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Theorem dmrecnq 8739
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 8688 . . . . . 6  |-  *Q  =  ( `'  .Q  " { 1Q } )
2 cnvimass 5136 . . . . . 6  |-  ( `'  .Q  " { 1Q } )  C_  dom  .Q
31, 2eqsstri 3294 . . . . 5  |-  *Q  C_  dom  .Q
4 mulnqf 8720 . . . . . 6  |-  .Q  :
( Q.  X.  Q. )
--> Q.
54fdmi 5500 . . . . 5  |-  dom  .Q  =  ( Q.  X.  Q. )
63, 5sseqtri 3296 . . . 4  |-  *Q  C_  ( Q.  X.  Q. )
7 dmss 4981 . . . 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 5001 . . 3  |-  dom  ( Q.  X.  Q. )  =  Q.
108, 9sseqtri 3296 . 2  |-  dom  *Q  C_ 
Q.
11 recclnq 8737 . . . . . . . 8  |-  ( x  e.  Q.  ->  ( *Q `  x )  e. 
Q. )
12 opelxpi 4824 . . . . . . . 8  |-  ( ( x  e.  Q.  /\  ( *Q `  x )  e.  Q. )  ->  <. x ,  ( *Q
`  x ) >.  e.  ( Q.  X.  Q. ) )
1311, 12mpdan 649 . . . . . . 7  |-  ( x  e.  Q.  ->  <. x ,  ( *Q `  x ) >.  e.  ( Q.  X.  Q. )
)
14 df-ov 5984 . . . . . . . 8  |-  ( x  .Q  ( *Q `  x ) )  =  (  .Q  `  <. x ,  ( *Q `  x ) >. )
15 recidnq 8736 . . . . . . . 8  |-  ( x  e.  Q.  ->  (
x  .Q  ( *Q
`  x ) )  =  1Q )
1614, 15syl5eqr 2412 . . . . . . 7  |-  ( x  e.  Q.  ->  (  .Q  `  <. x ,  ( *Q `  x )
>. )  =  1Q )
17 ffn 5495 . . . . . . . 8  |-  (  .Q  : ( Q.  X.  Q. ) --> Q.  ->  .Q  Fn  ( Q.  X.  Q. )
)
18 fniniseg 5753 . . . . . . . 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 9 . . . . . . 7  |-  ( <.
x ,  ( *Q
`  x ) >.  e.  ( `'  .Q  " { 1Q } )  <->  ( <. x ,  ( *Q `  x ) >.  e.  ( Q.  X.  Q. )  /\  (  .Q  `  <. x ,  ( *Q `  x ) >. )  =  1Q ) )
2013, 16, 19sylanbrc 645 . . . . . 6  |-  ( x  e.  Q.  ->  <. x ,  ( *Q `  x ) >.  e.  ( `'  .Q  " { 1Q } ) )
2120, 1syl6eleqr 2457 . . . . 5  |-  ( x  e.  Q.  ->  <. x ,  ( *Q `  x ) >.  e.  *Q )
22 df-br 4126 . . . . 5  |-  ( x *Q ( *Q `  x )  <->  <. x ,  ( *Q `  x
) >.  e.  *Q )
2321, 22sylibr 203 . . . 4  |-  ( x  e.  Q.  ->  x *Q ( *Q `  x
) )
24 vex 2876 . . . . 5  |-  x  e. 
_V
25 fvex 5646 . . . . 5  |-  ( *Q
`  x )  e. 
_V
2624, 25breldm 4986 . . . 4  |-  ( x *Q ( *Q `  x )  ->  x  e.  dom  *Q )
2723, 26syl 15 . . 3  |-  ( x  e.  Q.  ->  x  e.  dom  *Q )
2827ssriv 3270 . 2  |-  Q.  C_  dom  *Q
2910, 28eqssi 3281 1  |-  dom  *Q  =  Q.
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    = wceq 1647    e. wcel 1715    C_ wss 3238   {csn 3729   <.cop 3732   class class class wbr 4125    X. cxp 4790   `'ccnv 4791   dom cdm 4792   "cima 4795    Fn wfn 5353   -->wf 5354   ` cfv 5358  (class class class)co 5981   Q.cnq 8621   1Qc1q 8622    .Q cmq 8625   *Qcrq 8626
This theorem is referenced by:  ltrnq  8750  reclem2pr  8819
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 936  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-reu 2635  df-rmo 2636  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-pss 3254  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-tp 3737  df-op 3738  df-uni 3930  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-tr 4216  df-eprel 4408  df-id 4412  df-po 4417  df-so 4418  df-fr 4455  df-we 4457  df-ord 4498  df-on 4499  df-lim 4500  df-suc 4501  df-om 4760  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-1st 6249  df-2nd 6250  df-recs 6530  df-rdg 6565  df-1o 6621  df-oadd 6625  df-omul 6626  df-er 6802  df-ni 8643  df-mi 8645  df-lti 8646  df-mpq 8680  df-enq 8682  df-nq 8683  df-erq 8684  df-mq 8686  df-1nq 8687  df-rq 8688
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