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Theorem mulidnq 9373
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 9338 . . 3  |-  1Q  e.  Q.
2 mulpqnq 9351 . . 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 4933 . . . . . . 7  |-  Rel  ( N.  X.  N. )
5 elpqn 9335 . . . . . . 7  |-  ( A  e.  Q.  ->  A  e.  ( N.  X.  N. ) )
6 1st2nd 6832 . . . . . . 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 9326 . . . . . . 7  |-  1Q  =  <. 1o ,  1o >.
98a1i 11 . . . . . 6  |-  ( A  e.  Q.  ->  1Q  =  <. 1o ,  1o >. )
107, 9oveq12d 6298 . . . . 5  |-  ( A  e.  Q.  ->  ( A  .pQ  1Q )  =  ( <. ( 1st `  A
) ,  ( 2nd `  A ) >.  .pQ  <. 1o ,  1o >. ) )
11 xp1st 6816 . . . . . . 7  |-  ( A  e.  ( N.  X.  N. )  ->  ( 1st `  A )  e.  N. )
125, 11syl 17 . . . . . 6  |-  ( A  e.  Q.  ->  ( 1st `  A )  e. 
N. )
13 xp2nd 6817 . . . . . . 7  |-  ( A  e.  ( N.  X.  N. )  ->  ( 2nd `  A )  e.  N. )
145, 13syl 17 . . . . . 6  |-  ( A  e.  Q.  ->  ( 2nd `  A )  e. 
N. )
15 1pi 9293 . . . . . . 7  |-  1o  e.  N.
1615a1i 11 . . . . . 6  |-  ( A  e.  Q.  ->  1o  e.  N. )
17 mulpipq 9350 . . . . . 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 1233 . . . . 5  |-  ( A  e.  Q.  ->  ( <. ( 1st `  A
) ,  ( 2nd `  A ) >.  .pQ  <. 1o ,  1o >. )  =  <. ( ( 1st `  A
)  .N  1o ) ,  ( ( 2nd `  A )  .N  1o ) >. )
19 mulidpi 9296 . . . . . . . 8  |-  ( ( 1st `  A )  e.  N.  ->  (
( 1st `  A
)  .N  1o )  =  ( 1st `  A
) )
2011, 19syl 17 . . . . . . 7  |-  ( A  e.  ( N.  X.  N. )  ->  ( ( 1st `  A )  .N  1o )  =  ( 1st `  A
) )
21 mulidpi 9296 . . . . . . . 8  |-  ( ( 2nd `  A )  e.  N.  ->  (
( 2nd `  A
)  .N  1o )  =  ( 2nd `  A
) )
2213, 21syl 17 . . . . . . 7  |-  ( A  e.  ( N.  X.  N. )  ->  ( ( 2nd `  A )  .N  1o )  =  ( 2nd `  A
) )
2320, 22opeq12d 4169 . . . . . 6  |-  ( A  e.  ( N.  X.  N. )  ->  <. (
( 1st `  A
)  .N  1o ) ,  ( ( 2nd `  A )  .N  1o ) >.  =  <. ( 1st `  A ) ,  ( 2nd `  A
) >. )
245, 23syl 17 . . . . 5  |-  ( A  e.  Q.  ->  <. (
( 1st `  A
)  .N  1o ) ,  ( ( 2nd `  A )  .N  1o ) >.  =  <. ( 1st `  A ) ,  ( 2nd `  A
) >. )
2510, 18, 243eqtrd 2449 . . . 4  |-  ( A  e.  Q.  ->  ( A  .pQ  1Q )  = 
<. ( 1st `  A
) ,  ( 2nd `  A ) >. )
2625, 7eqtr4d 2448 . . 3  |-  ( A  e.  Q.  ->  ( A  .pQ  1Q )  =  A )
2726fveq2d 5855 . 2  |-  ( A  e.  Q.  ->  ( /Q `  ( A  .pQ  1Q ) )  =  ( /Q `  A ) )
28 nqerid 9343 . 2  |-  ( A  e.  Q.  ->  ( /Q `  A )  =  A )
293, 27, 283eqtrd 2449 1  |-  ( A  e.  Q.  ->  ( A  .Q  1Q )  =  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1407    e. wcel 1844   <.cop 3980    X. cxp 4823   Rel wrel 4830   ` cfv 5571  (class class class)co 6280   1stc1st 6784   2ndc2nd 6785   1oc1o 7162   N.cnpi 9254    .N cmi 9256    .pQ cmpq 9259   Q.cnq 9262   1Qc1q 9263   /Qcerq 9264    .Q cmq 9266
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632  ax-un 6576
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3or 977  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3063  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-tp 3979  df-op 3981  df-uni 4194  df-iun 4275  df-br 4398  df-opab 4456  df-mpt 4457  df-tr 4492  df-eprel 4736  df-id 4740  df-po 4746  df-so 4747  df-fr 4784  df-we 4786  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-pred 5369  df-ord 5415  df-on 5416  df-lim 5417  df-suc 5418  df-iota 5535  df-fun 5573  df-fn 5574  df-f 5575  df-f1 5576  df-fo 5577  df-f1o 5578  df-fv 5579  df-ov 6283  df-oprab 6284  df-mpt2 6285  df-om 6686  df-1st 6786  df-2nd 6787  df-wrecs 7015  df-recs 7077  df-rdg 7115  df-1o 7169  df-oadd 7173  df-omul 7174  df-er 7350  df-ni 9282  df-mi 9284  df-lti 9285  df-mpq 9319  df-enq 9321  df-nq 9322  df-erq 9323  df-mq 9325  df-1nq 9326
This theorem is referenced by:  recmulnq  9374  ltaddnq  9384  halfnq  9386  ltrnq  9389  addclprlem1  9426  addclprlem2  9427  mulclprlem  9429  1idpr  9439  prlem934  9443  prlem936  9457  reclem3pr  9459
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