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Theorem ltasrg 7546
Description: Ordering property of addition. (Contributed by NM, 10-May-1996.)
Assertion
Ref Expression
ltasrg  |-  ( ( A  e.  R.  /\  B  e.  R.  /\  C  e.  R. )  ->  ( A  <R  B  <->  ( C  +R  A )  <R  ( C  +R  B ) ) )

Proof of Theorem ltasrg
Dummy variables  x  y  z  w  v  u  s  r  t are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-nr 7503 . . 3  |-  R.  =  ( ( P.  X.  P. ) /.  ~R  )
2 oveq1 5749 . . . . 5  |-  ( [
<. v ,  u >. ]  ~R  =  C  -> 
( [ <. v ,  u >. ]  ~R  +R  [
<. x ,  y >. ]  ~R  )  =  ( C  +R  [ <. x ,  y >. ]  ~R  ) )
3 oveq1 5749 . . . . 5  |-  ( [
<. v ,  u >. ]  ~R  =  C  -> 
( [ <. v ,  u >. ]  ~R  +R  [
<. z ,  w >. ]  ~R  )  =  ( C  +R  [ <. z ,  w >. ]  ~R  ) )
42, 3breq12d 3912 . . . 4  |-  ( [
<. v ,  u >. ]  ~R  =  C  -> 
( ( [ <. v ,  u >. ]  ~R  +R  [ <. x ,  y
>. ]  ~R  )  <R 
( [ <. v ,  u >. ]  ~R  +R  [
<. z ,  w >. ]  ~R  )  <->  ( C  +R  [ <. x ,  y
>. ]  ~R  )  <R 
( C  +R  [ <. z ,  w >. ]  ~R  ) ) )
54bibi2d 231 . . 3  |-  ( [
<. v ,  u >. ]  ~R  =  C  -> 
( ( [ <. x ,  y >. ]  ~R  <R  [ <. z ,  w >. ]  ~R  <->  ( [ <. v ,  u >. ]  ~R  +R  [ <. x ,  y >. ]  ~R  )  <R  ( [ <. v ,  u >. ]  ~R  +R  [ <. z ,  w >. ]  ~R  ) )  <-> 
( [ <. x ,  y >. ]  ~R  <R  [ <. z ,  w >. ]  ~R  <->  ( C  +R  [ <. x ,  y
>. ]  ~R  )  <R 
( C  +R  [ <. z ,  w >. ]  ~R  ) ) ) )
6 breq1 3902 . . . 4  |-  ( [
<. x ,  y >. ]  ~R  =  A  -> 
( [ <. x ,  y >. ]  ~R  <R  [ <. z ,  w >. ]  ~R  <->  A  <R  [
<. z ,  w >. ]  ~R  ) )
7 oveq2 5750 . . . . 5  |-  ( [
<. x ,  y >. ]  ~R  =  A  -> 
( C  +R  [ <. x ,  y >. ]  ~R  )  =  ( C  +R  A ) )
87breq1d 3909 . . . 4  |-  ( [
<. x ,  y >. ]  ~R  =  A  -> 
( ( C  +R  [
<. x ,  y >. ]  ~R  )  <R  ( C  +R  [ <. z ,  w >. ]  ~R  )  <->  ( C  +R  A ) 
<R  ( C  +R  [ <. z ,  w >. ]  ~R  ) ) )
96, 8bibi12d 234 . . 3  |-  ( [
<. x ,  y >. ]  ~R  =  A  -> 
( ( [ <. x ,  y >. ]  ~R  <R  [ <. z ,  w >. ]  ~R  <->  ( C  +R  [ <. x ,  y
>. ]  ~R  )  <R 
( C  +R  [ <. z ,  w >. ]  ~R  ) )  <->  ( A  <R  [ <. z ,  w >. ]  ~R  <->  ( C  +R  A )  <R  ( C  +R  [ <. z ,  w >. ]  ~R  )
) ) )
10 breq2 3903 . . . 4  |-  ( [
<. z ,  w >. ]  ~R  =  B  -> 
( A  <R  [ <. z ,  w >. ]  ~R  <->  A 
<R  B ) )
11 oveq2 5750 . . . . 5  |-  ( [
<. z ,  w >. ]  ~R  =  B  -> 
( C  +R  [ <. z ,  w >. ]  ~R  )  =  ( C  +R  B ) )
1211breq2d 3911 . . . 4  |-  ( [
<. z ,  w >. ]  ~R  =  B  -> 
( ( C  +R  A )  <R  ( C  +R  [ <. z ,  w >. ]  ~R  )  <->  ( C  +R  A ) 
<R  ( C  +R  B
) ) )
1310, 12bibi12d 234 . . 3  |-  ( [
<. z ,  w >. ]  ~R  =  B  -> 
( ( A  <R  [
<. z ,  w >. ]  ~R  <->  ( C  +R  A )  <R  ( C  +R  [ <. z ,  w >. ]  ~R  )
)  <->  ( A  <R  B  <-> 
( C  +R  A
)  <R  ( C  +R  B ) ) ) )
14 simp2l 992 . . . . . . 7  |-  ( ( ( v  e.  P.  /\  u  e.  P. )  /\  ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  ->  x  e.  P. )
15 simp3r 995 . . . . . . 7  |-  ( ( ( v  e.  P.  /\  u  e.  P. )  /\  ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  ->  w  e.  P. )
16 addclpr 7313 . . . . . . 7  |-  ( ( x  e.  P.  /\  w  e.  P. )  ->  ( x  +P.  w
)  e.  P. )
1714, 15, 16syl2anc 408 . . . . . 6  |-  ( ( ( v  e.  P.  /\  u  e.  P. )  /\  ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  ->  ( x  +P.  w )  e.  P. )
18 simp2r 993 . . . . . . 7  |-  ( ( ( v  e.  P.  /\  u  e.  P. )  /\  ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  ->  y  e.  P. )
19 simp3l 994 . . . . . . 7  |-  ( ( ( v  e.  P.  /\  u  e.  P. )  /\  ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  ->  z  e.  P. )
20 addclpr 7313 . . . . . . 7  |-  ( ( y  e.  P.  /\  z  e.  P. )  ->  ( y  +P.  z
)  e.  P. )
2118, 19, 20syl2anc 408 . . . . . 6  |-  ( ( ( v  e.  P.  /\  u  e.  P. )  /\  ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  ->  ( y  +P.  z )  e.  P. )
22 addclpr 7313 . . . . . . 7  |-  ( ( v  e.  P.  /\  u  e.  P. )  ->  ( v  +P.  u
)  e.  P. )
23223ad2ant1 987 . . . . . 6  |-  ( ( ( v  e.  P.  /\  u  e.  P. )  /\  ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  ->  ( v  +P.  u )  e.  P. )
24 ltaprg 7395 . . . . . 6  |-  ( ( ( x  +P.  w
)  e.  P.  /\  ( y  +P.  z
)  e.  P.  /\  ( v  +P.  u
)  e.  P. )  ->  ( ( x  +P.  w )  <P  (
y  +P.  z )  <->  ( ( v  +P.  u
)  +P.  ( x  +P.  w ) )  <P 
( ( v  +P.  u )  +P.  (
y  +P.  z )
) ) )
2517, 21, 23, 24syl3anc 1201 . . . . 5  |-  ( ( ( v  e.  P.  /\  u  e.  P. )  /\  ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  ->  ( (
x  +P.  w )  <P  ( y  +P.  z
)  <->  ( ( v  +P.  u )  +P.  ( x  +P.  w
) )  <P  (
( v  +P.  u
)  +P.  ( y  +P.  z ) ) ) )
26 ltsrprg 7523 . . . . . 6  |-  ( ( ( 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 )
) )
27263adant1 984 . . . . 5  |-  ( ( ( v  e.  P.  /\  u  e.  P. )  /\  ( 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 )
) )
28 simp1l 990 . . . . . . . 8  |-  ( ( ( v  e.  P.  /\  u  e.  P. )  /\  ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  ->  v  e.  P. )
29 addclpr 7313 . . . . . . . 8  |-  ( ( v  e.  P.  /\  x  e.  P. )  ->  ( v  +P.  x
)  e.  P. )
3028, 14, 29syl2anc 408 . . . . . . 7  |-  ( ( ( v  e.  P.  /\  u  e.  P. )  /\  ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  ->  ( v  +P.  x )  e.  P. )
31 simp1r 991 . . . . . . . 8  |-  ( ( ( v  e.  P.  /\  u  e.  P. )  /\  ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  ->  u  e.  P. )
32 addclpr 7313 . . . . . . . 8  |-  ( ( u  e.  P.  /\  y  e.  P. )  ->  ( u  +P.  y
)  e.  P. )
3331, 18, 32syl2anc 408 . . . . . . 7  |-  ( ( ( v  e.  P.  /\  u  e.  P. )  /\  ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  ->  ( u  +P.  y )  e.  P. )
34 addclpr 7313 . . . . . . . 8  |-  ( ( v  e.  P.  /\  z  e.  P. )  ->  ( v  +P.  z
)  e.  P. )
3528, 19, 34syl2anc 408 . . . . . . 7  |-  ( ( ( v  e.  P.  /\  u  e.  P. )  /\  ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  ->  ( v  +P.  z )  e.  P. )
36 addclpr 7313 . . . . . . . 8  |-  ( ( u  e.  P.  /\  w  e.  P. )  ->  ( u  +P.  w
)  e.  P. )
3731, 15, 36syl2anc 408 . . . . . . 7  |-  ( ( ( v  e.  P.  /\  u  e.  P. )  /\  ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  ->  ( u  +P.  w )  e.  P. )
38 ltsrprg 7523 . . . . . . 7  |-  ( ( ( ( v  +P.  x )  e.  P.  /\  ( u  +P.  y
)  e.  P. )  /\  ( ( v  +P.  z )  e.  P.  /\  ( u  +P.  w
)  e.  P. )
)  ->  ( [ <. ( v  +P.  x
) ,  ( u  +P.  y ) >. ]  ~R  <R  [ <. (
v  +P.  z ) ,  ( u  +P.  w ) >. ]  ~R  <->  ( ( v  +P.  x
)  +P.  ( u  +P.  w ) )  <P 
( ( u  +P.  y )  +P.  (
v  +P.  z )
) ) )
3930, 33, 35, 37, 38syl22anc 1202 . . . . . 6  |-  ( ( ( v  e.  P.  /\  u  e.  P. )  /\  ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  ->  ( [ <. ( v  +P.  x
) ,  ( u  +P.  y ) >. ]  ~R  <R  [ <. (
v  +P.  z ) ,  ( u  +P.  w ) >. ]  ~R  <->  ( ( v  +P.  x
)  +P.  ( u  +P.  w ) )  <P 
( ( u  +P.  y )  +P.  (
v  +P.  z )
) ) )
40 addcomprg 7354 . . . . . . . . 9  |-  ( ( r  e.  P.  /\  s  e.  P. )  ->  ( r  +P.  s
)  =  ( s  +P.  r ) )
4140adantl 275 . . . . . . . 8  |-  ( ( ( ( v  e. 
P.  /\  u  e.  P. )  /\  (
x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  /\  ( r  e.  P.  /\  s  e. 
P. ) )  -> 
( r  +P.  s
)  =  ( s  +P.  r ) )
42 addassprg 7355 . . . . . . . . 9  |-  ( ( r  e.  P.  /\  s  e.  P.  /\  t  e.  P. )  ->  (
( r  +P.  s
)  +P.  t )  =  ( r  +P.  ( s  +P.  t
) ) )
4342adantl 275 . . . . . . . 8  |-  ( ( ( ( v  e. 
P.  /\  u  e.  P. )  /\  (
x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  /\  ( r  e.  P.  /\  s  e. 
P.  /\  t  e.  P. ) )  ->  (
( r  +P.  s
)  +P.  t )  =  ( r  +P.  ( s  +P.  t
) ) )
44 addclpr 7313 . . . . . . . . 9  |-  ( ( r  e.  P.  /\  s  e.  P. )  ->  ( r  +P.  s
)  e.  P. )
4544adantl 275 . . . . . . . 8  |-  ( ( ( ( v  e. 
P.  /\  u  e.  P. )  /\  (
x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  /\  ( r  e.  P.  /\  s  e. 
P. ) )  -> 
( r  +P.  s
)  e.  P. )
4628, 14, 31, 41, 43, 15, 45caov4d 5923 . . . . . . 7  |-  ( ( ( v  e.  P.  /\  u  e.  P. )  /\  ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  ->  ( (
v  +P.  x )  +P.  ( u  +P.  w
) )  =  ( ( v  +P.  u
)  +P.  ( x  +P.  w ) ) )
4741, 33, 35caovcomd 5895 . . . . . . . 8  |-  ( ( ( v  e.  P.  /\  u  e.  P. )  /\  ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  ->  ( (
u  +P.  y )  +P.  ( v  +P.  z
) )  =  ( ( v  +P.  z
)  +P.  ( u  +P.  y ) ) )
4828, 19, 31, 41, 43, 18, 45caov42d 5925 . . . . . . . 8  |-  ( ( ( v  e.  P.  /\  u  e.  P. )  /\  ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  ->  ( (
v  +P.  z )  +P.  ( u  +P.  y
) )  =  ( ( v  +P.  u
)  +P.  ( y  +P.  z ) ) )
4947, 48eqtrd 2150 . . . . . . 7  |-  ( ( ( v  e.  P.  /\  u  e.  P. )  /\  ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  ->  ( (
u  +P.  y )  +P.  ( v  +P.  z
) )  =  ( ( v  +P.  u
)  +P.  ( y  +P.  z ) ) )
5046, 49breq12d 3912 . . . . . 6  |-  ( ( ( v  e.  P.  /\  u  e.  P. )  /\  ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  ->  ( (
( v  +P.  x
)  +P.  ( u  +P.  w ) )  <P 
( ( u  +P.  y )  +P.  (
v  +P.  z )
)  <->  ( ( v  +P.  u )  +P.  ( x  +P.  w
) )  <P  (
( v  +P.  u
)  +P.  ( y  +P.  z ) ) ) )
5139, 50bitrd 187 . . . . 5  |-  ( ( ( v  e.  P.  /\  u  e.  P. )  /\  ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  ->  ( [ <. ( v  +P.  x
) ,  ( u  +P.  y ) >. ]  ~R  <R  [ <. (
v  +P.  z ) ,  ( u  +P.  w ) >. ]  ~R  <->  ( ( v  +P.  u
)  +P.  ( x  +P.  w ) )  <P 
( ( v  +P.  u )  +P.  (
y  +P.  z )
) ) )
5225, 27, 513bitr4d 219 . . . 4  |-  ( ( ( v  e.  P.  /\  u  e.  P. )  /\  ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  ->  ( [ <. x ,  y >. ]  ~R  <R  [ <. z ,  w >. ]  ~R  <->  [ <. (
v  +P.  x ) ,  ( u  +P.  y ) >. ]  ~R  <R  [ <. ( v  +P.  z ) ,  ( u  +P.  w )
>. ]  ~R  ) )
53 addsrpr 7521 . . . . . 6  |-  ( ( ( v  e.  P.  /\  u  e.  P. )  /\  ( x  e.  P.  /\  y  e.  P. )
)  ->  ( [ <. v ,  u >. ]  ~R  +R  [ <. x ,  y >. ]  ~R  )  =  [ <. (
v  +P.  x ) ,  ( u  +P.  y ) >. ]  ~R  )
54533adant3 986 . . . . 5  |-  ( ( ( v  e.  P.  /\  u  e.  P. )  /\  ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  ->  ( [ <. v ,  u >. ]  ~R  +R  [ <. x ,  y >. ]  ~R  )  =  [ <. (
v  +P.  x ) ,  ( u  +P.  y ) >. ]  ~R  )
55 addsrpr 7521 . . . . . 6  |-  ( ( ( v  e.  P.  /\  u  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  ->  ( [ <. v ,  u >. ]  ~R  +R  [ <. z ,  w >. ]  ~R  )  =  [ <. (
v  +P.  z ) ,  ( u  +P.  w ) >. ]  ~R  )
56553adant2 985 . . . . 5  |-  ( ( ( v  e.  P.  /\  u  e.  P. )  /\  ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  ->  ( [ <. v ,  u >. ]  ~R  +R  [ <. z ,  w >. ]  ~R  )  =  [ <. (
v  +P.  z ) ,  ( u  +P.  w ) >. ]  ~R  )
5754, 56breq12d 3912 . . . 4  |-  ( ( ( v  e.  P.  /\  u  e.  P. )  /\  ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  ->  ( ( [ <. v ,  u >. ]  ~R  +R  [ <. x ,  y >. ]  ~R  )  <R  ( [ <. v ,  u >. ]  ~R  +R  [ <. z ,  w >. ]  ~R  )  <->  [ <. (
v  +P.  x ) ,  ( u  +P.  y ) >. ]  ~R  <R  [ <. ( v  +P.  z ) ,  ( u  +P.  w )
>. ]  ~R  ) )
5852, 57bitr4d 190 . . 3  |-  ( ( ( v  e.  P.  /\  u  e.  P. )  /\  ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  ->  ( [ <. x ,  y >. ]  ~R  <R  [ <. z ,  w >. ]  ~R  <->  ( [ <. v ,  u >. ]  ~R  +R  [ <. x ,  y >. ]  ~R  )  <R  ( [ <. v ,  u >. ]  ~R  +R  [ <. z ,  w >. ]  ~R  ) ) )
591, 5, 9, 13, 583ecoptocl 6486 . 2  |-  ( ( C  e.  R.  /\  A  e.  R.  /\  B  e.  R. )  ->  ( A  <R  B  <->  ( C  +R  A )  <R  ( C  +R  B ) ) )
60593coml 1173 1  |-  ( ( A  e.  R.  /\  B  e.  R.  /\  C  e.  R. )  ->  ( A  <R  B  <->  ( C  +R  A )  <R  ( C  +R  B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    /\ 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 cltp 7071    ~R cer 7072   R.cnr 7073    +R cplr 7077    <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-iplp 7244  df-iltp 7246  df-enr 7502  df-nr 7503  df-plr 7504  df-ltr 7506
This theorem is referenced by:  addgt0sr  7551  ltadd1sr  7552  caucvgsrlemoffcau  7574  caucvgsrlemoffgt1  7575  caucvgsrlemoffres  7576  caucvgsr  7578  ltpsrprg  7579  mappsrprg  7580  map2psrprg  7581  suplocsrlempr  7583  axpre-ltadd  7662
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