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Theorem mulidnq 9353
Description: Multiplication identity element for positive fractions. (Contributed by NM, 3-Mar-1996.) (Revised by Mario Carneiro, 28-Apr-2013.) (New usage is discouraged.)
Assertion
Ref Expression
mulidnq  |-  ( A  e.  Q.  ->  ( A  .Q  1Q )  =  A )

Proof of Theorem mulidnq
StepHypRef Expression
1 1nq 9318 . . 3  |-  1Q  e.  Q.
2 mulpqnq 9331 . . 3  |-  ( ( A  e.  Q.  /\  1Q  e.  Q. )  -> 
( A  .Q  1Q )  =  ( /Q `  ( A  .pQ  1Q ) ) )
31, 2mpan2 671 . 2  |-  ( A  e.  Q.  ->  ( A  .Q  1Q )  =  ( /Q `  ( A  .pQ  1Q ) ) )
4 relxp 5116 . . . . . . 7  |-  Rel  ( N.  X.  N. )
5 elpqn 9315 . . . . . . 7  |-  ( A  e.  Q.  ->  A  e.  ( N.  X.  N. ) )
6 1st2nd 6841 . . . . . . 7  |-  ( ( Rel  ( N.  X.  N. )  /\  A  e.  ( N.  X.  N. ) )  ->  A  =  <. ( 1st `  A
) ,  ( 2nd `  A ) >. )
74, 5, 6sylancr 663 . . . . . 6  |-  ( A  e.  Q.  ->  A  =  <. ( 1st `  A
) ,  ( 2nd `  A ) >. )
8 df-1nq 9306 . . . . . . 7  |-  1Q  =  <. 1o ,  1o >.
98a1i 11 . . . . . 6  |-  ( A  e.  Q.  ->  1Q  =  <. 1o ,  1o >. )
107, 9oveq12d 6313 . . . . 5  |-  ( A  e.  Q.  ->  ( A  .pQ  1Q )  =  ( <. ( 1st `  A
) ,  ( 2nd `  A ) >.  .pQ  <. 1o ,  1o >. ) )
11 xp1st 6825 . . . . . . 7  |-  ( A  e.  ( N.  X.  N. )  ->  ( 1st `  A )  e.  N. )
125, 11syl 16 . . . . . 6  |-  ( A  e.  Q.  ->  ( 1st `  A )  e. 
N. )
13 xp2nd 6826 . . . . . . 7  |-  ( A  e.  ( N.  X.  N. )  ->  ( 2nd `  A )  e.  N. )
145, 13syl 16 . . . . . 6  |-  ( A  e.  Q.  ->  ( 2nd `  A )  e. 
N. )
15 1pi 9273 . . . . . . 7  |-  1o  e.  N.
1615a1i 11 . . . . . 6  |-  ( A  e.  Q.  ->  1o  e.  N. )
17 mulpipq 9330 . . . . . 6  |-  ( ( ( ( 1st `  A
)  e.  N.  /\  ( 2nd `  A )  e.  N. )  /\  ( 1o  e.  N.  /\  1o  e.  N. )
)  ->  ( <. ( 1st `  A ) ,  ( 2nd `  A
) >.  .pQ  <. 1o ,  1o >. )  =  <. ( ( 1st `  A
)  .N  1o ) ,  ( ( 2nd `  A )  .N  1o ) >. )
1812, 14, 16, 16, 17syl22anc 1229 . . . . 5  |-  ( A  e.  Q.  ->  ( <. ( 1st `  A
) ,  ( 2nd `  A ) >.  .pQ  <. 1o ,  1o >. )  =  <. ( ( 1st `  A
)  .N  1o ) ,  ( ( 2nd `  A )  .N  1o ) >. )
19 mulidpi 9276 . . . . . . . 8  |-  ( ( 1st `  A )  e.  N.  ->  (
( 1st `  A
)  .N  1o )  =  ( 1st `  A
) )
2011, 19syl 16 . . . . . . 7  |-  ( A  e.  ( N.  X.  N. )  ->  ( ( 1st `  A )  .N  1o )  =  ( 1st `  A
) )
21 mulidpi 9276 . . . . . . . 8  |-  ( ( 2nd `  A )  e.  N.  ->  (
( 2nd `  A
)  .N  1o )  =  ( 2nd `  A
) )
2213, 21syl 16 . . . . . . 7  |-  ( A  e.  ( N.  X.  N. )  ->  ( ( 2nd `  A )  .N  1o )  =  ( 2nd `  A
) )
2320, 22opeq12d 4227 . . . . . 6  |-  ( A  e.  ( N.  X.  N. )  ->  <. (
( 1st `  A
)  .N  1o ) ,  ( ( 2nd `  A )  .N  1o ) >.  =  <. ( 1st `  A ) ,  ( 2nd `  A
) >. )
245, 23syl 16 . . . . 5  |-  ( A  e.  Q.  ->  <. (
( 1st `  A
)  .N  1o ) ,  ( ( 2nd `  A )  .N  1o ) >.  =  <. ( 1st `  A ) ,  ( 2nd `  A
) >. )
2510, 18, 243eqtrd 2512 . . . 4  |-  ( A  e.  Q.  ->  ( A  .pQ  1Q )  = 
<. ( 1st `  A
) ,  ( 2nd `  A ) >. )
2625, 7eqtr4d 2511 . . 3  |-  ( A  e.  Q.  ->  ( A  .pQ  1Q )  =  A )
2726fveq2d 5876 . 2  |-  ( A  e.  Q.  ->  ( /Q `  ( A  .pQ  1Q ) )  =  ( /Q `  A ) )
28 nqerid 9323 . 2  |-  ( A  e.  Q.  ->  ( /Q `  A )  =  A )
293, 27, 283eqtrd 2512 1  |-  ( A  e.  Q.  ->  ( A  .Q  1Q )  =  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    e. wcel 1767   <.cop 4039    X. cxp 5003   Rel wrel 5010   ` cfv 5594  (class class class)co 6295   1stc1st 6793   2ndc2nd 6794   1oc1o 7135   N.cnpi 9234    .N cmi 9236    .pQ cmpq 9239   Q.cnq 9242   1Qc1q 9243   /Qcerq 9244    .Q cmq 9246
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-1o 7142  df-oadd 7146  df-omul 7147  df-er 7323  df-ni 9262  df-mi 9264  df-lti 9265  df-mpq 9299  df-enq 9301  df-nq 9302  df-erq 9303  df-mq 9305  df-1nq 9306
This theorem is referenced by:  recmulnq  9354  ltaddnq  9364  halfnq  9366  ltrnq  9369  addclprlem1  9406  addclprlem2  9407  mulclprlem  9409  1idpr  9419  prlem934  9423  prlem936  9437  reclem3pr  9439
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