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Theorem dmrecnq 8592
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 8541 . . . . . 6  |-  *Q  =  ( `'  .Q  " { 1Q } )
2 cnvimass 5033 . . . . . 6  |-  ( `'  .Q  " { 1Q } )  C_  dom  .Q
31, 2eqsstri 3208 . . . . 5  |-  *Q  C_  dom  .Q
4 mulnqf 8573 . . . . . 6  |-  .Q  :
( Q.  X.  Q. )
--> Q.
54fdmi 5394 . . . . 5  |-  dom  .Q  =  ( Q.  X.  Q. )
63, 5sseqtri 3210 . . . 4  |-  *Q  C_  ( Q.  X.  Q. )
7 dmss 4878 . . . 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 4898 . . 3  |-  dom  ( Q.  X.  Q. )  =  Q.
108, 9sseqtri 3210 . 2  |-  dom  *Q  C_ 
Q.
11 recclnq 8590 . . . . . . . 8  |-  ( x  e.  Q.  ->  ( *Q `  x )  e. 
Q. )
12 opelxpi 4721 . . . . . . . 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 5861 . . . . . . . 8  |-  ( x  .Q  ( *Q `  x ) )  =  (  .Q  `  <. x ,  ( *Q `  x ) >. )
15 recidnq 8589 . . . . . . . 8  |-  ( x  e.  Q.  ->  (
x  .Q  ( *Q
`  x ) )  =  1Q )
1614, 15syl5eqr 2329 . . . . . . 7  |-  ( x  e.  Q.  ->  (  .Q  `  <. x ,  ( *Q `  x )
>. )  =  1Q )
17 ffn 5389 . . . . . . . 8  |-  (  .Q  : ( Q.  X.  Q. ) --> Q.  ->  .Q  Fn  ( Q.  X.  Q. )
)
18 fniniseg 5646 . . . . . . . 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 2374 . . . . 5  |-  ( x  e.  Q.  ->  <. x ,  ( *Q `  x ) >.  e.  *Q )
22 df-br 4024 . . . . 5  |-  ( x *Q ( *Q `  x )  <->  <. x ,  ( *Q `  x
) >.  e.  *Q )
2321, 22sylibr 203 . . . 4  |-  ( x  e.  Q.  ->  x *Q ( *Q `  x
) )
24 vex 2791 . . . . 5  |-  x  e. 
_V
25 fvex 5539 . . . . 5  |-  ( *Q
`  x )  e. 
_V
2624, 25breldm 4883 . . . 4  |-  ( x *Q ( *Q `  x )  ->  x  e.  dom  *Q )
2723, 26syl 15 . . 3  |-  ( x  e.  Q.  ->  x  e.  dom  *Q )
2827ssriv 3184 . 2  |-  Q.  C_  dom  *Q
2910, 28eqssi 3195 1  |-  dom  *Q  =  Q.
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684    C_ wss 3152   {csn 3640   <.cop 3643   class class class wbr 4023    X. cxp 4687   `'ccnv 4688   dom cdm 4689   "cima 4692    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858   Q.cnq 8474   1Qc1q 8475    .Q cmq 8478   *Qcrq 8479
This theorem is referenced by:  ltrnq  8603  reclem2pr  8672
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-omul 6484  df-er 6660  df-ni 8496  df-mi 8498  df-lti 8499  df-mpq 8533  df-enq 8535  df-nq 8536  df-erq 8537  df-mq 8539  df-1nq 8540  df-rq 8541
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