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Theorem dmrecnq 8594
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 8543 . . . . . 6  |-  *Q  =  ( `'  .Q  " { 1Q } )
2 cnvimass 5035 . . . . . 6  |-  ( `'  .Q  " { 1Q } )  C_  dom  .Q
31, 2eqsstri 3210 . . . . 5  |-  *Q  C_  dom  .Q
4 mulnqf 8575 . . . . . 6  |-  .Q  :
( Q.  X.  Q. )
--> Q.
54fdmi 5396 . . . . 5  |-  dom  .Q  =  ( Q.  X.  Q. )
63, 5sseqtri 3212 . . . 4  |-  *Q  C_  ( Q.  X.  Q. )
7 dmss 4880 . . . 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 4900 . . 3  |-  dom  ( Q.  X.  Q. )  =  Q.
108, 9sseqtri 3212 . 2  |-  dom  *Q  C_ 
Q.
11 recclnq 8592 . . . . . . . 8  |-  ( x  e.  Q.  ->  ( *Q `  x )  e. 
Q. )
12 opelxpi 4723 . . . . . . . 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 5863 . . . . . . . 8  |-  ( x  .Q  ( *Q `  x ) )  =  (  .Q  `  <. x ,  ( *Q `  x ) >. )
15 recidnq 8591 . . . . . . . 8  |-  ( x  e.  Q.  ->  (
x  .Q  ( *Q
`  x ) )  =  1Q )
1614, 15syl5eqr 2331 . . . . . . 7  |-  ( x  e.  Q.  ->  (  .Q  `  <. x ,  ( *Q `  x )
>. )  =  1Q )
17 ffn 5391 . . . . . . . 8  |-  (  .Q  : ( Q.  X.  Q. ) --> Q.  ->  .Q  Fn  ( Q.  X.  Q. )
)
18 fniniseg 5648 . . . . . . . 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 2376 . . . . 5  |-  ( x  e.  Q.  ->  <. x ,  ( *Q `  x ) >.  e.  *Q )
22 df-br 4026 . . . . 5  |-  ( x *Q ( *Q `  x )  <->  <. x ,  ( *Q `  x
) >.  e.  *Q )
2321, 22sylibr 203 . . . 4  |-  ( x  e.  Q.  ->  x *Q ( *Q `  x
) )
24 vex 2793 . . . . 5  |-  x  e. 
_V
25 fvex 5541 . . . . 5  |-  ( *Q
`  x )  e. 
_V
2624, 25breldm 4885 . . . 4  |-  ( x *Q ( *Q `  x )  ->  x  e.  dom  *Q )
2723, 26syl 15 . . 3  |-  ( x  e.  Q.  ->  x  e.  dom  *Q )
2827ssriv 3186 . 2  |-  Q.  C_  dom  *Q
2910, 28eqssi 3197 1  |-  dom  *Q  =  Q.
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    = wceq 1625    e. wcel 1686    C_ wss 3154   {csn 3642   <.cop 3645   class class class wbr 4025    X. cxp 4689   `'ccnv 4690   dom cdm 4691   "cima 4694    Fn wfn 5252   -->wf 5253   ` cfv 5257  (class class class)co 5860   Q.cnq 8476   1Qc1q 8477    .Q cmq 8480   *Qcrq 8481
This theorem is referenced by:  ltrnq  8605  reclem2pr  8674
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514
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 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-reu 2552  df-rmo 2553  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  df-eprel 4307  df-id 4311  df-po 4316  df-so 4317  df-fr 4354  df-we 4356  df-ord 4397  df-on 4398  df-lim 4399  df-suc 4400  df-om 4659  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-1st 6124  df-2nd 6125  df-recs 6390  df-rdg 6425  df-1o 6481  df-oadd 6485  df-omul 6486  df-er 6662  df-ni 8498  df-mi 8500  df-lti 8501  df-mpq 8535  df-enq 8537  df-nq 8538  df-erq 8539  df-mq 8541  df-1nq 8542  df-rq 8543
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