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Theorem mulextsr1 6865
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 6812 . 2  |-  R.  =  ( ( P.  X.  P. ) /.  ~R  )
2 oveq1 5519 . . . 4  |-  ( [
<. x ,  y >. ]  ~R  =  A  -> 
( [ <. x ,  y >. ]  ~R  .R 
[ <. u ,  v
>. ]  ~R  )  =  ( A  .R  [ <. u ,  v >. ]  ~R  ) )
32breq1d 3774 . . 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 3767 . . . 4  |-  ( [
<. x ,  y >. ]  ~R  =  A  -> 
( [ <. x ,  y >. ]  ~R  <R  [ <. z ,  w >. ]  ~R  <->  A  <R  [
<. z ,  w >. ]  ~R  ) )
5 breq2 3768 . . . 4  |-  ( [
<. x ,  y >. ]  ~R  =  A  -> 
( [ <. z ,  w >. ]  ~R  <R  [
<. x ,  y >. ]  ~R  <->  [ <. z ,  w >. ]  ~R  <R  A ) )
64, 5orbi12d 707 . . 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 223 . 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 5519 . . . 4  |-  ( [
<. z ,  w >. ]  ~R  =  B  -> 
( [ <. z ,  w >. ]  ~R  .R  [
<. u ,  v >. ]  ~R  )  =  ( B  .R  [ <. u ,  v >. ]  ~R  ) )
98breq2d 3776 . . 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 3768 . . . 4  |-  ( [
<. z ,  w >. ]  ~R  =  B  -> 
( A  <R  [ <. z ,  w >. ]  ~R  <->  A 
<R  B ) )
11 breq1 3767 . . . 4  |-  ( [
<. z ,  w >. ]  ~R  =  B  -> 
( [ <. z ,  w >. ]  ~R  <R  A  <-> 
B  <R  A ) )
1210, 11orbi12d 707 . . 3  |-  ( [
<. z ,  w >. ]  ~R  =  B  -> 
( ( A  <R  [
<. z ,  w >. ]  ~R  \/  [ <. z ,  w >. ]  ~R  <R  A )  <->  ( A  <R  B  \/  B  <R  A ) ) )
139, 12imbi12d 223 . 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 5520 . . . 4  |-  ( [
<. u ,  v >. ]  ~R  =  C  -> 
( A  .R  [ <. u ,  v >. ]  ~R  )  =  ( A  .R  C ) )
15 oveq2 5520 . . . 4  |-  ( [
<. u ,  v >. ]  ~R  =  C  -> 
( B  .R  [ <. u ,  v >. ]  ~R  )  =  ( B  .R  C ) )
1614, 15breq12d 3777 . . 3  |-  ( [
<. u ,  v >. ]  ~R  =  C  -> 
( ( A  .R  [
<. u ,  v >. ]  ~R  )  <R  ( B  .R  [ <. u ,  v >. ]  ~R  ) 
<->  ( A  .R  C
)  <R  ( B  .R  C ) ) )
1716imbi1d 220 . 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 6864 . . 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 6831 . . . . . 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 923 . . . . 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 6831 . . . . . 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 922 . . . . 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 3777 . . . 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 928 . . . . . . 7  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )  /\  ( u  e.  P.  /\  v  e.  P. )
)  ->  x  e.  P. )
25 simp3l 932 . . . . . . 7  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )  /\  ( u  e.  P.  /\  v  e.  P. )
)  ->  u  e.  P. )
26 mulclpr 6670 . . . . . . 7  |-  ( ( x  e.  P.  /\  u  e.  P. )  ->  ( x  .P.  u
)  e.  P. )
2724, 25, 26syl2anc 391 . . . . . 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 929 . . . . . . 7  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )  /\  ( u  e.  P.  /\  v  e.  P. )
)  ->  y  e.  P. )
29 simp3r 933 . . . . . . 7  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )  /\  ( u  e.  P.  /\  v  e.  P. )
)  ->  v  e.  P. )
30 mulclpr 6670 . . . . . . 7  |-  ( ( y  e.  P.  /\  v  e.  P. )  ->  ( y  .P.  v
)  e.  P. )
3128, 29, 30syl2anc 391 . . . . . 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 6635 . . . . . 6  |-  ( ( ( x  .P.  u
)  e.  P.  /\  ( y  .P.  v
)  e.  P. )  ->  ( ( x  .P.  u )  +P.  (
y  .P.  v )
)  e.  P. )
3327, 31, 32syl2anc 391 . . . . 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 6670 . . . . . . 7  |-  ( ( x  e.  P.  /\  v  e.  P. )  ->  ( x  .P.  v
)  e.  P. )
3524, 29, 34syl2anc 391 . . . . . 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 6670 . . . . . . 7  |-  ( ( y  e.  P.  /\  u  e.  P. )  ->  ( y  .P.  u
)  e.  P. )
3728, 25, 36syl2anc 391 . . . . . 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 6635 . . . . . 6  |-  ( ( ( x  .P.  v
)  e.  P.  /\  ( y  .P.  u
)  e.  P. )  ->  ( ( x  .P.  v )  +P.  (
y  .P.  u )
)  e.  P. )
3935, 37, 38syl2anc 391 . . . . 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 930 . . . . . . 7  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )  /\  ( u  e.  P.  /\  v  e.  P. )
)  ->  z  e.  P. )
41 mulclpr 6670 . . . . . . 7  |-  ( ( z  e.  P.  /\  u  e.  P. )  ->  ( z  .P.  u
)  e.  P. )
4240, 25, 41syl2anc 391 . . . . . 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 931 . . . . . . 7  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )  /\  ( u  e.  P.  /\  v  e.  P. )
)  ->  w  e.  P. )
44 mulclpr 6670 . . . . . . 7  |-  ( ( w  e.  P.  /\  v  e.  P. )  ->  ( w  .P.  v
)  e.  P. )
4543, 29, 44syl2anc 391 . . . . . 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 6635 . . . . . 6  |-  ( ( ( z  .P.  u
)  e.  P.  /\  ( w  .P.  v )  e.  P. )  -> 
( ( z  .P.  u )  +P.  (
w  .P.  v )
)  e.  P. )
4742, 45, 46syl2anc 391 . . . . 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 6670 . . . . . . 7  |-  ( ( z  e.  P.  /\  v  e.  P. )  ->  ( z  .P.  v
)  e.  P. )
4940, 29, 48syl2anc 391 . . . . . 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 6670 . . . . . . 7  |-  ( ( w  e.  P.  /\  u  e.  P. )  ->  ( w  .P.  u
)  e.  P. )
5143, 25, 50syl2anc 391 . . . . . 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 6635 . . . . . 6  |-  ( ( ( z  .P.  v
)  e.  P.  /\  ( w  .P.  u )  e.  P. )  -> 
( ( z  .P.  v )  +P.  (
w  .P.  u )
)  e.  P. )
5349, 51, 52syl2anc 391 . . . . 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 6832 . . . . 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 1136 . . . 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 177 . . 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 6832 . . . . 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 924 . . . 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 6832 . . . . . 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 255 . . . . 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 924 . . . 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 707 . . 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 192 . 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 6195 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 97    <-> wb 98    \/ wo 629    /\ w3a 885    = wceq 1243    e. wcel 1393   <.cop 3378   class class class wbr 3764  (class class class)co 5512   [cec 6104   P.cnp 6389    +P. cpp 6391    .P. cmp 6392    <P cltp 6393    ~R cer 6394   R.cnr 6395    .R cmr 6400    <R cltr 6401
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311
This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-eprel 4026  df-id 4030  df-po 4033  df-iso 4034  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-1o 6001  df-2o 6002  df-oadd 6005  df-omul 6006  df-er 6106  df-ec 6108  df-qs 6112  df-ni 6402  df-pli 6403  df-mi 6404  df-lti 6405  df-plpq 6442  df-mpq 6443  df-enq 6445  df-nqqs 6446  df-plqqs 6447  df-mqqs 6448  df-1nqqs 6449  df-rq 6450  df-ltnqqs 6451  df-enq0 6522  df-nq0 6523  df-0nq0 6524  df-plq0 6525  df-mq0 6526  df-inp 6564  df-i1p 6565  df-iplp 6566  df-imp 6567  df-iltp 6568  df-enr 6811  df-nr 6812  df-mr 6814  df-ltr 6815
This theorem is referenced by:  axpre-mulext  6962
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