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Theorem mulassnq 8583
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 8521 . . . . . . 7  |-  ( ( ( 1st `  A
)  .N  ( 1st `  B ) )  .N  ( 1st `  C
) )  =  ( ( 1st `  A
)  .N  ( ( 1st `  B )  .N  ( 1st `  C
) ) )
2 mulasspi 8521 . . . . . . 7  |-  ( ( ( 2nd `  A
)  .N  ( 2nd `  B ) )  .N  ( 2nd `  C
) )  =  ( ( 2nd `  A
)  .N  ( ( 2nd `  B )  .N  ( 2nd `  C
) ) )
31, 2opeq12i 3801 . . . . . 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 8549 . . . . . . . . . 10  |-  ( A  e.  Q.  ->  A  e.  ( N.  X.  N. ) )
543ad2ant1 976 . . . . . . . . 9  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  A  e.  ( N.  X.  N. ) )
6 elpqn 8549 . . . . . . . . . 10  |-  ( B  e.  Q.  ->  B  e.  ( N.  X.  N. ) )
763ad2ant2 977 . . . . . . . . 9  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  B  e.  ( N.  X.  N. ) )
8 mulpipq2 8563 . . . . . . . . 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 642 . . . . . . . 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 4794 . . . . . . . . 9  |-  Rel  ( N.  X.  N. )
11 elpqn 8549 . . . . . . . . . 10  |-  ( C  e.  Q.  ->  C  e.  ( N.  X.  N. ) )
12113ad2ant3 978 . . . . . . . . 9  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  C  e.  ( N.  X.  N. ) )
13 1st2nd 6166 . . . . . . . . 9  |-  ( ( Rel  ( N.  X.  N. )  /\  C  e.  ( N.  X.  N. ) )  ->  C  =  <. ( 1st `  C
) ,  ( 2nd `  C ) >. )
1410, 12, 13sylancr 644 . . . . . . . 8  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  C  =  <. ( 1st `  C
) ,  ( 2nd `  C ) >. )
159, 14oveq12d 5876 . . . . . . 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 6149 . . . . . . . . . 10  |-  ( A  e.  ( N.  X.  N. )  ->  ( 1st `  A )  e.  N. )
175, 16syl 15 . . . . . . . . 9  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( 1st `  A )  e. 
N. )
18 xp1st 6149 . . . . . . . . . 10  |-  ( B  e.  ( N.  X.  N. )  ->  ( 1st `  B )  e.  N. )
197, 18syl 15 . . . . . . . . 9  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( 1st `  B )  e. 
N. )
20 mulclpi 8517 . . . . . . . . 9  |-  ( ( ( 1st `  A
)  e.  N.  /\  ( 1st `  B )  e.  N. )  -> 
( ( 1st `  A
)  .N  ( 1st `  B ) )  e. 
N. )
2117, 19, 20syl2anc 642 . . . . . . . 8  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  (
( 1st `  A
)  .N  ( 1st `  B ) )  e. 
N. )
22 xp2nd 6150 . . . . . . . . . 10  |-  ( A  e.  ( N.  X.  N. )  ->  ( 2nd `  A )  e.  N. )
235, 22syl 15 . . . . . . . . 9  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( 2nd `  A )  e. 
N. )
24 xp2nd 6150 . . . . . . . . . 10  |-  ( B  e.  ( N.  X.  N. )  ->  ( 2nd `  B )  e.  N. )
257, 24syl 15 . . . . . . . . 9  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( 2nd `  B )  e. 
N. )
26 mulclpi 8517 . . . . . . . . 9  |-  ( ( ( 2nd `  A
)  e.  N.  /\  ( 2nd `  B )  e.  N. )  -> 
( ( 2nd `  A
)  .N  ( 2nd `  B ) )  e. 
N. )
2723, 25, 26syl2anc 642 . . . . . . . 8  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  (
( 2nd `  A
)  .N  ( 2nd `  B ) )  e. 
N. )
28 xp1st 6149 . . . . . . . . 9  |-  ( C  e.  ( N.  X.  N. )  ->  ( 1st `  C )  e.  N. )
2912, 28syl 15 . . . . . . . 8  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( 1st `  C )  e. 
N. )
30 xp2nd 6150 . . . . . . . . 9  |-  ( C  e.  ( N.  X.  N. )  ->  ( 2nd `  C )  e.  N. )
3112, 30syl 15 . . . . . . . 8  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( 2nd `  C )  e. 
N. )
32 mulpipq 8564 . . . . . . . 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 1183 . . . . . . 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 2315 . . . . . 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 6166 . . . . . . . . 9  |-  ( ( Rel  ( N.  X.  N. )  /\  A  e.  ( N.  X.  N. ) )  ->  A  =  <. ( 1st `  A
) ,  ( 2nd `  A ) >. )
3610, 5, 35sylancr 644 . . . . . . . 8  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  A  =  <. ( 1st `  A
) ,  ( 2nd `  A ) >. )
37 mulpipq2 8563 . . . . . . . . 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 642 . . . . . . . 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 5876 . . . . . . 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 8517 . . . . . . . . 9  |-  ( ( ( 1st `  B
)  e.  N.  /\  ( 1st `  C )  e.  N. )  -> 
( ( 1st `  B
)  .N  ( 1st `  C ) )  e. 
N. )
4119, 29, 40syl2anc 642 . . . . . . . 8  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  (
( 1st `  B
)  .N  ( 1st `  C ) )  e. 
N. )
42 mulclpi 8517 . . . . . . . . 9  |-  ( ( ( 2nd `  B
)  e.  N.  /\  ( 2nd `  C )  e.  N. )  -> 
( ( 2nd `  B
)  .N  ( 2nd `  C ) )  e. 
N. )
4325, 31, 42syl2anc 642 . . . . . . . 8  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  (
( 2nd `  B
)  .N  ( 2nd `  C ) )  e. 
N. )
44 mulpipq 8564 . . . . . . . 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 1183 . . . . . . 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 2315 . . . . . 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 2341 . . . . 5  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  (
( A  .pQ  B
)  .pQ  C )  =  ( A  .pQ  ( B  .pQ  C ) ) )
4847fveq2d 5529 . . . 4  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( /Q `  ( ( A 
.pQ  B )  .pQ  C ) )  =  ( /Q `  ( A 
.pQ  ( B  .pQ  C ) ) ) )
49 mulerpq 8581 . . . 4  |-  ( ( /Q `  ( A 
.pQ  B ) )  .Q  ( /Q `  C ) )  =  ( /Q `  (
( A  .pQ  B
)  .pQ  C )
)
50 mulerpq 8581 . . . 4  |-  ( ( /Q `  A )  .Q  ( /Q `  ( B  .pQ  C ) ) )  =  ( /Q `  ( A 
.pQ  ( B  .pQ  C ) ) )
5148, 49, 503eqtr4g 2340 . . 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 8565 . . . . 5  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  ( A  .Q  B
)  =  ( /Q
`  ( A  .pQ  B ) ) )
53523adant3 975 . . . 4  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( A  .Q  B )  =  ( /Q `  ( A  .pQ  B ) ) )
54 nqerid 8557 . . . . . 6  |-  ( C  e.  Q.  ->  ( /Q `  C )  =  C )
5554eqcomd 2288 . . . . 5  |-  ( C  e.  Q.  ->  C  =  ( /Q `  C ) )
56553ad2ant3 978 . . . 4  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  C  =  ( /Q `  C ) )
5753, 56oveq12d 5876 . . 3  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  (
( A  .Q  B
)  .Q  C )  =  ( ( /Q
`  ( A  .pQ  B ) )  .Q  ( /Q `  C ) ) )
58 nqerid 8557 . . . . . 6  |-  ( A  e.  Q.  ->  ( /Q `  A )  =  A )
5958eqcomd 2288 . . . . 5  |-  ( A  e.  Q.  ->  A  =  ( /Q `  A ) )
60593ad2ant1 976 . . . 4  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  A  =  ( /Q `  A ) )
61 mulpqnq 8565 . . . . 5  |-  ( ( B  e.  Q.  /\  C  e.  Q. )  ->  ( B  .Q  C
)  =  ( /Q
`  ( B  .pQ  C ) ) )
62613adant1 973 . . . 4  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( B  .Q  C )  =  ( /Q `  ( B  .pQ  C ) ) )
6360, 62oveq12d 5876 . . 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 2325 . 2  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  (
( A  .Q  B
)  .Q  C )  =  ( A  .Q  ( B  .Q  C
) ) )
65 mulnqf 8573 . . . 4  |-  .Q  :
( Q.  X.  Q. )
--> Q.
6665fdmi 5394 . . 3  |-  dom  .Q  =  ( Q.  X.  Q. )
67 0nnq 8548 . . 3  |-  -.  (/)  e.  Q.
6866, 67ndmovass 6008 . 2  |-  ( -.  ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( ( A  .Q  B )  .Q  C
)  =  ( A  .Q  ( B  .Q  C ) ) )
6964, 68pm2.61i 156 1  |-  ( ( A  .Q  B )  .Q  C )  =  ( A  .Q  ( B  .Q  C ) )
Colors of variables: wff set class
Syntax hints:    /\ w3a 934    = wceq 1623    e. wcel 1684   <.cop 3643    X. cxp 4687   Rel wrel 4694   ` cfv 5255  (class class class)co 5858   1stc1st 6120   2ndc2nd 6121   N.cnpi 8466    .N cmi 8468    .pQ cmpq 8471   Q.cnq 8474   /Qcerq 8476    .Q cmq 8478
This theorem is referenced by:  recmulnq  8588  halfnq  8600  ltrnq  8603  addclprlem2  8641  mulclprlem  8643  mulasspr  8648  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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