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Theorem mulidnq 8587
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 8552 . . 3  |-  1Q  e.  Q.
2 mulpqnq 8565 . . 3  |-  ( ( A  e.  Q.  /\  1Q  e.  Q. )  -> 
( A  .Q  1Q )  =  ( /Q `  ( A  .pQ  1Q ) ) )
31, 2mpan2 652 . 2  |-  ( A  e.  Q.  ->  ( A  .Q  1Q )  =  ( /Q `  ( A  .pQ  1Q ) ) )
4 relxp 4794 . . . . . . 7  |-  Rel  ( N.  X.  N. )
5 elpqn 8549 . . . . . . 7  |-  ( A  e.  Q.  ->  A  e.  ( N.  X.  N. ) )
6 1st2nd 6166 . . . . . . 7  |-  ( ( Rel  ( N.  X.  N. )  /\  A  e.  ( N.  X.  N. ) )  ->  A  =  <. ( 1st `  A
) ,  ( 2nd `  A ) >. )
74, 5, 6sylancr 644 . . . . . 6  |-  ( A  e.  Q.  ->  A  =  <. ( 1st `  A
) ,  ( 2nd `  A ) >. )
8 df-1nq 8540 . . . . . . 7  |-  1Q  =  <. 1o ,  1o >.
98a1i 10 . . . . . 6  |-  ( A  e.  Q.  ->  1Q  =  <. 1o ,  1o >. )
107, 9oveq12d 5876 . . . . 5  |-  ( A  e.  Q.  ->  ( A  .pQ  1Q )  =  ( <. ( 1st `  A
) ,  ( 2nd `  A ) >.  .pQ  <. 1o ,  1o >. ) )
11 xp1st 6149 . . . . . . 7  |-  ( A  e.  ( N.  X.  N. )  ->  ( 1st `  A )  e.  N. )
125, 11syl 15 . . . . . 6  |-  ( A  e.  Q.  ->  ( 1st `  A )  e. 
N. )
13 xp2nd 6150 . . . . . . 7  |-  ( A  e.  ( N.  X.  N. )  ->  ( 2nd `  A )  e.  N. )
145, 13syl 15 . . . . . 6  |-  ( A  e.  Q.  ->  ( 2nd `  A )  e. 
N. )
15 1pi 8507 . . . . . . 7  |-  1o  e.  N.
1615a1i 10 . . . . . 6  |-  ( A  e.  Q.  ->  1o  e.  N. )
17 mulpipq 8564 . . . . . 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 1183 . . . . 5  |-  ( A  e.  Q.  ->  ( <. ( 1st `  A
) ,  ( 2nd `  A ) >.  .pQ  <. 1o ,  1o >. )  =  <. ( ( 1st `  A
)  .N  1o ) ,  ( ( 2nd `  A )  .N  1o ) >. )
19 mulidpi 8510 . . . . . . . 8  |-  ( ( 1st `  A )  e.  N.  ->  (
( 1st `  A
)  .N  1o )  =  ( 1st `  A
) )
2011, 19syl 15 . . . . . . 7  |-  ( A  e.  ( N.  X.  N. )  ->  ( ( 1st `  A )  .N  1o )  =  ( 1st `  A
) )
21 mulidpi 8510 . . . . . . . 8  |-  ( ( 2nd `  A )  e.  N.  ->  (
( 2nd `  A
)  .N  1o )  =  ( 2nd `  A
) )
2213, 21syl 15 . . . . . . 7  |-  ( A  e.  ( N.  X.  N. )  ->  ( ( 2nd `  A )  .N  1o )  =  ( 2nd `  A
) )
2320, 22opeq12d 3804 . . . . . 6  |-  ( A  e.  ( N.  X.  N. )  ->  <. (
( 1st `  A
)  .N  1o ) ,  ( ( 2nd `  A )  .N  1o ) >.  =  <. ( 1st `  A ) ,  ( 2nd `  A
) >. )
245, 23syl 15 . . . . 5  |-  ( A  e.  Q.  ->  <. (
( 1st `  A
)  .N  1o ) ,  ( ( 2nd `  A )  .N  1o ) >.  =  <. ( 1st `  A ) ,  ( 2nd `  A
) >. )
2510, 18, 243eqtrd 2319 . . . 4  |-  ( A  e.  Q.  ->  ( A  .pQ  1Q )  = 
<. ( 1st `  A
) ,  ( 2nd `  A ) >. )
2625, 7eqtr4d 2318 . . 3  |-  ( A  e.  Q.  ->  ( A  .pQ  1Q )  =  A )
2726fveq2d 5529 . 2  |-  ( A  e.  Q.  ->  ( /Q `  ( A  .pQ  1Q ) )  =  ( /Q `  A ) )
28 nqerid 8557 . 2  |-  ( A  e.  Q.  ->  ( /Q `  A )  =  A )
293, 27, 283eqtrd 2319 1  |-  ( A  e.  Q.  ->  ( A  .Q  1Q )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   <.cop 3643    X. cxp 4687   Rel wrel 4694   ` cfv 5255  (class class class)co 5858   1stc1st 6120   2ndc2nd 6121   1oc1o 6472   N.cnpi 8466    .N cmi 8468    .pQ cmpq 8471   Q.cnq 8474   1Qc1q 8475   /Qcerq 8476    .Q cmq 8478
This theorem is referenced by:  recmulnq  8588  ltaddnq  8598  halfnq  8600  ltrnq  8603  addclprlem1  8640  addclprlem2  8641  mulclprlem  8643  1idpr  8653  prlem934  8657  prlem936  8671  reclem3pr  8673
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
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