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Theorem recexgt0sr 7545
Description: The reciprocal of a positive signed real exists and is positive. (Contributed by Jim Kingdon, 6-Feb-2020.)
Assertion
Ref Expression
recexgt0sr  |-  ( 0R 
<R  A  ->  E. x  e.  R.  ( 0R  <R  x  /\  ( A  .R  x )  =  1R ) )
Distinct variable group:    x, A

Proof of Theorem recexgt0sr
Dummy variables  y  z  w  v  f  g  h are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ltrelsr 7510 . . . 4  |-  <R  C_  ( R.  X.  R. )
21brel 4559 . . 3  |-  ( 0R 
<R  A  ->  ( 0R  e.  R.  /\  A  e.  R. ) )
32simprd 113 . 2  |-  ( 0R 
<R  A  ->  A  e. 
R. )
4 df-nr 7499 . . 3  |-  R.  =  ( ( P.  X.  P. ) /.  ~R  )
5 breq2 3901 . . . 4  |-  ( [
<. y ,  z >. ]  ~R  =  A  -> 
( 0R  <R  [ <. y ,  z >. ]  ~R  <->  0R 
<R  A ) )
6 oveq1 5747 . . . . . . 7  |-  ( [
<. y ,  z >. ]  ~R  =  A  -> 
( [ <. y ,  z >. ]  ~R  .R  x )  =  ( A  .R  x ) )
76eqeq1d 2124 . . . . . 6  |-  ( [
<. y ,  z >. ]  ~R  =  A  -> 
( ( [ <. y ,  z >. ]  ~R  .R  x )  =  1R  <->  ( A  .R  x )  =  1R ) )
87anbi2d 457 . . . . 5  |-  ( [
<. y ,  z >. ]  ~R  =  A  -> 
( ( 0R  <R  x  /\  ( [ <. y ,  z >. ]  ~R  .R  x )  =  1R ) 
<->  ( 0R  <R  x  /\  ( A  .R  x
)  =  1R )
) )
98rexbidv 2413 . . . 4  |-  ( [
<. y ,  z >. ]  ~R  =  A  -> 
( E. x  e. 
R.  ( 0R  <R  x  /\  ( [ <. y ,  z >. ]  ~R  .R  x )  =  1R ) 
<->  E. x  e.  R.  ( 0R  <R  x  /\  ( A  .R  x
)  =  1R )
) )
105, 9imbi12d 233 . . 3  |-  ( [
<. y ,  z >. ]  ~R  =  A  -> 
( ( 0R  <R  [
<. y ,  z >. ]  ~R  ->  E. x  e.  R.  ( 0R  <R  x  /\  ( [ <. y ,  z >. ]  ~R  .R  x )  =  1R ) )  <->  ( 0R  <R  A  ->  E. x  e.  R.  ( 0R  <R  x  /\  ( A  .R  x )  =  1R ) ) ) )
11 gt0srpr 7520 . . . . 5  |-  ( 0R 
<R  [ <. y ,  z
>. ]  ~R  <->  z  <P  y )
12 ltexpri 7385 . . . . 5  |-  ( z 
<P  y  ->  E. w  e.  P.  ( z  +P.  w )  =  y )
1311, 12sylbi 120 . . . 4  |-  ( 0R 
<R  [ <. y ,  z
>. ]  ~R  ->  E. w  e.  P.  ( z  +P.  w )  =  y )
14 recexpr 7410 . . . . . . 7  |-  ( w  e.  P.  ->  E. v  e.  P.  ( w  .P.  v )  =  1P )
1514adantl 273 . . . . . 6  |-  ( ( ( y  e.  P.  /\  z  e.  P. )  /\  w  e.  P. )  ->  E. v  e.  P.  ( w  .P.  v )  =  1P )
16 1pr 7326 . . . . . . . . . . . . . 14  |-  1P  e.  P.
17 addclpr 7309 . . . . . . . . . . . . . 14  |-  ( ( v  e.  P.  /\  1P  e.  P. )  -> 
( v  +P.  1P )  e.  P. )
1816, 17mpan2 419 . . . . . . . . . . . . 13  |-  ( v  e.  P.  ->  (
v  +P.  1P )  e.  P. )
19 enrex 7509 . . . . . . . . . . . . . 14  |-  ~R  e.  _V
2019, 4ecopqsi 6450 . . . . . . . . . . . . 13  |-  ( ( ( v  +P.  1P )  e.  P.  /\  1P  e.  P. )  ->  [ <. ( v  +P.  1P ) ,  1P >. ]  ~R  e.  R. )
2118, 16, 20sylancl 407 . . . . . . . . . . . 12  |-  ( v  e.  P.  ->  [ <. ( v  +P.  1P ) ,  1P >. ]  ~R  e.  R. )
2221adantl 273 . . . . . . . . . . 11  |-  ( ( w  e.  P.  /\  v  e.  P. )  ->  [ <. ( v  +P. 
1P ) ,  1P >. ]  ~R  e.  R. )
2322ad2antlr 478 . . . . . . . . . 10  |-  ( ( ( ( y  e. 
P.  /\  z  e.  P. )  /\  (
w  e.  P.  /\  v  e.  P. )
)  /\  ( (
w  .P.  v )  =  1P  /\  (
z  +P.  w )  =  y ) )  ->  [ <. (
v  +P.  1P ) ,  1P >. ]  ~R  e.  R. )
24 simprr 504 . . . . . . . . . . . 12  |-  ( ( ( y  e.  P.  /\  z  e.  P. )  /\  ( w  e.  P.  /\  v  e.  P. )
)  ->  v  e.  P. )
2524adantr 272 . . . . . . . . . . 11  |-  ( ( ( ( y  e. 
P.  /\  z  e.  P. )  /\  (
w  e.  P.  /\  v  e.  P. )
)  /\  ( (
w  .P.  v )  =  1P  /\  (
z  +P.  w )  =  y ) )  ->  v  e.  P. )
26 ltaddpr 7369 . . . . . . . . . . . . . 14  |-  ( ( 1P  e.  P.  /\  v  e.  P. )  ->  1P  <P  ( 1P  +P.  v ) )
2716, 26mpan 418 . . . . . . . . . . . . 13  |-  ( v  e.  P.  ->  1P  <P  ( 1P  +P.  v
) )
28 addcomprg 7350 . . . . . . . . . . . . . 14  |-  ( ( 1P  e.  P.  /\  v  e.  P. )  ->  ( 1P  +P.  v
)  =  ( v  +P.  1P ) )
2916, 28mpan 418 . . . . . . . . . . . . 13  |-  ( v  e.  P.  ->  ( 1P  +P.  v )  =  ( v  +P.  1P ) )
3027, 29breqtrd 3922 . . . . . . . . . . . 12  |-  ( v  e.  P.  ->  1P  <P  ( v  +P.  1P ) )
31 gt0srpr 7520 . . . . . . . . . . . 12  |-  ( 0R 
<R  [ <. ( v  +P. 
1P ) ,  1P >. ]  ~R  <->  1P  <P  ( v  +P.  1P ) )
3230, 31sylibr 133 . . . . . . . . . . 11  |-  ( v  e.  P.  ->  0R  <R  [ <. ( v  +P. 
1P ) ,  1P >. ]  ~R  )
3325, 32syl 14 . . . . . . . . . 10  |-  ( ( ( ( y  e. 
P.  /\  z  e.  P. )  /\  (
w  e.  P.  /\  v  e.  P. )
)  /\  ( (
w  .P.  v )  =  1P  /\  (
z  +P.  w )  =  y ) )  ->  0R  <R  [ <. ( v  +P.  1P ) ,  1P >. ]  ~R  )
3418, 16jctir 309 . . . . . . . . . . . . . . . 16  |-  ( v  e.  P.  ->  (
( v  +P.  1P )  e.  P.  /\  1P  e.  P. ) )
3534anim2i 337 . . . . . . . . . . . . . . 15  |-  ( ( ( y  e.  P.  /\  z  e.  P. )  /\  v  e.  P. )  ->  ( ( y  e.  P.  /\  z  e.  P. )  /\  (
( v  +P.  1P )  e.  P.  /\  1P  e.  P. ) ) )
3635adantr 272 . . . . . . . . . . . . . 14  |-  ( ( ( ( y  e. 
P.  /\  z  e.  P. )  /\  v  e.  P. )  /\  (
( w  .P.  v
)  =  1P  /\  ( z  +P.  w
)  =  y ) )  ->  ( (
y  e.  P.  /\  z  e.  P. )  /\  ( ( v  +P. 
1P )  e.  P.  /\  1P  e.  P. )
) )
37 mulsrpr 7518 . . . . . . . . . . . . . 14  |-  ( ( ( y  e.  P.  /\  z  e.  P. )  /\  ( ( v  +P. 
1P )  e.  P.  /\  1P  e.  P. )
)  ->  ( [ <. y ,  z >. ]  ~R  .R  [ <. ( v  +P.  1P ) ,  1P >. ]  ~R  )  =  [ <. (
( y  .P.  (
v  +P.  1P )
)  +P.  ( z  .P.  1P ) ) ,  ( ( y  .P. 
1P )  +P.  (
z  .P.  ( v  +P.  1P ) ) )
>. ]  ~R  )
3836, 37syl 14 . . . . . . . . . . . . 13  |-  ( ( ( ( y  e. 
P.  /\  z  e.  P. )  /\  v  e.  P. )  /\  (
( w  .P.  v
)  =  1P  /\  ( z  +P.  w
)  =  y ) )  ->  ( [ <. y ,  z >. ]  ~R  .R  [ <. ( v  +P.  1P ) ,  1P >. ]  ~R  )  =  [ <. (
( y  .P.  (
v  +P.  1P )
)  +P.  ( z  .P.  1P ) ) ,  ( ( y  .P. 
1P )  +P.  (
z  .P.  ( v  +P.  1P ) ) )
>. ]  ~R  )
3938adantlrl 471 . . . . . . . . . . . 12  |-  ( ( ( ( y  e. 
P.  /\  z  e.  P. )  /\  (
w  e.  P.  /\  v  e.  P. )
)  /\  ( (
w  .P.  v )  =  1P  /\  (
z  +P.  w )  =  y ) )  ->  ( [ <. y ,  z >. ]  ~R  .R 
[ <. ( v  +P. 
1P ) ,  1P >. ]  ~R  )  =  [ <. ( ( y  .P.  ( v  +P. 
1P ) )  +P.  ( z  .P.  1P ) ) ,  ( ( y  .P.  1P )  +P.  ( z  .P.  ( v  +P.  1P ) ) ) >. ]  ~R  )
40 oveq1 5747 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( z  +P.  w )  =  y  ->  (
( z  +P.  w
)  .P.  v )  =  ( y  .P.  v ) )
4140eqcomd 2121 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( z  +P.  w )  =  y  ->  (
y  .P.  v )  =  ( ( z  +P.  w )  .P.  v ) )
4241ad2antll 480 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( y  e. 
P.  /\  z  e.  P. )  /\  (
w  e.  P.  /\  v  e.  P. )
)  /\  ( (
w  .P.  v )  =  1P  /\  (
z  +P.  w )  =  y ) )  ->  ( y  .P.  v )  =  ( ( z  +P.  w
)  .P.  v )
)
43 mulcomprg 7352 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( f  e.  P.  /\  h  e.  P. )  ->  ( f  .P.  h
)  =  ( h  .P.  f ) )
44433adant2 983 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( f  e.  P.  /\  g  e.  P.  /\  h  e.  P. )  ->  (
f  .P.  h )  =  ( h  .P.  f ) )
45 mulcomprg 7352 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( g  e.  P.  /\  h  e.  P. )  ->  ( g  .P.  h
)  =  ( h  .P.  g ) )
46453adant1 982 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( f  e.  P.  /\  g  e.  P.  /\  h  e.  P. )  ->  (
g  .P.  h )  =  ( h  .P.  g ) )
4744, 46oveq12d 5758 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( f  e.  P.  /\  g  e.  P.  /\  h  e.  P. )  ->  (
( f  .P.  h
)  +P.  ( g  .P.  h ) )  =  ( ( h  .P.  f )  +P.  (
h  .P.  g )
) )
48 distrprg 7360 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( h  e.  P.  /\  f  e.  P.  /\  g  e.  P. )  ->  (
h  .P.  ( f  +P.  g ) )  =  ( ( h  .P.  f )  +P.  (
h  .P.  g )
) )
49483coml 1171 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( f  e.  P.  /\  g  e.  P.  /\  h  e.  P. )  ->  (
h  .P.  ( f  +P.  g ) )  =  ( ( h  .P.  f )  +P.  (
h  .P.  g )
) )
50 simp3 966 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( f  e.  P.  /\  g  e.  P.  /\  h  e.  P. )  ->  h  e.  P. )
51 addclpr 7309 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( f  e.  P.  /\  g  e.  P. )  ->  ( f  +P.  g
)  e.  P. )
52513adant3 984 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( f  e.  P.  /\  g  e.  P.  /\  h  e.  P. )  ->  (
f  +P.  g )  e.  P. )
53 mulcomprg 7352 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( h  e.  P.  /\  ( f  +P.  g
)  e.  P. )  ->  ( h  .P.  (
f  +P.  g )
)  =  ( ( f  +P.  g )  .P.  h ) )
5450, 52, 53syl2anc 406 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( f  e.  P.  /\  g  e.  P.  /\  h  e.  P. )  ->  (
h  .P.  ( f  +P.  g ) )  =  ( ( f  +P.  g )  .P.  h
) )
5547, 49, 543eqtr2rd 2155 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( f  e.  P.  /\  g  e.  P.  /\  h  e.  P. )  ->  (
( f  +P.  g
)  .P.  h )  =  ( ( f  .P.  h )  +P.  ( g  .P.  h
) ) )
5655adantl 273 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( ( y  e. 
P.  /\  z  e.  P. )  /\  (
w  e.  P.  /\  v  e.  P. )
)  /\  ( f  e.  P.  /\  g  e. 
P.  /\  h  e.  P. ) )  ->  (
( f  +P.  g
)  .P.  h )  =  ( ( f  .P.  h )  +P.  ( g  .P.  h
) ) )
57 simplr 502 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( y  e.  P.  /\  z  e.  P. )  /\  ( w  e.  P.  /\  v  e.  P. )
)  ->  z  e.  P. )
58 simprl 503 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( y  e.  P.  /\  z  e.  P. )  /\  ( w  e.  P.  /\  v  e.  P. )
)  ->  w  e.  P. )
5956, 57, 58, 24caovdird 5915 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( y  e.  P.  /\  z  e.  P. )  /\  ( w  e.  P.  /\  v  e.  P. )
)  ->  ( (
z  +P.  w )  .P.  v )  =  ( ( z  .P.  v
)  +P.  ( w  .P.  v ) ) )
60 oveq2 5748 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( w  .P.  v )  =  1P  ->  (
( z  .P.  v
)  +P.  ( w  .P.  v ) )  =  ( ( z  .P.  v )  +P.  1P ) )
6159, 60sylan9eq 2168 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ( y  e. 
P.  /\  z  e.  P. )  /\  (
w  e.  P.  /\  v  e.  P. )
)  /\  ( w  .P.  v )  =  1P )  ->  ( (
z  +P.  w )  .P.  v )  =  ( ( z  .P.  v
)  +P.  1P )
)
6261adantrr 468 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( y  e. 
P.  /\  z  e.  P. )  /\  (
w  e.  P.  /\  v  e.  P. )
)  /\  ( (
w  .P.  v )  =  1P  /\  (
z  +P.  w )  =  y ) )  ->  ( ( z  +P.  w )  .P.  v )  =  ( ( z  .P.  v
)  +P.  1P )
)
6342, 62eqtrd 2148 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( y  e. 
P.  /\  z  e.  P. )  /\  (
w  e.  P.  /\  v  e.  P. )
)  /\  ( (
w  .P.  v )  =  1P  /\  (
z  +P.  w )  =  y ) )  ->  ( y  .P.  v )  =  ( ( z  .P.  v
)  +P.  1P )
)
6463oveq1d 5755 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( y  e. 
P.  /\  z  e.  P. )  /\  (
w  e.  P.  /\  v  e.  P. )
)  /\  ( (
w  .P.  v )  =  1P  /\  (
z  +P.  w )  =  y ) )  ->  ( ( y  .P.  v )  +P.  ( ( y  .P. 
1P )  +P.  (
z  .P.  1P )
) )  =  ( ( ( z  .P.  v )  +P.  1P )  +P.  ( ( y  .P.  1P )  +P.  ( z  .P.  1P ) ) ) )
65 mulclpr 7344 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( z  e.  P.  /\  v  e.  P. )  ->  ( z  .P.  v
)  e.  P. )
6657, 24, 65syl2anc 406 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( y  e.  P.  /\  z  e.  P. )  /\  ( w  e.  P.  /\  v  e.  P. )
)  ->  ( z  .P.  v )  e.  P. )
6716a1i 9 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( y  e.  P.  /\  z  e.  P. )  /\  ( w  e.  P.  /\  v  e.  P. )
)  ->  1P  e.  P. )
68 simpll 501 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( y  e.  P.  /\  z  e.  P. )  /\  ( w  e.  P.  /\  v  e.  P. )
)  ->  y  e.  P. )
69 mulclpr 7344 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( y  e.  P.  /\  1P  e.  P. )  -> 
( y  .P.  1P )  e.  P. )
7068, 16, 69sylancl 407 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( y  e.  P.  /\  z  e.  P. )  /\  ( w  e.  P.  /\  v  e.  P. )
)  ->  ( y  .P.  1P )  e.  P. )
71 mulclpr 7344 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( z  e.  P.  /\  1P  e.  P. )  -> 
( z  .P.  1P )  e.  P. )
7257, 16, 71sylancl 407 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( y  e.  P.  /\  z  e.  P. )  /\  ( w  e.  P.  /\  v  e.  P. )
)  ->  ( z  .P.  1P )  e.  P. )
73 addclpr 7309 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( y  .P.  1P )  e.  P.  /\  (
z  .P.  1P )  e.  P. )  ->  (
( y  .P.  1P )  +P.  ( z  .P. 
1P ) )  e. 
P. )
7470, 72, 73syl2anc 406 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( y  e.  P.  /\  z  e.  P. )  /\  ( w  e.  P.  /\  v  e.  P. )
)  ->  ( (
y  .P.  1P )  +P.  ( z  .P.  1P ) )  e.  P. )
75 addcomprg 7350 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( f  e.  P.  /\  g  e.  P. )  ->  ( f  +P.  g
)  =  ( g  +P.  f ) )
7675adantl 273 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( y  e. 
P.  /\  z  e.  P. )  /\  (
w  e.  P.  /\  v  e.  P. )
)  /\  ( f  e.  P.  /\  g  e. 
P. ) )  -> 
( f  +P.  g
)  =  ( g  +P.  f ) )
77 addassprg 7351 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( f  e.  P.  /\  g  e.  P.  /\  h  e.  P. )  ->  (
( f  +P.  g
)  +P.  h )  =  ( f  +P.  ( g  +P.  h
) ) )
7877adantl 273 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( y  e. 
P.  /\  z  e.  P. )  /\  (
w  e.  P.  /\  v  e.  P. )
)  /\  ( f  e.  P.  /\  g  e. 
P.  /\  h  e.  P. ) )  ->  (
( f  +P.  g
)  +P.  h )  =  ( f  +P.  ( g  +P.  h
) ) )
7966, 67, 74, 76, 78caov32d 5917 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( y  e.  P.  /\  z  e.  P. )  /\  ( w  e.  P.  /\  v  e.  P. )
)  ->  ( (
( z  .P.  v
)  +P.  1P )  +P.  ( ( y  .P. 
1P )  +P.  (
z  .P.  1P )
) )  =  ( ( ( z  .P.  v )  +P.  (
( y  .P.  1P )  +P.  ( z  .P. 
1P ) ) )  +P.  1P ) )
8079adantr 272 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( y  e. 
P.  /\  z  e.  P. )  /\  (
w  e.  P.  /\  v  e.  P. )
)  /\  ( (
w  .P.  v )  =  1P  /\  (
z  +P.  w )  =  y ) )  ->  ( ( ( z  .P.  v )  +P.  1P )  +P.  ( ( y  .P. 
1P )  +P.  (
z  .P.  1P )
) )  =  ( ( ( z  .P.  v )  +P.  (
( y  .P.  1P )  +P.  ( z  .P. 
1P ) ) )  +P.  1P ) )
8164, 80eqtrd 2148 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( y  e. 
P.  /\  z  e.  P. )  /\  (
w  e.  P.  /\  v  e.  P. )
)  /\  ( (
w  .P.  v )  =  1P  /\  (
z  +P.  w )  =  y ) )  ->  ( ( y  .P.  v )  +P.  ( ( y  .P. 
1P )  +P.  (
z  .P.  1P )
) )  =  ( ( ( z  .P.  v )  +P.  (
( y  .P.  1P )  +P.  ( z  .P. 
1P ) ) )  +P.  1P ) )
8281oveq1d 5755 . . . . . . . . . . . . . . 15  |-  ( ( ( ( y  e. 
P.  /\  z  e.  P. )  /\  (
w  e.  P.  /\  v  e.  P. )
)  /\  ( (
w  .P.  v )  =  1P  /\  (
z  +P.  w )  =  y ) )  ->  ( ( ( y  .P.  v )  +P.  ( ( y  .P.  1P )  +P.  ( z  .P.  1P ) ) )  +P. 
1P )  =  ( ( ( ( z  .P.  v )  +P.  ( ( y  .P. 
1P )  +P.  (
z  .P.  1P )
) )  +P.  1P )  +P.  1P ) )
83 addclpr 7309 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( z  .P.  v
)  e.  P.  /\  ( ( y  .P. 
1P )  +P.  (
z  .P.  1P )
)  e.  P. )  ->  ( ( z  .P.  v )  +P.  (
( y  .P.  1P )  +P.  ( z  .P. 
1P ) ) )  e.  P. )
8466, 74, 83syl2anc 406 . . . . . . . . . . . . . . . . 17  |-  ( ( ( y  e.  P.  /\  z  e.  P. )  /\  ( w  e.  P.  /\  v  e.  P. )
)  ->  ( (
z  .P.  v )  +P.  ( ( y  .P. 
1P )  +P.  (
z  .P.  1P )
) )  e.  P. )
8584adantr 272 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( y  e. 
P.  /\  z  e.  P. )  /\  (
w  e.  P.  /\  v  e.  P. )
)  /\  ( (
w  .P.  v )  =  1P  /\  (
z  +P.  w )  =  y ) )  ->  ( ( z  .P.  v )  +P.  ( ( y  .P. 
1P )  +P.  (
z  .P.  1P )
) )  e.  P. )
8616a1i 9 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( y  e. 
P.  /\  z  e.  P. )  /\  (
w  e.  P.  /\  v  e.  P. )
)  /\  ( (
w  .P.  v )  =  1P  /\  (
z  +P.  w )  =  y ) )  ->  1P  e.  P. )
87 addassprg 7351 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( z  .P.  v )  +P.  (
( y  .P.  1P )  +P.  ( z  .P. 
1P ) ) )  e.  P.  /\  1P  e.  P.  /\  1P  e.  P. )  ->  ( ( ( ( z  .P.  v )  +P.  (
( y  .P.  1P )  +P.  ( z  .P. 
1P ) ) )  +P.  1P )  +P. 
1P )  =  ( ( ( z  .P.  v )  +P.  (
( y  .P.  1P )  +P.  ( z  .P. 
1P ) ) )  +P.  ( 1P  +P.  1P ) ) )
8885, 86, 86, 87syl3anc 1199 . . . . . . . . . . . . . . 15  |-  ( ( ( ( y  e. 
P.  /\  z  e.  P. )  /\  (
w  e.  P.  /\  v  e.  P. )
)  /\  ( (
w  .P.  v )  =  1P  /\  (
z  +P.  w )  =  y ) )  ->  ( ( ( ( z  .P.  v
)  +P.  ( (
y  .P.  1P )  +P.  ( z  .P.  1P ) ) )  +P. 
1P )  +P.  1P )  =  ( (
( z  .P.  v
)  +P.  ( (
y  .P.  1P )  +P.  ( z  .P.  1P ) ) )  +P.  ( 1P  +P.  1P ) ) )
8982, 88eqtrd 2148 . . . . . . . . . . . . . 14  |-  ( ( ( ( y  e. 
P.  /\  z  e.  P. )  /\  (
w  e.  P.  /\  v  e.  P. )
)  /\  ( (
w  .P.  v )  =  1P  /\  (
z  +P.  w )  =  y ) )  ->  ( ( ( y  .P.  v )  +P.  ( ( y  .P.  1P )  +P.  ( z  .P.  1P ) ) )  +P. 
1P )  =  ( ( ( z  .P.  v )  +P.  (
( y  .P.  1P )  +P.  ( z  .P. 
1P ) ) )  +P.  ( 1P  +P.  1P ) ) )
90 distrprg 7360 . . . . . . . . . . . . . . . . . . 19  |-  ( ( y  e.  P.  /\  v  e.  P.  /\  1P  e.  P. )  ->  (
y  .P.  ( v  +P.  1P ) )  =  ( ( y  .P.  v )  +P.  (
y  .P.  1P )
) )
9168, 24, 67, 90syl3anc 1199 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( y  e.  P.  /\  z  e.  P. )  /\  ( w  e.  P.  /\  v  e.  P. )
)  ->  ( y  .P.  ( v  +P.  1P ) )  =  ( ( y  .P.  v
)  +P.  ( y  .P.  1P ) ) )
9291oveq1d 5755 . . . . . . . . . . . . . . . . 17  |-  ( ( ( y  e.  P.  /\  z  e.  P. )  /\  ( w  e.  P.  /\  v  e.  P. )
)  ->  ( (
y  .P.  ( v  +P.  1P ) )  +P.  ( z  .P.  1P ) )  =  ( ( ( y  .P.  v )  +P.  (
y  .P.  1P )
)  +P.  ( z  .P.  1P ) ) )
93 mulclpr 7344 . . . . . . . . . . . . . . . . . . 19  |-  ( ( y  e.  P.  /\  v  e.  P. )  ->  ( y  .P.  v
)  e.  P. )
9468, 24, 93syl2anc 406 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( y  e.  P.  /\  z  e.  P. )  /\  ( w  e.  P.  /\  v  e.  P. )
)  ->  ( y  .P.  v )  e.  P. )
95 addassprg 7351 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( y  .P.  v
)  e.  P.  /\  ( y  .P.  1P )  e.  P.  /\  (
z  .P.  1P )  e.  P. )  ->  (
( ( y  .P.  v )  +P.  (
y  .P.  1P )
)  +P.  ( z  .P.  1P ) )  =  ( ( y  .P.  v )  +P.  (
( y  .P.  1P )  +P.  ( z  .P. 
1P ) ) ) )
9694, 70, 72, 95syl3anc 1199 . . . . . . . . . . . . . . . . 17  |-  ( ( ( y  e.  P.  /\  z  e.  P. )  /\  ( w  e.  P.  /\  v  e.  P. )
)  ->  ( (
( y  .P.  v
)  +P.  ( y  .P.  1P ) )  +P.  ( z  .P.  1P ) )  =  ( ( y  .P.  v
)  +P.  ( (
y  .P.  1P )  +P.  ( z  .P.  1P ) ) ) )
9792, 96eqtrd 2148 . . . . . . . . . . . . . . . 16  |-  ( ( ( y  e.  P.  /\  z  e.  P. )  /\  ( w  e.  P.  /\  v  e.  P. )
)  ->  ( (
y  .P.  ( v  +P.  1P ) )  +P.  ( z  .P.  1P ) )  =  ( ( y  .P.  v
)  +P.  ( (
y  .P.  1P )  +P.  ( z  .P.  1P ) ) ) )
9897oveq1d 5755 . . . . . . . . . . . . . . 15  |-  ( ( ( y  e.  P.  /\  z  e.  P. )  /\  ( w  e.  P.  /\  v  e.  P. )
)  ->  ( (
( y  .P.  (
v  +P.  1P )
)  +P.  ( z  .P.  1P ) )  +P. 
1P )  =  ( ( ( y  .P.  v )  +P.  (
( y  .P.  1P )  +P.  ( z  .P. 
1P ) ) )  +P.  1P ) )
9998adantr 272 . . . . . . . . . . . . . 14  |-  ( ( ( ( y  e. 
P.  /\  z  e.  P. )  /\  (
w  e.  P.  /\  v  e.  P. )
)  /\  ( (
w  .P.  v )  =  1P  /\  (
z  +P.  w )  =  y ) )  ->  ( ( ( y  .P.  ( v  +P.  1P ) )  +P.  ( z  .P. 
1P ) )  +P. 
1P )  =  ( ( ( y  .P.  v )  +P.  (
( y  .P.  1P )  +P.  ( z  .P. 
1P ) ) )  +P.  1P ) )
100 distrprg 7360 . . . . . . . . . . . . . . . . . . 19  |-  ( ( z  e.  P.  /\  v  e.  P.  /\  1P  e.  P. )  ->  (
z  .P.  ( v  +P.  1P ) )  =  ( ( z  .P.  v )  +P.  (
z  .P.  1P )
) )
10157, 24, 67, 100syl3anc 1199 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( y  e.  P.  /\  z  e.  P. )  /\  ( w  e.  P.  /\  v  e.  P. )
)  ->  ( z  .P.  ( v  +P.  1P ) )  =  ( ( z  .P.  v
)  +P.  ( z  .P.  1P ) ) )
102101oveq2d 5756 . . . . . . . . . . . . . . . . 17  |-  ( ( ( y  e.  P.  /\  z  e.  P. )  /\  ( w  e.  P.  /\  v  e.  P. )
)  ->  ( (
y  .P.  1P )  +P.  ( z  .P.  (
v  +P.  1P )
) )  =  ( ( y  .P.  1P )  +P.  ( ( z  .P.  v )  +P.  ( z  .P.  1P ) ) ) )
10370, 66, 72, 76, 78caov12d 5918 . . . . . . . . . . . . . . . . 17  |-  ( ( ( y  e.  P.  /\  z  e.  P. )  /\  ( w  e.  P.  /\  v  e.  P. )
)  ->  ( (
y  .P.  1P )  +P.  ( ( z  .P.  v )  +P.  (
z  .P.  1P )
) )  =  ( ( z  .P.  v
)  +P.  ( (
y  .P.  1P )  +P.  ( z  .P.  1P ) ) ) )
104102, 103eqtrd 2148 . . . . . . . . . . . . . . . 16  |-  ( ( ( y  e.  P.  /\  z  e.  P. )  /\  ( w  e.  P.  /\  v  e.  P. )
)  ->  ( (
y  .P.  1P )  +P.  ( z  .P.  (
v  +P.  1P )
) )  =  ( ( z  .P.  v
)  +P.  ( (
y  .P.  1P )  +P.  ( z  .P.  1P ) ) ) )
105104oveq1d 5755 . . . . . . . . . . . . . . 15  |-  ( ( ( y  e.  P.  /\  z  e.  P. )  /\  ( w  e.  P.  /\  v  e.  P. )
)  ->  ( (
( y  .P.  1P )  +P.  ( z  .P.  ( v  +P.  1P ) ) )  +P.  ( 1P  +P.  1P ) )  =  ( ( ( z  .P.  v )  +P.  (
( y  .P.  1P )  +P.  ( z  .P. 
1P ) ) )  +P.  ( 1P  +P.  1P ) ) )
106105adantr 272 . . . . . . . . . . . . . 14  |-  ( ( ( ( y  e. 
P.  /\  z  e.  P. )  /\  (
w  e.  P.  /\  v  e.  P. )
)  /\  ( (
w  .P.  v )  =  1P  /\  (
z  +P.  w )  =  y ) )  ->  ( ( ( y  .P.  1P )  +P.  ( z  .P.  ( v  +P.  1P ) ) )  +P.  ( 1P  +P.  1P ) )  =  ( ( ( z  .P.  v )  +P.  (
( y  .P.  1P )  +P.  ( z  .P. 
1P ) ) )  +P.  ( 1P  +P.  1P ) ) )
10789, 99, 1063eqtr4d 2158 . . . . . . . . . . . . 13  |-  ( ( ( ( y  e. 
P.  /\  z  e.  P. )  /\  (
w  e.  P.  /\  v  e.  P. )
)  /\  ( (
w  .P.  v )  =  1P  /\  (
z  +P.  w )  =  y ) )  ->  ( ( ( y  .P.  ( v  +P.  1P ) )  +P.  ( z  .P. 
1P ) )  +P. 
1P )  =  ( ( ( y  .P. 
1P )  +P.  (
z  .P.  ( v  +P.  1P ) ) )  +P.  ( 1P  +P.  1P ) ) )
10824, 16, 17sylancl 407 . . . . . . . . . . . . . . . . 17  |-  ( ( ( y  e.  P.  /\  z  e.  P. )  /\  ( w  e.  P.  /\  v  e.  P. )
)  ->  ( v  +P.  1P )  e.  P. )
109 mulclpr 7344 . . . . . . . . . . . . . . . . 17  |-  ( ( y  e.  P.  /\  ( v  +P.  1P )  e.  P. )  ->  ( y  .P.  (
v  +P.  1P )
)  e.  P. )
11068, 108, 109syl2anc 406 . . . . . . . . . . . . . . . 16  |-  ( ( ( y  e.  P.  /\  z  e.  P. )  /\  ( w  e.  P.  /\  v  e.  P. )
)  ->  ( y  .P.  ( v  +P.  1P ) )  e.  P. )
111 addclpr 7309 . . . . . . . . . . . . . . . 16  |-  ( ( ( y  .P.  (
v  +P.  1P )
)  e.  P.  /\  ( z  .P.  1P )  e.  P. )  ->  ( ( y  .P.  ( v  +P.  1P ) )  +P.  (
z  .P.  1P )
)  e.  P. )
112110, 72, 111syl2anc 406 . . . . . . . . . . . . . . 15  |-  ( ( ( y  e.  P.  /\  z  e.  P. )  /\  ( w  e.  P.  /\  v  e.  P. )
)  ->  ( (
y  .P.  ( v  +P.  1P ) )  +P.  ( z  .P.  1P ) )  e.  P. )
113104, 84eqeltrd 2192 . . . . . . . . . . . . . . 15  |-  ( ( ( y  e.  P.  /\  z  e.  P. )  /\  ( w  e.  P.  /\  v  e.  P. )
)  ->  ( (
y  .P.  1P )  +P.  ( z  .P.  (
v  +P.  1P )
) )  e.  P. )
114 addclpr 7309 . . . . . . . . . . . . . . . . 17  |-  ( ( 1P  e.  P.  /\  1P  e.  P. )  -> 
( 1P  +P.  1P )  e.  P. )
11516, 16, 114mp2an 420 . . . . . . . . . . . . . . . 16  |-  ( 1P 
+P.  1P )  e.  P.
116115a1i 9 . . . . . . . . . . . . . . 15  |-  ( ( ( y  e.  P.  /\  z  e.  P. )  /\  ( w  e.  P.  /\  v  e.  P. )
)  ->  ( 1P  +P.  1P )  e.  P. )
117 enreceq 7508 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( y  .P.  ( v  +P. 
1P ) )  +P.  ( z  .P.  1P ) )  e.  P.  /\  ( ( y  .P. 
1P )  +P.  (
z  .P.  ( v  +P.  1P ) ) )  e.  P. )  /\  ( ( 1P  +P.  1P )  e.  P.  /\  1P  e.  P. ) )  ->  ( [ <. ( ( y  .P.  (
v  +P.  1P )
)  +P.  ( z  .P.  1P ) ) ,  ( ( y  .P. 
1P )  +P.  (
z  .P.  ( v  +P.  1P ) ) )
>. ]  ~R  =  [ <. ( 1P  +P.  1P ) ,  1P >. ]  ~R  <->  ( ( ( y  .P.  ( v  +P.  1P ) )  +P.  (
z  .P.  1P )
)  +P.  1P )  =  ( ( ( y  .P.  1P )  +P.  ( z  .P.  ( v  +P.  1P ) ) )  +P.  ( 1P  +P.  1P ) ) ) )
118112, 113, 116, 67, 117syl22anc 1200 . . . . . . . . . . . . . 14  |-  ( ( ( y  e.  P.  /\  z  e.  P. )  /\  ( w  e.  P.  /\  v  e.  P. )
)  ->  ( [ <. ( ( y  .P.  ( v  +P.  1P ) )  +P.  (
z  .P.  1P )
) ,  ( ( y  .P.  1P )  +P.  ( z  .P.  ( v  +P.  1P ) ) ) >. ]  ~R  =  [ <. ( 1P  +P.  1P ) ,  1P >. ]  ~R  <->  ( ( ( y  .P.  ( v  +P.  1P ) )  +P.  (
z  .P.  1P )
)  +P.  1P )  =  ( ( ( y  .P.  1P )  +P.  ( z  .P.  ( v  +P.  1P ) ) )  +P.  ( 1P  +P.  1P ) ) ) )
119118adantr 272 . . . . . . . . . . . . 13  |-  ( ( ( ( y  e. 
P.  /\  z  e.  P. )  /\  (
w  e.  P.  /\  v  e.  P. )
)  /\  ( (
w  .P.  v )  =  1P  /\  (
z  +P.  w )  =  y ) )  ->  ( [ <. ( ( y  .P.  (
v  +P.  1P )
)  +P.  ( z  .P.  1P ) ) ,  ( ( y  .P. 
1P )  +P.  (
z  .P.  ( v  +P.  1P ) ) )
>. ]  ~R  =  [ <. ( 1P  +P.  1P ) ,  1P >. ]  ~R  <->  ( ( ( y  .P.  ( v  +P.  1P ) )  +P.  (
z  .P.  1P )
)  +P.  1P )  =  ( ( ( y  .P.  1P )  +P.  ( z  .P.  ( v  +P.  1P ) ) )  +P.  ( 1P  +P.  1P ) ) ) )
120107, 119mpbird 166 . . . . . . . . . . . 12  |-  ( ( ( ( y  e. 
P.  /\  z  e.  P. )  /\  (
w  e.  P.  /\  v  e.  P. )
)  /\  ( (
w  .P.  v )  =  1P  /\  (
z  +P.  w )  =  y ) )  ->  [ <. (
( y  .P.  (
v  +P.  1P )
)  +P.  ( z  .P.  1P ) ) ,  ( ( y  .P. 
1P )  +P.  (
z  .P.  ( v  +P.  1P ) ) )
>. ]  ~R  =  [ <. ( 1P  +P.  1P ) ,  1P >. ]  ~R  )
12139, 120eqtrd 2148 . . . . . . . . . . 11  |-  ( ( ( ( y  e. 
P.  /\  z  e.  P. )  /\  (
w  e.  P.  /\  v  e.  P. )
)  /\  ( (
w  .P.  v )  =  1P  /\  (
z  +P.  w )  =  y ) )  ->  ( [ <. y ,  z >. ]  ~R  .R 
[ <. ( v  +P. 
1P ) ,  1P >. ]  ~R  )  =  [ <. ( 1P  +P.  1P ) ,  1P >. ]  ~R  )
122 df-1r 7504 . . . . . . . . . . 11  |-  1R  =  [ <. ( 1P  +P.  1P ) ,  1P >. ]  ~R
123121, 122syl6eqr 2166 . . . . . . . . . 10  |-  ( ( ( ( y  e. 
P.  /\  z  e.  P. )  /\  (
w  e.  P.  /\  v  e.  P. )
)  /\  ( (
w  .P.  v )  =  1P  /\  (
z  +P.  w )  =  y ) )  ->  ( [ <. y ,  z >. ]  ~R  .R 
[ <. ( v  +P. 
1P ) ,  1P >. ]  ~R  )  =  1R )
124 breq2 3901 . . . . . . . . . . . 12  |-  ( x  =  [ <. (
v  +P.  1P ) ,  1P >. ]  ~R  ->  ( 0R  <R  x  <->  0R  <R  [
<. ( v  +P.  1P ) ,  1P >. ]  ~R  ) )
125 oveq2 5748 . . . . . . . . . . . . 13  |-  ( x  =  [ <. (
v  +P.  1P ) ,  1P >. ]  ~R  ->  ( [ <. y ,  z
>. ]  ~R  .R  x
)  =  ( [
<. y ,  z >. ]  ~R  .R  [ <. ( v  +P.  1P ) ,  1P >. ]  ~R  ) )
126125eqeq1d 2124 . . . . . . . . . . . 12  |-  ( x  =  [ <. (
v  +P.  1P ) ,  1P >. ]  ~R  ->  ( ( [ <. y ,  z >. ]  ~R  .R  x )  =  1R  <->  ( [ <. y ,  z
>. ]  ~R  .R  [ <. ( v  +P.  1P ) ,  1P >. ]  ~R  )  =  1R )
)
127124, 126anbi12d 462 . . . . . . . . . . 11  |-  ( x  =  [ <. (
v  +P.  1P ) ,  1P >. ]  ~R  ->  ( ( 0R  <R  x  /\  ( [ <. y ,  z >. ]  ~R  .R  x )  =  1R ) 
<->  ( 0R  <R  [ <. ( v  +P.  1P ) ,  1P >. ]  ~R  /\  ( [ <. y ,  z >. ]  ~R  .R 
[ <. ( v  +P. 
1P ) ,  1P >. ]  ~R  )  =  1R ) ) )
128127rspcev 2761 . . . . . . . . . 10  |-  ( ( [ <. ( v  +P. 
1P ) ,  1P >. ]  ~R  e.  R.  /\  ( 0R  <R  [ <. ( v  +P.  1P ) ,  1P >. ]  ~R  /\  ( [ <. y ,  z >. ]  ~R  .R 
[ <. ( v  +P. 
1P ) ,  1P >. ]  ~R  )  =  1R ) )  ->  E. x  e.  R.  ( 0R  <R  x  /\  ( [ <. y ,  z
>. ]  ~R  .R  x
)  =  1R )
)
12923, 33, 123, 128syl12anc 1197 . . . . . . . . 9  |-  ( ( ( ( y  e. 
P.  /\  z  e.  P. )  /\  (
w  e.  P.  /\  v  e.  P. )
)  /\  ( (
w  .P.  v )  =  1P  /\  (
z  +P.  w )  =  y ) )  ->  E. x  e.  R.  ( 0R  <R  x  /\  ( [ <. y ,  z
>. ]  ~R  .R  x
)  =  1R )
)
130129exp32 360 . . . . . . . 8  |-  ( ( ( y  e.  P.  /\  z  e.  P. )  /\  ( w  e.  P.  /\  v  e.  P. )
)  ->  ( (
w  .P.  v )  =  1P  ->  ( ( z  +P.  w )  =  y  ->  E. x  e.  R.  ( 0R  <R  x  /\  ( [ <. y ,  z >. ]  ~R  .R  x )  =  1R ) ) ) )
131130anassrs 395 . . . . . . 7  |-  ( ( ( ( y  e. 
P.  /\  z  e.  P. )  /\  w  e.  P. )  /\  v  e.  P. )  ->  (
( w  .P.  v
)  =  1P  ->  ( ( z  +P.  w
)  =  y  ->  E. x  e.  R.  ( 0R  <R  x  /\  ( [ <. y ,  z
>. ]  ~R  .R  x
)  =  1R )
) ) )
132131rexlimdva 2524 . . . . . 6  |-  ( ( ( y  e.  P.  /\  z  e.  P. )  /\  w  e.  P. )  ->  ( E. v  e.  P.  ( w  .P.  v )  =  1P 
->  ( ( z  +P.  w )  =  y  ->  E. x  e.  R.  ( 0R  <R  x  /\  ( [ <. y ,  z
>. ]  ~R  .R  x
)  =  1R )
) ) )
13315, 132mpd 13 . . . . 5  |-  ( ( ( y  e.  P.  /\  z  e.  P. )  /\  w  e.  P. )  ->  ( ( z  +P.  w )  =  y  ->  E. x  e.  R.  ( 0R  <R  x  /\  ( [ <. y ,  z >. ]  ~R  .R  x )  =  1R ) ) )
134133rexlimdva 2524 . . . 4  |-  ( ( y  e.  P.  /\  z  e.  P. )  ->  ( E. w  e. 
P.  ( z  +P.  w )  =  y  ->  E. x  e.  R.  ( 0R  <R  x  /\  ( [ <. y ,  z
>. ]  ~R  .R  x
)  =  1R )
) )
13513, 134syl5 32 . . 3  |-  ( ( y  e.  P.  /\  z  e.  P. )  ->  ( 0R  <R  [ <. y ,  z >. ]  ~R  ->  E. x  e.  R.  ( 0R  <R  x  /\  ( [ <. y ,  z
>. ]  ~R  .R  x
)  =  1R )
) )
1364, 10, 135ecoptocl 6482 . 2  |-  ( A  e.  R.  ->  ( 0R  <R  A  ->  E. x  e.  R.  ( 0R  <R  x  /\  ( A  .R  x )  =  1R ) ) )
1373, 136mpcom 36 1  |-  ( 0R 
<R  A  ->  E. x  e.  R.  ( 0R  <R  x  /\  ( A  .R  x )  =  1R ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    /\ w3a 945    = wceq 1314    e. wcel 1463   E.wrex 2392   <.cop 3498   class class class wbr 3897  (class class class)co 5740   [cec 6393   P.cnp 7063   1Pc1p 7064    +P. cpp 7065    .P. cmp 7066    <P cltp 7067    ~R cer 7068   R.cnr 7069   0Rc0r 7070   1Rc1r 7071    .R cmr 7074    <R cltr 7075
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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4011  ax-sep 4014  ax-nul 4022  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420  ax-iinf 4470
This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-ral 2396  df-rex 2397  df-reu 2398  df-rab 2400  df-v 2660  df-sbc 2881  df-csb 2974  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-int 3740  df-iun 3783  df-br 3898  df-opab 3958  df-mpt 3959  df-tr 3995  df-eprel 4179  df-id 4183  df-po 4186  df-iso 4187  df-iord 4256  df-on 4258  df-suc 4261  df-iom 4473  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099  df-ov 5743  df-oprab 5744  df-mpo 5745  df-1st 6004  df-2nd 6005  df-recs 6168  df-irdg 6233  df-1o 6279  df-2o 6280  df-oadd 6283  df-omul 6284  df-er 6395  df-ec 6397  df-qs 6401  df-ni 7076  df-pli 7077  df-mi 7078  df-lti 7079  df-plpq 7116  df-mpq 7117  df-enq 7119  df-nqqs 7120  df-plqqs 7121  df-mqqs 7122  df-1nqqs 7123  df-rq 7124  df-ltnqqs 7125  df-enq0 7196  df-nq0 7197  df-0nq0 7198  df-plq0 7199  df-mq0 7200  df-inp 7238  df-i1p 7239  df-iplp 7240  df-imp 7241  df-iltp 7242  df-enr 7498  df-nr 7499  df-mr 7501  df-ltr 7502  df-0r 7503  df-1r 7504
This theorem is referenced by:  recexsrlem  7546  axprecex  7652
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