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Theorem mulidnq 9144
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 9109 . . 3  |-  1Q  e.  Q.
2 mulpqnq 9122 . . 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 4959 . . . . . . 7  |-  Rel  ( N.  X.  N. )
5 elpqn 9106 . . . . . . 7  |-  ( A  e.  Q.  ->  A  e.  ( N.  X.  N. ) )
6 1st2nd 6632 . . . . . . 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 9097 . . . . . . 7  |-  1Q  =  <. 1o ,  1o >.
98a1i 11 . . . . . 6  |-  ( A  e.  Q.  ->  1Q  =  <. 1o ,  1o >. )
107, 9oveq12d 6121 . . . . 5  |-  ( A  e.  Q.  ->  ( A  .pQ  1Q )  =  ( <. ( 1st `  A
) ,  ( 2nd `  A ) >.  .pQ  <. 1o ,  1o >. ) )
11 xp1st 6618 . . . . . . 7  |-  ( A  e.  ( N.  X.  N. )  ->  ( 1st `  A )  e.  N. )
125, 11syl 16 . . . . . 6  |-  ( A  e.  Q.  ->  ( 1st `  A )  e. 
N. )
13 xp2nd 6619 . . . . . . 7  |-  ( A  e.  ( N.  X.  N. )  ->  ( 2nd `  A )  e.  N. )
145, 13syl 16 . . . . . 6  |-  ( A  e.  Q.  ->  ( 2nd `  A )  e. 
N. )
15 1pi 9064 . . . . . . 7  |-  1o  e.  N.
1615a1i 11 . . . . . 6  |-  ( A  e.  Q.  ->  1o  e.  N. )
17 mulpipq 9121 . . . . . 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 1219 . . . . 5  |-  ( A  e.  Q.  ->  ( <. ( 1st `  A
) ,  ( 2nd `  A ) >.  .pQ  <. 1o ,  1o >. )  =  <. ( ( 1st `  A
)  .N  1o ) ,  ( ( 2nd `  A )  .N  1o ) >. )
19 mulidpi 9067 . . . . . . . 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 9067 . . . . . . . 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 4079 . . . . . 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 2479 . . . 4  |-  ( A  e.  Q.  ->  ( A  .pQ  1Q )  = 
<. ( 1st `  A
) ,  ( 2nd `  A ) >. )
2625, 7eqtr4d 2478 . . 3  |-  ( A  e.  Q.  ->  ( A  .pQ  1Q )  =  A )
2726fveq2d 5707 . 2  |-  ( A  e.  Q.  ->  ( /Q `  ( A  .pQ  1Q ) )  =  ( /Q `  A ) )
28 nqerid 9114 . 2  |-  ( A  e.  Q.  ->  ( /Q `  A )  =  A )
293, 27, 283eqtrd 2479 1  |-  ( A  e.  Q.  ->  ( A  .Q  1Q )  =  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1369    e. wcel 1756   <.cop 3895    X. cxp 4850   Rel wrel 4857   ` cfv 5430  (class class class)co 6103   1stc1st 6587   2ndc2nd 6588   1oc1o 6925   N.cnpi 9023    .N cmi 9025    .pQ cmpq 9028   Q.cnq 9031   1Qc1q 9032   /Qcerq 9033    .Q cmq 9035
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-ral 2732  df-rex 2733  df-reu 2734  df-rmo 2735  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-pss 3356  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-tp 3894  df-op 3896  df-uni 4104  df-iun 4185  df-br 4305  df-opab 4363  df-mpt 4364  df-tr 4398  df-eprel 4644  df-id 4648  df-po 4653  df-so 4654  df-fr 4691  df-we 4693  df-ord 4734  df-on 4735  df-lim 4736  df-suc 4737  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-om 6489  df-1st 6589  df-2nd 6590  df-recs 6844  df-rdg 6878  df-1o 6932  df-oadd 6936  df-omul 6937  df-er 7113  df-ni 9053  df-mi 9055  df-lti 9056  df-mpq 9090  df-enq 9092  df-nq 9093  df-erq 9094  df-mq 9096  df-1nq 9097
This theorem is referenced by:  recmulnq  9145  ltaddnq  9155  halfnq  9157  ltrnq  9160  addclprlem1  9197  addclprlem2  9198  mulclprlem  9200  1idpr  9210  prlem934  9214  prlem936  9228  reclem3pr  9230
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