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Theorem recexsrlem 9483
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 9448 . . . 4  |-  <R  C_  ( R.  X.  R. )
21brel 5038 . . 3  |-  ( 0R 
<R  A  ->  ( 0R  e.  R.  /\  A  e.  R. ) )
32simprd 463 . 2  |-  ( 0R 
<R  A  ->  A  e. 
R. )
4 df-nr 9437 . . 3  |-  R.  =  ( ( P.  X.  P. ) /.  ~R  )
5 breq2 4441 . . . 4  |-  ( [
<. y ,  z >. ]  ~R  =  A  -> 
( 0R  <R  [ <. y ,  z >. ]  ~R  <->  0R 
<R  A ) )
6 oveq1 6288 . . . . . 6  |-  ( [
<. y ,  z >. ]  ~R  =  A  -> 
( [ <. y ,  z >. ]  ~R  .R  x )  =  ( A  .R  x ) )
76eqeq1d 2445 . . . . 5  |-  ( [
<. y ,  z >. ]  ~R  =  A  -> 
( ( [ <. y ,  z >. ]  ~R  .R  x )  =  1R  <->  ( A  .R  x )  =  1R ) )
87rexbidv 2954 . . . 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 9458 . . . . 5  |-  ( 0R 
<R  [ <. y ,  z
>. ]  ~R  <->  z  <P  y )
11 ltexpri 9424 . . . . 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 9432 . . . . . 6  |-  ( w  e.  P.  ->  E. v  e.  P.  ( w  .P.  v )  =  1P )
14 1pr 9396 . . . . . . . . . . . 12  |-  1P  e.  P.
15 addclpr 9399 . . . . . . . . . . . 12  |-  ( ( v  e.  P.  /\  1P  e.  P. )  -> 
( v  +P.  1P )  e.  P. )
1614, 15mpan2 671 . . . . . . . . . . 11  |-  ( v  e.  P.  ->  (
v  +P.  1P )  e.  P. )
17 enrex 9447 . . . . . . . . . . . 12  |-  ~R  e.  _V
1817, 4ecopqsi 7370 . . . . . . . . . . 11  |-  ( ( ( v  +P.  1P )  e.  P.  /\  1P  e.  P. )  ->  [ <. ( v  +P.  1P ) ,  1P >. ]  ~R  e.  R. )
1916, 14, 18sylancl 662 . . . . . . . . . 10  |-  ( v  e.  P.  ->  [ <. ( v  +P.  1P ) ,  1P >. ]  ~R  e.  R. )
2019ad2antlr 726 . . . . . . . . 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 538 . . . . . . . . . . . . . 14  |-  ( v  e.  P.  ->  (
( v  +P.  1P )  e.  P.  /\  1P  e.  P. ) )
2221anim2i 569 . . . . . . . . . . . . 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 465 . . . . . . . . . . . 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 9456 . . . . . . . . . . . 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 6288 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( z  +P.  w )  =  y  ->  (
( z  +P.  w
)  .P.  v )  =  ( y  .P.  v ) )
2726eqcomd 2451 . . . . . . . . . . . . . . . . . . 19  |-  ( ( z  +P.  w )  =  y  ->  (
y  .P.  v )  =  ( ( z  +P.  w )  .P.  v ) )
28 vex 3098 . . . . . . . . . . . . . . . . . . . . 21  |-  z  e. 
_V
29 vex 3098 . . . . . . . . . . . . . . . . . . . . 21  |-  w  e. 
_V
30 vex 3098 . . . . . . . . . . . . . . . . . . . . 21  |-  v  e. 
_V
31 mulcompr 9404 . . . . . . . . . . . . . . . . . . . . 21  |-  ( u  .P.  f )  =  ( f  .P.  u
)
32 distrpr 9409 . . . . . . . . . . . . . . . . . . . . 21  |-  ( u  .P.  ( f  +P.  x ) )  =  ( ( u  .P.  f )  +P.  (
u  .P.  x )
)
3328, 29, 30, 31, 32caovdir 6494 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( z  +P.  w )  .P.  v )  =  ( ( z  .P.  v )  +P.  (
w  .P.  v )
)
34 oveq2 6289 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( w  .P.  v )  =  1P  ->  (
( z  .P.  v
)  +P.  ( w  .P.  v ) )  =  ( ( z  .P.  v )  +P.  1P ) )
3533, 34syl5eq 2496 . . . . . . . . . . . . . . . . . . 19  |-  ( ( w  .P.  v )  =  1P  ->  (
( z  +P.  w
)  .P.  v )  =  ( ( z  .P.  v )  +P. 
1P ) )
3627, 35sylan9eqr 2506 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( w  .P.  v
)  =  1P  /\  ( z  +P.  w
)  =  y )  ->  ( y  .P.  v )  =  ( ( z  .P.  v
)  +P.  1P )
)
3736oveq1d 6296 . . . . . . . . . . . . . . . . 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 6309 . . . . . . . . . . . . . . . . . 18  |-  ( z  .P.  v )  e. 
_V
3914elexi 3105 . . . . . . . . . . . . . . . . . 18  |-  1P  e.  _V
40 ovex 6309 . . . . . . . . . . . . . . . . . 18  |-  ( ( y  .P.  1P )  +P.  ( z  .P. 
1P ) )  e. 
_V
41 addcompr 9402 . . . . . . . . . . . . . . . . . 18  |-  ( u  +P.  f )  =  ( f  +P.  u
)
42 addasspr 9403 . . . . . . . . . . . . . . . . . 18  |-  ( ( u  +P.  f )  +P.  x )  =  ( u  +P.  (
f  +P.  x )
)
4338, 39, 40, 41, 42caov32 6487 . . . . . . . . . . . . . . . . 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 2500 . . . . . . . . . . . . . . . 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 6296 . . . . . . . . . . . . . . 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 9403 . . . . . . . . . . . . . . 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 2500 . . . . . . . . . . . . . 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 9409 . . . . . . . . . . . . . . . . 17  |-  ( y  .P.  ( v  +P. 
1P ) )  =  ( ( y  .P.  v )  +P.  (
y  .P.  1P )
)
4948oveq1i 6291 . . . . . . . . . . . . . . . 16  |-  ( ( y  .P.  ( v  +P.  1P ) )  +P.  ( z  .P. 
1P ) )  =  ( ( ( y  .P.  v )  +P.  ( y  .P.  1P ) )  +P.  (
z  .P.  1P )
)
50 addasspr 9403 . . . . . . . . . . . . . . . 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 2472 . . . . . . . . . . . . . . 15  |-  ( ( y  .P.  ( v  +P.  1P ) )  +P.  ( z  .P. 
1P ) )  =  ( ( y  .P.  v )  +P.  (
( y  .P.  1P )  +P.  ( z  .P. 
1P ) ) )
5251oveq1i 6291 . . . . . . . . . . . . . 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 9409 . . . . . . . . . . . . . . . . 17  |-  ( z  .P.  ( v  +P. 
1P ) )  =  ( ( z  .P.  v )  +P.  (
z  .P.  1P )
)
5453oveq2i 6292 . . . . . . . . . . . . . . . 16  |-  ( ( y  .P.  1P )  +P.  ( z  .P.  ( v  +P.  1P ) ) )  =  ( ( y  .P. 
1P )  +P.  (
( z  .P.  v
)  +P.  ( z  .P.  1P ) ) )
55 ovex 6309 . . . . . . . . . . . . . . . . 17  |-  ( y  .P.  1P )  e. 
_V
56 ovex 6309 . . . . . . . . . . . . . . . . 17  |-  ( z  .P.  1P )  e. 
_V
5755, 38, 56, 41, 42caov12 6488 . . . . . . . . . . . . . . . 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 2472 . . . . . . . . . . . . . . 15  |-  ( ( y  .P.  1P )  +P.  ( z  .P.  ( v  +P.  1P ) ) )  =  ( ( z  .P.  v )  +P.  (
( y  .P.  1P )  +P.  ( z  .P. 
1P ) ) )
5958oveq1i 6291 . . . . . . . . . . . . . 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 2509 . . . . . . . . . . . . 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 9401 . . . . . . . . . . . . . . . . . 18  |-  ( ( y  e.  P.  /\  ( v  +P.  1P )  e.  P. )  ->  ( y  .P.  (
v  +P.  1P )
)  e.  P. )
6216, 61sylan2 474 . . . . . . . . . . . . . . . . 17  |-  ( ( y  e.  P.  /\  v  e.  P. )  ->  ( y  .P.  (
v  +P.  1P )
)  e.  P. )
63 mulclpr 9401 . . . . . . . . . . . . . . . . . 18  |-  ( ( z  e.  P.  /\  1P  e.  P. )  -> 
( z  .P.  1P )  e.  P. )
6414, 63mpan2 671 . . . . . . . . . . . . . . . . 17  |-  ( z  e.  P.  ->  (
z  .P.  1P )  e.  P. )
65 addclpr 9399 . . . . . . . . . . . . . . . . 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 477 . . . . . . . . . . . . . . . 16  |-  ( ( ( y  e.  P.  /\  v  e.  P. )  /\  z  e.  P. )  ->  ( ( y  .P.  ( v  +P. 
1P ) )  +P.  ( z  .P.  1P ) )  e.  P. )
6766an32s 804 . . . . . . . . . . . . . . 15  |-  ( ( ( y  e.  P.  /\  z  e.  P. )  /\  v  e.  P. )  ->  ( ( y  .P.  ( v  +P. 
1P ) )  +P.  ( z  .P.  1P ) )  e.  P. )
68 mulclpr 9401 . . . . . . . . . . . . . . . . . 18  |-  ( ( y  e.  P.  /\  1P  e.  P. )  -> 
( y  .P.  1P )  e.  P. )
6914, 68mpan2 671 . . . . . . . . . . . . . . . . 17  |-  ( y  e.  P.  ->  (
y  .P.  1P )  e.  P. )
70 mulclpr 9401 . . . . . . . . . . . . . . . . . 18  |-  ( ( z  e.  P.  /\  ( v  +P.  1P )  e.  P. )  ->  ( z  .P.  (
v  +P.  1P )
)  e.  P. )
7116, 70sylan2 474 . . . . . . . . . . . . . . . . 17  |-  ( ( z  e.  P.  /\  v  e.  P. )  ->  ( z  .P.  (
v  +P.  1P )
)  e.  P. )
72 addclpr 9399 . . . . . . . . . . . . . . . . 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 477 . . . . . . . . . . . . . . . 16  |-  ( ( y  e.  P.  /\  ( z  e.  P.  /\  v  e.  P. )
)  ->  ( (
y  .P.  1P )  +P.  ( z  .P.  (
v  +P.  1P )
) )  e.  P. )
7473anassrs 648 . . . . . . . . . . . . . . 15  |-  ( ( ( y  e.  P.  /\  z  e.  P. )  /\  v  e.  P. )  ->  ( ( y  .P.  1P )  +P.  ( z  .P.  (
v  +P.  1P )
) )  e.  P. )
7567, 74jca 532 . . . . . . . . . . . . . 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 9399 . . . . . . . . . . . . . . . 16  |-  ( ( 1P  e.  P.  /\  1P  e.  P. )  -> 
( 1P  +P.  1P )  e.  P. )
7714, 14, 76mp2an 672 . . . . . . . . . . . . . . 15  |-  ( 1P 
+P.  1P )  e.  P.
7877, 14pm3.2i 455 . . . . . . . . . . . . . 14  |-  ( ( 1P  +P.  1P )  e.  P.  /\  1P  e.  P. )
79 enreceq 9446 . . . . . . . . . . . . . 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 662 . . . . . . . . . . . . 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 2484 . . . . . . . . . 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 9442 . . . . . . . . . 10  |-  1R  =  [ <. ( 1P  +P.  1P ) ,  1P >. ]  ~R
8583, 84syl6eqr 2502 . . . . . . . . 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 6289 . . . . . . . . . . 11  |-  ( x  =  [ <. (
v  +P.  1P ) ,  1P >. ]  ~R  ->  ( [ <. y ,  z
>. ]  ~R  .R  x
)  =  ( [
<. y ,  z >. ]  ~R  .R  [ <. ( v  +P.  1P ) ,  1P >. ]  ~R  ) )
8786eqeq1d 2445 . . . . . . . . . 10  |-  ( x  =  [ <. (
v  +P.  1P ) ,  1P >. ]  ~R  ->  ( ( [ <. y ,  z >. ]  ~R  .R  x )  =  1R  <->  ( [ <. y ,  z
>. ]  ~R  .R  [ <. ( v  +P.  1P ) ,  1P >. ]  ~R  )  =  1R )
)
8887rspcev 3196 . . . . . . . . 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 661 . . . . . . . 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 612 . . . . . . 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 2933 . . . . . 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 2933 . . . 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 7403 . 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 1383    e. wcel 1804   E.wrex 2794   <.cop 4020   class class class wbr 4437  (class class class)co 6281   [cec 7311   P.cnp 9240   1Pc1p 9241    +P. cpp 9242    .P. cmp 9243    <P cltp 9244    ~R cer 9245   R.cnr 9246   0Rc0r 9247   1Rc1r 9248    .R cmr 9251    <R cltr 9252
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-inf2 8061
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7044  df-rdg 7078  df-1o 7132  df-oadd 7136  df-omul 7137  df-er 7313  df-ec 7315  df-qs 7319  df-ni 9253  df-pli 9254  df-mi 9255  df-lti 9256  df-plpq 9289  df-mpq 9290  df-ltpq 9291  df-enq 9292  df-nq 9293  df-erq 9294  df-plq 9295  df-mq 9296  df-1nq 9297  df-rq 9298  df-ltnq 9299  df-np 9362  df-1p 9363  df-plp 9364  df-mp 9365  df-ltp 9366  df-enr 9436  df-nr 9437  df-mr 9439  df-ltr 9440  df-0r 9441  df-1r 9442
This theorem is referenced by:  recexsr  9487
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