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Theorem recexpr 7195
Description: The reciprocal of a positive real exists. Part of Proposition 9-3.7(v) of [Gleason] p. 124. (Contributed by NM, 15-May-1996.) (Revised by Mario Carneiro, 12-Jun-2013.)
Assertion
Ref Expression
recexpr  |-  ( A  e.  P.  ->  E. x  e.  P.  ( A  .P.  x )  =  1P )
Distinct variable group:    x, A

Proof of Theorem recexpr
Dummy variables  u  v  w  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 breq12 3850 . . . . . . 7  |-  ( ( z  =  u  /\  w  =  v )  ->  ( z  <Q  w  <->  u 
<Q  v ) )
2 simpr 108 . . . . . . . . 9  |-  ( ( z  =  u  /\  w  =  v )  ->  w  =  v )
32fveq2d 5309 . . . . . . . 8  |-  ( ( z  =  u  /\  w  =  v )  ->  ( *Q `  w
)  =  ( *Q
`  v ) )
43eleq1d 2156 . . . . . . 7  |-  ( ( z  =  u  /\  w  =  v )  ->  ( ( *Q `  w )  e.  ( 2nd `  A )  <-> 
( *Q `  v
)  e.  ( 2nd `  A ) ) )
51, 4anbi12d 457 . . . . . 6  |-  ( ( z  =  u  /\  w  =  v )  ->  ( ( z  <Q  w  /\  ( *Q `  w )  e.  ( 2nd `  A ) )  <->  ( u  <Q  v  /\  ( *Q `  v )  e.  ( 2nd `  A ) ) ) )
65cbvexdva 1852 . . . . 5  |-  ( z  =  u  ->  ( E. w ( z  <Q  w  /\  ( *Q `  w )  e.  ( 2nd `  A ) )  <->  E. v ( u 
<Q  v  /\  ( *Q `  v )  e.  ( 2nd `  A
) ) ) )
76cbvabv 2211 . . . 4  |-  { z  |  E. w ( z  <Q  w  /\  ( *Q `  w )  e.  ( 2nd `  A
) ) }  =  { u  |  E. v ( u  <Q  v  /\  ( *Q `  v )  e.  ( 2nd `  A ) ) }
8 simpl 107 . . . . . . . 8  |-  ( ( z  =  u  /\  w  =  v )  ->  z  =  u )
92, 8breq12d 3858 . . . . . . 7  |-  ( ( z  =  u  /\  w  =  v )  ->  ( w  <Q  z  <->  v 
<Q  u ) )
103eleq1d 2156 . . . . . . 7  |-  ( ( z  =  u  /\  w  =  v )  ->  ( ( *Q `  w )  e.  ( 1st `  A )  <-> 
( *Q `  v
)  e.  ( 1st `  A ) ) )
119, 10anbi12d 457 . . . . . 6  |-  ( ( z  =  u  /\  w  =  v )  ->  ( ( w  <Q  z  /\  ( *Q `  w )  e.  ( 1st `  A ) )  <->  ( v  <Q  u  /\  ( *Q `  v )  e.  ( 1st `  A ) ) ) )
1211cbvexdva 1852 . . . . 5  |-  ( z  =  u  ->  ( E. w ( w  <Q  z  /\  ( *Q `  w )  e.  ( 1st `  A ) )  <->  E. v ( v 
<Q  u  /\  ( *Q `  v )  e.  ( 1st `  A
) ) ) )
1312cbvabv 2211 . . . 4  |-  { z  |  E. w ( w  <Q  z  /\  ( *Q `  w )  e.  ( 1st `  A
) ) }  =  { u  |  E. v ( v  <Q  u  /\  ( *Q `  v )  e.  ( 1st `  A ) ) }
147, 13opeq12i 3627 . . 3  |-  <. { z  |  E. w ( z  <Q  w  /\  ( *Q `  w )  e.  ( 2nd `  A
) ) } ,  { z  |  E. w ( w  <Q  z  /\  ( *Q `  w )  e.  ( 1st `  A ) ) } >.  =  <. { u  |  E. v
( u  <Q  v  /\  ( *Q `  v
)  e.  ( 2nd `  A ) ) } ,  { u  |  E. v ( v 
<Q  u  /\  ( *Q `  v )  e.  ( 1st `  A
) ) } >.
1514recexprlempr 7189 . 2  |-  ( A  e.  P.  ->  <. { z  |  E. w ( z  <Q  w  /\  ( *Q `  w )  e.  ( 2nd `  A
) ) } ,  { z  |  E. w ( w  <Q  z  /\  ( *Q `  w )  e.  ( 1st `  A ) ) } >.  e.  P. )
1614recexprlemex 7194 . 2  |-  ( A  e.  P.  ->  ( A  .P.  <. { z  |  E. w ( z 
<Q  w  /\  ( *Q `  w )  e.  ( 2nd `  A
) ) } ,  { z  |  E. w ( w  <Q  z  /\  ( *Q `  w )  e.  ( 1st `  A ) ) } >. )  =  1P )
17 oveq2 5660 . . . 4  |-  ( x  =  <. { z  |  E. w ( z 
<Q  w  /\  ( *Q `  w )  e.  ( 2nd `  A
) ) } ,  { z  |  E. w ( w  <Q  z  /\  ( *Q `  w )  e.  ( 1st `  A ) ) } >.  ->  ( A  .P.  x )  =  ( A  .P.  <. { z  |  E. w
( z  <Q  w  /\  ( *Q `  w
)  e.  ( 2nd `  A ) ) } ,  { z  |  E. w ( w 
<Q  z  /\  ( *Q `  w )  e.  ( 1st `  A
) ) } >. ) )
1817eqeq1d 2096 . . 3  |-  ( x  =  <. { z  |  E. w ( z 
<Q  w  /\  ( *Q `  w )  e.  ( 2nd `  A
) ) } ,  { z  |  E. w ( w  <Q  z  /\  ( *Q `  w )  e.  ( 1st `  A ) ) } >.  ->  (
( A  .P.  x
)  =  1P  <->  ( A  .P.  <. { z  |  E. w ( z 
<Q  w  /\  ( *Q `  w )  e.  ( 2nd `  A
) ) } ,  { z  |  E. w ( w  <Q  z  /\  ( *Q `  w )  e.  ( 1st `  A ) ) } >. )  =  1P ) )
1918rspcev 2722 . 2  |-  ( (
<. { z  |  E. w ( z  <Q  w  /\  ( *Q `  w )  e.  ( 2nd `  A ) ) } ,  {
z  |  E. w
( w  <Q  z  /\  ( *Q `  w
)  e.  ( 1st `  A ) ) }
>.  e.  P.  /\  ( A  .P.  <. { z  |  E. w ( z 
<Q  w  /\  ( *Q `  w )  e.  ( 2nd `  A
) ) } ,  { z  |  E. w ( w  <Q  z  /\  ( *Q `  w )  e.  ( 1st `  A ) ) } >. )  =  1P )  ->  E. x  e.  P.  ( A  .P.  x )  =  1P )
2015, 16, 19syl2anc 403 1  |-  ( A  e.  P.  ->  E. x  e.  P.  ( A  .P.  x )  =  1P )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1289   E.wex 1426    e. wcel 1438   {cab 2074   E.wrex 2360   <.cop 3449   class class class wbr 3845   ` cfv 5015  (class class class)co 5652   1stc1st 5909   2ndc2nd 5910   *Qcrq 6841    <Q cltq 6842   P.cnp 6848   1Pc1p 6849    .P. cmp 6851
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3954  ax-sep 3957  ax-nul 3965  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-setind 4353  ax-iinf 4403
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2841  df-csb 2934  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-int 3689  df-iun 3732  df-br 3846  df-opab 3900  df-mpt 3901  df-tr 3937  df-eprel 4116  df-id 4120  df-po 4123  df-iso 4124  df-iord 4193  df-on 4195  df-suc 4198  df-iom 4406  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-f1 5020  df-fo 5021  df-f1o 5022  df-fv 5023  df-ov 5655  df-oprab 5656  df-mpt2 5657  df-1st 5911  df-2nd 5912  df-recs 6070  df-irdg 6135  df-1o 6181  df-2o 6182  df-oadd 6185  df-omul 6186  df-er 6290  df-ec 6292  df-qs 6296  df-ni 6861  df-pli 6862  df-mi 6863  df-lti 6864  df-plpq 6901  df-mpq 6902  df-enq 6904  df-nqqs 6905  df-plqqs 6906  df-mqqs 6907  df-1nqqs 6908  df-rq 6909  df-ltnqqs 6910  df-enq0 6981  df-nq0 6982  df-0nq0 6983  df-plq0 6984  df-mq0 6985  df-inp 7023  df-i1p 7024  df-imp 7026
This theorem is referenced by:  ltmprr  7199  recexgt0sr  7317
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