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Theorem mulassnq 8820
Description: Multiplication of positive fractions is associative. (Contributed by NM, 1-Sep-1995.) (Revised by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)
Assertion
Ref Expression
mulassnq  |-  ( ( A  .Q  B )  .Q  C )  =  ( A  .Q  ( B  .Q  C ) )

Proof of Theorem mulassnq
StepHypRef Expression
1 mulasspi 8758 . . . . . . 7  |-  ( ( ( 1st `  A
)  .N  ( 1st `  B ) )  .N  ( 1st `  C
) )  =  ( ( 1st `  A
)  .N  ( ( 1st `  B )  .N  ( 1st `  C
) ) )
2 mulasspi 8758 . . . . . . 7  |-  ( ( ( 2nd `  A
)  .N  ( 2nd `  B ) )  .N  ( 2nd `  C
) )  =  ( ( 2nd `  A
)  .N  ( ( 2nd `  B )  .N  ( 2nd `  C
) ) )
31, 2opeq12i 3976 . . . . . 6  |-  <. (
( ( 1st `  A
)  .N  ( 1st `  B ) )  .N  ( 1st `  C
) ) ,  ( ( ( 2nd `  A
)  .N  ( 2nd `  B ) )  .N  ( 2nd `  C
) ) >.  =  <. ( ( 1st `  A
)  .N  ( ( 1st `  B )  .N  ( 1st `  C
) ) ) ,  ( ( 2nd `  A
)  .N  ( ( 2nd `  B )  .N  ( 2nd `  C
) ) ) >.
4 elpqn 8786 . . . . . . . . . 10  |-  ( A  e.  Q.  ->  A  e.  ( N.  X.  N. ) )
543ad2ant1 978 . . . . . . . . 9  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  A  e.  ( N.  X.  N. ) )
6 elpqn 8786 . . . . . . . . . 10  |-  ( B  e.  Q.  ->  B  e.  ( N.  X.  N. ) )
763ad2ant2 979 . . . . . . . . 9  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  B  e.  ( N.  X.  N. ) )
8 mulpipq2 8800 . . . . . . . . 9  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. ) )  ->  ( A  .pQ  B )  = 
<. ( ( 1st `  A
)  .N  ( 1st `  B ) ) ,  ( ( 2nd `  A
)  .N  ( 2nd `  B ) ) >.
)
95, 7, 8syl2anc 643 . . . . . . . 8  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( A  .pQ  B )  = 
<. ( ( 1st `  A
)  .N  ( 1st `  B ) ) ,  ( ( 2nd `  A
)  .N  ( 2nd `  B ) ) >.
)
10 relxp 4969 . . . . . . . . 9  |-  Rel  ( N.  X.  N. )
11 elpqn 8786 . . . . . . . . . 10  |-  ( C  e.  Q.  ->  C  e.  ( N.  X.  N. ) )
12113ad2ant3 980 . . . . . . . . 9  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  C  e.  ( N.  X.  N. ) )
13 1st2nd 6379 . . . . . . . . 9  |-  ( ( Rel  ( N.  X.  N. )  /\  C  e.  ( N.  X.  N. ) )  ->  C  =  <. ( 1st `  C
) ,  ( 2nd `  C ) >. )
1410, 12, 13sylancr 645 . . . . . . . 8  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  C  =  <. ( 1st `  C
) ,  ( 2nd `  C ) >. )
159, 14oveq12d 6085 . . . . . . 7  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  (
( A  .pQ  B
)  .pQ  C )  =  ( <. (
( 1st `  A
)  .N  ( 1st `  B ) ) ,  ( ( 2nd `  A
)  .N  ( 2nd `  B ) ) >.  .pQ  <. ( 1st `  C
) ,  ( 2nd `  C ) >. )
)
16 xp1st 6362 . . . . . . . . . 10  |-  ( A  e.  ( N.  X.  N. )  ->  ( 1st `  A )  e.  N. )
175, 16syl 16 . . . . . . . . 9  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( 1st `  A )  e. 
N. )
18 xp1st 6362 . . . . . . . . . 10  |-  ( B  e.  ( N.  X.  N. )  ->  ( 1st `  B )  e.  N. )
197, 18syl 16 . . . . . . . . 9  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( 1st `  B )  e. 
N. )
20 mulclpi 8754 . . . . . . . . 9  |-  ( ( ( 1st `  A
)  e.  N.  /\  ( 1st `  B )  e.  N. )  -> 
( ( 1st `  A
)  .N  ( 1st `  B ) )  e. 
N. )
2117, 19, 20syl2anc 643 . . . . . . . 8  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  (
( 1st `  A
)  .N  ( 1st `  B ) )  e. 
N. )
22 xp2nd 6363 . . . . . . . . . 10  |-  ( A  e.  ( N.  X.  N. )  ->  ( 2nd `  A )  e.  N. )
235, 22syl 16 . . . . . . . . 9  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( 2nd `  A )  e. 
N. )
24 xp2nd 6363 . . . . . . . . . 10  |-  ( B  e.  ( N.  X.  N. )  ->  ( 2nd `  B )  e.  N. )
257, 24syl 16 . . . . . . . . 9  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( 2nd `  B )  e. 
N. )
26 mulclpi 8754 . . . . . . . . 9  |-  ( ( ( 2nd `  A
)  e.  N.  /\  ( 2nd `  B )  e.  N. )  -> 
( ( 2nd `  A
)  .N  ( 2nd `  B ) )  e. 
N. )
2723, 25, 26syl2anc 643 . . . . . . . 8  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  (
( 2nd `  A
)  .N  ( 2nd `  B ) )  e. 
N. )
28 xp1st 6362 . . . . . . . . 9  |-  ( C  e.  ( N.  X.  N. )  ->  ( 1st `  C )  e.  N. )
2912, 28syl 16 . . . . . . . 8  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( 1st `  C )  e. 
N. )
30 xp2nd 6363 . . . . . . . . 9  |-  ( C  e.  ( N.  X.  N. )  ->  ( 2nd `  C )  e.  N. )
3112, 30syl 16 . . . . . . . 8  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( 2nd `  C )  e. 
N. )
32 mulpipq 8801 . . . . . . . 8  |-  ( ( ( ( ( 1st `  A )  .N  ( 1st `  B ) )  e.  N.  /\  (
( 2nd `  A
)  .N  ( 2nd `  B ) )  e. 
N. )  /\  (
( 1st `  C
)  e.  N.  /\  ( 2nd `  C )  e.  N. ) )  ->  ( <. (
( 1st `  A
)  .N  ( 1st `  B ) ) ,  ( ( 2nd `  A
)  .N  ( 2nd `  B ) ) >.  .pQ  <. ( 1st `  C
) ,  ( 2nd `  C ) >. )  =  <. ( ( ( 1st `  A )  .N  ( 1st `  B
) )  .N  ( 1st `  C ) ) ,  ( ( ( 2nd `  A )  .N  ( 2nd `  B
) )  .N  ( 2nd `  C ) )
>. )
3321, 27, 29, 31, 32syl22anc 1185 . . . . . . 7  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( <. ( ( 1st `  A
)  .N  ( 1st `  B ) ) ,  ( ( 2nd `  A
)  .N  ( 2nd `  B ) ) >.  .pQ  <. ( 1st `  C
) ,  ( 2nd `  C ) >. )  =  <. ( ( ( 1st `  A )  .N  ( 1st `  B
) )  .N  ( 1st `  C ) ) ,  ( ( ( 2nd `  A )  .N  ( 2nd `  B
) )  .N  ( 2nd `  C ) )
>. )
3415, 33eqtrd 2462 . . . . . 6  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  (
( A  .pQ  B
)  .pQ  C )  =  <. ( ( ( 1st `  A )  .N  ( 1st `  B
) )  .N  ( 1st `  C ) ) ,  ( ( ( 2nd `  A )  .N  ( 2nd `  B
) )  .N  ( 2nd `  C ) )
>. )
35 1st2nd 6379 . . . . . . . . 9  |-  ( ( Rel  ( N.  X.  N. )  /\  A  e.  ( N.  X.  N. ) )  ->  A  =  <. ( 1st `  A
) ,  ( 2nd `  A ) >. )
3610, 5, 35sylancr 645 . . . . . . . 8  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  A  =  <. ( 1st `  A
) ,  ( 2nd `  A ) >. )
37 mulpipq2 8800 . . . . . . . . 9  |-  ( ( B  e.  ( N. 
X.  N. )  /\  C  e.  ( N.  X.  N. ) )  ->  ( B  .pQ  C )  = 
<. ( ( 1st `  B
)  .N  ( 1st `  C ) ) ,  ( ( 2nd `  B
)  .N  ( 2nd `  C ) ) >.
)
387, 12, 37syl2anc 643 . . . . . . . 8  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( B  .pQ  C )  = 
<. ( ( 1st `  B
)  .N  ( 1st `  C ) ) ,  ( ( 2nd `  B
)  .N  ( 2nd `  C ) ) >.
)
3936, 38oveq12d 6085 . . . . . . 7  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( A  .pQ  ( B  .pQ  C ) )  =  (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  .pQ  <. (
( 1st `  B
)  .N  ( 1st `  C ) ) ,  ( ( 2nd `  B
)  .N  ( 2nd `  C ) ) >.
) )
40 mulclpi 8754 . . . . . . . . 9  |-  ( ( ( 1st `  B
)  e.  N.  /\  ( 1st `  C )  e.  N. )  -> 
( ( 1st `  B
)  .N  ( 1st `  C ) )  e. 
N. )
4119, 29, 40syl2anc 643 . . . . . . . 8  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  (
( 1st `  B
)  .N  ( 1st `  C ) )  e. 
N. )
42 mulclpi 8754 . . . . . . . . 9  |-  ( ( ( 2nd `  B
)  e.  N.  /\  ( 2nd `  C )  e.  N. )  -> 
( ( 2nd `  B
)  .N  ( 2nd `  C ) )  e. 
N. )
4325, 31, 42syl2anc 643 . . . . . . . 8  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  (
( 2nd `  B
)  .N  ( 2nd `  C ) )  e. 
N. )
44 mulpipq 8801 . . . . . . . 8  |-  ( ( ( ( 1st `  A
)  e.  N.  /\  ( 2nd `  A )  e.  N. )  /\  ( ( ( 1st `  B )  .N  ( 1st `  C ) )  e.  N.  /\  (
( 2nd `  B
)  .N  ( 2nd `  C ) )  e. 
N. ) )  -> 
( <. ( 1st `  A
) ,  ( 2nd `  A ) >.  .pQ  <. (
( 1st `  B
)  .N  ( 1st `  C ) ) ,  ( ( 2nd `  B
)  .N  ( 2nd `  C ) ) >.
)  =  <. (
( 1st `  A
)  .N  ( ( 1st `  B )  .N  ( 1st `  C
) ) ) ,  ( ( 2nd `  A
)  .N  ( ( 2nd `  B )  .N  ( 2nd `  C
) ) ) >.
)
4517, 23, 41, 43, 44syl22anc 1185 . . . . . . 7  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( <. ( 1st `  A
) ,  ( 2nd `  A ) >.  .pQ  <. (
( 1st `  B
)  .N  ( 1st `  C ) ) ,  ( ( 2nd `  B
)  .N  ( 2nd `  C ) ) >.
)  =  <. (
( 1st `  A
)  .N  ( ( 1st `  B )  .N  ( 1st `  C
) ) ) ,  ( ( 2nd `  A
)  .N  ( ( 2nd `  B )  .N  ( 2nd `  C
) ) ) >.
)
4639, 45eqtrd 2462 . . . . . 6  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( A  .pQ  ( B  .pQ  C ) )  =  <. ( ( 1st `  A
)  .N  ( ( 1st `  B )  .N  ( 1st `  C
) ) ) ,  ( ( 2nd `  A
)  .N  ( ( 2nd `  B )  .N  ( 2nd `  C
) ) ) >.
)
473, 34, 463eqtr4a 2488 . . . . 5  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  (
( A  .pQ  B
)  .pQ  C )  =  ( A  .pQ  ( B  .pQ  C ) ) )
4847fveq2d 5718 . . . 4  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( /Q `  ( ( A 
.pQ  B )  .pQ  C ) )  =  ( /Q `  ( A 
.pQ  ( B  .pQ  C ) ) ) )
49 mulerpq 8818 . . . 4  |-  ( ( /Q `  ( A 
.pQ  B ) )  .Q  ( /Q `  C ) )  =  ( /Q `  (
( A  .pQ  B
)  .pQ  C )
)
50 mulerpq 8818 . . . 4  |-  ( ( /Q `  A )  .Q  ( /Q `  ( B  .pQ  C ) ) )  =  ( /Q `  ( A 
.pQ  ( B  .pQ  C ) ) )
5148, 49, 503eqtr4g 2487 . . 3  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  (
( /Q `  ( A  .pQ  B ) )  .Q  ( /Q `  C ) )  =  ( ( /Q `  A )  .Q  ( /Q `  ( B  .pQ  C ) ) ) )
52 mulpqnq 8802 . . . . 5  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  ( A  .Q  B
)  =  ( /Q
`  ( A  .pQ  B ) ) )
53523adant3 977 . . . 4  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( A  .Q  B )  =  ( /Q `  ( A  .pQ  B ) ) )
54 nqerid 8794 . . . . . 6  |-  ( C  e.  Q.  ->  ( /Q `  C )  =  C )
5554eqcomd 2435 . . . . 5  |-  ( C  e.  Q.  ->  C  =  ( /Q `  C ) )
56553ad2ant3 980 . . . 4  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  C  =  ( /Q `  C ) )
5753, 56oveq12d 6085 . . 3  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  (
( A  .Q  B
)  .Q  C )  =  ( ( /Q
`  ( A  .pQ  B ) )  .Q  ( /Q `  C ) ) )
58 nqerid 8794 . . . . . 6  |-  ( A  e.  Q.  ->  ( /Q `  A )  =  A )
5958eqcomd 2435 . . . . 5  |-  ( A  e.  Q.  ->  A  =  ( /Q `  A ) )
60593ad2ant1 978 . . . 4  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  A  =  ( /Q `  A ) )
61 mulpqnq 8802 . . . . 5  |-  ( ( B  e.  Q.  /\  C  e.  Q. )  ->  ( B  .Q  C
)  =  ( /Q
`  ( B  .pQ  C ) ) )
62613adant1 975 . . . 4  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( B  .Q  C )  =  ( /Q `  ( B  .pQ  C ) ) )
6360, 62oveq12d 6085 . . 3  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( A  .Q  ( B  .Q  C ) )  =  ( ( /Q `  A )  .Q  ( /Q `  ( B  .pQ  C ) ) ) )
6451, 57, 633eqtr4d 2472 . 2  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  (
( A  .Q  B
)  .Q  C )  =  ( A  .Q  ( B  .Q  C
) ) )
65 mulnqf 8810 . . . 4  |-  .Q  :
( Q.  X.  Q. )
--> Q.
6665fdmi 5582 . . 3  |-  dom  .Q  =  ( Q.  X.  Q. )
67 0nnq 8785 . . 3  |-  -.  (/)  e.  Q.
6866, 67ndmovass 6221 . 2  |-  ( -.  ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( ( A  .Q  B )  .Q  C
)  =  ( A  .Q  ( B  .Q  C ) ) )
6964, 68pm2.61i 158 1  |-  ( ( A  .Q  B )  .Q  C )  =  ( A  .Q  ( B  .Q  C ) )
Colors of variables: wff set class
Syntax hints:    /\ w3a 936    = wceq 1652    e. wcel 1725   <.cop 3804    X. cxp 4862   Rel wrel 4869   ` cfv 5440  (class class class)co 6067   1stc1st 6333   2ndc2nd 6334   N.cnpi 8703    .N cmi 8705    .pQ cmpq 8708   Q.cnq 8711   /Qcerq 8713    .Q cmq 8715
This theorem is referenced by:  recmulnq  8825  halfnq  8837  ltrnq  8840  addclprlem2  8878  mulclprlem  8880  mulasspr  8885  1idpr  8890  prlem934  8894  prlem936  8908  reclem3pr  8910
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2411  ax-sep 4317  ax-nul 4325  ax-pow 4364  ax-pr 4390  ax-un 4687
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2417  df-cleq 2423  df-clel 2426  df-nfc 2555  df-ne 2595  df-ral 2697  df-rex 2698  df-reu 2699  df-rmo 2700  df-rab 2701  df-v 2945  df-sbc 3149  df-csb 3239  df-dif 3310  df-un 3312  df-in 3314  df-ss 3321  df-pss 3323  df-nul 3616  df-if 3727  df-pw 3788  df-sn 3807  df-pr 3808  df-tp 3809  df-op 3810  df-uni 4003  df-iun 4082  df-br 4200  df-opab 4254  df-mpt 4255  df-tr 4290  df-eprel 4481  df-id 4485  df-po 4490  df-so 4491  df-fr 4528  df-we 4530  df-ord 4571  df-on 4572  df-lim 4573  df-suc 4574  df-om 4832  df-xp 4870  df-rel 4871  df-cnv 4872  df-co 4873  df-dm 4874  df-rn 4875  df-res 4876  df-ima 4877  df-iota 5404  df-fun 5442  df-fn 5443  df-f 5444  df-f1 5445  df-fo 5446  df-f1o 5447  df-fv 5448  df-ov 6070  df-oprab 6071  df-mpt2 6072  df-1st 6335  df-2nd 6336  df-recs 6619  df-rdg 6654  df-1o 6710  df-oadd 6714  df-omul 6715  df-er 6891  df-ni 8733  df-mi 8735  df-lti 8736  df-mpq 8770  df-enq 8772  df-nq 8773  df-erq 8774  df-mq 8776  df-1nq 8777
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