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Theorem recexpr 6794
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 3797 . . . . . . 7  |-  ( ( z  =  u  /\  w  =  v )  ->  ( z  <Q  w  <->  u 
<Q  v ) )
2 simpr 107 . . . . . . . . 9  |-  ( ( z  =  u  /\  w  =  v )  ->  w  =  v )
32fveq2d 5210 . . . . . . . 8  |-  ( ( z  =  u  /\  w  =  v )  ->  ( *Q `  w
)  =  ( *Q
`  v ) )
43eleq1d 2122 . . . . . . 7  |-  ( ( z  =  u  /\  w  =  v )  ->  ( ( *Q `  w )  e.  ( 2nd `  A )  <-> 
( *Q `  v
)  e.  ( 2nd `  A ) ) )
51, 4anbi12d 450 . . . . . 6  |-  ( ( z  =  u  /\  w  =  v )  ->  ( ( z  <Q  w  /\  ( *Q `  w )  e.  ( 2nd `  A ) )  <->  ( u  <Q  v  /\  ( *Q `  v )  e.  ( 2nd `  A ) ) ) )
65cbvexdva 1820 . . . . 5  |-  ( z  =  u  ->  ( E. w ( z  <Q  w  /\  ( *Q `  w )  e.  ( 2nd `  A ) )  <->  E. v ( u 
<Q  v  /\  ( *Q `  v )  e.  ( 2nd `  A
) ) ) )
76cbvabv 2177 . . . 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 106 . . . . . . . 8  |-  ( ( z  =  u  /\  w  =  v )  ->  z  =  u )
92, 8breq12d 3805 . . . . . . 7  |-  ( ( z  =  u  /\  w  =  v )  ->  ( w  <Q  z  <->  v 
<Q  u ) )
103eleq1d 2122 . . . . . . 7  |-  ( ( z  =  u  /\  w  =  v )  ->  ( ( *Q `  w )  e.  ( 1st `  A )  <-> 
( *Q `  v
)  e.  ( 1st `  A ) ) )
119, 10anbi12d 450 . . . . . 6  |-  ( ( z  =  u  /\  w  =  v )  ->  ( ( w  <Q  z  /\  ( *Q `  w )  e.  ( 1st `  A ) )  <->  ( v  <Q  u  /\  ( *Q `  v )  e.  ( 1st `  A ) ) ) )
1211cbvexdva 1820 . . . . 5  |-  ( z  =  u  ->  ( E. w ( w  <Q  z  /\  ( *Q `  w )  e.  ( 1st `  A ) )  <->  E. v ( v 
<Q  u  /\  ( *Q `  v )  e.  ( 1st `  A
) ) ) )
1312cbvabv 2177 . . . 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 3582 . . 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 6788 . 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 6793 . 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 5548 . . . 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 2064 . . 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 2673 . 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 397 1  |-  ( A  e.  P.  ->  E. x  e.  P.  ( A  .P.  x )  =  1P )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101    = wceq 1259   E.wex 1397    e. wcel 1409   {cab 2042   E.wrex 2324   <.cop 3406   class class class wbr 3792   ` cfv 4930  (class class class)co 5540   1stc1st 5793   2ndc2nd 5794   *Qcrq 6440    <Q cltq 6441   P.cnp 6447   1Pc1p 6448    .P. cmp 6450
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3900  ax-sep 3903  ax-nul 3911  ax-pow 3955  ax-pr 3972  ax-un 4198  ax-setind 4290  ax-iinf 4339
This theorem depends on definitions:  df-bi 114  df-dc 754  df-3or 897  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2788  df-csb 2881  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-nul 3253  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-int 3644  df-iun 3687  df-br 3793  df-opab 3847  df-mpt 3848  df-tr 3883  df-eprel 4054  df-id 4058  df-po 4061  df-iso 4062  df-iord 4131  df-on 4133  df-suc 4136  df-iom 4342  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-f1 4935  df-fo 4936  df-f1o 4937  df-fv 4938  df-ov 5543  df-oprab 5544  df-mpt2 5545  df-1st 5795  df-2nd 5796  df-recs 5951  df-irdg 5988  df-1o 6032  df-2o 6033  df-oadd 6036  df-omul 6037  df-er 6137  df-ec 6139  df-qs 6143  df-ni 6460  df-pli 6461  df-mi 6462  df-lti 6463  df-plpq 6500  df-mpq 6501  df-enq 6503  df-nqqs 6504  df-plqqs 6505  df-mqqs 6506  df-1nqqs 6507  df-rq 6508  df-ltnqqs 6509  df-enq0 6580  df-nq0 6581  df-0nq0 6582  df-plq0 6583  df-mq0 6584  df-inp 6622  df-i1p 6623  df-imp 6625
This theorem is referenced by:  ltmprr  6798  recexgt0sr  6916
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