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Theorem recexsrlem 8725
Description: The reciprocal of a positive signed real exists. Part of Proposition 9-4.3 of [Gleason] p. 126. (Contributed by NM, 15-May-1996.) (New usage is discouraged.)
Assertion
Ref Expression
recexsrlem  |-  ( 0R 
<R  A  ->  E. x  e.  R.  ( A  .R  x )  =  1R )
Distinct variable group:    x, A

Proof of Theorem recexsrlem
Dummy variables  y 
z  w  v  u  f are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ltrelsr 8693 . . . 4  |-  <R  C_  ( R.  X.  R. )
21brel 4737 . . 3  |-  ( 0R 
<R  A  ->  ( 0R  e.  R.  /\  A  e.  R. ) )
32simprd 449 . 2  |-  ( 0R 
<R  A  ->  A  e. 
R. )
4 df-nr 8682 . . 3  |-  R.  =  ( ( P.  X.  P. ) /.  ~R  )
5 breq2 4027 . . . 4  |-  ( [
<. y ,  z >. ]  ~R  =  A  -> 
( 0R  <R  [ <. y ,  z >. ]  ~R  <->  0R 
<R  A ) )
6 oveq1 5865 . . . . . 6  |-  ( [
<. y ,  z >. ]  ~R  =  A  -> 
( [ <. y ,  z >. ]  ~R  .R  x )  =  ( A  .R  x ) )
76eqeq1d 2291 . . . . 5  |-  ( [
<. y ,  z >. ]  ~R  =  A  -> 
( ( [ <. y ,  z >. ]  ~R  .R  x )  =  1R  <->  ( A  .R  x )  =  1R ) )
87rexbidv 2564 . . . 4  |-  ( [
<. y ,  z >. ]  ~R  =  A  -> 
( E. x  e. 
R.  ( [ <. y ,  z >. ]  ~R  .R  x )  =  1R  <->  E. x  e.  R.  ( A  .R  x )  =  1R ) )
95, 8imbi12d 311 . . 3  |-  ( [
<. y ,  z >. ]  ~R  =  A  -> 
( ( 0R  <R  [
<. y ,  z >. ]  ~R  ->  E. x  e.  R.  ( [ <. y ,  z >. ]  ~R  .R  x )  =  1R ) 
<->  ( 0R  <R  A  ->  E. x  e.  R.  ( A  .R  x
)  =  1R )
) )
10 gt0srpr 8700 . . . . 5  |-  ( 0R 
<R  [ <. y ,  z
>. ]  ~R  <->  z  <P  y )
11 ltexpri 8667 . . . . 5  |-  ( z 
<P  y  ->  E. w  e.  P.  ( z  +P.  w )  =  y )
1210, 11sylbi 187 . . . 4  |-  ( 0R 
<R  [ <. y ,  z
>. ]  ~R  ->  E. w  e.  P.  ( z  +P.  w )  =  y )
13 recexpr 8675 . . . . . 6  |-  ( w  e.  P.  ->  E. v  e.  P.  ( w  .P.  v )  =  1P )
14 1pr 8639 . . . . . . . . . . . 12  |-  1P  e.  P.
15 addclpr 8642 . . . . . . . . . . . 12  |-  ( ( v  e.  P.  /\  1P  e.  P. )  -> 
( v  +P.  1P )  e.  P. )
1614, 15mpan2 652 . . . . . . . . . . 11  |-  ( v  e.  P.  ->  (
v  +P.  1P )  e.  P. )
17 enrex 8692 . . . . . . . . . . . 12  |-  ~R  e.  _V
1817, 4ecopqsi 6716 . . . . . . . . . . 11  |-  ( ( ( v  +P.  1P )  e.  P.  /\  1P  e.  P. )  ->  [ <. ( v  +P.  1P ) ,  1P >. ]  ~R  e.  R. )
1916, 14, 18sylancl 643 . . . . . . . . . 10  |-  ( v  e.  P.  ->  [ <. ( v  +P.  1P ) ,  1P >. ]  ~R  e.  R. )
2019ad2antlr 707 . . . . . . . . 9  |-  ( ( ( ( y  e. 
P.  /\  z  e.  P. )  /\  v  e.  P. )  /\  (
( w  .P.  v
)  =  1P  /\  ( z  +P.  w
)  =  y ) )  ->  [ <. (
v  +P.  1P ) ,  1P >. ]  ~R  e.  R. )
2116, 14jctir 524 . . . . . . . . . . . . . 14  |-  ( v  e.  P.  ->  (
( v  +P.  1P )  e.  P.  /\  1P  e.  P. ) )
2221anim2i 552 . . . . . . . . . . . . 13  |-  ( ( ( y  e.  P.  /\  z  e.  P. )  /\  v  e.  P. )  ->  ( ( y  e.  P.  /\  z  e.  P. )  /\  (
( v  +P.  1P )  e.  P.  /\  1P  e.  P. ) ) )
2322adantr 451 . . . . . . . . . . . 12  |-  ( ( ( ( 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. )
) )
24 mulsrpr 8698 . . . . . . . . . . . 12  |-  ( ( ( 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  )
2523, 24syl 15 . . . . . . . . . . 11  |-  ( ( ( ( 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  )
26 oveq1 5865 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( z  +P.  w )  =  y  ->  (
( z  +P.  w
)  .P.  v )  =  ( y  .P.  v ) )
2726eqcomd 2288 . . . . . . . . . . . . . . . . . . 19  |-  ( ( z  +P.  w )  =  y  ->  (
y  .P.  v )  =  ( ( z  +P.  w )  .P.  v ) )
28 vex 2791 . . . . . . . . . . . . . . . . . . . . 21  |-  z  e. 
_V
29 vex 2791 . . . . . . . . . . . . . . . . . . . . 21  |-  w  e. 
_V
30 vex 2791 . . . . . . . . . . . . . . . . . . . . 21  |-  v  e. 
_V
31 mulcompr 8647 . . . . . . . . . . . . . . . . . . . . 21  |-  ( u  .P.  f )  =  ( f  .P.  u
)
32 distrpr 8652 . . . . . . . . . . . . . . . . . . . . 21  |-  ( u  .P.  ( f  +P.  x ) )  =  ( ( u  .P.  f )  +P.  (
u  .P.  x )
)
3328, 29, 30, 31, 32caovdir 6054 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( z  +P.  w )  .P.  v )  =  ( ( z  .P.  v )  +P.  (
w  .P.  v )
)
34 oveq2 5866 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( w  .P.  v )  =  1P  ->  (
( z  .P.  v
)  +P.  ( w  .P.  v ) )  =  ( ( z  .P.  v )  +P.  1P ) )
3533, 34syl5eq 2327 . . . . . . . . . . . . . . . . . . 19  |-  ( ( w  .P.  v )  =  1P  ->  (
( z  +P.  w
)  .P.  v )  =  ( ( z  .P.  v )  +P. 
1P ) )
3627, 35sylan9eqr 2337 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( w  .P.  v
)  =  1P  /\  ( z  +P.  w
)  =  y )  ->  ( y  .P.  v )  =  ( ( z  .P.  v
)  +P.  1P )
)
3736oveq1d 5873 . . . . . . . . . . . . . . . . 17  |-  ( ( ( 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 ) ) ) )
38 ovex 5883 . . . . . . . . . . . . . . . . . 18  |-  ( z  .P.  v )  e. 
_V
3914elexi 2797 . . . . . . . . . . . . . . . . . 18  |-  1P  e.  _V
40 ovex 5883 . . . . . . . . . . . . . . . . . 18  |-  ( ( y  .P.  1P )  +P.  ( z  .P. 
1P ) )  e. 
_V
41 addcompr 8645 . . . . . . . . . . . . . . . . . 18  |-  ( u  +P.  f )  =  ( f  +P.  u
)
42 addasspr 8646 . . . . . . . . . . . . . . . . . 18  |-  ( ( u  +P.  f )  +P.  x )  =  ( u  +P.  (
f  +P.  x )
)
4338, 39, 40, 41, 42caov32 6047 . . . . . . . . . . . . . . . . 17  |-  ( ( ( 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 )
4437, 43syl6eq 2331 . . . . . . . . . . . . . . . 16  |-  ( ( ( 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 ) )
4544oveq1d 5873 . . . . . . . . . . . . . . 15  |-  ( ( ( 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 ) )
46 addasspr 8646 . . . . . . . . . . . . . . 15  |-  ( ( ( ( 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 ) )
4745, 46syl6eq 2331 . . . . . . . . . . . . . 14  |-  ( ( ( 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 ) ) )
48 distrpr 8652 . . . . . . . . . . . . . . . . 17  |-  ( y  .P.  ( v  +P. 
1P ) )  =  ( ( y  .P.  v )  +P.  (
y  .P.  1P )
)
4948oveq1i 5868 . . . . . . . . . . . . . . . 16  |-  ( ( y  .P.  ( v  +P.  1P ) )  +P.  ( z  .P. 
1P ) )  =  ( ( ( y  .P.  v )  +P.  ( y  .P.  1P ) )  +P.  (
z  .P.  1P )
)
50 addasspr 8646 . . . . . . . . . . . . . . . 16  |-  ( ( ( y  .P.  v
)  +P.  ( y  .P.  1P ) )  +P.  ( z  .P.  1P ) )  =  ( ( y  .P.  v
)  +P.  ( (
y  .P.  1P )  +P.  ( z  .P.  1P ) ) )
5149, 50eqtri 2303 . . . . . . . . . . . . . . 15  |-  ( ( y  .P.  ( v  +P.  1P ) )  +P.  ( z  .P. 
1P ) )  =  ( ( y  .P.  v )  +P.  (
( y  .P.  1P )  +P.  ( z  .P. 
1P ) ) )
5251oveq1i 5868 . . . . . . . . . . . . . 14  |-  ( ( ( y  .P.  (
v  +P.  1P )
)  +P.  ( z  .P.  1P ) )  +P. 
1P )  =  ( ( ( y  .P.  v )  +P.  (
( y  .P.  1P )  +P.  ( z  .P. 
1P ) ) )  +P.  1P )
53 distrpr 8652 . . . . . . . . . . . . . . . . 17  |-  ( z  .P.  ( v  +P. 
1P ) )  =  ( ( z  .P.  v )  +P.  (
z  .P.  1P )
)
5453oveq2i 5869 . . . . . . . . . . . . . . . 16  |-  ( ( y  .P.  1P )  +P.  ( z  .P.  ( v  +P.  1P ) ) )  =  ( ( y  .P. 
1P )  +P.  (
( z  .P.  v
)  +P.  ( z  .P.  1P ) ) )
55 ovex 5883 . . . . . . . . . . . . . . . . 17  |-  ( y  .P.  1P )  e. 
_V
56 ovex 5883 . . . . . . . . . . . . . . . . 17  |-  ( z  .P.  1P )  e. 
_V
5755, 38, 56, 41, 42caov12 6048 . . . . . . . . . . . . . . . 16  |-  ( ( y  .P.  1P )  +P.  ( ( z  .P.  v )  +P.  ( z  .P.  1P ) ) )  =  ( ( z  .P.  v )  +P.  (
( y  .P.  1P )  +P.  ( z  .P. 
1P ) ) )
5854, 57eqtri 2303 . . . . . . . . . . . . . . 15  |-  ( ( y  .P.  1P )  +P.  ( z  .P.  ( v  +P.  1P ) ) )  =  ( ( z  .P.  v )  +P.  (
( y  .P.  1P )  +P.  ( z  .P. 
1P ) ) )
5958oveq1i 5868 . . . . . . . . . . . . . 14  |-  ( ( ( 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 ) )
6047, 52, 593eqtr4g 2340 . . . . . . . . . . . . 13  |-  ( ( ( 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 ) ) )
61 mulclpr 8644 . . . . . . . . . . . . . . . . . 18  |-  ( ( y  e.  P.  /\  ( v  +P.  1P )  e.  P. )  ->  ( y  .P.  (
v  +P.  1P )
)  e.  P. )
6216, 61sylan2 460 . . . . . . . . . . . . . . . . 17  |-  ( ( y  e.  P.  /\  v  e.  P. )  ->  ( y  .P.  (
v  +P.  1P )
)  e.  P. )
63 mulclpr 8644 . . . . . . . . . . . . . . . . . 18  |-  ( ( z  e.  P.  /\  1P  e.  P. )  -> 
( z  .P.  1P )  e.  P. )
6414, 63mpan2 652 . . . . . . . . . . . . . . . . 17  |-  ( z  e.  P.  ->  (
z  .P.  1P )  e.  P. )
65 addclpr 8642 . . . . . . . . . . . . . . . . 17  |-  ( ( ( y  .P.  (
v  +P.  1P )
)  e.  P.  /\  ( z  .P.  1P )  e.  P. )  ->  ( ( y  .P.  ( v  +P.  1P ) )  +P.  (
z  .P.  1P )
)  e.  P. )
6662, 64, 65syl2an 463 . . . . . . . . . . . . . . . 16  |-  ( ( ( y  e.  P.  /\  v  e.  P. )  /\  z  e.  P. )  ->  ( ( y  .P.  ( v  +P. 
1P ) )  +P.  ( z  .P.  1P ) )  e.  P. )
6766an32s 779 . . . . . . . . . . . . . . 15  |-  ( ( ( y  e.  P.  /\  z  e.  P. )  /\  v  e.  P. )  ->  ( ( y  .P.  ( v  +P. 
1P ) )  +P.  ( z  .P.  1P ) )  e.  P. )
68 mulclpr 8644 . . . . . . . . . . . . . . . . . 18  |-  ( ( y  e.  P.  /\  1P  e.  P. )  -> 
( y  .P.  1P )  e.  P. )
6914, 68mpan2 652 . . . . . . . . . . . . . . . . 17  |-  ( y  e.  P.  ->  (
y  .P.  1P )  e.  P. )
70 mulclpr 8644 . . . . . . . . . . . . . . . . . 18  |-  ( ( z  e.  P.  /\  ( v  +P.  1P )  e.  P. )  ->  ( z  .P.  (
v  +P.  1P )
)  e.  P. )
7116, 70sylan2 460 . . . . . . . . . . . . . . . . 17  |-  ( ( z  e.  P.  /\  v  e.  P. )  ->  ( z  .P.  (
v  +P.  1P )
)  e.  P. )
72 addclpr 8642 . . . . . . . . . . . . . . . . 17  |-  ( ( ( y  .P.  1P )  e.  P.  /\  (
z  .P.  ( v  +P.  1P ) )  e. 
P. )  ->  (
( y  .P.  1P )  +P.  ( z  .P.  ( v  +P.  1P ) ) )  e. 
P. )
7369, 71, 72syl2an 463 . . . . . . . . . . . . . . . 16  |-  ( ( y  e.  P.  /\  ( z  e.  P.  /\  v  e.  P. )
)  ->  ( (
y  .P.  1P )  +P.  ( z  .P.  (
v  +P.  1P )
) )  e.  P. )
7473anassrs 629 . . . . . . . . . . . . . . 15  |-  ( ( ( y  e.  P.  /\  z  e.  P. )  /\  v  e.  P. )  ->  ( ( y  .P.  1P )  +P.  ( z  .P.  (
v  +P.  1P )
) )  e.  P. )
7567, 74jca 518 . . . . . . . . . . . . . 14  |-  ( ( ( y  e.  P.  /\  z  e.  P. )  /\  v  e.  P. )  ->  ( ( ( y  .P.  ( v  +P.  1P ) )  +P.  ( z  .P. 
1P ) )  e. 
P.  /\  ( (
y  .P.  1P )  +P.  ( z  .P.  (
v  +P.  1P )
) )  e.  P. ) )
76 addclpr 8642 . . . . . . . . . . . . . . . 16  |-  ( ( 1P  e.  P.  /\  1P  e.  P. )  -> 
( 1P  +P.  1P )  e.  P. )
7714, 14, 76mp2an 653 . . . . . . . . . . . . . . 15  |-  ( 1P 
+P.  1P )  e.  P.
7877, 14pm3.2i 441 . . . . . . . . . . . . . 14  |-  ( ( 1P  +P.  1P )  e.  P.  /\  1P  e.  P. )
79 enreceq 8691 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( 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 ) ) ) )
8075, 78, 79sylancl 643 . . . . . . . . . . . . 13  |-  ( ( ( y  e.  P.  /\  z  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 ) ) ) )
8160, 80syl5ibr 212 . . . . . . . . . . . 12  |-  ( ( ( y  e.  P.  /\  z  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  ) )
8281imp 418 . . . . . . . . . . 11  |-  ( ( ( ( y  e. 
P.  /\  z  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  )
8325, 82eqtrd 2315 . . . . . . . . . 10  |-  ( ( ( ( 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  )  =  [ <. ( 1P  +P.  1P ) ,  1P >. ]  ~R  )
84 df-1r 8687 . . . . . . . . . 10  |-  1R  =  [ <. ( 1P  +P.  1P ) ,  1P >. ]  ~R
8583, 84syl6eqr 2333 . . . . . . . . 9  |-  ( ( ( ( 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  )  =  1R )
86 oveq2 5866 . . . . . . . . . . 11  |-  ( x  =  [ <. (
v  +P.  1P ) ,  1P >. ]  ~R  ->  ( [ <. y ,  z
>. ]  ~R  .R  x
)  =  ( [
<. y ,  z >. ]  ~R  .R  [ <. ( v  +P.  1P ) ,  1P >. ]  ~R  ) )
8786eqeq1d 2291 . . . . . . . . . 10  |-  ( x  =  [ <. (
v  +P.  1P ) ,  1P >. ]  ~R  ->  ( ( [ <. y ,  z >. ]  ~R  .R  x )  =  1R  <->  ( [ <. y ,  z
>. ]  ~R  .R  [ <. ( v  +P.  1P ) ,  1P >. ]  ~R  )  =  1R )
)
8887rspcev 2884 . . . . . . . . 9  |-  ( ( [ <. ( v  +P. 
1P ) ,  1P >. ]  ~R  e.  R.  /\  ( [ <. y ,  z >. ]  ~R  .R 
[ <. ( v  +P. 
1P ) ,  1P >. ]  ~R  )  =  1R )  ->  E. x  e.  R.  ( [ <. y ,  z >. ]  ~R  .R  x )  =  1R )
8920, 85, 88syl2anc 642 . . . . . . . 8  |-  ( ( ( ( y  e. 
P.  /\  z  e.  P. )  /\  v  e.  P. )  /\  (
( w  .P.  v
)  =  1P  /\  ( z  +P.  w
)  =  y ) )  ->  E. x  e.  R.  ( [ <. y ,  z >. ]  ~R  .R  x )  =  1R )
9089exp43 595 . . . . . . 7  |-  ( ( y  e.  P.  /\  z  e.  P. )  ->  ( v  e.  P.  ->  ( ( w  .P.  v )  =  1P 
->  ( ( z  +P.  w )  =  y  ->  E. x  e.  R.  ( [ <. y ,  z
>. ]  ~R  .R  x
)  =  1R )
) ) )
9190rexlimdv 2666 . . . . . 6  |-  ( ( y  e.  P.  /\  z  e.  P. )  ->  ( E. v  e. 
P.  ( w  .P.  v )  =  1P 
->  ( ( z  +P.  w )  =  y  ->  E. x  e.  R.  ( [ <. y ,  z
>. ]  ~R  .R  x
)  =  1R )
) )
9213, 91syl5 28 . . . . 5  |-  ( ( y  e.  P.  /\  z  e.  P. )  ->  ( w  e.  P.  ->  ( ( z  +P.  w )  =  y  ->  E. x  e.  R.  ( [ <. y ,  z
>. ]  ~R  .R  x
)  =  1R )
) )
9392rexlimdv 2666 . . . 4  |-  ( ( y  e.  P.  /\  z  e.  P. )  ->  ( E. w  e. 
P.  ( z  +P.  w )  =  y  ->  E. x  e.  R.  ( [ <. y ,  z
>. ]  ~R  .R  x
)  =  1R )
)
9412, 93syl5 28 . . 3  |-  ( ( y  e.  P.  /\  z  e.  P. )  ->  ( 0R  <R  [ <. y ,  z >. ]  ~R  ->  E. x  e.  R.  ( [ <. y ,  z
>. ]  ~R  .R  x
)  =  1R )
)
954, 9, 94ecoptocl 6748 . 2  |-  ( A  e.  R.  ->  ( 0R  <R  A  ->  E. x  e.  R.  ( A  .R  x )  =  1R ) )
963, 95mpcom 32 1  |-  ( 0R 
<R  A  ->  E. x  e.  R.  ( A  .R  x )  =  1R )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   E.wrex 2544   <.cop 3643   class class class wbr 4023  (class class class)co 5858   [cec 6658   P.cnp 8481   1Pc1p 8482    +P. cpp 8483    .P. cmp 8484    <P cltp 8485    ~R cer 8488   R.cnr 8489   0Rc0r 8490   1Rc1r 8491    .R cmr 8494    <R cltr 8495
This theorem is referenced by:  recexsr  8729
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-omul 6484  df-er 6660  df-ec 6662  df-qs 6666  df-ni 8496  df-pli 8497  df-mi 8498  df-lti 8499  df-plpq 8532  df-mpq 8533  df-ltpq 8534  df-enq 8535  df-nq 8536  df-erq 8537  df-plq 8538  df-mq 8539  df-1nq 8540  df-rq 8541  df-ltnq 8542  df-np 8605  df-1p 8606  df-plp 8607  df-mp 8608  df-ltp 8609  df-mpr 8680  df-enr 8681  df-nr 8682  df-mr 8684  df-ltr 8685  df-0r 8686  df-1r 8687
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