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Theorem dmrecnq 8847
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 8796 . . . . . 6  |-  *Q  =  ( `'  .Q  " { 1Q } )
2 cnvimass 5226 . . . . . 6  |-  ( `'  .Q  " { 1Q } )  C_  dom  .Q
31, 2eqsstri 3380 . . . . 5  |-  *Q  C_  dom  .Q
4 mulnqf 8828 . . . . . 6  |-  .Q  :
( Q.  X.  Q. )
--> Q.
54fdmi 5598 . . . . 5  |-  dom  .Q  =  ( Q.  X.  Q. )
63, 5sseqtri 3382 . . . 4  |-  *Q  C_  ( Q.  X.  Q. )
7 dmss 5071 . . . 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 5091 . . 3  |-  dom  ( Q.  X.  Q. )  =  Q.
108, 9sseqtri 3382 . 2  |-  dom  *Q  C_ 
Q.
11 recclnq 8845 . . . . . . . 8  |-  ( x  e.  Q.  ->  ( *Q `  x )  e. 
Q. )
12 opelxpi 4912 . . . . . . . 8  |-  ( ( x  e.  Q.  /\  ( *Q `  x )  e.  Q. )  ->  <. x ,  ( *Q
`  x ) >.  e.  ( Q.  X.  Q. ) )
1311, 12mpdan 651 . . . . . . 7  |-  ( x  e.  Q.  ->  <. x ,  ( *Q `  x ) >.  e.  ( Q.  X.  Q. )
)
14 df-ov 6086 . . . . . . . 8  |-  ( x  .Q  ( *Q `  x ) )  =  (  .Q  `  <. x ,  ( *Q `  x ) >. )
15 recidnq 8844 . . . . . . . 8  |-  ( x  e.  Q.  ->  (
x  .Q  ( *Q
`  x ) )  =  1Q )
1614, 15syl5eqr 2484 . . . . . . 7  |-  ( x  e.  Q.  ->  (  .Q  `  <. x ,  ( *Q `  x )
>. )  =  1Q )
17 ffn 5593 . . . . . . . 8  |-  (  .Q  : ( Q.  X.  Q. ) --> Q.  ->  .Q  Fn  ( Q.  X.  Q. )
)
18 fniniseg 5853 . . . . . . . 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 647 . . . . . 6  |-  ( x  e.  Q.  ->  <. x ,  ( *Q `  x ) >.  e.  ( `'  .Q  " { 1Q } ) )
2120, 1syl6eleqr 2529 . . . . 5  |-  ( x  e.  Q.  ->  <. x ,  ( *Q `  x ) >.  e.  *Q )
22 df-br 4215 . . . . 5  |-  ( x *Q ( *Q `  x )  <->  <. x ,  ( *Q `  x
) >.  e.  *Q )
2321, 22sylibr 205 . . . 4  |-  ( x  e.  Q.  ->  x *Q ( *Q `  x
) )
24 vex 2961 . . . . 5  |-  x  e. 
_V
25 fvex 5744 . . . . 5  |-  ( *Q
`  x )  e. 
_V
2624, 25breldm 5076 . . . 4  |-  ( x *Q ( *Q `  x )  ->  x  e.  dom  *Q )
2723, 26syl 16 . . 3  |-  ( x  e.  Q.  ->  x  e.  dom  *Q )
2827ssriv 3354 . 2  |-  Q.  C_  dom  *Q
2910, 28eqssi 3366 1  |-  dom  *Q  =  Q.
Colors of variables: wff set class
Syntax hints:    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726    C_ wss 3322   {csn 3816   <.cop 3819   class class class wbr 4214    X. cxp 4878   `'ccnv 4879   dom cdm 4880   "cima 4883    Fn wfn 5451   -->wf 5452   ` cfv 5456  (class class class)co 6083   Q.cnq 8729   1Qc1q 8730    .Q cmq 8733   *Qcrq 8734
This theorem is referenced by:  ltrnq  8858  reclem2pr  8927
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-recs 6635  df-rdg 6670  df-1o 6726  df-oadd 6730  df-omul 6731  df-er 6907  df-ni 8751  df-mi 8753  df-lti 8754  df-mpq 8788  df-enq 8790  df-nq 8791  df-erq 8792  df-mq 8794  df-1nq 8795  df-rq 8796
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