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Theorem trirec0 13410
Description: Every real number having a reciprocal or equaling zero is equivalent to real number trichotomy.

This is the key part of the definition of what is known as a discrete field, so "the real numbers are a discrete field" can be taken as an equivalent way to state real trichotomy (see further discussion at trilpo 13409). (Contributed by Jim Kingdon, 10-Jun-2024.)

Assertion
Ref Expression
trirec0  |-  ( A. x  e.  RR  A. y  e.  RR  ( x  < 
y  \/  x  =  y  \/  y  < 
x )  <->  A. x  e.  RR  ( E. z  e.  RR  ( x  x.  z )  =  1  \/  x  =  0 ) )
Distinct variable group:    x, y, z

Proof of Theorem trirec0
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 simpll 519 . . . . . 6  |-  ( ( ( x  e.  RR  /\ 
A. y  e.  RR  ( x  <  y  \/  x  =  y  \/  y  <  x ) )  /\  x  <  0 )  ->  x  e.  RR )
2 simpr 109 . . . . . . 7  |-  ( ( ( x  e.  RR  /\ 
A. y  e.  RR  ( x  <  y  \/  x  =  y  \/  y  <  x ) )  /\  x  <  0 )  ->  x  <  0 )
31, 2lt0ap0d 8434 . . . . . 6  |-  ( ( ( x  e.  RR  /\ 
A. y  e.  RR  ( x  <  y  \/  x  =  y  \/  y  <  x ) )  /\  x  <  0 )  ->  x #  0 )
4 rerecclap 8513 . . . . . . 7  |-  ( ( x  e.  RR  /\  x #  0 )  ->  (
1  /  x )  e.  RR )
5 recn 7776 . . . . . . . 8  |-  ( x  e.  RR  ->  x  e.  CC )
6 recidap 8469 . . . . . . . 8  |-  ( ( x  e.  CC  /\  x #  0 )  ->  (
x  x.  ( 1  /  x ) )  =  1 )
75, 6sylan 281 . . . . . . 7  |-  ( ( x  e.  RR  /\  x #  0 )  ->  (
x  x.  ( 1  /  x ) )  =  1 )
8 oveq2 5789 . . . . . . . . 9  |-  ( z  =  ( 1  /  x )  ->  (
x  x.  z )  =  ( x  x.  ( 1  /  x
) ) )
98eqeq1d 2149 . . . . . . . 8  |-  ( z  =  ( 1  /  x )  ->  (
( x  x.  z
)  =  1  <->  (
x  x.  ( 1  /  x ) )  =  1 ) )
109rspcev 2792 . . . . . . 7  |-  ( ( ( 1  /  x
)  e.  RR  /\  ( x  x.  (
1  /  x ) )  =  1 )  ->  E. z  e.  RR  ( x  x.  z
)  =  1 )
114, 7, 10syl2anc 409 . . . . . 6  |-  ( ( x  e.  RR  /\  x #  0 )  ->  E. z  e.  RR  ( x  x.  z )  =  1 )
121, 3, 11syl2anc 409 . . . . 5  |-  ( ( ( x  e.  RR  /\ 
A. y  e.  RR  ( x  <  y  \/  x  =  y  \/  y  <  x ) )  /\  x  <  0 )  ->  E. z  e.  RR  ( x  x.  z )  =  1 )
1312orcd 723 . . . 4  |-  ( ( ( x  e.  RR  /\ 
A. y  e.  RR  ( x  <  y  \/  x  =  y  \/  y  <  x ) )  /\  x  <  0 )  ->  ( E. z  e.  RR  ( x  x.  z
)  =  1  \/  x  =  0 ) )
14 simpr 109 . . . . 5  |-  ( ( ( x  e.  RR  /\ 
A. y  e.  RR  ( x  <  y  \/  x  =  y  \/  y  <  x ) )  /\  x  =  0 )  ->  x  =  0 )
1514olcd 724 . . . 4  |-  ( ( ( x  e.  RR  /\ 
A. y  e.  RR  ( x  <  y  \/  x  =  y  \/  y  <  x ) )  /\  x  =  0 )  ->  ( E. z  e.  RR  ( x  x.  z
)  =  1  \/  x  =  0 ) )
16 simpll 519 . . . . . 6  |-  ( ( ( x  e.  RR  /\ 
A. y  e.  RR  ( x  <  y  \/  x  =  y  \/  y  <  x ) )  /\  0  < 
x )  ->  x  e.  RR )
17 simpr 109 . . . . . . 7  |-  ( ( ( x  e.  RR  /\ 
A. y  e.  RR  ( x  <  y  \/  x  =  y  \/  y  <  x ) )  /\  0  < 
x )  ->  0  <  x )
1816, 17gt0ap0d 8414 . . . . . 6  |-  ( ( ( x  e.  RR  /\ 
A. y  e.  RR  ( x  <  y  \/  x  =  y  \/  y  <  x ) )  /\  0  < 
x )  ->  x #  0 )
1916, 18, 11syl2anc 409 . . . . 5  |-  ( ( ( x  e.  RR  /\ 
A. y  e.  RR  ( x  <  y  \/  x  =  y  \/  y  <  x ) )  /\  0  < 
x )  ->  E. z  e.  RR  ( x  x.  z )  =  1 )
2019orcd 723 . . . 4  |-  ( ( ( x  e.  RR  /\ 
A. y  e.  RR  ( x  <  y  \/  x  =  y  \/  y  <  x ) )  /\  0  < 
x )  ->  ( E. z  e.  RR  ( x  x.  z
)  =  1  \/  x  =  0 ) )
21 0re 7789 . . . . . 6  |-  0  e.  RR
22 breq2 3940 . . . . . . . 8  |-  ( y  =  0  ->  (
x  <  y  <->  x  <  0 ) )
23 eqeq2 2150 . . . . . . . 8  |-  ( y  =  0  ->  (
x  =  y  <->  x  = 
0 ) )
24 breq1 3939 . . . . . . . 8  |-  ( y  =  0  ->  (
y  <  x  <->  0  <  x ) )
2522, 23, 243orbi123d 1290 . . . . . . 7  |-  ( y  =  0  ->  (
( x  <  y  \/  x  =  y  \/  y  <  x )  <-> 
( x  <  0  \/  x  =  0  \/  0  <  x ) ) )
2625rspcv 2788 . . . . . 6  |-  ( 0  e.  RR  ->  ( A. y  e.  RR  ( x  <  y  \/  x  =  y  \/  y  <  x )  ->  ( x  <  0  \/  x  =  0  \/  0  < 
x ) ) )
2721, 26ax-mp 5 . . . . 5  |-  ( A. y  e.  RR  (
x  <  y  \/  x  =  y  \/  y  <  x )  -> 
( x  <  0  \/  x  =  0  \/  0  <  x ) )
2827adantl 275 . . . 4  |-  ( ( x  e.  RR  /\  A. y  e.  RR  (
x  <  y  \/  x  =  y  \/  y  <  x ) )  ->  ( x  <  0  \/  x  =  0  \/  0  < 
x ) )
2913, 15, 20, 28mpjao3dan 1286 . . 3  |-  ( ( x  e.  RR  /\  A. y  e.  RR  (
x  <  y  \/  x  =  y  \/  y  <  x ) )  ->  ( E. z  e.  RR  ( x  x.  z )  =  1  \/  x  =  0 ) )
3029ralimiaa 2497 . 2  |-  ( A. x  e.  RR  A. y  e.  RR  ( x  < 
y  \/  x  =  y  \/  y  < 
x )  ->  A. x  e.  RR  ( E. z  e.  RR  ( x  x.  z )  =  1  \/  x  =  0 ) )
31 oveq1 5788 . . . . . . 7  |-  ( x  =  w  ->  (
x  x.  z )  =  ( w  x.  z ) )
3231eqeq1d 2149 . . . . . 6  |-  ( x  =  w  ->  (
( x  x.  z
)  =  1  <->  (
w  x.  z )  =  1 ) )
3332rexbidv 2439 . . . . 5  |-  ( x  =  w  ->  ( E. z  e.  RR  ( x  x.  z
)  =  1  <->  E. z  e.  RR  (
w  x.  z )  =  1 ) )
34 eqeq1 2147 . . . . 5  |-  ( x  =  w  ->  (
x  =  0  <->  w  =  0 ) )
3533, 34orbi12d 783 . . . 4  |-  ( x  =  w  ->  (
( E. z  e.  RR  ( x  x.  z )  =  1  \/  x  =  0 )  <->  ( E. z  e.  RR  ( w  x.  z )  =  1  \/  w  =  0 ) ) )
3635cbvralv 2657 . . 3  |-  ( A. x  e.  RR  ( E. z  e.  RR  ( x  x.  z
)  =  1  \/  x  =  0 )  <->  A. w  e.  RR  ( E. z  e.  RR  ( w  x.  z
)  =  1  \/  w  =  0 ) )
37 nfcv 2282 . . . . . . . . 9  |-  F/_ z RR
38 nfre1 2479 . . . . . . . . . 10  |-  F/ z E. z  e.  RR  ( w  x.  z
)  =  1
39 nfv 1509 . . . . . . . . . 10  |-  F/ z  w  =  0
4038, 39nfor 1554 . . . . . . . . 9  |-  F/ z ( E. z  e.  RR  ( w  x.  z )  =  1  \/  w  =  0 )
4137, 40nfralya 2476 . . . . . . . 8  |-  F/ z A. w  e.  RR  ( E. z  e.  RR  ( w  x.  z
)  =  1  \/  w  =  0 )
42 nfv 1509 . . . . . . . 8  |-  F/ z ( x  e.  RR  /\  y  e.  RR )
4341, 42nfan 1545 . . . . . . 7  |-  F/ z ( A. w  e.  RR  ( E. z  e.  RR  ( w  x.  z )  =  1  \/  w  =  0 )  /\  ( x  e.  RR  /\  y  e.  RR ) )
44 nfv 1509 . . . . . . 7  |-  F/ z ( x  <  y  \/  x  =  y  \/  y  <  x )
45 simpr 109 . . . . . . . . . . 11  |-  ( ( ( ( ( A. w  e.  RR  ( E. z  e.  RR  ( w  x.  z
)  =  1  \/  w  =  0 )  /\  ( x  e.  RR  /\  y  e.  RR ) )  /\  z  e.  RR )  /\  ( ( y  -  x )  x.  z
)  =  1 )  /\  ( y  -  x )  <  0
)  ->  ( y  -  x )  <  0
)
46 simprr 522 . . . . . . . . . . . . . 14  |-  ( ( A. w  e.  RR  ( E. z  e.  RR  ( w  x.  z
)  =  1  \/  w  =  0 )  /\  ( x  e.  RR  /\  y  e.  RR ) )  -> 
y  e.  RR )
4746ad2antrr 480 . . . . . . . . . . . . 13  |-  ( ( ( ( A. w  e.  RR  ( E. z  e.  RR  ( w  x.  z )  =  1  \/  w  =  0 )  /\  ( x  e.  RR  /\  y  e.  RR ) )  /\  z  e.  RR )  /\  ( ( y  -  x )  x.  z
)  =  1 )  ->  y  e.  RR )
4847adantr 274 . . . . . . . . . . . 12  |-  ( ( ( ( ( A. w  e.  RR  ( E. z  e.  RR  ( w  x.  z
)  =  1  \/  w  =  0 )  /\  ( x  e.  RR  /\  y  e.  RR ) )  /\  z  e.  RR )  /\  ( ( y  -  x )  x.  z
)  =  1 )  /\  ( y  -  x )  <  0
)  ->  y  e.  RR )
49 simprl 521 . . . . . . . . . . . . . 14  |-  ( ( A. w  e.  RR  ( E. z  e.  RR  ( w  x.  z
)  =  1  \/  w  =  0 )  /\  ( x  e.  RR  /\  y  e.  RR ) )  ->  x  e.  RR )
5049ad2antrr 480 . . . . . . . . . . . . 13  |-  ( ( ( ( A. w  e.  RR  ( E. z  e.  RR  ( w  x.  z )  =  1  \/  w  =  0 )  /\  ( x  e.  RR  /\  y  e.  RR ) )  /\  z  e.  RR )  /\  ( ( y  -  x )  x.  z
)  =  1 )  ->  x  e.  RR )
5150adantr 274 . . . . . . . . . . . 12  |-  ( ( ( ( ( A. w  e.  RR  ( E. z  e.  RR  ( w  x.  z
)  =  1  \/  w  =  0 )  /\  ( x  e.  RR  /\  y  e.  RR ) )  /\  z  e.  RR )  /\  ( ( y  -  x )  x.  z
)  =  1 )  /\  ( y  -  x )  <  0
)  ->  x  e.  RR )
5248, 51sublt0d 8355 . . . . . . . . . . 11  |-  ( ( ( ( ( A. w  e.  RR  ( E. z  e.  RR  ( w  x.  z
)  =  1  \/  w  =  0 )  /\  ( x  e.  RR  /\  y  e.  RR ) )  /\  z  e.  RR )  /\  ( ( y  -  x )  x.  z
)  =  1 )  /\  ( y  -  x )  <  0
)  ->  ( (
y  -  x )  <  0  <->  y  <  x ) )
5345, 52mpbid 146 . . . . . . . . . 10  |-  ( ( ( ( ( A. w  e.  RR  ( E. z  e.  RR  ( w  x.  z
)  =  1  \/  w  =  0 )  /\  ( x  e.  RR  /\  y  e.  RR ) )  /\  z  e.  RR )  /\  ( ( y  -  x )  x.  z
)  =  1 )  /\  ( y  -  x )  <  0
)  ->  y  <  x )
54533mix3d 1159 . . . . . . . . 9  |-  ( ( ( ( ( A. w  e.  RR  ( E. z  e.  RR  ( w  x.  z
)  =  1  \/  w  =  0 )  /\  ( x  e.  RR  /\  y  e.  RR ) )  /\  z  e.  RR )  /\  ( ( y  -  x )  x.  z
)  =  1 )  /\  ( y  -  x )  <  0
)  ->  ( x  <  y  \/  x  =  y  \/  y  < 
x ) )
55 simpr 109 . . . . . . . . . . 11  |-  ( ( ( ( ( A. w  e.  RR  ( E. z  e.  RR  ( w  x.  z
)  =  1  \/  w  =  0 )  /\  ( x  e.  RR  /\  y  e.  RR ) )  /\  z  e.  RR )  /\  ( ( y  -  x )  x.  z
)  =  1 )  /\  0  <  (
y  -  x ) )  ->  0  <  ( y  -  x ) )
5650adantr 274 . . . . . . . . . . . 12  |-  ( ( ( ( ( A. w  e.  RR  ( E. z  e.  RR  ( w  x.  z
)  =  1  \/  w  =  0 )  /\  ( x  e.  RR  /\  y  e.  RR ) )  /\  z  e.  RR )  /\  ( ( y  -  x )  x.  z
)  =  1 )  /\  0  <  (
y  -  x ) )  ->  x  e.  RR )
5747adantr 274 . . . . . . . . . . . 12  |-  ( ( ( ( ( A. w  e.  RR  ( E. z  e.  RR  ( w  x.  z
)  =  1  \/  w  =  0 )  /\  ( x  e.  RR  /\  y  e.  RR ) )  /\  z  e.  RR )  /\  ( ( y  -  x )  x.  z
)  =  1 )  /\  0  <  (
y  -  x ) )  ->  y  e.  RR )
5856, 57posdifd 8317 . . . . . . . . . . 11  |-  ( ( ( ( ( A. w  e.  RR  ( E. z  e.  RR  ( w  x.  z
)  =  1  \/  w  =  0 )  /\  ( x  e.  RR  /\  y  e.  RR ) )  /\  z  e.  RR )  /\  ( ( y  -  x )  x.  z
)  =  1 )  /\  0  <  (
y  -  x ) )  ->  ( x  <  y  <->  0  <  (
y  -  x ) ) )
5955, 58mpbird 166 . . . . . . . . . 10  |-  ( ( ( ( ( A. w  e.  RR  ( E. z  e.  RR  ( w  x.  z
)  =  1  \/  w  =  0 )  /\  ( x  e.  RR  /\  y  e.  RR ) )  /\  z  e.  RR )  /\  ( ( y  -  x )  x.  z
)  =  1 )  /\  0  <  (
y  -  x ) )  ->  x  <  y )
60593mix1d 1157 . . . . . . . . 9  |-  ( ( ( ( ( A. w  e.  RR  ( E. z  e.  RR  ( w  x.  z
)  =  1  \/  w  =  0 )  /\  ( x  e.  RR  /\  y  e.  RR ) )  /\  z  e.  RR )  /\  ( ( y  -  x )  x.  z
)  =  1 )  /\  0  <  (
y  -  x ) )  ->  ( x  <  y  \/  x  =  y  \/  y  < 
x ) )
6147recnd 7817 . . . . . . . . . . . 12  |-  ( ( ( ( A. w  e.  RR  ( E. z  e.  RR  ( w  x.  z )  =  1  \/  w  =  0 )  /\  ( x  e.  RR  /\  y  e.  RR ) )  /\  z  e.  RR )  /\  ( ( y  -  x )  x.  z
)  =  1 )  ->  y  e.  CC )
6250recnd 7817 . . . . . . . . . . . 12  |-  ( ( ( ( A. w  e.  RR  ( E. z  e.  RR  ( w  x.  z )  =  1  \/  w  =  0 )  /\  ( x  e.  RR  /\  y  e.  RR ) )  /\  z  e.  RR )  /\  ( ( y  -  x )  x.  z
)  =  1 )  ->  x  e.  CC )
6361, 62subcld 8096 . . . . . . . . . . 11  |-  ( ( ( ( A. w  e.  RR  ( E. z  e.  RR  ( w  x.  z )  =  1  \/  w  =  0 )  /\  ( x  e.  RR  /\  y  e.  RR ) )  /\  z  e.  RR )  /\  ( ( y  -  x )  x.  z
)  =  1 )  ->  ( y  -  x )  e.  CC )
64 simplr 520 . . . . . . . . . . . 12  |-  ( ( ( ( A. w  e.  RR  ( E. z  e.  RR  ( w  x.  z )  =  1  \/  w  =  0 )  /\  ( x  e.  RR  /\  y  e.  RR ) )  /\  z  e.  RR )  /\  ( ( y  -  x )  x.  z
)  =  1 )  ->  z  e.  RR )
6564recnd 7817 . . . . . . . . . . 11  |-  ( ( ( ( A. w  e.  RR  ( E. z  e.  RR  ( w  x.  z )  =  1  \/  w  =  0 )  /\  ( x  e.  RR  /\  y  e.  RR ) )  /\  z  e.  RR )  /\  ( ( y  -  x )  x.  z
)  =  1 )  ->  z  e.  CC )
66 simpr 109 . . . . . . . . . . . 12  |-  ( ( ( ( A. w  e.  RR  ( E. z  e.  RR  ( w  x.  z )  =  1  \/  w  =  0 )  /\  ( x  e.  RR  /\  y  e.  RR ) )  /\  z  e.  RR )  /\  ( ( y  -  x )  x.  z
)  =  1 )  ->  ( ( y  -  x )  x.  z )  =  1 )
67 1ap0 8375 . . . . . . . . . . . 12  |-  1 #  0
6866, 67eqbrtrdi 3974 . . . . . . . . . . 11  |-  ( ( ( ( A. w  e.  RR  ( E. z  e.  RR  ( w  x.  z )  =  1  \/  w  =  0 )  /\  ( x  e.  RR  /\  y  e.  RR ) )  /\  z  e.  RR )  /\  ( ( y  -  x )  x.  z
)  =  1 )  ->  ( ( y  -  x )  x.  z ) #  0 )
6963, 65, 68mulap0bad 8443 . . . . . . . . . 10  |-  ( ( ( ( A. w  e.  RR  ( E. z  e.  RR  ( w  x.  z )  =  1  \/  w  =  0 )  /\  ( x  e.  RR  /\  y  e.  RR ) )  /\  z  e.  RR )  /\  ( ( y  -  x )  x.  z
)  =  1 )  ->  ( y  -  x ) #  0 )
7046, 49resubcld 8166 . . . . . . . . . . . 12  |-  ( ( A. w  e.  RR  ( E. z  e.  RR  ( w  x.  z
)  =  1  \/  w  =  0 )  /\  ( x  e.  RR  /\  y  e.  RR ) )  -> 
( y  -  x
)  e.  RR )
7170ad2antrr 480 . . . . . . . . . . 11  |-  ( ( ( ( A. w  e.  RR  ( E. z  e.  RR  ( w  x.  z )  =  1  \/  w  =  0 )  /\  ( x  e.  RR  /\  y  e.  RR ) )  /\  z  e.  RR )  /\  ( ( y  -  x )  x.  z
)  =  1 )  ->  ( y  -  x )  e.  RR )
72 reaplt 8373 . . . . . . . . . . 11  |-  ( ( ( y  -  x
)  e.  RR  /\  0  e.  RR )  ->  ( ( y  -  x ) #  0  <->  ( (
y  -  x )  <  0  \/  0  <  ( y  -  x ) ) ) )
7371, 21, 72sylancl 410 . . . . . . . . . 10  |-  ( ( ( ( A. w  e.  RR  ( E. z  e.  RR  ( w  x.  z )  =  1  \/  w  =  0 )  /\  ( x  e.  RR  /\  y  e.  RR ) )  /\  z  e.  RR )  /\  ( ( y  -  x )  x.  z
)  =  1 )  ->  ( ( y  -  x ) #  0  <-> 
( ( y  -  x )  <  0  \/  0  <  ( y  -  x ) ) ) )
7469, 73mpbid 146 . . . . . . . . 9  |-  ( ( ( ( A. w  e.  RR  ( E. z  e.  RR  ( w  x.  z )  =  1  \/  w  =  0 )  /\  ( x  e.  RR  /\  y  e.  RR ) )  /\  z  e.  RR )  /\  ( ( y  -  x )  x.  z
)  =  1 )  ->  ( ( y  -  x )  <  0  \/  0  < 
( y  -  x
) ) )
7554, 60, 74mpjaodan 788 . . . . . . . 8  |-  ( ( ( ( A. w  e.  RR  ( E. z  e.  RR  ( w  x.  z )  =  1  \/  w  =  0 )  /\  ( x  e.  RR  /\  y  e.  RR ) )  /\  z  e.  RR )  /\  ( ( y  -  x )  x.  z
)  =  1 )  ->  ( x  < 
y  \/  x  =  y  \/  y  < 
x ) )
7675exp31 362 . . . . . . 7  |-  ( ( A. w  e.  RR  ( E. z  e.  RR  ( w  x.  z
)  =  1  \/  w  =  0 )  /\  ( x  e.  RR  /\  y  e.  RR ) )  -> 
( z  e.  RR  ->  ( ( ( y  -  x )  x.  z )  =  1  ->  ( x  < 
y  \/  x  =  y  \/  y  < 
x ) ) ) )
7743, 44, 76rexlimd 2549 . . . . . 6  |-  ( ( A. w  e.  RR  ( E. z  e.  RR  ( w  x.  z
)  =  1  \/  w  =  0 )  /\  ( x  e.  RR  /\  y  e.  RR ) )  -> 
( E. z  e.  RR  ( ( y  -  x )  x.  z )  =  1  ->  ( x  < 
y  \/  x  =  y  \/  y  < 
x ) ) )
7877imp 123 . . . . 5  |-  ( ( ( A. w  e.  RR  ( E. z  e.  RR  ( w  x.  z )  =  1  \/  w  =  0 )  /\  ( x  e.  RR  /\  y  e.  RR ) )  /\  E. z  e.  RR  (
( y  -  x
)  x.  z )  =  1 )  -> 
( x  <  y  \/  x  =  y  \/  y  <  x ) )
7946recnd 7817 . . . . . . . . 9  |-  ( ( A. w  e.  RR  ( E. z  e.  RR  ( w  x.  z
)  =  1  \/  w  =  0 )  /\  ( x  e.  RR  /\  y  e.  RR ) )  -> 
y  e.  CC )
8079adantr 274 . . . . . . . 8  |-  ( ( ( A. w  e.  RR  ( E. z  e.  RR  ( w  x.  z )  =  1  \/  w  =  0 )  /\  ( x  e.  RR  /\  y  e.  RR ) )  /\  ( y  -  x
)  =  0 )  ->  y  e.  CC )
8149recnd 7817 . . . . . . . . 9  |-  ( ( A. w  e.  RR  ( E. z  e.  RR  ( w  x.  z
)  =  1  \/  w  =  0 )  /\  ( x  e.  RR  /\  y  e.  RR ) )  ->  x  e.  CC )
8281adantr 274 . . . . . . . 8  |-  ( ( ( A. w  e.  RR  ( E. z  e.  RR  ( w  x.  z )  =  1  \/  w  =  0 )  /\  ( x  e.  RR  /\  y  e.  RR ) )  /\  ( y  -  x
)  =  0 )  ->  x  e.  CC )
83 simpr 109 . . . . . . . 8  |-  ( ( ( A. w  e.  RR  ( E. z  e.  RR  ( w  x.  z )  =  1  \/  w  =  0 )  /\  ( x  e.  RR  /\  y  e.  RR ) )  /\  ( y  -  x
)  =  0 )  ->  ( y  -  x )  =  0 )
8480, 82, 83subeq0d 8104 . . . . . . 7  |-  ( ( ( A. w  e.  RR  ( E. z  e.  RR  ( w  x.  z )  =  1  \/  w  =  0 )  /\  ( x  e.  RR  /\  y  e.  RR ) )  /\  ( y  -  x
)  =  0 )  ->  y  =  x )
8584equcomd 1684 . . . . . 6  |-  ( ( ( A. w  e.  RR  ( E. z  e.  RR  ( w  x.  z )  =  1  \/  w  =  0 )  /\  ( x  e.  RR  /\  y  e.  RR ) )  /\  ( y  -  x
)  =  0 )  ->  x  =  y )
86853mix2d 1158 . . . . 5  |-  ( ( ( A. w  e.  RR  ( E. z  e.  RR  ( w  x.  z )  =  1  \/  w  =  0 )  /\  ( x  e.  RR  /\  y  e.  RR ) )  /\  ( y  -  x
)  =  0 )  ->  ( x  < 
y  \/  x  =  y  \/  y  < 
x ) )
87 oveq1 5788 . . . . . . . . 9  |-  ( w  =  ( y  -  x )  ->  (
w  x.  z )  =  ( ( y  -  x )  x.  z ) )
8887eqeq1d 2149 . . . . . . . 8  |-  ( w  =  ( y  -  x )  ->  (
( w  x.  z
)  =  1  <->  (
( y  -  x
)  x.  z )  =  1 ) )
8988rexbidv 2439 . . . . . . 7  |-  ( w  =  ( y  -  x )  ->  ( E. z  e.  RR  ( w  x.  z
)  =  1  <->  E. z  e.  RR  (
( y  -  x
)  x.  z )  =  1 ) )
90 eqeq1 2147 . . . . . . 7  |-  ( w  =  ( y  -  x )  ->  (
w  =  0  <->  (
y  -  x )  =  0 ) )
9189, 90orbi12d 783 . . . . . 6  |-  ( w  =  ( y  -  x )  ->  (
( E. z  e.  RR  ( w  x.  z )  =  1  \/  w  =  0 )  <->  ( E. z  e.  RR  ( ( y  -  x )  x.  z )  =  1  \/  ( y  -  x )  =  0 ) ) )
92 simpl 108 . . . . . 6  |-  ( ( A. w  e.  RR  ( E. z  e.  RR  ( w  x.  z
)  =  1  \/  w  =  0 )  /\  ( x  e.  RR  /\  y  e.  RR ) )  ->  A. w  e.  RR  ( E. z  e.  RR  ( w  x.  z
)  =  1  \/  w  =  0 ) )
9391, 92, 70rspcdva 2797 . . . . 5  |-  ( ( A. w  e.  RR  ( E. z  e.  RR  ( w  x.  z
)  =  1  \/  w  =  0 )  /\  ( x  e.  RR  /\  y  e.  RR ) )  -> 
( E. z  e.  RR  ( ( y  -  x )  x.  z )  =  1  \/  ( y  -  x )  =  0 ) )
9478, 86, 93mpjaodan 788 . . . 4  |-  ( ( A. w  e.  RR  ( E. z  e.  RR  ( w  x.  z
)  =  1  \/  w  =  0 )  /\  ( x  e.  RR  /\  y  e.  RR ) )  -> 
( x  <  y  \/  x  =  y  \/  y  <  x ) )
9594ralrimivva 2517 . . 3  |-  ( A. w  e.  RR  ( E. z  e.  RR  ( w  x.  z
)  =  1  \/  w  =  0 )  ->  A. x  e.  RR  A. y  e.  RR  (
x  <  y  \/  x  =  y  \/  y  <  x ) )
9636, 95sylbi 120 . 2  |-  ( A. x  e.  RR  ( E. z  e.  RR  ( x  x.  z
)  =  1  \/  x  =  0 )  ->  A. x  e.  RR  A. y  e.  RR  (
x  <  y  \/  x  =  y  \/  y  <  x ) )
9730, 96impbii 125 1  |-  ( A. x  e.  RR  A. y  e.  RR  ( x  < 
y  \/  x  =  y  \/  y  < 
x )  <->  A. x  e.  RR  ( E. z  e.  RR  ( x  x.  z )  =  1  \/  x  =  0 ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    \/ wo 698    \/ w3o 962    = wceq 1332    e. wcel 1481   A.wral 2417   E.wrex 2418   class class class wbr 3936  (class class class)co 5781   CCcc 7641   RRcr 7642   0cc0 7643   1c1 7644    x. cmul 7648    < clt 7823    - cmin 7956   # cap 8366    / cdiv 8455
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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4053  ax-pow 4105  ax-pr 4138  ax-un 4362  ax-setind 4459  ax-cnex 7734  ax-resscn 7735  ax-1cn 7736  ax-1re 7737  ax-icn 7738  ax-addcl 7739  ax-addrcl 7740  ax-mulcl 7741  ax-mulrcl 7742  ax-addcom 7743  ax-mulcom 7744  ax-addass 7745  ax-mulass 7746  ax-distr 7747  ax-i2m1 7748  ax-0lt1 7749  ax-1rid 7750  ax-0id 7751  ax-rnegex 7752  ax-precex 7753  ax-cnre 7754  ax-pre-ltirr 7755  ax-pre-ltwlin 7756  ax-pre-lttrn 7757  ax-pre-apti 7758  ax-pre-ltadd 7759  ax-pre-mulgt0 7760  ax-pre-mulext 7761
This theorem depends on definitions:  df-bi 116  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-reu 2424  df-rmo 2425  df-rab 2426  df-v 2691  df-sbc 2913  df-dif 3077  df-un 3079  df-in 3081  df-ss 3088  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-br 3937  df-opab 3997  df-id 4222  df-po 4225  df-iso 4226  df-xp 4552  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-iota 5095  df-fun 5132  df-fv 5138  df-riota 5737  df-ov 5784  df-oprab 5785  df-mpo 5786  df-pnf 7825  df-mnf 7826  df-xr 7827  df-ltxr 7828  df-le 7829  df-sub 7958  df-neg 7959  df-reap 8360  df-ap 8367  df-div 8456
This theorem is referenced by:  trirec0xor  13411
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