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Theorem recexsrlem 9266
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 9234 . . . 4  |-  <R  C_  ( R.  X.  R. )
21brel 4883 . . 3  |-  ( 0R 
<R  A  ->  ( 0R  e.  R.  /\  A  e.  R. ) )
32simprd 460 . 2  |-  ( 0R 
<R  A  ->  A  e. 
R. )
4 df-nr 9223 . . 3  |-  R.  =  ( ( P.  X.  P. ) /.  ~R  )
5 breq2 4293 . . . 4  |-  ( [
<. y ,  z >. ]  ~R  =  A  -> 
( 0R  <R  [ <. y ,  z >. ]  ~R  <->  0R 
<R  A ) )
6 oveq1 6097 . . . . . 6  |-  ( [
<. y ,  z >. ]  ~R  =  A  -> 
( [ <. y ,  z >. ]  ~R  .R  x )  =  ( A  .R  x ) )
76eqeq1d 2449 . . . . 5  |-  ( [
<. y ,  z >. ]  ~R  =  A  -> 
( ( [ <. y ,  z >. ]  ~R  .R  x )  =  1R  <->  ( A  .R  x )  =  1R ) )
87rexbidv 2734 . . . 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 320 . . 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 9241 . . . . 5  |-  ( 0R 
<R  [ <. y ,  z
>. ]  ~R  <->  z  <P  y )
11 ltexpri 9208 . . . . 5  |-  ( z 
<P  y  ->  E. w  e.  P.  ( z  +P.  w )  =  y )
1210, 11sylbi 195 . . . 4  |-  ( 0R 
<R  [ <. y ,  z
>. ]  ~R  ->  E. w  e.  P.  ( z  +P.  w )  =  y )
13 recexpr 9216 . . . . . 6  |-  ( w  e.  P.  ->  E. v  e.  P.  ( w  .P.  v )  =  1P )
14 1pr 9180 . . . . . . . . . . . 12  |-  1P  e.  P.
15 addclpr 9183 . . . . . . . . . . . 12  |-  ( ( v  e.  P.  /\  1P  e.  P. )  -> 
( v  +P.  1P )  e.  P. )
1614, 15mpan2 666 . . . . . . . . . . 11  |-  ( v  e.  P.  ->  (
v  +P.  1P )  e.  P. )
17 enrex 9233 . . . . . . . . . . . 12  |-  ~R  e.  _V
1817, 4ecopqsi 7153 . . . . . . . . . . 11  |-  ( ( ( v  +P.  1P )  e.  P.  /\  1P  e.  P. )  ->  [ <. ( v  +P.  1P ) ,  1P >. ]  ~R  e.  R. )
1916, 14, 18sylancl 657 . . . . . . . . . 10  |-  ( v  e.  P.  ->  [ <. ( v  +P.  1P ) ,  1P >. ]  ~R  e.  R. )
2019ad2antlr 721 . . . . . . . . 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 535 . . . . . . . . . . . . . 14  |-  ( v  e.  P.  ->  (
( v  +P.  1P )  e.  P.  /\  1P  e.  P. ) )
2221anim2i 566 . . . . . . . . . . . . 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 462 . . . . . . . . . . . 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 9239 . . . . . . . . . . . 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 16 . . . . . . . . . . 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 6097 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( z  +P.  w )  =  y  ->  (
( z  +P.  w
)  .P.  v )  =  ( y  .P.  v ) )
2726eqcomd 2446 . . . . . . . . . . . . . . . . . . 19  |-  ( ( z  +P.  w )  =  y  ->  (
y  .P.  v )  =  ( ( z  +P.  w )  .P.  v ) )
28 vex 2973 . . . . . . . . . . . . . . . . . . . . 21  |-  z  e. 
_V
29 vex 2973 . . . . . . . . . . . . . . . . . . . . 21  |-  w  e. 
_V
30 vex 2973 . . . . . . . . . . . . . . . . . . . . 21  |-  v  e. 
_V
31 mulcompr 9188 . . . . . . . . . . . . . . . . . . . . 21  |-  ( u  .P.  f )  =  ( f  .P.  u
)
32 distrpr 9193 . . . . . . . . . . . . . . . . . . . . 21  |-  ( u  .P.  ( f  +P.  x ) )  =  ( ( u  .P.  f )  +P.  (
u  .P.  x )
)
3328, 29, 30, 31, 32caovdir 6296 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( z  +P.  w )  .P.  v )  =  ( ( z  .P.  v )  +P.  (
w  .P.  v )
)
34 oveq2 6098 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( w  .P.  v )  =  1P  ->  (
( z  .P.  v
)  +P.  ( w  .P.  v ) )  =  ( ( z  .P.  v )  +P.  1P ) )
3533, 34syl5eq 2485 . . . . . . . . . . . . . . . . . . 19  |-  ( ( w  .P.  v )  =  1P  ->  (
( z  +P.  w
)  .P.  v )  =  ( ( z  .P.  v )  +P. 
1P ) )
3627, 35sylan9eqr 2495 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( w  .P.  v
)  =  1P  /\  ( z  +P.  w
)  =  y )  ->  ( y  .P.  v )  =  ( ( z  .P.  v
)  +P.  1P )
)
3736oveq1d 6105 . . . . . . . . . . . . . . . . 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 6115 . . . . . . . . . . . . . . . . . 18  |-  ( z  .P.  v )  e. 
_V
3914elexi 2980 . . . . . . . . . . . . . . . . . 18  |-  1P  e.  _V
40 ovex 6115 . . . . . . . . . . . . . . . . . 18  |-  ( ( y  .P.  1P )  +P.  ( z  .P. 
1P ) )  e. 
_V
41 addcompr 9186 . . . . . . . . . . . . . . . . . 18  |-  ( u  +P.  f )  =  ( f  +P.  u
)
42 addasspr 9187 . . . . . . . . . . . . . . . . . 18  |-  ( ( u  +P.  f )  +P.  x )  =  ( u  +P.  (
f  +P.  x )
)
4338, 39, 40, 41, 42caov32 6289 . . . . . . . . . . . . . . . . 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 2489 . . . . . . . . . . . . . . . 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 6105 . . . . . . . . . . . . . . 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 9187 . . . . . . . . . . . . . . 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 2489 . . . . . . . . . . . . . 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 9193 . . . . . . . . . . . . . . . . 17  |-  ( y  .P.  ( v  +P. 
1P ) )  =  ( ( y  .P.  v )  +P.  (
y  .P.  1P )
)
4948oveq1i 6100 . . . . . . . . . . . . . . . 16  |-  ( ( y  .P.  ( v  +P.  1P ) )  +P.  ( z  .P. 
1P ) )  =  ( ( ( y  .P.  v )  +P.  ( y  .P.  1P ) )  +P.  (
z  .P.  1P )
)
50 addasspr 9187 . . . . . . . . . . . . . . . 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 2461 . . . . . . . . . . . . . . 15  |-  ( ( y  .P.  ( v  +P.  1P ) )  +P.  ( z  .P. 
1P ) )  =  ( ( y  .P.  v )  +P.  (
( y  .P.  1P )  +P.  ( z  .P. 
1P ) ) )
5251oveq1i 6100 . . . . . . . . . . . . . 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 9193 . . . . . . . . . . . . . . . . 17  |-  ( z  .P.  ( v  +P. 
1P ) )  =  ( ( z  .P.  v )  +P.  (
z  .P.  1P )
)
5453oveq2i 6101 . . . . . . . . . . . . . . . 16  |-  ( ( y  .P.  1P )  +P.  ( z  .P.  ( v  +P.  1P ) ) )  =  ( ( y  .P. 
1P )  +P.  (
( z  .P.  v
)  +P.  ( z  .P.  1P ) ) )
55 ovex 6115 . . . . . . . . . . . . . . . . 17  |-  ( y  .P.  1P )  e. 
_V
56 ovex 6115 . . . . . . . . . . . . . . . . 17  |-  ( z  .P.  1P )  e. 
_V
5755, 38, 56, 41, 42caov12 6290 . . . . . . . . . . . . . . . 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 2461 . . . . . . . . . . . . . . 15  |-  ( ( y  .P.  1P )  +P.  ( z  .P.  ( v  +P.  1P ) ) )  =  ( ( z  .P.  v )  +P.  (
( y  .P.  1P )  +P.  ( z  .P. 
1P ) ) )
5958oveq1i 6100 . . . . . . . . . . . . . 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 2498 . . . . . . . . . . . . 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 9185 . . . . . . . . . . . . . . . . . 18  |-  ( ( y  e.  P.  /\  ( v  +P.  1P )  e.  P. )  ->  ( y  .P.  (
v  +P.  1P )
)  e.  P. )
6216, 61sylan2 471 . . . . . . . . . . . . . . . . 17  |-  ( ( y  e.  P.  /\  v  e.  P. )  ->  ( y  .P.  (
v  +P.  1P )
)  e.  P. )
63 mulclpr 9185 . . . . . . . . . . . . . . . . . 18  |-  ( ( z  e.  P.  /\  1P  e.  P. )  -> 
( z  .P.  1P )  e.  P. )
6414, 63mpan2 666 . . . . . . . . . . . . . . . . 17  |-  ( z  e.  P.  ->  (
z  .P.  1P )  e.  P. )
65 addclpr 9183 . . . . . . . . . . . . . . . . 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 474 . . . . . . . . . . . . . . . 16  |-  ( ( ( y  e.  P.  /\  v  e.  P. )  /\  z  e.  P. )  ->  ( ( y  .P.  ( v  +P. 
1P ) )  +P.  ( z  .P.  1P ) )  e.  P. )
6766an32s 797 . . . . . . . . . . . . . . 15  |-  ( ( ( y  e.  P.  /\  z  e.  P. )  /\  v  e.  P. )  ->  ( ( y  .P.  ( v  +P. 
1P ) )  +P.  ( z  .P.  1P ) )  e.  P. )
68 mulclpr 9185 . . . . . . . . . . . . . . . . . 18  |-  ( ( y  e.  P.  /\  1P  e.  P. )  -> 
( y  .P.  1P )  e.  P. )
6914, 68mpan2 666 . . . . . . . . . . . . . . . . 17  |-  ( y  e.  P.  ->  (
y  .P.  1P )  e.  P. )
70 mulclpr 9185 . . . . . . . . . . . . . . . . . 18  |-  ( ( z  e.  P.  /\  ( v  +P.  1P )  e.  P. )  ->  ( z  .P.  (
v  +P.  1P )
)  e.  P. )
7116, 70sylan2 471 . . . . . . . . . . . . . . . . 17  |-  ( ( z  e.  P.  /\  v  e.  P. )  ->  ( z  .P.  (
v  +P.  1P )
)  e.  P. )
72 addclpr 9183 . . . . . . . . . . . . . . . . 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 474 . . . . . . . . . . . . . . . 16  |-  ( ( y  e.  P.  /\  ( z  e.  P.  /\  v  e.  P. )
)  ->  ( (
y  .P.  1P )  +P.  ( z  .P.  (
v  +P.  1P )
) )  e.  P. )
7473anassrs 643 . . . . . . . . . . . . . . 15  |-  ( ( ( y  e.  P.  /\  z  e.  P. )  /\  v  e.  P. )  ->  ( ( y  .P.  1P )  +P.  ( z  .P.  (
v  +P.  1P )
) )  e.  P. )
7567, 74jca 529 . . . . . . . . . . . . . 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 9183 . . . . . . . . . . . . . . . 16  |-  ( ( 1P  e.  P.  /\  1P  e.  P. )  -> 
( 1P  +P.  1P )  e.  P. )
7714, 14, 76mp2an 667 . . . . . . . . . . . . . . 15  |-  ( 1P 
+P.  1P )  e.  P.
7877, 14pm3.2i 452 . . . . . . . . . . . . . 14  |-  ( ( 1P  +P.  1P )  e.  P.  /\  1P  e.  P. )
79 enreceq 9232 . . . . . . . . . . . . . 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 657 . . . . . . . . . . . . 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 221 . . . . . . . . . . . 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 429 . . . . . . . . . . 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 2473 . . . . . . . . . 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 9228 . . . . . . . . . 10  |-  1R  =  [ <. ( 1P  +P.  1P ) ,  1P >. ]  ~R
8583, 84syl6eqr 2491 . . . . . . . . 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 6098 . . . . . . . . . . 11  |-  ( x  =  [ <. (
v  +P.  1P ) ,  1P >. ]  ~R  ->  ( [ <. y ,  z
>. ]  ~R  .R  x
)  =  ( [
<. y ,  z >. ]  ~R  .R  [ <. ( v  +P.  1P ) ,  1P >. ]  ~R  ) )
8786eqeq1d 2449 . . . . . . . . . 10  |-  ( x  =  [ <. (
v  +P.  1P ) ,  1P >. ]  ~R  ->  ( ( [ <. y ,  z >. ]  ~R  .R  x )  =  1R  <->  ( [ <. y ,  z
>. ]  ~R  .R  [ <. ( v  +P.  1P ) ,  1P >. ]  ~R  )  =  1R )
)
8887rspcev 3070 . . . . . . . . 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 656 . . . . . . . 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 609 . . . . . . 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 2838 . . . . . 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 32 . . . . 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 2838 . . . 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 32 . . 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 7186 . 2  |-  ( A  e.  R.  ->  ( 0R  <R  A  ->  E. x  e.  R.  ( A  .R  x )  =  1R ) )
963, 95mpcom 36 1  |-  ( 0R 
<R  A  ->  E. x  e.  R.  ( A  .R  x )  =  1R )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1364    e. wcel 1761   E.wrex 2714   <.cop 3880   class class class wbr 4289  (class class class)co 6090   [cec 7095   P.cnp 9022   1Pc1p 9023    +P. cpp 9024    .P. cmp 9025    <P cltp 9026    ~R cer 9029   R.cnr 9030   0Rc0r 9031   1Rc1r 9032    .R cmr 9035    <R cltr 9036
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-inf2 7843
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-1o 6916  df-oadd 6920  df-omul 6921  df-er 7097  df-ec 7099  df-qs 7103  df-ni 9037  df-pli 9038  df-mi 9039  df-lti 9040  df-plpq 9073  df-mpq 9074  df-ltpq 9075  df-enq 9076  df-nq 9077  df-erq 9078  df-plq 9079  df-mq 9080  df-1nq 9081  df-rq 9082  df-ltnq 9083  df-np 9146  df-1p 9147  df-plp 9148  df-mp 9149  df-ltp 9150  df-mpr 9221  df-enr 9222  df-nr 9223  df-mr 9225  df-ltr 9226  df-0r 9227  df-1r 9228
This theorem is referenced by:  recexsr  9270
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