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Theorem dmrecnq 8801
Description: Domain of reciprocal on positive fractions. (Contributed by Mario Carneiro, 6-Mar-1996.) (Revised by Mario Carneiro, 10-Jul-2014.) (New usage is discouraged.)
Assertion
Ref Expression
dmrecnq  |-  dom  *Q  =  Q.

Proof of Theorem dmrecnq
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 df-rq 8750 . . . . . 6  |-  *Q  =  ( `'  .Q  " { 1Q } )
2 cnvimass 5183 . . . . . 6  |-  ( `'  .Q  " { 1Q } )  C_  dom  .Q
31, 2eqsstri 3338 . . . . 5  |-  *Q  C_  dom  .Q
4 mulnqf 8782 . . . . . 6  |-  .Q  :
( Q.  X.  Q. )
--> Q.
54fdmi 5555 . . . . 5  |-  dom  .Q  =  ( Q.  X.  Q. )
63, 5sseqtri 3340 . . . 4  |-  *Q  C_  ( Q.  X.  Q. )
7 dmss 5028 . . . 4  |-  ( *Q  C_  ( Q.  X.  Q. )  ->  dom  *Q  C_  dom  ( Q.  X.  Q. )
)
86, 7ax-mp 8 . . 3  |-  dom  *Q  C_ 
dom  ( Q.  X.  Q. )
9 dmxpid 5048 . . 3  |-  dom  ( Q.  X.  Q. )  =  Q.
108, 9sseqtri 3340 . 2  |-  dom  *Q  C_ 
Q.
11 recclnq 8799 . . . . . . . 8  |-  ( x  e.  Q.  ->  ( *Q `  x )  e. 
Q. )
12 opelxpi 4869 . . . . . . . 8  |-  ( ( x  e.  Q.  /\  ( *Q `  x )  e.  Q. )  ->  <. x ,  ( *Q
`  x ) >.  e.  ( Q.  X.  Q. ) )
1311, 12mpdan 650 . . . . . . 7  |-  ( x  e.  Q.  ->  <. x ,  ( *Q `  x ) >.  e.  ( Q.  X.  Q. )
)
14 df-ov 6043 . . . . . . . 8  |-  ( x  .Q  ( *Q `  x ) )  =  (  .Q  `  <. x ,  ( *Q `  x ) >. )
15 recidnq 8798 . . . . . . . 8  |-  ( x  e.  Q.  ->  (
x  .Q  ( *Q
`  x ) )  =  1Q )
1614, 15syl5eqr 2450 . . . . . . 7  |-  ( x  e.  Q.  ->  (  .Q  `  <. x ,  ( *Q `  x )
>. )  =  1Q )
17 ffn 5550 . . . . . . . 8  |-  (  .Q  : ( Q.  X.  Q. ) --> Q.  ->  .Q  Fn  ( Q.  X.  Q. )
)
18 fniniseg 5810 . . . . . . . 8  |-  (  .Q  Fn  ( Q.  X.  Q. )  ->  ( <.
x ,  ( *Q
`  x ) >.  e.  ( `'  .Q  " { 1Q } )  <->  ( <. x ,  ( *Q `  x ) >.  e.  ( Q.  X.  Q. )  /\  (  .Q  `  <. x ,  ( *Q `  x ) >. )  =  1Q ) ) )
194, 17, 18mp2b 10 . . . . . . 7  |-  ( <.
x ,  ( *Q
`  x ) >.  e.  ( `'  .Q  " { 1Q } )  <->  ( <. x ,  ( *Q `  x ) >.  e.  ( Q.  X.  Q. )  /\  (  .Q  `  <. x ,  ( *Q `  x ) >. )  =  1Q ) )
2013, 16, 19sylanbrc 646 . . . . . 6  |-  ( x  e.  Q.  ->  <. x ,  ( *Q `  x ) >.  e.  ( `'  .Q  " { 1Q } ) )
2120, 1syl6eleqr 2495 . . . . 5  |-  ( x  e.  Q.  ->  <. x ,  ( *Q `  x ) >.  e.  *Q )
22 df-br 4173 . . . . 5  |-  ( x *Q ( *Q `  x )  <->  <. x ,  ( *Q `  x
) >.  e.  *Q )
2321, 22sylibr 204 . . . 4  |-  ( x  e.  Q.  ->  x *Q ( *Q `  x
) )
24 vex 2919 . . . . 5  |-  x  e. 
_V
25 fvex 5701 . . . . 5  |-  ( *Q
`  x )  e. 
_V
2624, 25breldm 5033 . . . 4  |-  ( x *Q ( *Q `  x )  ->  x  e.  dom  *Q )
2723, 26syl 16 . . 3  |-  ( x  e.  Q.  ->  x  e.  dom  *Q )
2827ssriv 3312 . 2  |-  Q.  C_  dom  *Q
2910, 28eqssi 3324 1  |-  dom  *Q  =  Q.
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721    C_ wss 3280   {csn 3774   <.cop 3777   class class class wbr 4172    X. cxp 4835   `'ccnv 4836   dom cdm 4837   "cima 4840    Fn wfn 5408   -->wf 5409   ` cfv 5413  (class class class)co 6040   Q.cnq 8683   1Qc1q 8684    .Q cmq 8687   *Qcrq 8688
This theorem is referenced by:  ltrnq  8812  reclem2pr  8881
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-omul 6688  df-er 6864  df-ni 8705  df-mi 8707  df-lti 8708  df-mpq 8742  df-enq 8744  df-nq 8745  df-erq 8746  df-mq 8748  df-1nq 8749  df-rq 8750
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