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Theorem mulassnq 9402
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 9340 . . . . . . 7  |-  ( ( ( 1st `  A
)  .N  ( 1st `  B ) )  .N  ( 1st `  C
) )  =  ( ( 1st `  A
)  .N  ( ( 1st `  B )  .N  ( 1st `  C
) ) )
2 mulasspi 9340 . . . . . . 7  |-  ( ( ( 2nd `  A
)  .N  ( 2nd `  B ) )  .N  ( 2nd `  C
) )  =  ( ( 2nd `  A
)  .N  ( ( 2nd `  B )  .N  ( 2nd `  C
) ) )
31, 2opeq12i 4163 . . . . . 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 9368 . . . . . . . . . 10  |-  ( A  e.  Q.  ->  A  e.  ( N.  X.  N. ) )
543ad2ant1 1051 . . . . . . . . 9  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  A  e.  ( N.  X.  N. ) )
6 elpqn 9368 . . . . . . . . . 10  |-  ( B  e.  Q.  ->  B  e.  ( N.  X.  N. ) )
763ad2ant2 1052 . . . . . . . . 9  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  B  e.  ( N.  X.  N. ) )
8 mulpipq2 9382 . . . . . . . . 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 673 . . . . . . . 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 4947 . . . . . . . . 9  |-  Rel  ( N.  X.  N. )
11 elpqn 9368 . . . . . . . . . 10  |-  ( C  e.  Q.  ->  C  e.  ( N.  X.  N. ) )
12113ad2ant3 1053 . . . . . . . . 9  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  C  e.  ( N.  X.  N. ) )
13 1st2nd 6858 . . . . . . . . 9  |-  ( ( Rel  ( N.  X.  N. )  /\  C  e.  ( N.  X.  N. ) )  ->  C  =  <. ( 1st `  C
) ,  ( 2nd `  C ) >. )
1410, 12, 13sylancr 676 . . . . . . . 8  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  C  =  <. ( 1st `  C
) ,  ( 2nd `  C ) >. )
159, 14oveq12d 6326 . . . . . . 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 6842 . . . . . . . . . 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 6842 . . . . . . . . . 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 9336 . . . . . . . . 9  |-  ( ( ( 1st `  A
)  e.  N.  /\  ( 1st `  B )  e.  N. )  -> 
( ( 1st `  A
)  .N  ( 1st `  B ) )  e. 
N. )
2117, 19, 20syl2anc 673 . . . . . . . 8  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  (
( 1st `  A
)  .N  ( 1st `  B ) )  e. 
N. )
22 xp2nd 6843 . . . . . . . . . 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 6843 . . . . . . . . . 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 9336 . . . . . . . . 9  |-  ( ( ( 2nd `  A
)  e.  N.  /\  ( 2nd `  B )  e.  N. )  -> 
( ( 2nd `  A
)  .N  ( 2nd `  B ) )  e. 
N. )
2723, 25, 26syl2anc 673 . . . . . . . 8  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  (
( 2nd `  A
)  .N  ( 2nd `  B ) )  e. 
N. )
28 xp1st 6842 . . . . . . . . 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 6843 . . . . . . . . 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 9383 . . . . . . . 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 1293 . . . . . . 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 2505 . . . . . 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 6858 . . . . . . . . 9  |-  ( ( Rel  ( N.  X.  N. )  /\  A  e.  ( N.  X.  N. ) )  ->  A  =  <. ( 1st `  A
) ,  ( 2nd `  A ) >. )
3610, 5, 35sylancr 676 . . . . . . . 8  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  A  =  <. ( 1st `  A
) ,  ( 2nd `  A ) >. )
37 mulpipq2 9382 . . . . . . . . 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 673 . . . . . . . 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 6326 . . . . . . 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 9336 . . . . . . . . 9  |-  ( ( ( 1st `  B
)  e.  N.  /\  ( 1st `  C )  e.  N. )  -> 
( ( 1st `  B
)  .N  ( 1st `  C ) )  e. 
N. )
4119, 29, 40syl2anc 673 . . . . . . . 8  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  (
( 1st `  B
)  .N  ( 1st `  C ) )  e. 
N. )
42 mulclpi 9336 . . . . . . . . 9  |-  ( ( ( 2nd `  B
)  e.  N.  /\  ( 2nd `  C )  e.  N. )  -> 
( ( 2nd `  B
)  .N  ( 2nd `  C ) )  e. 
N. )
4325, 31, 42syl2anc 673 . . . . . . . 8  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  (
( 2nd `  B
)  .N  ( 2nd `  C ) )  e. 
N. )
44 mulpipq 9383 . . . . . . . 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 1293 . . . . . . 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 2505 . . . . . 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 2531 . . . . 5  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  (
( A  .pQ  B
)  .pQ  C )  =  ( A  .pQ  ( B  .pQ  C ) ) )
4847fveq2d 5883 . . . 4  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( /Q `  ( ( A 
.pQ  B )  .pQ  C ) )  =  ( /Q `  ( A 
.pQ  ( B  .pQ  C ) ) ) )
49 mulerpq 9400 . . . 4  |-  ( ( /Q `  ( A 
.pQ  B ) )  .Q  ( /Q `  C ) )  =  ( /Q `  (
( A  .pQ  B
)  .pQ  C )
)
50 mulerpq 9400 . . . 4  |-  ( ( /Q `  A )  .Q  ( /Q `  ( B  .pQ  C ) ) )  =  ( /Q `  ( A 
.pQ  ( B  .pQ  C ) ) )
5148, 49, 503eqtr4g 2530 . . 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 9384 . . . . 5  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  ( A  .Q  B
)  =  ( /Q
`  ( A  .pQ  B ) ) )
53523adant3 1050 . . . 4  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( A  .Q  B )  =  ( /Q `  ( A  .pQ  B ) ) )
54 nqerid 9376 . . . . . 6  |-  ( C  e.  Q.  ->  ( /Q `  C )  =  C )
5554eqcomd 2477 . . . . 5  |-  ( C  e.  Q.  ->  C  =  ( /Q `  C ) )
56553ad2ant3 1053 . . . 4  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  C  =  ( /Q `  C ) )
5753, 56oveq12d 6326 . . 3  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  (
( A  .Q  B
)  .Q  C )  =  ( ( /Q
`  ( A  .pQ  B ) )  .Q  ( /Q `  C ) ) )
58 nqerid 9376 . . . . . 6  |-  ( A  e.  Q.  ->  ( /Q `  A )  =  A )
5958eqcomd 2477 . . . . 5  |-  ( A  e.  Q.  ->  A  =  ( /Q `  A ) )
60593ad2ant1 1051 . . . 4  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  A  =  ( /Q `  A ) )
61 mulpqnq 9384 . . . . 5  |-  ( ( B  e.  Q.  /\  C  e.  Q. )  ->  ( B  .Q  C
)  =  ( /Q
`  ( B  .pQ  C ) ) )
62613adant1 1048 . . . 4  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( B  .Q  C )  =  ( /Q `  ( B  .pQ  C ) ) )
6360, 62oveq12d 6326 . . 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 2515 . 2  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  (
( A  .Q  B
)  .Q  C )  =  ( A  .Q  ( B  .Q  C
) ) )
65 mulnqf 9392 . . . 4  |-  .Q  :
( Q.  X.  Q. )
--> Q.
6665fdmi 5746 . . 3  |-  dom  .Q  =  ( Q.  X.  Q. )
67 0nnq 9367 . . 3  |-  -.  (/)  e.  Q.
6866, 67ndmovass 6476 . 2  |-  ( -.  ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( ( A  .Q  B )  .Q  C
)  =  ( A  .Q  ( B  .Q  C ) ) )
6964, 68pm2.61i 169 1  |-  ( ( A  .Q  B )  .Q  C )  =  ( A  .Q  ( B  .Q  C ) )
Colors of variables: wff setvar class
Syntax hints:    /\ w3a 1007    = wceq 1452    e. wcel 1904   <.cop 3965    X. cxp 4837   Rel wrel 4844   ` cfv 5589  (class class class)co 6308   1stc1st 6810   2ndc2nd 6811   N.cnpi 9287    .N cmi 9289    .pQ cmpq 9292   Q.cnq 9295   /Qcerq 9297    .Q cmq 9299
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-omul 7205  df-er 7381  df-ni 9315  df-mi 9317  df-lti 9318  df-mpq 9352  df-enq 9354  df-nq 9355  df-erq 9356  df-mq 9358  df-1nq 9359
This theorem is referenced by:  recmulnq  9407  halfnq  9419  ltrnq  9422  addclprlem2  9460  mulclprlem  9462  mulasspr  9467  1idpr  9472  prlem934  9476  prlem936  9490  reclem3pr  9492
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