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Theorem mulextsr1 7557
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 7503 . 2  |-  R.  =  ( ( P.  X.  P. ) /.  ~R  )
2 oveq1 5749 . . . 4  |-  ( [
<. x ,  y >. ]  ~R  =  A  -> 
( [ <. x ,  y >. ]  ~R  .R 
[ <. u ,  v
>. ]  ~R  )  =  ( A  .R  [ <. u ,  v >. ]  ~R  ) )
32breq1d 3909 . . 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 3902 . . . 4  |-  ( [
<. x ,  y >. ]  ~R  =  A  -> 
( [ <. x ,  y >. ]  ~R  <R  [ <. z ,  w >. ]  ~R  <->  A  <R  [
<. z ,  w >. ]  ~R  ) )
5 breq2 3903 . . . 4  |-  ( [
<. x ,  y >. ]  ~R  =  A  -> 
( [ <. z ,  w >. ]  ~R  <R  [
<. x ,  y >. ]  ~R  <->  [ <. z ,  w >. ]  ~R  <R  A ) )
64, 5orbi12d 767 . . 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 233 . 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 5749 . . . 4  |-  ( [
<. z ,  w >. ]  ~R  =  B  -> 
( [ <. z ,  w >. ]  ~R  .R  [
<. u ,  v >. ]  ~R  )  =  ( B  .R  [ <. u ,  v >. ]  ~R  ) )
98breq2d 3911 . . 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 3903 . . . 4  |-  ( [
<. z ,  w >. ]  ~R  =  B  -> 
( A  <R  [ <. z ,  w >. ]  ~R  <->  A 
<R  B ) )
11 breq1 3902 . . . 4  |-  ( [
<. z ,  w >. ]  ~R  =  B  -> 
( [ <. z ,  w >. ]  ~R  <R  A  <-> 
B  <R  A ) )
1210, 11orbi12d 767 . . 3  |-  ( [
<. z ,  w >. ]  ~R  =  B  -> 
( ( A  <R  [
<. z ,  w >. ]  ~R  \/  [ <. z ,  w >. ]  ~R  <R  A )  <->  ( A  <R  B  \/  B  <R  A ) ) )
139, 12imbi12d 233 . 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 5750 . . . 4  |-  ( [
<. u ,  v >. ]  ~R  =  C  -> 
( A  .R  [ <. u ,  v >. ]  ~R  )  =  ( A  .R  C ) )
15 oveq2 5750 . . . 4  |-  ( [
<. u ,  v >. ]  ~R  =  C  -> 
( B  .R  [ <. u ,  v >. ]  ~R  )  =  ( B  .R  C ) )
1614, 15breq12d 3912 . . 3  |-  ( [
<. u ,  v >. ]  ~R  =  C  -> 
( ( A  .R  [
<. u ,  v >. ]  ~R  )  <R  ( B  .R  [ <. u ,  v >. ]  ~R  ) 
<->  ( A  .R  C
)  <R  ( B  .R  C ) ) )
1716imbi1d 230 . 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 7556 . . 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 7522 . . . . . 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 985 . . . . 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 7522 . . . . . 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 984 . . . . 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 3912 . . . 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 990 . . . . . . 7  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )  /\  ( u  e.  P.  /\  v  e.  P. )
)  ->  x  e.  P. )
25 simp3l 994 . . . . . . 7  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )  /\  ( u  e.  P.  /\  v  e.  P. )
)  ->  u  e.  P. )
26 mulclpr 7348 . . . . . . 7  |-  ( ( x  e.  P.  /\  u  e.  P. )  ->  ( x  .P.  u
)  e.  P. )
2724, 25, 26syl2anc 408 . . . . . 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 991 . . . . . . 7  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )  /\  ( u  e.  P.  /\  v  e.  P. )
)  ->  y  e.  P. )
29 simp3r 995 . . . . . . 7  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )  /\  ( u  e.  P.  /\  v  e.  P. )
)  ->  v  e.  P. )
30 mulclpr 7348 . . . . . . 7  |-  ( ( y  e.  P.  /\  v  e.  P. )  ->  ( y  .P.  v
)  e.  P. )
3128, 29, 30syl2anc 408 . . . . . 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 7313 . . . . . 6  |-  ( ( ( x  .P.  u
)  e.  P.  /\  ( y  .P.  v
)  e.  P. )  ->  ( ( x  .P.  u )  +P.  (
y  .P.  v )
)  e.  P. )
3327, 31, 32syl2anc 408 . . . . 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 7348 . . . . . . 7  |-  ( ( x  e.  P.  /\  v  e.  P. )  ->  ( x  .P.  v
)  e.  P. )
3524, 29, 34syl2anc 408 . . . . . 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 7348 . . . . . . 7  |-  ( ( y  e.  P.  /\  u  e.  P. )  ->  ( y  .P.  u
)  e.  P. )
3728, 25, 36syl2anc 408 . . . . . 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 7313 . . . . . 6  |-  ( ( ( x  .P.  v
)  e.  P.  /\  ( y  .P.  u
)  e.  P. )  ->  ( ( x  .P.  v )  +P.  (
y  .P.  u )
)  e.  P. )
3935, 37, 38syl2anc 408 . . . . 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 992 . . . . . . 7  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )  /\  ( u  e.  P.  /\  v  e.  P. )
)  ->  z  e.  P. )
41 mulclpr 7348 . . . . . . 7  |-  ( ( z  e.  P.  /\  u  e.  P. )  ->  ( z  .P.  u
)  e.  P. )
4240, 25, 41syl2anc 408 . . . . . 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 993 . . . . . . 7  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )  /\  ( u  e.  P.  /\  v  e.  P. )
)  ->  w  e.  P. )
44 mulclpr 7348 . . . . . . 7  |-  ( ( w  e.  P.  /\  v  e.  P. )  ->  ( w  .P.  v
)  e.  P. )
4543, 29, 44syl2anc 408 . . . . . 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 7313 . . . . . 6  |-  ( ( ( z  .P.  u
)  e.  P.  /\  ( w  .P.  v )  e.  P. )  -> 
( ( z  .P.  u )  +P.  (
w  .P.  v )
)  e.  P. )
4742, 45, 46syl2anc 408 . . . . 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 7348 . . . . . . 7  |-  ( ( z  e.  P.  /\  v  e.  P. )  ->  ( z  .P.  v
)  e.  P. )
4940, 29, 48syl2anc 408 . . . . . 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 7348 . . . . . . 7  |-  ( ( w  e.  P.  /\  u  e.  P. )  ->  ( w  .P.  u
)  e.  P. )
5143, 25, 50syl2anc 408 . . . . . 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 7313 . . . . . 6  |-  ( ( ( z  .P.  v
)  e.  P.  /\  ( w  .P.  u )  e.  P. )  -> 
( ( z  .P.  v )  +P.  (
w  .P.  u )
)  e.  P. )
5349, 51, 52syl2anc 408 . . . . 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 7523 . . . . 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 1202 . . . 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 187 . . 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 7523 . . . . 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 986 . . . 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 7523 . . . . . 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 266 . . . . 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 986 . . . 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 767 . . 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 202 . 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 6486 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 103    <-> wb 104    \/ wo 682    /\ w3a 947    = wceq 1316    e. wcel 1465   <.cop 3500   class class class wbr 3899  (class class class)co 5742   [cec 6395   P.cnp 7067    +P. cpp 7069    .P. cmp 7070    <P cltp 7071    ~R cer 7072   R.cnr 7073    .R cmr 7078    <R cltr 7079
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-coll 4013  ax-sep 4016  ax-nul 4024  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-iinf 4472
This theorem depends on definitions:  df-bi 116  df-dc 805  df-3or 948  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-tr 3997  df-eprel 4181  df-id 4185  df-po 4188  df-iso 4189  df-iord 4258  df-on 4260  df-suc 4263  df-iom 4475  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-ov 5745  df-oprab 5746  df-mpo 5747  df-1st 6006  df-2nd 6007  df-recs 6170  df-irdg 6235  df-1o 6281  df-2o 6282  df-oadd 6285  df-omul 6286  df-er 6397  df-ec 6399  df-qs 6403  df-ni 7080  df-pli 7081  df-mi 7082  df-lti 7083  df-plpq 7120  df-mpq 7121  df-enq 7123  df-nqqs 7124  df-plqqs 7125  df-mqqs 7126  df-1nqqs 7127  df-rq 7128  df-ltnqqs 7129  df-enq0 7200  df-nq0 7201  df-0nq0 7202  df-plq0 7203  df-mq0 7204  df-inp 7242  df-i1p 7243  df-iplp 7244  df-imp 7245  df-iltp 7246  df-enr 7502  df-nr 7503  df-mr 7505  df-ltr 7506
This theorem is referenced by:  axpre-mulext  7664
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