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Theorem recexsrlem 9545
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 9510 . . . 4  |-  <R  C_  ( R.  X.  R. )
21brel 4888 . . 3  |-  ( 0R 
<R  A  ->  ( 0R  e.  R.  /\  A  e.  R. ) )
32simprd 470 . 2  |-  ( 0R 
<R  A  ->  A  e. 
R. )
4 df-nr 9499 . . 3  |-  R.  =  ( ( P.  X.  P. ) /.  ~R  )
5 breq2 4399 . . . 4  |-  ( [
<. y ,  z >. ]  ~R  =  A  -> 
( 0R  <R  [ <. y ,  z >. ]  ~R  <->  0R 
<R  A ) )
6 oveq1 6315 . . . . . 6  |-  ( [
<. y ,  z >. ]  ~R  =  A  -> 
( [ <. y ,  z >. ]  ~R  .R  x )  =  ( A  .R  x ) )
76eqeq1d 2473 . . . . 5  |-  ( [
<. y ,  z >. ]  ~R  =  A  -> 
( ( [ <. y ,  z >. ]  ~R  .R  x )  =  1R  <->  ( A  .R  x )  =  1R ) )
87rexbidv 2892 . . . 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 327 . . 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 9520 . . . . 5  |-  ( 0R 
<R  [ <. y ,  z
>. ]  ~R  <->  z  <P  y )
11 ltexpri 9486 . . . . 5  |-  ( z 
<P  y  ->  E. w  e.  P.  ( z  +P.  w )  =  y )
1210, 11sylbi 200 . . . 4  |-  ( 0R 
<R  [ <. y ,  z
>. ]  ~R  ->  E. w  e.  P.  ( z  +P.  w )  =  y )
13 recexpr 9494 . . . . . 6  |-  ( w  e.  P.  ->  E. v  e.  P.  ( w  .P.  v )  =  1P )
14 1pr 9458 . . . . . . . . . . . 12  |-  1P  e.  P.
15 addclpr 9461 . . . . . . . . . . . 12  |-  ( ( v  e.  P.  /\  1P  e.  P. )  -> 
( v  +P.  1P )  e.  P. )
1614, 15mpan2 685 . . . . . . . . . . 11  |-  ( v  e.  P.  ->  (
v  +P.  1P )  e.  P. )
17 enrex 9509 . . . . . . . . . . . 12  |-  ~R  e.  _V
1817, 4ecopqsi 7438 . . . . . . . . . . 11  |-  ( ( ( v  +P.  1P )  e.  P.  /\  1P  e.  P. )  ->  [ <. ( v  +P.  1P ) ,  1P >. ]  ~R  e.  R. )
1916, 14, 18sylancl 675 . . . . . . . . . 10  |-  ( v  e.  P.  ->  [ <. ( v  +P.  1P ) ,  1P >. ]  ~R  e.  R. )
2019ad2antlr 741 . . . . . . . . 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 547 . . . . . . . . . . . . . 14  |-  ( v  e.  P.  ->  (
( v  +P.  1P )  e.  P.  /\  1P  e.  P. ) )
2221anim2i 579 . . . . . . . . . . . . 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 472 . . . . . . . . . . . 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 9518 . . . . . . . . . . . 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 17 . . . . . . . . . . 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 6315 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( z  +P.  w )  =  y  ->  (
( z  +P.  w
)  .P.  v )  =  ( y  .P.  v ) )
2726eqcomd 2477 . . . . . . . . . . . . . . . . . . 19  |-  ( ( z  +P.  w )  =  y  ->  (
y  .P.  v )  =  ( ( z  +P.  w )  .P.  v ) )
28 vex 3034 . . . . . . . . . . . . . . . . . . . . 21  |-  z  e. 
_V
29 vex 3034 . . . . . . . . . . . . . . . . . . . . 21  |-  w  e. 
_V
30 vex 3034 . . . . . . . . . . . . . . . . . . . . 21  |-  v  e. 
_V
31 mulcompr 9466 . . . . . . . . . . . . . . . . . . . . 21  |-  ( u  .P.  f )  =  ( f  .P.  u
)
32 distrpr 9471 . . . . . . . . . . . . . . . . . . . . 21  |-  ( u  .P.  ( f  +P.  x ) )  =  ( ( u  .P.  f )  +P.  (
u  .P.  x )
)
3328, 29, 30, 31, 32caovdir 6522 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( z  +P.  w )  .P.  v )  =  ( ( z  .P.  v )  +P.  (
w  .P.  v )
)
34 oveq2 6316 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( w  .P.  v )  =  1P  ->  (
( z  .P.  v
)  +P.  ( w  .P.  v ) )  =  ( ( z  .P.  v )  +P.  1P ) )
3533, 34syl5eq 2517 . . . . . . . . . . . . . . . . . . 19  |-  ( ( w  .P.  v )  =  1P  ->  (
( z  +P.  w
)  .P.  v )  =  ( ( z  .P.  v )  +P. 
1P ) )
3627, 35sylan9eqr 2527 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( w  .P.  v
)  =  1P  /\  ( z  +P.  w
)  =  y )  ->  ( y  .P.  v )  =  ( ( z  .P.  v
)  +P.  1P )
)
3736oveq1d 6323 . . . . . . . . . . . . . . . . 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 6336 . . . . . . . . . . . . . . . . . 18  |-  ( z  .P.  v )  e. 
_V
3914elexi 3041 . . . . . . . . . . . . . . . . . 18  |-  1P  e.  _V
40 ovex 6336 . . . . . . . . . . . . . . . . . 18  |-  ( ( y  .P.  1P )  +P.  ( z  .P. 
1P ) )  e. 
_V
41 addcompr 9464 . . . . . . . . . . . . . . . . . 18  |-  ( u  +P.  f )  =  ( f  +P.  u
)
42 addasspr 9465 . . . . . . . . . . . . . . . . . 18  |-  ( ( u  +P.  f )  +P.  x )  =  ( u  +P.  (
f  +P.  x )
)
4338, 39, 40, 41, 42caov32 6515 . . . . . . . . . . . . . . . . 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 2521 . . . . . . . . . . . . . . . 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 6323 . . . . . . . . . . . . . . 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 9465 . . . . . . . . . . . . . . 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 2521 . . . . . . . . . . . . . 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 9471 . . . . . . . . . . . . . . . . 17  |-  ( y  .P.  ( v  +P. 
1P ) )  =  ( ( y  .P.  v )  +P.  (
y  .P.  1P )
)
4948oveq1i 6318 . . . . . . . . . . . . . . . 16  |-  ( ( y  .P.  ( v  +P.  1P ) )  +P.  ( z  .P. 
1P ) )  =  ( ( ( y  .P.  v )  +P.  ( y  .P.  1P ) )  +P.  (
z  .P.  1P )
)
50 addasspr 9465 . . . . . . . . . . . . . . . 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 2493 . . . . . . . . . . . . . . 15  |-  ( ( y  .P.  ( v  +P.  1P ) )  +P.  ( z  .P. 
1P ) )  =  ( ( y  .P.  v )  +P.  (
( y  .P.  1P )  +P.  ( z  .P. 
1P ) ) )
5251oveq1i 6318 . . . . . . . . . . . . . 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 9471 . . . . . . . . . . . . . . . . 17  |-  ( z  .P.  ( v  +P. 
1P ) )  =  ( ( z  .P.  v )  +P.  (
z  .P.  1P )
)
5453oveq2i 6319 . . . . . . . . . . . . . . . 16  |-  ( ( y  .P.  1P )  +P.  ( z  .P.  ( v  +P.  1P ) ) )  =  ( ( y  .P. 
1P )  +P.  (
( z  .P.  v
)  +P.  ( z  .P.  1P ) ) )
55 ovex 6336 . . . . . . . . . . . . . . . . 17  |-  ( y  .P.  1P )  e. 
_V
56 ovex 6336 . . . . . . . . . . . . . . . . 17  |-  ( z  .P.  1P )  e. 
_V
5755, 38, 56, 41, 42caov12 6516 . . . . . . . . . . . . . . . 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 2493 . . . . . . . . . . . . . . 15  |-  ( ( y  .P.  1P )  +P.  ( z  .P.  ( v  +P.  1P ) ) )  =  ( ( z  .P.  v )  +P.  (
( y  .P.  1P )  +P.  ( z  .P. 
1P ) ) )
5958oveq1i 6318 . . . . . . . . . . . . . 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 2530 . . . . . . . . . . . . 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 9463 . . . . . . . . . . . . . . . . . 18  |-  ( ( y  e.  P.  /\  ( v  +P.  1P )  e.  P. )  ->  ( y  .P.  (
v  +P.  1P )
)  e.  P. )
6216, 61sylan2 482 . . . . . . . . . . . . . . . . 17  |-  ( ( y  e.  P.  /\  v  e.  P. )  ->  ( y  .P.  (
v  +P.  1P )
)  e.  P. )
63 mulclpr 9463 . . . . . . . . . . . . . . . . . 18  |-  ( ( z  e.  P.  /\  1P  e.  P. )  -> 
( z  .P.  1P )  e.  P. )
6414, 63mpan2 685 . . . . . . . . . . . . . . . . 17  |-  ( z  e.  P.  ->  (
z  .P.  1P )  e.  P. )
65 addclpr 9461 . . . . . . . . . . . . . . . . 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 485 . . . . . . . . . . . . . . . 16  |-  ( ( ( y  e.  P.  /\  v  e.  P. )  /\  z  e.  P. )  ->  ( ( y  .P.  ( v  +P. 
1P ) )  +P.  ( z  .P.  1P ) )  e.  P. )
6766an32s 821 . . . . . . . . . . . . . . 15  |-  ( ( ( y  e.  P.  /\  z  e.  P. )  /\  v  e.  P. )  ->  ( ( y  .P.  ( v  +P. 
1P ) )  +P.  ( z  .P.  1P ) )  e.  P. )
68 mulclpr 9463 . . . . . . . . . . . . . . . . . 18  |-  ( ( y  e.  P.  /\  1P  e.  P. )  -> 
( y  .P.  1P )  e.  P. )
6914, 68mpan2 685 . . . . . . . . . . . . . . . . 17  |-  ( y  e.  P.  ->  (
y  .P.  1P )  e.  P. )
70 mulclpr 9463 . . . . . . . . . . . . . . . . . 18  |-  ( ( z  e.  P.  /\  ( v  +P.  1P )  e.  P. )  ->  ( z  .P.  (
v  +P.  1P )
)  e.  P. )
7116, 70sylan2 482 . . . . . . . . . . . . . . . . 17  |-  ( ( z  e.  P.  /\  v  e.  P. )  ->  ( z  .P.  (
v  +P.  1P )
)  e.  P. )
72 addclpr 9461 . . . . . . . . . . . . . . . . 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 485 . . . . . . . . . . . . . . . 16  |-  ( ( y  e.  P.  /\  ( z  e.  P.  /\  v  e.  P. )
)  ->  ( (
y  .P.  1P )  +P.  ( z  .P.  (
v  +P.  1P )
) )  e.  P. )
7473anassrs 660 . . . . . . . . . . . . . . 15  |-  ( ( ( y  e.  P.  /\  z  e.  P. )  /\  v  e.  P. )  ->  ( ( y  .P.  1P )  +P.  ( z  .P.  (
v  +P.  1P )
) )  e.  P. )
7567, 74jca 541 . . . . . . . . . . . . . 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 9461 . . . . . . . . . . . . . . . 16  |-  ( ( 1P  e.  P.  /\  1P  e.  P. )  -> 
( 1P  +P.  1P )  e.  P. )
7714, 14, 76mp2an 686 . . . . . . . . . . . . . . 15  |-  ( 1P 
+P.  1P )  e.  P.
7877, 14pm3.2i 462 . . . . . . . . . . . . . 14  |-  ( ( 1P  +P.  1P )  e.  P.  /\  1P  e.  P. )
79 enreceq 9508 . . . . . . . . . . . . . 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 675 . . . . . . . . . . . . 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 229 . . . . . . . . . . . 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 436 . . . . . . . . . . 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 2505 . . . . . . . . . 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 9504 . . . . . . . . . 10  |-  1R  =  [ <. ( 1P  +P.  1P ) ,  1P >. ]  ~R
8583, 84syl6eqr 2523 . . . . . . . . 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 6316 . . . . . . . . . . 11  |-  ( x  =  [ <. (
v  +P.  1P ) ,  1P >. ]  ~R  ->  ( [ <. y ,  z
>. ]  ~R  .R  x
)  =  ( [
<. y ,  z >. ]  ~R  .R  [ <. ( v  +P.  1P ) ,  1P >. ]  ~R  ) )
8786eqeq1d 2473 . . . . . . . . . 10  |-  ( x  =  [ <. (
v  +P.  1P ) ,  1P >. ]  ~R  ->  ( ( [ <. y ,  z >. ]  ~R  .R  x )  =  1R  <->  ( [ <. y ,  z
>. ]  ~R  .R  [ <. ( v  +P.  1P ) ,  1P >. ]  ~R  )  =  1R )
)
8887rspcev 3136 . . . . . . . . 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 673 . . . . . . . 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 623 . . . . . . 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 2870 . . . . . 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 2870 . . . 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 7471 . 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 189    /\ wa 376    = wceq 1452    e. wcel 1904   E.wrex 2757   <.cop 3965   class class class wbr 4395  (class class class)co 6308   [cec 7379   P.cnp 9302   1Pc1p 9303    +P. cpp 9304    .P. cmp 9305    <P cltp 9306    ~R cer 9307   R.cnr 9308   0Rc0r 9309   1Rc1r 9310    .R cmr 9313    <R cltr 9314
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-omul 7205  df-er 7381  df-ec 7383  df-qs 7387  df-ni 9315  df-pli 9316  df-mi 9317  df-lti 9318  df-plpq 9351  df-mpq 9352  df-ltpq 9353  df-enq 9354  df-nq 9355  df-erq 9356  df-plq 9357  df-mq 9358  df-1nq 9359  df-rq 9360  df-ltnq 9361  df-np 9424  df-1p 9425  df-plp 9426  df-mp 9427  df-ltp 9428  df-enr 9498  df-nr 9499  df-mr 9501  df-ltr 9502  df-0r 9503  df-1r 9504
This theorem is referenced by:  recexsr  9549
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