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Theorem recexpr 7414
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 3904 . . . . . . 7  |-  ( ( z  =  u  /\  w  =  v )  ->  ( z  <Q  w  <->  u 
<Q  v ) )
2 simpr 109 . . . . . . . . 9  |-  ( ( z  =  u  /\  w  =  v )  ->  w  =  v )
32fveq2d 5393 . . . . . . . 8  |-  ( ( z  =  u  /\  w  =  v )  ->  ( *Q `  w
)  =  ( *Q
`  v ) )
43eleq1d 2186 . . . . . . 7  |-  ( ( z  =  u  /\  w  =  v )  ->  ( ( *Q `  w )  e.  ( 2nd `  A )  <-> 
( *Q `  v
)  e.  ( 2nd `  A ) ) )
51, 4anbi12d 464 . . . . . 6  |-  ( ( z  =  u  /\  w  =  v )  ->  ( ( z  <Q  w  /\  ( *Q `  w )  e.  ( 2nd `  A ) )  <->  ( u  <Q  v  /\  ( *Q `  v )  e.  ( 2nd `  A ) ) ) )
65cbvexdva 1881 . . . . 5  |-  ( z  =  u  ->  ( E. w ( z  <Q  w  /\  ( *Q `  w )  e.  ( 2nd `  A ) )  <->  E. v ( u 
<Q  v  /\  ( *Q `  v )  e.  ( 2nd `  A
) ) ) )
76cbvabv 2241 . . . 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 108 . . . . . . . 8  |-  ( ( z  =  u  /\  w  =  v )  ->  z  =  u )
92, 8breq12d 3912 . . . . . . 7  |-  ( ( z  =  u  /\  w  =  v )  ->  ( w  <Q  z  <->  v 
<Q  u ) )
103eleq1d 2186 . . . . . . 7  |-  ( ( z  =  u  /\  w  =  v )  ->  ( ( *Q `  w )  e.  ( 1st `  A )  <-> 
( *Q `  v
)  e.  ( 1st `  A ) ) )
119, 10anbi12d 464 . . . . . 6  |-  ( ( z  =  u  /\  w  =  v )  ->  ( ( w  <Q  z  /\  ( *Q `  w )  e.  ( 1st `  A ) )  <->  ( v  <Q  u  /\  ( *Q `  v )  e.  ( 1st `  A ) ) ) )
1211cbvexdva 1881 . . . . 5  |-  ( z  =  u  ->  ( E. w ( w  <Q  z  /\  ( *Q `  w )  e.  ( 1st `  A ) )  <->  E. v ( v 
<Q  u  /\  ( *Q `  v )  e.  ( 1st `  A
) ) ) )
1312cbvabv 2241 . . . 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 3680 . . 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 7408 . 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 7413 . 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 5750 . . . 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 2126 . . 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 2763 . 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 408 1  |-  ( A  e.  P.  ->  E. x  e.  P.  ( A  .P.  x )  =  1P )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1316   E.wex 1453    e. wcel 1465   {cab 2103   E.wrex 2394   <.cop 3500   class class class wbr 3899   ` cfv 5093  (class class class)co 5742   1stc1st 6004   2ndc2nd 6005   *Qcrq 7060    <Q cltq 7061   P.cnp 7067   1Pc1p 7068    .P. cmp 7070
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-coll 4013  ax-sep 4016  ax-nul 4024  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-iinf 4472
This theorem depends on definitions:  df-bi 116  df-dc 805  df-3or 948  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-tr 3997  df-eprel 4181  df-id 4185  df-po 4188  df-iso 4189  df-iord 4258  df-on 4260  df-suc 4263  df-iom 4475  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-ov 5745  df-oprab 5746  df-mpo 5747  df-1st 6006  df-2nd 6007  df-recs 6170  df-irdg 6235  df-1o 6281  df-2o 6282  df-oadd 6285  df-omul 6286  df-er 6397  df-ec 6399  df-qs 6403  df-ni 7080  df-pli 7081  df-mi 7082  df-lti 7083  df-plpq 7120  df-mpq 7121  df-enq 7123  df-nqqs 7124  df-plqqs 7125  df-mqqs 7126  df-1nqqs 7127  df-rq 7128  df-ltnqqs 7129  df-enq0 7200  df-nq0 7201  df-0nq0 7202  df-plq0 7203  df-mq0 7204  df-inp 7242  df-i1p 7243  df-imp 7245
This theorem is referenced by:  ltmprr  7418  recexgt0sr  7549
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