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Theorem mulextsr1 7779
Description: Strong extensionality of multiplication of signed reals. (Contributed by Jim Kingdon, 18-Feb-2020.)
Assertion
Ref Expression
mulextsr1  |-  ( ( A  e.  R.  /\  B  e.  R.  /\  C  e.  R. )  ->  (
( A  .R  C
)  <R  ( B  .R  C )  ->  ( A  <R  B  \/  B  <R  A ) ) )

Proof of Theorem mulextsr1
Dummy variables  u  v  w  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-nr 7725 . 2  |-  R.  =  ( ( P.  X.  P. ) /.  ~R  )
2 oveq1 5881 . . . 4  |-  ( [
<. x ,  y >. ]  ~R  =  A  -> 
( [ <. x ,  y >. ]  ~R  .R 
[ <. u ,  v
>. ]  ~R  )  =  ( A  .R  [ <. u ,  v >. ]  ~R  ) )
32breq1d 4013 . . 3  |-  ( [
<. x ,  y >. ]  ~R  =  A  -> 
( ( [ <. x ,  y >. ]  ~R  .R 
[ <. u ,  v
>. ]  ~R  )  <R 
( [ <. z ,  w >. ]  ~R  .R  [
<. u ,  v >. ]  ~R  )  <->  ( A  .R  [ <. u ,  v
>. ]  ~R  )  <R 
( [ <. z ,  w >. ]  ~R  .R  [
<. u ,  v >. ]  ~R  ) ) )
4 breq1 4006 . . . 4  |-  ( [
<. x ,  y >. ]  ~R  =  A  -> 
( [ <. x ,  y >. ]  ~R  <R  [ <. z ,  w >. ]  ~R  <->  A  <R  [
<. z ,  w >. ]  ~R  ) )
5 breq2 4007 . . . 4  |-  ( [
<. x ,  y >. ]  ~R  =  A  -> 
( [ <. z ,  w >. ]  ~R  <R  [
<. x ,  y >. ]  ~R  <->  [ <. z ,  w >. ]  ~R  <R  A ) )
64, 5orbi12d 793 . . 3  |-  ( [
<. x ,  y >. ]  ~R  =  A  -> 
( ( [ <. x ,  y >. ]  ~R  <R  [ <. z ,  w >. ]  ~R  \/  [ <. z ,  w >. ]  ~R  <R  [ <. x ,  y >. ]  ~R  ) 
<->  ( A  <R  [ <. z ,  w >. ]  ~R  \/  [ <. z ,  w >. ]  ~R  <R  A ) ) )
73, 6imbi12d 234 . 2  |-  ( [
<. x ,  y >. ]  ~R  =  A  -> 
( ( ( [
<. x ,  y >. ]  ~R  .R  [ <. u ,  v >. ]  ~R  )  <R  ( [ <. z ,  w >. ]  ~R  .R 
[ <. u ,  v
>. ]  ~R  )  -> 
( [ <. x ,  y >. ]  ~R  <R  [ <. z ,  w >. ]  ~R  \/  [ <. z ,  w >. ]  ~R  <R  [ <. x ,  y >. ]  ~R  ) )  <->  ( ( A  .R  [ <. u ,  v >. ]  ~R  )  <R  ( [ <. z ,  w >. ]  ~R  .R 
[ <. u ,  v
>. ]  ~R  )  -> 
( A  <R  [ <. z ,  w >. ]  ~R  \/  [ <. z ,  w >. ]  ~R  <R  A ) ) ) )
8 oveq1 5881 . . . 4  |-  ( [
<. z ,  w >. ]  ~R  =  B  -> 
( [ <. z ,  w >. ]  ~R  .R  [
<. u ,  v >. ]  ~R  )  =  ( B  .R  [ <. u ,  v >. ]  ~R  ) )
98breq2d 4015 . . 3  |-  ( [
<. z ,  w >. ]  ~R  =  B  -> 
( ( A  .R  [
<. u ,  v >. ]  ~R  )  <R  ( [ <. z ,  w >. ]  ~R  .R  [ <. u ,  v >. ]  ~R  )  <->  ( A  .R  [ <. u ,  v
>. ]  ~R  )  <R 
( B  .R  [ <. u ,  v >. ]  ~R  ) ) )
10 breq2 4007 . . . 4  |-  ( [
<. z ,  w >. ]  ~R  =  B  -> 
( A  <R  [ <. z ,  w >. ]  ~R  <->  A 
<R  B ) )
11 breq1 4006 . . . 4  |-  ( [
<. z ,  w >. ]  ~R  =  B  -> 
( [ <. z ,  w >. ]  ~R  <R  A  <-> 
B  <R  A ) )
1210, 11orbi12d 793 . . 3  |-  ( [
<. z ,  w >. ]  ~R  =  B  -> 
( ( A  <R  [
<. z ,  w >. ]  ~R  \/  [ <. z ,  w >. ]  ~R  <R  A )  <->  ( A  <R  B  \/  B  <R  A ) ) )
139, 12imbi12d 234 . 2  |-  ( [
<. z ,  w >. ]  ~R  =  B  -> 
( ( ( A  .R  [ <. u ,  v >. ]  ~R  )  <R  ( [ <. z ,  w >. ]  ~R  .R 
[ <. u ,  v
>. ]  ~R  )  -> 
( A  <R  [ <. z ,  w >. ]  ~R  \/  [ <. z ,  w >. ]  ~R  <R  A ) )  <->  ( ( A  .R  [ <. u ,  v >. ]  ~R  )  <R  ( B  .R  [
<. u ,  v >. ]  ~R  )  ->  ( A  <R  B  \/  B  <R  A ) ) ) )
14 oveq2 5882 . . . 4  |-  ( [
<. u ,  v >. ]  ~R  =  C  -> 
( A  .R  [ <. u ,  v >. ]  ~R  )  =  ( A  .R  C ) )
15 oveq2 5882 . . . 4  |-  ( [
<. u ,  v >. ]  ~R  =  C  -> 
( B  .R  [ <. u ,  v >. ]  ~R  )  =  ( B  .R  C ) )
1614, 15breq12d 4016 . . 3  |-  ( [
<. u ,  v >. ]  ~R  =  C  -> 
( ( A  .R  [
<. u ,  v >. ]  ~R  )  <R  ( B  .R  [ <. u ,  v >. ]  ~R  ) 
<->  ( A  .R  C
)  <R  ( B  .R  C ) ) )
1716imbi1d 231 . 2  |-  ( [
<. u ,  v >. ]  ~R  =  C  -> 
( ( ( A  .R  [ <. u ,  v >. ]  ~R  )  <R  ( B  .R  [
<. u ,  v >. ]  ~R  )  ->  ( A  <R  B  \/  B  <R  A ) )  <->  ( ( A  .R  C )  <R 
( B  .R  C
)  ->  ( A  <R  B  \/  B  <R  A ) ) ) )
18 mulextsr1lem 7778 . . 3  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )  /\  ( u  e.  P.  /\  v  e.  P. )
)  ->  ( (
( ( x  .P.  u )  +P.  (
y  .P.  v )
)  +P.  ( (
z  .P.  v )  +P.  ( w  .P.  u
) ) )  <P 
( ( ( x  .P.  v )  +P.  ( y  .P.  u
) )  +P.  (
( z  .P.  u
)  +P.  ( w  .P.  v ) ) )  ->  ( ( x  +P.  w )  <P 
( y  +P.  z
)  \/  ( z  +P.  y )  <P 
( w  +P.  x
) ) ) )
19 mulsrpr 7744 . . . . . 6  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( u  e.  P.  /\  v  e.  P. )
)  ->  ( [ <. x ,  y >. ]  ~R  .R  [ <. u ,  v >. ]  ~R  )  =  [ <. (
( x  .P.  u
)  +P.  ( y  .P.  v ) ) ,  ( ( x  .P.  v )  +P.  (
y  .P.  u )
) >. ]  ~R  )
20193adant2 1016 . . . . 5  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )  /\  ( u  e.  P.  /\  v  e.  P. )
)  ->  ( [ <. x ,  y >. ]  ~R  .R  [ <. u ,  v >. ]  ~R  )  =  [ <. (
( x  .P.  u
)  +P.  ( y  .P.  v ) ) ,  ( ( x  .P.  v )  +P.  (
y  .P.  u )
) >. ]  ~R  )
21 mulsrpr 7744 . . . . . 6  |-  ( ( ( z  e.  P.  /\  w  e.  P. )  /\  ( u  e.  P.  /\  v  e.  P. )
)  ->  ( [ <. z ,  w >. ]  ~R  .R  [ <. u ,  v >. ]  ~R  )  =  [ <. (
( z  .P.  u
)  +P.  ( w  .P.  v ) ) ,  ( ( z  .P.  v )  +P.  (
w  .P.  u )
) >. ]  ~R  )
22213adant1 1015 . . . . 5  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )  /\  ( u  e.  P.  /\  v  e.  P. )
)  ->  ( [ <. z ,  w >. ]  ~R  .R  [ <. u ,  v >. ]  ~R  )  =  [ <. (
( z  .P.  u
)  +P.  ( w  .P.  v ) ) ,  ( ( z  .P.  v )  +P.  (
w  .P.  u )
) >. ]  ~R  )
2320, 22breq12d 4016 . . . 4  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )  /\  ( u  e.  P.  /\  v  e.  P. )
)  ->  ( ( [ <. x ,  y
>. ]  ~R  .R  [ <. u ,  v >. ]  ~R  )  <R  ( [ <. z ,  w >. ]  ~R  .R  [ <. u ,  v >. ]  ~R  )  <->  [ <. (
( x  .P.  u
)  +P.  ( y  .P.  v ) ) ,  ( ( x  .P.  v )  +P.  (
y  .P.  u )
) >. ]  ~R  <R  [
<. ( ( z  .P.  u )  +P.  (
w  .P.  v )
) ,  ( ( z  .P.  v )  +P.  ( w  .P.  u ) ) >. ]  ~R  ) )
24 simp1l 1021 . . . . . . 7  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )  /\  ( u  e.  P.  /\  v  e.  P. )
)  ->  x  e.  P. )
25 simp3l 1025 . . . . . . 7  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )  /\  ( u  e.  P.  /\  v  e.  P. )
)  ->  u  e.  P. )
26 mulclpr 7570 . . . . . . 7  |-  ( ( x  e.  P.  /\  u  e.  P. )  ->  ( x  .P.  u
)  e.  P. )
2724, 25, 26syl2anc 411 . . . . . 6  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )  /\  ( u  e.  P.  /\  v  e.  P. )
)  ->  ( x  .P.  u )  e.  P. )
28 simp1r 1022 . . . . . . 7  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )  /\  ( u  e.  P.  /\  v  e.  P. )
)  ->  y  e.  P. )
29 simp3r 1026 . . . . . . 7  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )  /\  ( u  e.  P.  /\  v  e.  P. )
)  ->  v  e.  P. )
30 mulclpr 7570 . . . . . . 7  |-  ( ( y  e.  P.  /\  v  e.  P. )  ->  ( y  .P.  v
)  e.  P. )
3128, 29, 30syl2anc 411 . . . . . 6  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )  /\  ( u  e.  P.  /\  v  e.  P. )
)  ->  ( y  .P.  v )  e.  P. )
32 addclpr 7535 . . . . . 6  |-  ( ( ( x  .P.  u
)  e.  P.  /\  ( y  .P.  v
)  e.  P. )  ->  ( ( x  .P.  u )  +P.  (
y  .P.  v )
)  e.  P. )
3327, 31, 32syl2anc 411 . . . . 5  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )  /\  ( u  e.  P.  /\  v  e.  P. )
)  ->  ( (
x  .P.  u )  +P.  ( y  .P.  v
) )  e.  P. )
34 mulclpr 7570 . . . . . . 7  |-  ( ( x  e.  P.  /\  v  e.  P. )  ->  ( x  .P.  v
)  e.  P. )
3524, 29, 34syl2anc 411 . . . . . 6  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )  /\  ( u  e.  P.  /\  v  e.  P. )
)  ->  ( x  .P.  v )  e.  P. )
36 mulclpr 7570 . . . . . . 7  |-  ( ( y  e.  P.  /\  u  e.  P. )  ->  ( y  .P.  u
)  e.  P. )
3728, 25, 36syl2anc 411 . . . . . 6  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )  /\  ( u  e.  P.  /\  v  e.  P. )
)  ->  ( y  .P.  u )  e.  P. )
38 addclpr 7535 . . . . . 6  |-  ( ( ( x  .P.  v
)  e.  P.  /\  ( y  .P.  u
)  e.  P. )  ->  ( ( x  .P.  v )  +P.  (
y  .P.  u )
)  e.  P. )
3935, 37, 38syl2anc 411 . . . . 5  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )  /\  ( u  e.  P.  /\  v  e.  P. )
)  ->  ( (
x  .P.  v )  +P.  ( y  .P.  u
) )  e.  P. )
40 simp2l 1023 . . . . . . 7  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )  /\  ( u  e.  P.  /\  v  e.  P. )
)  ->  z  e.  P. )
41 mulclpr 7570 . . . . . . 7  |-  ( ( z  e.  P.  /\  u  e.  P. )  ->  ( z  .P.  u
)  e.  P. )
4240, 25, 41syl2anc 411 . . . . . 6  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )  /\  ( u  e.  P.  /\  v  e.  P. )
)  ->  ( z  .P.  u )  e.  P. )
43 simp2r 1024 . . . . . . 7  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )  /\  ( u  e.  P.  /\  v  e.  P. )
)  ->  w  e.  P. )
44 mulclpr 7570 . . . . . . 7  |-  ( ( w  e.  P.  /\  v  e.  P. )  ->  ( w  .P.  v
)  e.  P. )
4543, 29, 44syl2anc 411 . . . . . 6  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )  /\  ( u  e.  P.  /\  v  e.  P. )
)  ->  ( w  .P.  v )  e.  P. )
46 addclpr 7535 . . . . . 6  |-  ( ( ( z  .P.  u
)  e.  P.  /\  ( w  .P.  v )  e.  P. )  -> 
( ( z  .P.  u )  +P.  (
w  .P.  v )
)  e.  P. )
4742, 45, 46syl2anc 411 . . . . 5  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )  /\  ( u  e.  P.  /\  v  e.  P. )
)  ->  ( (
z  .P.  u )  +P.  ( w  .P.  v
) )  e.  P. )
48 mulclpr 7570 . . . . . . 7  |-  ( ( z  e.  P.  /\  v  e.  P. )  ->  ( z  .P.  v
)  e.  P. )
4940, 29, 48syl2anc 411 . . . . . 6  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )  /\  ( u  e.  P.  /\  v  e.  P. )
)  ->  ( z  .P.  v )  e.  P. )
50 mulclpr 7570 . . . . . . 7  |-  ( ( w  e.  P.  /\  u  e.  P. )  ->  ( w  .P.  u
)  e.  P. )
5143, 25, 50syl2anc 411 . . . . . 6  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )  /\  ( u  e.  P.  /\  v  e.  P. )
)  ->  ( w  .P.  u )  e.  P. )
52 addclpr 7535 . . . . . 6  |-  ( ( ( z  .P.  v
)  e.  P.  /\  ( w  .P.  u )  e.  P. )  -> 
( ( z  .P.  v )  +P.  (
w  .P.  u )
)  e.  P. )
5349, 51, 52syl2anc 411 . . . . 5  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )  /\  ( u  e.  P.  /\  v  e.  P. )
)  ->  ( (
z  .P.  v )  +P.  ( w  .P.  u
) )  e.  P. )
54 ltsrprg 7745 . . . . 5  |-  ( ( ( ( ( x  .P.  u )  +P.  ( y  .P.  v
) )  e.  P.  /\  ( ( x  .P.  v )  +P.  (
y  .P.  u )
)  e.  P. )  /\  ( ( ( z  .P.  u )  +P.  ( w  .P.  v
) )  e.  P.  /\  ( ( z  .P.  v )  +P.  (
w  .P.  u )
)  e.  P. )
)  ->  ( [ <. ( ( x  .P.  u )  +P.  (
y  .P.  v )
) ,  ( ( x  .P.  v )  +P.  ( y  .P.  u ) ) >. ]  ~R  <R  [ <. (
( z  .P.  u
)  +P.  ( w  .P.  v ) ) ,  ( ( z  .P.  v )  +P.  (
w  .P.  u )
) >. ]  ~R  <->  ( (
( x  .P.  u
)  +P.  ( y  .P.  v ) )  +P.  ( ( z  .P.  v )  +P.  (
w  .P.  u )
) )  <P  (
( ( x  .P.  v )  +P.  (
y  .P.  u )
)  +P.  ( (
z  .P.  u )  +P.  ( w  .P.  v
) ) ) ) )
5533, 39, 47, 53, 54syl22anc 1239 . . . 4  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )  /\  ( u  e.  P.  /\  v  e.  P. )
)  ->  ( [ <. ( ( x  .P.  u )  +P.  (
y  .P.  v )
) ,  ( ( x  .P.  v )  +P.  ( y  .P.  u ) ) >. ]  ~R  <R  [ <. (
( z  .P.  u
)  +P.  ( w  .P.  v ) ) ,  ( ( z  .P.  v )  +P.  (
w  .P.  u )
) >. ]  ~R  <->  ( (
( x  .P.  u
)  +P.  ( y  .P.  v ) )  +P.  ( ( z  .P.  v )  +P.  (
w  .P.  u )
) )  <P  (
( ( x  .P.  v )  +P.  (
y  .P.  u )
)  +P.  ( (
z  .P.  u )  +P.  ( w  .P.  v
) ) ) ) )
5623, 55bitrd 188 . . 3  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )  /\  ( u  e.  P.  /\  v  e.  P. )
)  ->  ( ( [ <. x ,  y
>. ]  ~R  .R  [ <. u ,  v >. ]  ~R  )  <R  ( [ <. z ,  w >. ]  ~R  .R  [ <. u ,  v >. ]  ~R  )  <->  ( (
( x  .P.  u
)  +P.  ( y  .P.  v ) )  +P.  ( ( z  .P.  v )  +P.  (
w  .P.  u )
) )  <P  (
( ( x  .P.  v )  +P.  (
y  .P.  u )
)  +P.  ( (
z  .P.  u )  +P.  ( w  .P.  v
) ) ) ) )
57 ltsrprg 7745 . . . . 5  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  ->  ( [ <. x ,  y >. ]  ~R  <R  [ <. z ,  w >. ]  ~R  <->  ( x  +P.  w )  <P  (
y  +P.  z )
) )
58573adant3 1017 . . . 4  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )  /\  ( u  e.  P.  /\  v  e.  P. )
)  ->  ( [ <. x ,  y >. ]  ~R  <R  [ <. z ,  w >. ]  ~R  <->  ( x  +P.  w )  <P  (
y  +P.  z )
) )
59 ltsrprg 7745 . . . . . 6  |-  ( ( ( z  e.  P.  /\  w  e.  P. )  /\  ( x  e.  P.  /\  y  e.  P. )
)  ->  ( [ <. z ,  w >. ]  ~R  <R  [ <. x ,  y >. ]  ~R  <->  ( z  +P.  y ) 
<P  ( w  +P.  x
) ) )
6059ancoms 268 . . . . 5  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  ->  ( [ <. z ,  w >. ]  ~R  <R  [ <. x ,  y >. ]  ~R  <->  ( z  +P.  y ) 
<P  ( w  +P.  x
) ) )
61603adant3 1017 . . . 4  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )  /\  ( u  e.  P.  /\  v  e.  P. )
)  ->  ( [ <. z ,  w >. ]  ~R  <R  [ <. x ,  y >. ]  ~R  <->  ( z  +P.  y ) 
<P  ( w  +P.  x
) ) )
6258, 61orbi12d 793 . . 3  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )  /\  ( u  e.  P.  /\  v  e.  P. )
)  ->  ( ( [ <. x ,  y
>. ]  ~R  <R  [ <. z ,  w >. ]  ~R  \/  [ <. z ,  w >. ]  ~R  <R  [ <. x ,  y >. ]  ~R  ) 
<->  ( ( x  +P.  w )  <P  (
y  +P.  z )  \/  ( z  +P.  y
)  <P  ( w  +P.  x ) ) ) )
6318, 56, 623imtr4d 203 . 2  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )  /\  ( u  e.  P.  /\  v  e.  P. )
)  ->  ( ( [ <. x ,  y
>. ]  ~R  .R  [ <. u ,  v >. ]  ~R  )  <R  ( [ <. z ,  w >. ]  ~R  .R  [ <. u ,  v >. ]  ~R  )  ->  ( [ <. x ,  y
>. ]  ~R  <R  [ <. z ,  w >. ]  ~R  \/  [ <. z ,  w >. ]  ~R  <R  [ <. x ,  y >. ]  ~R  ) ) )
641, 7, 13, 17, 633ecoptocl 6623 1  |-  ( ( A  e.  R.  /\  B  e.  R.  /\  C  e.  R. )  ->  (
( A  .R  C
)  <R  ( B  .R  C )  ->  ( A  <R  B  \/  B  <R  A ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    \/ wo 708    /\ w3a 978    = wceq 1353    e. wcel 2148   <.cop 3595   class class class wbr 4003  (class class class)co 5874   [cec 6532   P.cnp 7289    +P. cpp 7291    .P. cmp 7292    <P cltp 7293    ~R cer 7294   R.cnr 7295    .R cmr 7300    <R cltr 7301
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-coll 4118  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-iinf 4587
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-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-tr 4102  df-eprel 4289  df-id 4293  df-po 4296  df-iso 4297  df-iord 4366  df-on 4368  df-suc 4371  df-iom 4590  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-ov 5877  df-oprab 5878  df-mpo 5879  df-1st 6140  df-2nd 6141  df-recs 6305  df-irdg 6370  df-1o 6416  df-2o 6417  df-oadd 6420  df-omul 6421  df-er 6534  df-ec 6536  df-qs 6540  df-ni 7302  df-pli 7303  df-mi 7304  df-lti 7305  df-plpq 7342  df-mpq 7343  df-enq 7345  df-nqqs 7346  df-plqqs 7347  df-mqqs 7348  df-1nqqs 7349  df-rq 7350  df-ltnqqs 7351  df-enq0 7422  df-nq0 7423  df-0nq0 7424  df-plq0 7425  df-mq0 7426  df-inp 7464  df-i1p 7465  df-iplp 7466  df-imp 7467  df-iltp 7468  df-enr 7724  df-nr 7725  df-mr 7727  df-ltr 7728
This theorem is referenced by:  axpre-mulext  7886
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