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Theorem mulidnq 8774
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 8739 . . 3  |-  1Q  e.  Q.
2 mulpqnq 8752 . . 3  |-  ( ( A  e.  Q.  /\  1Q  e.  Q. )  -> 
( A  .Q  1Q )  =  ( /Q `  ( A  .pQ  1Q ) ) )
31, 2mpan2 653 . 2  |-  ( A  e.  Q.  ->  ( A  .Q  1Q )  =  ( /Q `  ( A  .pQ  1Q ) ) )
4 relxp 4924 . . . . . . 7  |-  Rel  ( N.  X.  N. )
5 elpqn 8736 . . . . . . 7  |-  ( A  e.  Q.  ->  A  e.  ( N.  X.  N. ) )
6 1st2nd 6333 . . . . . . 7  |-  ( ( Rel  ( N.  X.  N. )  /\  A  e.  ( N.  X.  N. ) )  ->  A  =  <. ( 1st `  A
) ,  ( 2nd `  A ) >. )
74, 5, 6sylancr 645 . . . . . 6  |-  ( A  e.  Q.  ->  A  =  <. ( 1st `  A
) ,  ( 2nd `  A ) >. )
8 df-1nq 8727 . . . . . . 7  |-  1Q  =  <. 1o ,  1o >.
98a1i 11 . . . . . 6  |-  ( A  e.  Q.  ->  1Q  =  <. 1o ,  1o >. )
107, 9oveq12d 6039 . . . . 5  |-  ( A  e.  Q.  ->  ( A  .pQ  1Q )  =  ( <. ( 1st `  A
) ,  ( 2nd `  A ) >.  .pQ  <. 1o ,  1o >. ) )
11 xp1st 6316 . . . . . . 7  |-  ( A  e.  ( N.  X.  N. )  ->  ( 1st `  A )  e.  N. )
125, 11syl 16 . . . . . 6  |-  ( A  e.  Q.  ->  ( 1st `  A )  e. 
N. )
13 xp2nd 6317 . . . . . . 7  |-  ( A  e.  ( N.  X.  N. )  ->  ( 2nd `  A )  e.  N. )
145, 13syl 16 . . . . . 6  |-  ( A  e.  Q.  ->  ( 2nd `  A )  e. 
N. )
15 1pi 8694 . . . . . . 7  |-  1o  e.  N.
1615a1i 11 . . . . . 6  |-  ( A  e.  Q.  ->  1o  e.  N. )
17 mulpipq 8751 . . . . . 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 1185 . . . . 5  |-  ( A  e.  Q.  ->  ( <. ( 1st `  A
) ,  ( 2nd `  A ) >.  .pQ  <. 1o ,  1o >. )  =  <. ( ( 1st `  A
)  .N  1o ) ,  ( ( 2nd `  A )  .N  1o ) >. )
19 mulidpi 8697 . . . . . . . 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 8697 . . . . . . . 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 3935 . . . . . 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 2424 . . . 4  |-  ( A  e.  Q.  ->  ( A  .pQ  1Q )  = 
<. ( 1st `  A
) ,  ( 2nd `  A ) >. )
2625, 7eqtr4d 2423 . . 3  |-  ( A  e.  Q.  ->  ( A  .pQ  1Q )  =  A )
2726fveq2d 5673 . 2  |-  ( A  e.  Q.  ->  ( /Q `  ( A  .pQ  1Q ) )  =  ( /Q `  A ) )
28 nqerid 8744 . 2  |-  ( A  e.  Q.  ->  ( /Q `  A )  =  A )
293, 27, 283eqtrd 2424 1  |-  ( A  e.  Q.  ->  ( A  .Q  1Q )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1717   <.cop 3761    X. cxp 4817   Rel wrel 4824   ` cfv 5395  (class class class)co 6021   1stc1st 6287   2ndc2nd 6288   1oc1o 6654   N.cnpi 8653    .N cmi 8655    .pQ cmpq 8658   Q.cnq 8661   1Qc1q 8662   /Qcerq 8663    .Q cmq 8665
This theorem is referenced by:  recmulnq  8775  ltaddnq  8785  halfnq  8787  ltrnq  8790  addclprlem1  8827  addclprlem2  8828  mulclprlem  8830  1idpr  8840  prlem934  8844  prlem936  8858  reclem3pr  8860
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-reu 2657  df-rmo 2658  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-1st 6289  df-2nd 6290  df-recs 6570  df-rdg 6605  df-1o 6661  df-oadd 6665  df-omul 6666  df-er 6842  df-ni 8683  df-mi 8685  df-lti 8686  df-mpq 8720  df-enq 8722  df-nq 8723  df-erq 8724  df-mq 8726  df-1nq 8727
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