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Theorem recexgt0sr 7693
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 7658 . . . 4  |-  <R  C_  ( R.  X.  R. )
21brel 4638 . . 3  |-  ( 0R 
<R  A  ->  ( 0R  e.  R.  /\  A  e.  R. ) )
32simprd 113 . 2  |-  ( 0R 
<R  A  ->  A  e. 
R. )
4 df-nr 7647 . . 3  |-  R.  =  ( ( P.  X.  P. ) /.  ~R  )
5 breq2 3969 . . . 4  |-  ( [
<. y ,  z >. ]  ~R  =  A  -> 
( 0R  <R  [ <. y ,  z >. ]  ~R  <->  0R 
<R  A ) )
6 oveq1 5831 . . . . . . 7  |-  ( [
<. y ,  z >. ]  ~R  =  A  -> 
( [ <. y ,  z >. ]  ~R  .R  x )  =  ( A  .R  x ) )
76eqeq1d 2166 . . . . . 6  |-  ( [
<. y ,  z >. ]  ~R  =  A  -> 
( ( [ <. y ,  z >. ]  ~R  .R  x )  =  1R  <->  ( A  .R  x )  =  1R ) )
87anbi2d 460 . . . . 5  |-  ( [
<. y ,  z >. ]  ~R  =  A  -> 
( ( 0R  <R  x  /\  ( [ <. y ,  z >. ]  ~R  .R  x )  =  1R ) 
<->  ( 0R  <R  x  /\  ( A  .R  x
)  =  1R )
) )
98rexbidv 2458 . . . 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 7668 . . . . 5  |-  ( 0R 
<R  [ <. y ,  z
>. ]  ~R  <->  z  <P  y )
12 ltexpri 7533 . . . . 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 7558 . . . . . . 7  |-  ( w  e.  P.  ->  E. v  e.  P.  ( w  .P.  v )  =  1P )
1514adantl 275 . . . . . 6  |-  ( ( ( y  e.  P.  /\  z  e.  P. )  /\  w  e.  P. )  ->  E. v  e.  P.  ( w  .P.  v )  =  1P )
16 1pr 7474 . . . . . . . . . . . . . 14  |-  1P  e.  P.
17 addclpr 7457 . . . . . . . . . . . . . 14  |-  ( ( v  e.  P.  /\  1P  e.  P. )  -> 
( v  +P.  1P )  e.  P. )
1816, 17mpan2 422 . . . . . . . . . . . . 13  |-  ( v  e.  P.  ->  (
v  +P.  1P )  e.  P. )
19 enrex 7657 . . . . . . . . . . . . . 14  |-  ~R  e.  _V
2019, 4ecopqsi 6535 . . . . . . . . . . . . 13  |-  ( ( ( v  +P.  1P )  e.  P.  /\  1P  e.  P. )  ->  [ <. ( v  +P.  1P ) ,  1P >. ]  ~R  e.  R. )
2118, 16, 20sylancl 410 . . . . . . . . . . . 12  |-  ( v  e.  P.  ->  [ <. ( v  +P.  1P ) ,  1P >. ]  ~R  e.  R. )
2221adantl 275 . . . . . . . . . . 11  |-  ( ( w  e.  P.  /\  v  e.  P. )  ->  [ <. ( v  +P. 
1P ) ,  1P >. ]  ~R  e.  R. )
2322ad2antlr 481 . . . . . . . . . 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 522 . . . . . . . . . . . 12  |-  ( ( ( y  e.  P.  /\  z  e.  P. )  /\  ( w  e.  P.  /\  v  e.  P. )
)  ->  v  e.  P. )
2524adantr 274 . . . . . . . . . . 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 7517 . . . . . . . . . . . . . 14  |-  ( ( 1P  e.  P.  /\  v  e.  P. )  ->  1P  <P  ( 1P  +P.  v ) )
2716, 26mpan 421 . . . . . . . . . . . . 13  |-  ( v  e.  P.  ->  1P  <P  ( 1P  +P.  v
) )
28 addcomprg 7498 . . . . . . . . . . . . . 14  |-  ( ( 1P  e.  P.  /\  v  e.  P. )  ->  ( 1P  +P.  v
)  =  ( v  +P.  1P ) )
2916, 28mpan 421 . . . . . . . . . . . . 13  |-  ( v  e.  P.  ->  ( 1P  +P.  v )  =  ( v  +P.  1P ) )
3027, 29breqtrd 3990 . . . . . . . . . . . 12  |-  ( v  e.  P.  ->  1P  <P  ( v  +P.  1P ) )
31 gt0srpr 7668 . . . . . . . . . . . 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 311 . . . . . . . . . . . . . . . 16  |-  ( v  e.  P.  ->  (
( v  +P.  1P )  e.  P.  /\  1P  e.  P. ) )
3534anim2i 340 . . . . . . . . . . . . . . 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 274 . . . . . . . . . . . . . 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 7666 . . . . . . . . . . . . . 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 474 . . . . . . . . . . . 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 5831 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( z  +P.  w )  =  y  ->  (
( z  +P.  w
)  .P.  v )  =  ( y  .P.  v ) )
4140eqcomd 2163 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( z  +P.  w )  =  y  ->  (
y  .P.  v )  =  ( ( z  +P.  w )  .P.  v ) )
4241ad2antll 483 . . . . . . . . . . . . . . . . . . 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 7500 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( f  e.  P.  /\  h  e.  P. )  ->  ( f  .P.  h
)  =  ( h  .P.  f ) )
44433adant2 1001 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( f  e.  P.  /\  g  e.  P.  /\  h  e.  P. )  ->  (
f  .P.  h )  =  ( h  .P.  f ) )
45 mulcomprg 7500 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( g  e.  P.  /\  h  e.  P. )  ->  ( g  .P.  h
)  =  ( h  .P.  g ) )
46453adant1 1000 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( f  e.  P.  /\  g  e.  P.  /\  h  e.  P. )  ->  (
g  .P.  h )  =  ( h  .P.  g ) )
4744, 46oveq12d 5842 . . . . . . . . . . . . . . . . . . . . . . . 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 7508 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( h  e.  P.  /\  f  e.  P.  /\  g  e.  P. )  ->  (
h  .P.  ( f  +P.  g ) )  =  ( ( h  .P.  f )  +P.  (
h  .P.  g )
) )
49483coml 1192 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( f  e.  P.  /\  g  e.  P.  /\  h  e.  P. )  ->  (
h  .P.  ( f  +P.  g ) )  =  ( ( h  .P.  f )  +P.  (
h  .P.  g )
) )
50 simp3 984 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( f  e.  P.  /\  g  e.  P.  /\  h  e.  P. )  ->  h  e.  P. )
51 addclpr 7457 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( f  e.  P.  /\  g  e.  P. )  ->  ( f  +P.  g
)  e.  P. )
52513adant3 1002 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( f  e.  P.  /\  g  e.  P.  /\  h  e.  P. )  ->  (
f  +P.  g )  e.  P. )
53 mulcomprg 7500 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( h  e.  P.  /\  ( f  +P.  g
)  e.  P. )  ->  ( h  .P.  (
f  +P.  g )
)  =  ( ( f  +P.  g )  .P.  h ) )
5450, 52, 53syl2anc 409 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( f  e.  P.  /\  g  e.  P.  /\  h  e.  P. )  ->  (
h  .P.  ( f  +P.  g ) )  =  ( ( f  +P.  g )  .P.  h
) )
5547, 49, 543eqtr2rd 2197 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( f  e.  P.  /\  g  e.  P.  /\  h  e.  P. )  ->  (
( f  +P.  g
)  .P.  h )  =  ( ( f  .P.  h )  +P.  ( g  .P.  h
) ) )
5655adantl 275 . . . . . . . . . . . . . . . . . . . . . 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 520 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( y  e.  P.  /\  z  e.  P. )  /\  ( w  e.  P.  /\  v  e.  P. )
)  ->  z  e.  P. )
58 simprl 521 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( y  e.  P.  /\  z  e.  P. )  /\  ( w  e.  P.  /\  v  e.  P. )
)  ->  w  e.  P. )
5956, 57, 58, 24caovdird 5999 . . . . . . . . . . . . . . . . . . . . 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 5832 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( w  .P.  v )  =  1P  ->  (
( z  .P.  v
)  +P.  ( w  .P.  v ) )  =  ( ( z  .P.  v )  +P.  1P ) )
6159, 60sylan9eq 2210 . . . . . . . . . . . . . . . . . . . 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 471 . . . . . . . . . . . . . . . . . . 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 2190 . . . . . . . . . . . . . . . . . 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 5839 . . . . . . . . . . . . . . . . 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 7492 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( z  e.  P.  /\  v  e.  P. )  ->  ( z  .P.  v
)  e.  P. )
6657, 24, 65syl2anc 409 . . . . . . . . . . . . . . . . . . 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 519 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( y  e.  P.  /\  z  e.  P. )  /\  ( w  e.  P.  /\  v  e.  P. )
)  ->  y  e.  P. )
69 mulclpr 7492 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( y  e.  P.  /\  1P  e.  P. )  -> 
( y  .P.  1P )  e.  P. )
7068, 16, 69sylancl 410 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( y  e.  P.  /\  z  e.  P. )  /\  ( w  e.  P.  /\  v  e.  P. )
)  ->  ( y  .P.  1P )  e.  P. )
71 mulclpr 7492 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( z  e.  P.  /\  1P  e.  P. )  -> 
( z  .P.  1P )  e.  P. )
7257, 16, 71sylancl 410 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( y  e.  P.  /\  z  e.  P. )  /\  ( w  e.  P.  /\  v  e.  P. )
)  ->  ( z  .P.  1P )  e.  P. )
73 addclpr 7457 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( y  .P.  1P )  e.  P.  /\  (
z  .P.  1P )  e.  P. )  ->  (
( y  .P.  1P )  +P.  ( z  .P. 
1P ) )  e. 
P. )
7470, 72, 73syl2anc 409 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( y  e.  P.  /\  z  e.  P. )  /\  ( w  e.  P.  /\  v  e.  P. )
)  ->  ( (
y  .P.  1P )  +P.  ( z  .P.  1P ) )  e.  P. )
75 addcomprg 7498 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( f  e.  P.  /\  g  e.  P. )  ->  ( f  +P.  g
)  =  ( g  +P.  f ) )
7675adantl 275 . . . . . . . . . . . . . . . . . . 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 7499 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( f  e.  P.  /\  g  e.  P.  /\  h  e.  P. )  ->  (
( f  +P.  g
)  +P.  h )  =  ( f  +P.  ( g  +P.  h
) ) )
7877adantl 275 . . . . . . . . . . . . . . . . . . 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 6001 . . . . . . . . . . . . . . . . . 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 274 . . . . . . . . . . . . . . . . 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 2190 . . . . . . . . . . . . . . . 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 5839 . . . . . . . . . . . . . . 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 7457 . . . . . . . . . . . . . . . . . 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 409 . . . . . . . . . . . . . . . . 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 274 . . . . . . . . . . . . . . . 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 7499 . . . . . . . . . . . . . . . 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 1220 . . . . . . . . . . . . . . 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 2190 . . . . . . . . . . . . . 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 7508 . . . . . . . . . . . . . . . . . . 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 1220 . . . . . . . . . . . . . . . . . 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 5839 . . . . . . . . . . . . . . . . 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 7492 . . . . . . . . . . . . . . . . . . 19  |-  ( ( y  e.  P.  /\  v  e.  P. )  ->  ( y  .P.  v
)  e.  P. )
9468, 24, 93syl2anc 409 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( y  e.  P.  /\  z  e.  P. )  /\  ( w  e.  P.  /\  v  e.  P. )
)  ->  ( y  .P.  v )  e.  P. )
95 addassprg 7499 . . . . . . . . . . . . . . . . . 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 1220 . . . . . . . . . . . . . . . . 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 2190 . . . . . . . . . . . . . . . 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 5839 . . . . . . . . . . . . . . 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 274 . . . . . . . . . . . . . 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 7508 . . . . . . . . . . . . . . . . . . 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 1220 . . . . . . . . . . . . . . . . . 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 5840 . . . . . . . . . . . . . . . . 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 6002 . . . . . . . . . . . . . . . . 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 2190 . . . . . . . . . . . . . . . 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 5839 . . . . . . . . . . . . . . 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 274 . . . . . . . . . . . . . 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 2200 . . . . . . . . . . . . 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 410 . . . . . . . . . . . . . . . . 17  |-  ( ( ( y  e.  P.  /\  z  e.  P. )  /\  ( w  e.  P.  /\  v  e.  P. )
)  ->  ( v  +P.  1P )  e.  P. )
109 mulclpr 7492 . . . . . . . . . . . . . . . . 17  |-  ( ( y  e.  P.  /\  ( v  +P.  1P )  e.  P. )  ->  ( y  .P.  (
v  +P.  1P )
)  e.  P. )
11068, 108, 109syl2anc 409 . . . . . . . . . . . . . . . 16  |-  ( ( ( y  e.  P.  /\  z  e.  P. )  /\  ( w  e.  P.  /\  v  e.  P. )
)  ->  ( y  .P.  ( v  +P.  1P ) )  e.  P. )
111 addclpr 7457 . . . . . . . . . . . . . . . 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 409 . . . . . . . . . . . . . . 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 2234 . . . . . . . . . . . . . . 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 7457 . . . . . . . . . . . . . . . . 17  |-  ( ( 1P  e.  P.  /\  1P  e.  P. )  -> 
( 1P  +P.  1P )  e.  P. )
11516, 16, 114mp2an 423 . . . . . . . . . . . . . . . 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 7656 . . . . . . . . . . . . . . 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 1221 . . . . . . . . . . . . . 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 274 . . . . . . . . . . . . 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 2190 . . . . . . . . . . 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 7652 . . . . . . . . . . 11  |-  1R  =  [ <. ( 1P  +P.  1P ) ,  1P >. ]  ~R
123121, 122eqtr4di 2208 . . . . . . . . . 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 3969 . . . . . . . . . . . 12  |-  ( x  =  [ <. (
v  +P.  1P ) ,  1P >. ]  ~R  ->  ( 0R  <R  x  <->  0R  <R  [
<. ( v  +P.  1P ) ,  1P >. ]  ~R  ) )
125 oveq2 5832 . . . . . . . . . . . . 13  |-  ( x  =  [ <. (
v  +P.  1P ) ,  1P >. ]  ~R  ->  ( [ <. y ,  z
>. ]  ~R  .R  x
)  =  ( [
<. y ,  z >. ]  ~R  .R  [ <. ( v  +P.  1P ) ,  1P >. ]  ~R  ) )
126125eqeq1d 2166 . . . . . . . . . . . 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 465 . . . . . . . . . . 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 2816 . . . . . . . . . 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 1218 . . . . . . . . 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 363 . . . . . . . 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 398 . . . . . . 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 2574 . . . . . 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 2574 . . . 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 6567 . 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 963    = wceq 1335    e. wcel 2128   E.wrex 2436   <.cop 3563   class class class wbr 3965  (class class class)co 5824   [cec 6478   P.cnp 7211   1Pc1p 7212    +P. cpp 7213    .P. cmp 7214    <P cltp 7215    ~R cer 7216   R.cnr 7217   0Rc0r 7218   1Rc1r 7219    .R cmr 7222    <R cltr 7223
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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4079  ax-sep 4082  ax-nul 4090  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4496  ax-iinf 4547
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-int 3808  df-iun 3851  df-br 3966  df-opab 4026  df-mpt 4027  df-tr 4063  df-eprel 4249  df-id 4253  df-po 4256  df-iso 4257  df-iord 4326  df-on 4328  df-suc 4331  df-iom 4550  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-rn 4597  df-res 4598  df-ima 4599  df-iota 5135  df-fun 5172  df-fn 5173  df-f 5174  df-f1 5175  df-fo 5176  df-f1o 5177  df-fv 5178  df-ov 5827  df-oprab 5828  df-mpo 5829  df-1st 6088  df-2nd 6089  df-recs 6252  df-irdg 6317  df-1o 6363  df-2o 6364  df-oadd 6367  df-omul 6368  df-er 6480  df-ec 6482  df-qs 6486  df-ni 7224  df-pli 7225  df-mi 7226  df-lti 7227  df-plpq 7264  df-mpq 7265  df-enq 7267  df-nqqs 7268  df-plqqs 7269  df-mqqs 7270  df-1nqqs 7271  df-rq 7272  df-ltnqqs 7273  df-enq0 7344  df-nq0 7345  df-0nq0 7346  df-plq0 7347  df-mq0 7348  df-inp 7386  df-i1p 7387  df-iplp 7388  df-imp 7389  df-iltp 7390  df-enr 7646  df-nr 7647  df-mr 7649  df-ltr 7650  df-0r 7651  df-1r 7652
This theorem is referenced by:  recexsrlem  7694  axprecex  7800
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