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Theorem mulassnq 8551
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 8489 . . . . . . 7  |-  ( ( ( 1st `  A
)  .N  ( 1st `  B ) )  .N  ( 1st `  C
) )  =  ( ( 1st `  A
)  .N  ( ( 1st `  B )  .N  ( 1st `  C
) ) )
2 mulasspi 8489 . . . . . . 7  |-  ( ( ( 2nd `  A
)  .N  ( 2nd `  B ) )  .N  ( 2nd `  C
) )  =  ( ( 2nd `  A
)  .N  ( ( 2nd `  B )  .N  ( 2nd `  C
) ) )
31, 2opeq12i 3775 . . . . . 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 8517 . . . . . . . . . 10  |-  ( A  e.  Q.  ->  A  e.  ( N.  X.  N. ) )
543ad2ant1 981 . . . . . . . . 9  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  A  e.  ( N.  X.  N. ) )
6 elpqn 8517 . . . . . . . . . 10  |-  ( B  e.  Q.  ->  B  e.  ( N.  X.  N. ) )
763ad2ant2 982 . . . . . . . . 9  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  B  e.  ( N.  X.  N. ) )
8 mulpipq2 8531 . . . . . . . . 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 645 . . . . . . . 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 4782 . . . . . . . . 9  |-  Rel  ( N.  X.  N. )
11 elpqn 8517 . . . . . . . . . 10  |-  ( C  e.  Q.  ->  C  e.  ( N.  X.  N. ) )
12113ad2ant3 983 . . . . . . . . 9  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  C  e.  ( N.  X.  N. ) )
13 1st2nd 6100 . . . . . . . . 9  |-  ( ( Rel  ( N.  X.  N. )  /\  C  e.  ( N.  X.  N. ) )  ->  C  =  <. ( 1st `  C
) ,  ( 2nd `  C ) >. )
1410, 12, 13sylancr 647 . . . . . . . 8  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  C  =  <. ( 1st `  C
) ,  ( 2nd `  C ) >. )
159, 14oveq12d 5810 . . . . . . 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 6083 . . . . . . . . . 10  |-  ( A  e.  ( N.  X.  N. )  ->  ( 1st `  A )  e.  N. )
175, 16syl 17 . . . . . . . . 9  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( 1st `  A )  e. 
N. )
18 xp1st 6083 . . . . . . . . . 10  |-  ( B  e.  ( N.  X.  N. )  ->  ( 1st `  B )  e.  N. )
197, 18syl 17 . . . . . . . . 9  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( 1st `  B )  e. 
N. )
20 mulclpi 8485 . . . . . . . . 9  |-  ( ( ( 1st `  A
)  e.  N.  /\  ( 1st `  B )  e.  N. )  -> 
( ( 1st `  A
)  .N  ( 1st `  B ) )  e. 
N. )
2117, 19, 20syl2anc 645 . . . . . . . 8  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  (
( 1st `  A
)  .N  ( 1st `  B ) )  e. 
N. )
22 xp2nd 6084 . . . . . . . . . 10  |-  ( A  e.  ( N.  X.  N. )  ->  ( 2nd `  A )  e.  N. )
235, 22syl 17 . . . . . . . . 9  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( 2nd `  A )  e. 
N. )
24 xp2nd 6084 . . . . . . . . . 10  |-  ( B  e.  ( N.  X.  N. )  ->  ( 2nd `  B )  e.  N. )
257, 24syl 17 . . . . . . . . 9  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( 2nd `  B )  e. 
N. )
26 mulclpi 8485 . . . . . . . . 9  |-  ( ( ( 2nd `  A
)  e.  N.  /\  ( 2nd `  B )  e.  N. )  -> 
( ( 2nd `  A
)  .N  ( 2nd `  B ) )  e. 
N. )
2723, 25, 26syl2anc 645 . . . . . . . 8  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  (
( 2nd `  A
)  .N  ( 2nd `  B ) )  e. 
N. )
28 xp1st 6083 . . . . . . . . 9  |-  ( C  e.  ( N.  X.  N. )  ->  ( 1st `  C )  e.  N. )
2912, 28syl 17 . . . . . . . 8  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( 1st `  C )  e. 
N. )
30 xp2nd 6084 . . . . . . . . 9  |-  ( C  e.  ( N.  X.  N. )  ->  ( 2nd `  C )  e.  N. )
3112, 30syl 17 . . . . . . . 8  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( 2nd `  C )  e. 
N. )
32 mulpipq 8532 . . . . . . . 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 1188 . . . . . . 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 2290 . . . . . 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 6100 . . . . . . . . 9  |-  ( ( Rel  ( N.  X.  N. )  /\  A  e.  ( N.  X.  N. ) )  ->  A  =  <. ( 1st `  A
) ,  ( 2nd `  A ) >. )
3610, 5, 35sylancr 647 . . . . . . . 8  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  A  =  <. ( 1st `  A
) ,  ( 2nd `  A ) >. )
37 mulpipq2 8531 . . . . . . . . 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 645 . . . . . . . 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 5810 . . . . . . 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 8485 . . . . . . . . 9  |-  ( ( ( 1st `  B
)  e.  N.  /\  ( 1st `  C )  e.  N. )  -> 
( ( 1st `  B
)  .N  ( 1st `  C ) )  e. 
N. )
4119, 29, 40syl2anc 645 . . . . . . . 8  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  (
( 1st `  B
)  .N  ( 1st `  C ) )  e. 
N. )
42 mulclpi 8485 . . . . . . . . 9  |-  ( ( ( 2nd `  B
)  e.  N.  /\  ( 2nd `  C )  e.  N. )  -> 
( ( 2nd `  B
)  .N  ( 2nd `  C ) )  e. 
N. )
4325, 31, 42syl2anc 645 . . . . . . . 8  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  (
( 2nd `  B
)  .N  ( 2nd `  C ) )  e. 
N. )
44 mulpipq 8532 . . . . . . . 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 1188 . . . . . . 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 2290 . . . . . 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 2316 . . . . 5  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  (
( A  .pQ  B
)  .pQ  C )  =  ( A  .pQ  ( B  .pQ  C ) ) )
4847fveq2d 5462 . . . 4  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( /Q `  ( ( A 
.pQ  B )  .pQ  C ) )  =  ( /Q `  ( A 
.pQ  ( B  .pQ  C ) ) ) )
49 mulerpq 8549 . . . 4  |-  ( ( /Q `  ( A 
.pQ  B ) )  .Q  ( /Q `  C ) )  =  ( /Q `  (
( A  .pQ  B
)  .pQ  C )
)
50 mulerpq 8549 . . . 4  |-  ( ( /Q `  A )  .Q  ( /Q `  ( B  .pQ  C ) ) )  =  ( /Q `  ( A 
.pQ  ( B  .pQ  C ) ) )
5148, 49, 503eqtr4g 2315 . . 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 8533 . . . . 5  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  ( A  .Q  B
)  =  ( /Q
`  ( A  .pQ  B ) ) )
53523adant3 980 . . . 4  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( A  .Q  B )  =  ( /Q `  ( A  .pQ  B ) ) )
54 nqerid 8525 . . . . . 6  |-  ( C  e.  Q.  ->  ( /Q `  C )  =  C )
5554eqcomd 2263 . . . . 5  |-  ( C  e.  Q.  ->  C  =  ( /Q `  C ) )
56553ad2ant3 983 . . . 4  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  C  =  ( /Q `  C ) )
5753, 56oveq12d 5810 . . 3  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  (
( A  .Q  B
)  .Q  C )  =  ( ( /Q
`  ( A  .pQ  B ) )  .Q  ( /Q `  C ) ) )
58 nqerid 8525 . . . . . 6  |-  ( A  e.  Q.  ->  ( /Q `  A )  =  A )
5958eqcomd 2263 . . . . 5  |-  ( A  e.  Q.  ->  A  =  ( /Q `  A ) )
60593ad2ant1 981 . . . 4  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  A  =  ( /Q `  A ) )
61 mulpqnq 8533 . . . . 5  |-  ( ( B  e.  Q.  /\  C  e.  Q. )  ->  ( B  .Q  C
)  =  ( /Q
`  ( B  .pQ  C ) ) )
62613adant1 978 . . . 4  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( B  .Q  C )  =  ( /Q `  ( B  .pQ  C ) ) )
6360, 62oveq12d 5810 . . 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 2300 . 2  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  (
( A  .Q  B
)  .Q  C )  =  ( A  .Q  ( B  .Q  C
) ) )
65 mulnqf 8541 . . . 4  |-  .Q  :
( Q.  X.  Q. )
--> Q.
6665fdmi 5332 . . 3  |-  dom  .Q  =  ( Q.  X.  Q. )
67 0nnq 8516 . . 3  |-  -.  (/)  e.  Q.
6866, 67ndmovass 5942 . 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 939    = wceq 1619    e. wcel 1621   <.cop 3617    X. cxp 4659   Rel wrel 4666   ` cfv 4673  (class class class)co 5792   1stc1st 6054   2ndc2nd 6055   N.cnpi 8434    .N cmi 8436    .pQ cmpq 8439   Q.cnq 8442   /Qcerq 8444    .Q cmq 8446
This theorem is referenced by:  recmulnq  8556  halfnq  8568  ltrnq  8571  addclprlem2  8609  mulclprlem  8611  mulasspr  8616  1idpr  8621  prlem934  8625  prlem936  8639  reclem3pr  8641
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-sep 4115  ax-nul 4123  ax-pr 4186  ax-un 4484
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-ral 2523  df-rex 2524  df-reu 2525  df-rmo 2526  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-pss 3143  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-tp 3622  df-op 3623  df-uni 3802  df-iun 3881  df-br 3998  df-opab 4052  df-mpt 4053  df-tr 4088  df-eprel 4277  df-id 4281  df-po 4286  df-so 4287  df-fr 4324  df-we 4326  df-ord 4367  df-on 4368  df-lim 4369  df-suc 4370  df-om 4629  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-1st 6056  df-2nd 6057  df-recs 6356  df-rdg 6391  df-1o 6447  df-oadd 6451  df-omul 6452  df-er 6628  df-ni 8464  df-mi 8466  df-lti 8467  df-mpq 8501  df-enq 8503  df-nq 8504  df-erq 8505  df-mq 8507  df-1nq 8508
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