Theorem mulpipqqs 6471

Description: Multiplication of positive fractions in terms of positive integers. (Contributed by NM, 28-Aug-1995.)

Assertion

Ref	Expression
mulpipqqs	|- ( ( ( A e. N. /\ B e. N. ) /\ ( C e. N. /\ D e. N. ) ) -> ( [ <. A , B >. ] ~Q .Q [ <. C , D >. ] ~Q ) = [ <. ( A .N C ) , ( B .N D ) >. ] ~Q )

Proof of Theorem mulpipqqs

Dummy variables x y z w v u t s f g h a b c d are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mulclpi 6426	. . . 4 |- ( ( A e. N. /\ C e. N. ) -> ( A .N C ) e. N. )
2		mulclpi 6426	. . . 4 |- ( ( B e. N. /\ D e. N. ) -> ( B .N D ) e. N. )
3		opelxpi 4376	. . . 4 |- ( ( ( A .N C ) e. N. /\ ( B .N D ) e. N. ) -> <. ( A .N C ) , ( B .N D ) >. e. ( N. X. N. ) )
4	1, 2, 3	syl2an 273	. . 3 |- ( ( ( A e. N. /\ C e. N. ) /\ ( B e. N. /\ D e. N. ) ) -> <. ( A .N C ) , ( B .N D ) >. e. ( N. X. N. ) )
5	4	an4s 522	. 2 |- ( ( ( A e. N. /\ B e. N. ) /\ ( C e. N. /\ D e. N. ) ) -> <. ( A .N C ) , ( B .N D ) >. e. ( N. X. N. ) )
6		mulclpi 6426	. . . 4 |- ( ( a e. N. /\ g e. N. ) -> ( a .N g ) e. N. )
7		mulclpi 6426	. . . 4 |- ( ( b e. N. /\ h e. N. ) -> ( b .N h ) e. N. )
8		opelxpi 4376	. . . 4 |- ( ( ( a .N g ) e. N. /\ ( b .N h ) e. N. ) -> <. ( a .N g ) , ( b .N h ) >. e. ( N. X. N. ) )
9	6, 7, 8	syl2an 273	. . 3 |- ( ( ( a e. N. /\ g e. N. ) /\ ( b e. N. /\ h e. N. ) ) -> <. ( a .N g ) , ( b .N h ) >. e. ( N. X. N. ) )
10	9	an4s 522	. 2 |- ( ( ( a e. N. /\ b e. N. ) /\ ( g e. N. /\ h e. N. ) ) -> <. ( a .N g ) , ( b .N h ) >. e. ( N. X. N. ) )
11		mulclpi 6426	. . . 4 |- ( ( c e. N. /\ t e. N. ) -> ( c .N t ) e. N. )
12		mulclpi 6426	. . . 4 |- ( ( d e. N. /\ s e. N. ) -> ( d .N s ) e. N. )
13		opelxpi 4376	. . . 4 |- ( ( ( c .N t ) e. N. /\ ( d .N s ) e. N. ) -> <. ( c .N t ) , ( d .N s ) >. e. ( N. X. N. ) )
14	11, 12, 13	syl2an 273	. . 3 |- ( ( ( c e. N. /\ t e. N. ) /\ ( d e. N. /\ s e. N. ) ) -> <. ( c .N t ) , ( d .N s ) >. e. ( N. X. N. ) )
15	14	an4s 522	. 2 |- ( ( ( c e. N. /\ d e. N. ) /\ ( t e. N. /\ s e. N. ) ) -> <. ( c .N t ) , ( d .N s ) >. e. ( N. X. N. ) )
16		enqex 6458	. 2 |- ~Q e. _V
17		enqer 6456	. 2 |- ~Q Er ( N. X. N. )
18		df-enq 6445	. 2 |- ~Q = { <. x , y >. | ( ( x e. ( N. X. N. ) /\ y e. ( N. X. N. ) ) /\ E. z E. w E. v E. u ( ( x = <. z , w >. /\ y = <. v , u >. ) /\ ( z .N u ) = ( w .N v ) ) ) }
19		simpll 481	. . . 4 |- ( ( ( z = a /\ w = b ) /\ ( v = c /\ u = d ) ) -> z = a )
20		simprr 484	. . . 4 |- ( ( ( z = a /\ w = b ) /\ ( v = c /\ u = d ) ) -> u = d )
21	19, 20	oveq12d 5530	. . 3 |- ( ( ( z = a /\ w = b ) /\ ( v = c /\ u = d ) ) -> ( z .N u ) = ( a .N d ) )
22		simplr 482	. . . 4 |- ( ( ( z = a /\ w = b ) /\ ( v = c /\ u = d ) ) -> w = b )
23		simprl 483	. . . 4 |- ( ( ( z = a /\ w = b ) /\ ( v = c /\ u = d ) ) -> v = c )
24	22, 23	oveq12d 5530	. . 3 |- ( ( ( z = a /\ w = b ) /\ ( v = c /\ u = d ) ) -> ( w .N v ) = ( b .N c ) )
25	21, 24	eqeq12d 2054	. 2 |- ( ( ( z = a /\ w = b ) /\ ( v = c /\ u = d ) ) -> ( ( z .N u ) = ( w .N v ) <-> ( a .N d ) = ( b .N c ) ) )
26		simpll 481	. . . 4 |- ( ( ( z = g /\ w = h ) /\ ( v = t /\ u = s ) ) -> z = g )
27		simprr 484	. . . 4 |- ( ( ( z = g /\ w = h ) /\ ( v = t /\ u = s ) ) -> u = s )
28	26, 27	oveq12d 5530	. . 3 |- ( ( ( z = g /\ w = h ) /\ ( v = t /\ u = s ) ) -> ( z .N u ) = ( g .N s ) )
29		simplr 482	. . . 4 |- ( ( ( z = g /\ w = h ) /\ ( v = t /\ u = s ) ) -> w = h )
30		simprl 483	. . . 4 |- ( ( ( z = g /\ w = h ) /\ ( v = t /\ u = s ) ) -> v = t )
31	29, 30	oveq12d 5530	. . 3 |- ( ( ( z = g /\ w = h ) /\ ( v = t /\ u = s ) ) -> ( w .N v ) = ( h .N t ) )
32	28, 31	eqeq12d 2054	. 2 |- ( ( ( z = g /\ w = h ) /\ ( v = t /\ u = s ) ) -> ( ( z .N u ) = ( w .N v ) <-> ( g .N s ) = ( h .N t ) ) )
33		dfmpq2 6453	. 2 |- .pQ = { <. <. x , y >. , z >. | ( ( x e. ( N. X. N. ) /\ y e. ( N. X. N. ) ) /\ E. w E. v E. u E. f ( ( x = <. w , v >. /\ y = <. u , f >. ) /\ z = <. ( w .N u ) , ( v .N f ) >. ) ) }
34		simpll 481	. . . 4 |- ( ( ( w = a /\ v = b ) /\ ( u = g /\ f = h ) ) -> w = a )
35		simprl 483	. . . 4 |- ( ( ( w = a /\ v = b ) /\ ( u = g /\ f = h ) ) -> u = g )
36	34, 35	oveq12d 5530	. . 3 |- ( ( ( w = a /\ v = b ) /\ ( u = g /\ f = h ) ) -> ( w .N u ) = ( a .N g ) )
37		simplr 482	. . . 4 |- ( ( ( w = a /\ v = b ) /\ ( u = g /\ f = h ) ) -> v = b )
38		simprr 484	. . . 4 |- ( ( ( w = a /\ v = b ) /\ ( u = g /\ f = h ) ) -> f = h )
39	37, 38	oveq12d 5530	. . 3 |- ( ( ( w = a /\ v = b ) /\ ( u = g /\ f = h ) ) -> ( v .N f ) = ( b .N h ) )
40	36, 39	opeq12d 3557	. 2 |- ( ( ( w = a /\ v = b ) /\ ( u = g /\ f = h ) ) -> <. ( w .N u ) , ( v .N f ) >. = <. ( a .N g ) , ( b .N h ) >. )
41		simpll 481	. . . 4 |- ( ( ( w = c /\ v = d ) /\ ( u = t /\ f = s ) ) -> w = c )
42		simprl 483	. . . 4 |- ( ( ( w = c /\ v = d ) /\ ( u = t /\ f = s ) ) -> u = t )
43	41, 42	oveq12d 5530	. . 3 |- ( ( ( w = c /\ v = d ) /\ ( u = t /\ f = s ) ) -> ( w .N u ) = ( c .N t ) )
44		simplr 482	. . . 4 |- ( ( ( w = c /\ v = d ) /\ ( u = t /\ f = s ) ) -> v = d )
45		simprr 484	. . . 4 |- ( ( ( w = c /\ v = d ) /\ ( u = t /\ f = s ) ) -> f = s )
46	44, 45	oveq12d 5530	. . 3 |- ( ( ( w = c /\ v = d ) /\ ( u = t /\ f = s ) ) -> ( v .N f ) = ( d .N s ) )
47	43, 46	opeq12d 3557	. 2 |- ( ( ( w = c /\ v = d ) /\ ( u = t /\ f = s ) ) -> <. ( w .N u ) , ( v .N f ) >. = <. ( c .N t ) , ( d .N s ) >. )
48		simpll 481	. . . 4 |- ( ( ( w = A /\ v = B ) /\ ( u = C /\ f = D ) ) -> w = A )
49		simprl 483	. . . 4 |- ( ( ( w = A /\ v = B ) /\ ( u = C /\ f = D ) ) -> u = C )
50	48, 49	oveq12d 5530	. . 3 |- ( ( ( w = A /\ v = B ) /\ ( u = C /\ f = D ) ) -> ( w .N u ) = ( A .N C ) )
51		simplr 482	. . . 4 |- ( ( ( w = A /\ v = B ) /\ ( u = C /\ f = D ) ) -> v = B )
52		simprr 484	. . . 4 |- ( ( ( w = A /\ v = B ) /\ ( u = C /\ f = D ) ) -> f = D )
53	51, 52	oveq12d 5530	. . 3 |- ( ( ( w = A /\ v = B ) /\ ( u = C /\ f = D ) ) -> ( v .N f ) = ( B .N D ) )
54	50, 53	opeq12d 3557	. 2 |- ( ( ( w = A /\ v = B ) /\ ( u = C /\ f = D ) ) -> <. ( w .N u ) , ( v .N f ) >. = <. ( A .N C ) , ( B .N D ) >. )
55		df-mqqs 6448	. 2 |- .Q = { <. <. x , y >. , z >. | ( ( x e. Q. /\ y e. Q. ) /\ E. a E. b E. c E. d ( ( x = [ <. a , b >. ] ~Q /\ y = [ <. c , d >. ] ~Q ) /\ z = [ ( <. a , b >. .pQ <. c , d >. ) ] ~Q ) ) }
56		df-nqqs 6446	. 2 |- Q. = ( ( N. X. N. ) /. ~Q )
57		mulcmpblnq 6466	. 2 |- ( ( ( ( a e. N. /\ b e. N. ) /\ ( c e. N. /\ d e. N. ) ) /\ ( ( g e. N. /\ h e. N. ) /\ ( t e. N. /\ s e. N. ) ) ) -> ( ( ( a .N d ) = ( b .N c ) /\ ( g .N s ) = ( h .N t ) ) -> <. ( a .N g ) , ( b .N h ) >. ~Q <. ( c .N t ) , ( d .N s ) >. ) )
58	5, 10, 15, 16, 17, 18, 25, 32, 33, 40, 47, 54, 55, 56, 57	oviec 6212	1 |- ( ( ( A e. N. /\ B e. N. ) /\ ( C e. N. /\ D e. N. ) ) -> ( [ <. A , B >. ] ~Q .Q [ <. C , D >. ] ~Q ) = [ <. ( A .N C ) , ( B .N D ) >. ] ~Q )

Syntax hints:   -> wi 4   /\ wa 97   = wceq 1243   e. wcel 1393   <.cop 3378   X. cxp 4343  (class class class)co 5512   [cec 6104   N.cnpi 6370   .N cmi 6372   .pQ cmpq 6375   ~Q ceq 6377   Q.cnq 6378   .Q cmq 6381

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311

This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-id 4030  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-oadd 6005  df-omul 6006  df-er 6106  df-ec 6108  df-qs 6112  df-ni 6402  df-mi 6404  df-mpq 6443  df-enq 6445  df-nqqs 6446  df-mqqs 6448

This theorem is referenced by:  mulclnq  6474  mulcomnqg  6481  mulassnqg  6482  distrnqg  6485  mulidnq  6487  recexnq  6488  ltmnqg  6499  nqnq0m  6553