z 10069 . . . 4  |-  1  e.  ZZ
25 flbi 10962 . . . 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 9124 . . . 4  |-  -u 2  e.  RR
294renegcli 9124 . . . 4  |-  -u (
3  /  2 )  e.  RR
304, 10ltnegi 9333 . . . . 5  |-  ( ( 3  /  2 )  <  2  <->  -u 2  <  -u ( 3  /  2
) )
3121, 30mpbi 199 . . . 4  |-  -u 2  <  -u ( 3  / 
2 )
3228, 29, 31ltleii 8957 . . 3  |-  -u 2  <_ 
-u ( 3  / 
2 )
335negcli 9130 . . . . . . 7  |-  -u 2  e.  CC
34 ax-1cn 8811 . . . . . . 7  |-  1  e.  CC
35 negdi2 9121 . . . . . . 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 9132 . . . . . . 7  |-  -u -u 2  =  2
3837oveq1i 5884 . . . . . 6  |-  ( -u -u 2  -  1 )  =  ( 2  -  1 )
3936, 38eqtri 2316 . . . . 5  |-  -u ( -u 2  +  1 )  =  ( 2  -  1 )
40 1p1e2 9856 . . . . . . 7  |-  ( 1  +  1 )  =  2
415, 34, 34, 40subaddrii 9151 . . . . . 6  |-  ( 2  -  1 )  =  1
4241, 13eqbrtri 4058 . . . . 5  |-  ( 2  -  1 )  < 
( 3  /  2
)
4339, 42eqbrtri 4058 . . . 4  |-  -u ( -u 2  +  1 )  <  ( 3  / 
2 )
4428, 1readdcli 8866 . . . . 5  |-  ( -u
2  +  1 )  e.  RR
4544, 4ltnegcon1i 9340 . . . 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 10070 . . . . 5  |-  2  e.  ZZ
48 znegcl 10071 . . . . 5  |-  ( 2  e.  ZZ  ->  -u 2  e.  ZZ )
4947, 48ax-mp 8 . . . 4  |-  -u 2  e.  ZZ
50 flbi 10962 . . . 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 1632    e. wcel 1696   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   CCcc 8751   RRcr 8752   0cc0 8753   1c1 8754    + caddc 8756    x. cmul 8758    < clt 8883    <_ cle 8884    - cmin 9053   -ucneg 9054    / cdiv 9439   2c2 9811   3c3 9812   4c4 9813   ZZcz 10040   |_cfl 10940
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-n0 9982  df-z 10041  df-uz 10247  df-fl 10941
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