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Theorem reap0 14726
Description: Real number trichotomy is equivalent to decidability of apartness from zero. (Contributed by Jim Kingdon, 27-Jul-2024.)
Assertion
Ref Expression
reap0  |-  ( A. x  e.  RR  A. y  e.  RR  ( x  < 
y  \/  x  =  y  \/  y  < 
x )  <->  A. z  e.  RR DECID  z #  0 )
Distinct variable group:    x, y, z

Proof of Theorem reap0
StepHypRef Expression
1 simpl 109 . . . . 5  |-  ( ( A. x  e.  RR  A. y  e.  RR  (
x  <  y  \/  x  =  y  \/  y  <  x )  /\  z  e.  RR )  ->  A. x  e.  RR  A. y  e.  RR  (
x  <  y  \/  x  =  y  \/  y  <  x ) )
2 simpr 110 . . . . . 6  |-  ( ( A. x  e.  RR  A. y  e.  RR  (
x  <  y  \/  x  =  y  \/  y  <  x )  /\  z  e.  RR )  ->  z  e.  RR )
3 0re 7956 . . . . . 6  |-  0  e.  RR
4 breq1 4006 . . . . . . . 8  |-  ( x  =  z  ->  (
x  <  y  <->  z  <  y ) )
5 equequ1 1712 . . . . . . . 8  |-  ( x  =  z  ->  (
x  =  y  <->  z  =  y ) )
6 breq2 4007 . . . . . . . 8  |-  ( x  =  z  ->  (
y  <  x  <->  y  <  z ) )
74, 5, 63orbi123d 1311 . . . . . . 7  |-  ( x  =  z  ->  (
( x  <  y  \/  x  =  y  \/  y  <  x )  <-> 
( z  <  y  \/  z  =  y  \/  y  <  z ) ) )
8 breq2 4007 . . . . . . . 8  |-  ( y  =  0  ->  (
z  <  y  <->  z  <  0 ) )
9 eqeq2 2187 . . . . . . . 8  |-  ( y  =  0  ->  (
z  =  y  <->  z  = 
0 ) )
10 breq1 4006 . . . . . . . 8  |-  ( y  =  0  ->  (
y  <  z  <->  0  <  z ) )
118, 9, 103orbi123d 1311 . . . . . . 7  |-  ( y  =  0  ->  (
( z  <  y  \/  z  =  y  \/  y  <  z )  <-> 
( z  <  0  \/  z  =  0  \/  0  <  z ) ) )
127, 11rspc2v 2854 . . . . . 6  |-  ( ( z  e.  RR  /\  0  e.  RR )  ->  ( A. x  e.  RR  A. y  e.  RR  ( x  < 
y  \/  x  =  y  \/  y  < 
x )  ->  (
z  <  0  \/  z  =  0  \/  0  <  z ) ) )
132, 3, 12sylancl 413 . . . . 5  |-  ( ( A. x  e.  RR  A. y  e.  RR  (
x  <  y  \/  x  =  y  \/  y  <  x )  /\  z  e.  RR )  ->  ( A. x  e.  RR  A. y  e.  RR  ( x  < 
y  \/  x  =  y  \/  y  < 
x )  ->  (
z  <  0  \/  z  =  0  \/  0  <  z ) ) )
141, 13mpd 13 . . . 4  |-  ( ( A. x  e.  RR  A. y  e.  RR  (
x  <  y  \/  x  =  y  \/  y  <  x )  /\  z  e.  RR )  ->  ( z  <  0  \/  z  =  0  \/  0  <  z ) )
15 triap 14697 . . . . 5  |-  ( ( z  e.  RR  /\  0  e.  RR )  ->  ( ( z  <  0  \/  z  =  0  \/  0  < 
z )  <-> DECID  z #  0 ) )
162, 3, 15sylancl 413 . . . 4  |-  ( ( A. x  e.  RR  A. y  e.  RR  (
x  <  y  \/  x  =  y  \/  y  <  x )  /\  z  e.  RR )  ->  ( ( z  <  0  \/  z  =  0  \/  0  < 
z )  <-> DECID  z #  0 ) )
1714, 16mpbid 147 . . 3  |-  ( ( A. x  e.  RR  A. y  e.  RR  (
x  <  y  \/  x  =  y  \/  y  <  x )  /\  z  e.  RR )  -> DECID  z #  0 )
1817ralrimiva 2550 . 2  |-  ( A. x  e.  RR  A. y  e.  RR  ( x  < 
y  \/  x  =  y  \/  y  < 
x )  ->  A. z  e.  RR DECID  z #  0 )
19 breq1 4006 . . . . . . 7  |-  ( z  =  ( x  -  y )  ->  (
z #  0  <->  ( x  -  y ) #  0 ) )
2019dcbid 838 . . . . . 6  |-  ( z  =  ( x  -  y )  ->  (DECID  z #  0 
<-> DECID  ( x  -  y ) #  0 ) )
21 simpl 109 . . . . . 6  |-  ( ( A. z  e.  RR DECID  z #  0  /\  ( x  e.  RR  /\  y  e.  RR ) )  ->  A. z  e.  RR DECID  z #  0 )
22 resubcl 8219 . . . . . . 7  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( x  -  y
)  e.  RR )
2322adantl 277 . . . . . 6  |-  ( ( A. z  e.  RR DECID  z #  0  /\  ( x  e.  RR  /\  y  e.  RR ) )  -> 
( x  -  y
)  e.  RR )
2420, 21, 23rspcdva 2846 . . . . 5  |-  ( ( A. z  e.  RR DECID  z #  0  /\  ( x  e.  RR  /\  y  e.  RR ) )  -> DECID  (
x  -  y ) #  0 )
25 simprl 529 . . . . . . . 8  |-  ( ( A. z  e.  RR DECID  z #  0  /\  ( x  e.  RR  /\  y  e.  RR ) )  ->  x  e.  RR )
2625recnd 7984 . . . . . . 7  |-  ( ( A. z  e.  RR DECID  z #  0  /\  ( x  e.  RR  /\  y  e.  RR ) )  ->  x  e.  CC )
27 simprr 531 . . . . . . . 8  |-  ( ( A. z  e.  RR DECID  z #  0  /\  ( x  e.  RR  /\  y  e.  RR ) )  -> 
y  e.  RR )
2827recnd 7984 . . . . . . 7  |-  ( ( A. z  e.  RR DECID  z #  0  /\  ( x  e.  RR  /\  y  e.  RR ) )  -> 
y  e.  CC )
29 subap0 8598 . . . . . . 7  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( ( x  -  y ) #  0  <->  x #  y
) )
3026, 28, 29syl2anc 411 . . . . . 6  |-  ( ( A. z  e.  RR DECID  z #  0  /\  ( x  e.  RR  /\  y  e.  RR ) )  -> 
( ( x  -  y ) #  0  <->  x #  y
) )
3130dcbid 838 . . . . 5  |-  ( ( A. z  e.  RR DECID  z #  0  /\  ( x  e.  RR  /\  y  e.  RR ) )  -> 
(DECID  ( x  -  y
) #  0  <-> DECID  x #  y )
)
3224, 31mpbid 147 . . . 4  |-  ( ( A. z  e.  RR DECID  z #  0  /\  ( x  e.  RR  /\  y  e.  RR ) )  -> DECID  x #  y )
33 triap 14697 . . . . 5  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( ( x  < 
y  \/  x  =  y  \/  y  < 
x )  <-> DECID  x #  y )
)
3433adantl 277 . . . 4  |-  ( ( A. z  e.  RR DECID  z #  0  /\  ( x  e.  RR  /\  y  e.  RR ) )  -> 
( ( x  < 
y  \/  x  =  y  \/  y  < 
x )  <-> DECID  x #  y )
)
3532, 34mpbird 167 . . 3  |-  ( ( A. z  e.  RR DECID  z #  0  /\  ( x  e.  RR  /\  y  e.  RR ) )  -> 
( x  <  y  \/  x  =  y  \/  y  <  x ) )
3635ralrimivva 2559 . 2  |-  ( A. z  e.  RR DECID  z #  0  ->  A. x  e.  RR  A. y  e.  RR  (
x  <  y  \/  x  =  y  \/  y  <  x ) )
3718, 36impbii 126 1  |-  ( A. x  e.  RR  A. y  e.  RR  ( x  < 
y  \/  x  =  y  \/  y  < 
x )  <->  A. z  e.  RR DECID  z #  0 )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105  DECID wdc 834    \/ w3o 977    = wceq 1353    e. wcel 2148   A.wral 2455   class class class wbr 4003  (class class class)co 5874   CCcc 7808   RRcr 7809   0cc0 7810    < clt 7990    - cmin 8126   # cap 8536
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-cnex 7901  ax-resscn 7902  ax-1cn 7903  ax-1re 7904  ax-icn 7905  ax-addcl 7906  ax-addrcl 7907  ax-mulcl 7908  ax-mulrcl 7909  ax-addcom 7910  ax-mulcom 7911  ax-addass 7912  ax-mulass 7913  ax-distr 7914  ax-i2m1 7915  ax-0lt1 7916  ax-1rid 7917  ax-0id 7918  ax-rnegex 7919  ax-precex 7920  ax-cnre 7921  ax-pre-ltirr 7922  ax-pre-lttrn 7924  ax-pre-apti 7925  ax-pre-ltadd 7926  ax-pre-mulgt0 7927
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4004  df-opab 4065  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-iota 5178  df-fun 5218  df-fv 5224  df-riota 5830  df-ov 5877  df-oprab 5878  df-mpo 5879  df-pnf 7992  df-mnf 7993  df-ltxr 7995  df-sub 8128  df-neg 8129  df-reap 8530  df-ap 8537
This theorem is referenced by:  dcapnconstALT  14729
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