Theorem mulpipqqs 7403

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 7358	. . . 4 |- ( ( A e. N. /\ C e. N. ) -> ( A .N C ) e. N. )
2		mulclpi 7358	. . . 4 |- ( ( B e. N. /\ D e. N. ) -> ( B .N D ) e. N. )
3		opelxpi 4676	. . . 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 289	. . 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 588	. 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 7358	. . . 4 |- ( ( a e. N. /\ g e. N. ) -> ( a .N g ) e. N. )
7		mulclpi 7358	. . . 4 |- ( ( b e. N. /\ h e. N. ) -> ( b .N h ) e. N. )
8		opelxpi 4676	. . . 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 289	. . 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 588	. 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 7358	. . . 4 |- ( ( c e. N. /\ t e. N. ) -> ( c .N t ) e. N. )
12		mulclpi 7358	. . . 4 |- ( ( d e. N. /\ s e. N. ) -> ( d .N s ) e. N. )
13		opelxpi 4676	. . . 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 289	. . 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 588	. 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 7390	. 2 |- ~Q e. _V
17		enqer 7388	. 2 |- ~Q Er ( N. X. N. )
18		df-enq 7377	. 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 527	. . . 4 |- ( ( ( z = a /\ w = b ) /\ ( v = c /\ u = d ) ) -> z = a )
20		simprr 531	. . . 4 |- ( ( ( z = a /\ w = b ) /\ ( v = c /\ u = d ) ) -> u = d )
21	19, 20	oveq12d 5915	. . 3 |- ( ( ( z = a /\ w = b ) /\ ( v = c /\ u = d ) ) -> ( z .N u ) = ( a .N d ) )
22		simplr 528	. . . 4 |- ( ( ( z = a /\ w = b ) /\ ( v = c /\ u = d ) ) -> w = b )
23		simprl 529	. . . 4 |- ( ( ( z = a /\ w = b ) /\ ( v = c /\ u = d ) ) -> v = c )
24	22, 23	oveq12d 5915	. . 3 |- ( ( ( z = a /\ w = b ) /\ ( v = c /\ u = d ) ) -> ( w .N v ) = ( b .N c ) )
25	21, 24	eqeq12d 2204	. 2 |- ( ( ( z = a /\ w = b ) /\ ( v = c /\ u = d ) ) -> ( ( z .N u ) = ( w .N v ) <-> ( a .N d ) = ( b .N c ) ) )
26		simpll 527	. . . 4 |- ( ( ( z = g /\ w = h ) /\ ( v = t /\ u = s ) ) -> z = g )
27		simprr 531	. . . 4 |- ( ( ( z = g /\ w = h ) /\ ( v = t /\ u = s ) ) -> u = s )
28	26, 27	oveq12d 5915	. . 3 |- ( ( ( z = g /\ w = h ) /\ ( v = t /\ u = s ) ) -> ( z .N u ) = ( g .N s ) )
29		simplr 528	. . . 4 |- ( ( ( z = g /\ w = h ) /\ ( v = t /\ u = s ) ) -> w = h )
30		simprl 529	. . . 4 |- ( ( ( z = g /\ w = h ) /\ ( v = t /\ u = s ) ) -> v = t )
31	29, 30	oveq12d 5915	. . 3 |- ( ( ( z = g /\ w = h ) /\ ( v = t /\ u = s ) ) -> ( w .N v ) = ( h .N t ) )
32	28, 31	eqeq12d 2204	. 2 |- ( ( ( z = g /\ w = h ) /\ ( v = t /\ u = s ) ) -> ( ( z .N u ) = ( w .N v ) <-> ( g .N s ) = ( h .N t ) ) )
33		dfmpq2 7385	. 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 527	. . . 4 |- ( ( ( w = a /\ v = b ) /\ ( u = g /\ f = h ) ) -> w = a )
35		simprl 529	. . . 4 |- ( ( ( w = a /\ v = b ) /\ ( u = g /\ f = h ) ) -> u = g )
36	34, 35	oveq12d 5915	. . 3 |- ( ( ( w = a /\ v = b ) /\ ( u = g /\ f = h ) ) -> ( w .N u ) = ( a .N g ) )
37		simplr 528	. . . 4 |- ( ( ( w = a /\ v = b ) /\ ( u = g /\ f = h ) ) -> v = b )
38		simprr 531	. . . 4 |- ( ( ( w = a /\ v = b ) /\ ( u = g /\ f = h ) ) -> f = h )
39	37, 38	oveq12d 5915	. . 3 |- ( ( ( w = a /\ v = b ) /\ ( u = g /\ f = h ) ) -> ( v .N f ) = ( b .N h ) )
40	36, 39	opeq12d 3801	. 2 |- ( ( ( w = a /\ v = b ) /\ ( u = g /\ f = h ) ) -> <. ( w .N u ) , ( v .N f ) >. = <. ( a .N g ) , ( b .N h ) >. )
41		simpll 527	. . . 4 |- ( ( ( w = c /\ v = d ) /\ ( u = t /\ f = s ) ) -> w = c )
42		simprl 529	. . . 4 |- ( ( ( w = c /\ v = d ) /\ ( u = t /\ f = s ) ) -> u = t )
43	41, 42	oveq12d 5915	. . 3 |- ( ( ( w = c /\ v = d ) /\ ( u = t /\ f = s ) ) -> ( w .N u ) = ( c .N t ) )
44		simplr 528	. . . 4 |- ( ( ( w = c /\ v = d ) /\ ( u = t /\ f = s ) ) -> v = d )
45		simprr 531	. . . 4 |- ( ( ( w = c /\ v = d ) /\ ( u = t /\ f = s ) ) -> f = s )
46	44, 45	oveq12d 5915	. . 3 |- ( ( ( w = c /\ v = d ) /\ ( u = t /\ f = s ) ) -> ( v .N f ) = ( d .N s ) )
47	43, 46	opeq12d 3801	. 2 |- ( ( ( w = c /\ v = d ) /\ ( u = t /\ f = s ) ) -> <. ( w .N u ) , ( v .N f ) >. = <. ( c .N t ) , ( d .N s ) >. )
48		simpll 527	. . . 4 |- ( ( ( w = A /\ v = B ) /\ ( u = C /\ f = D ) ) -> w = A )
49		simprl 529	. . . 4 |- ( ( ( w = A /\ v = B ) /\ ( u = C /\ f = D ) ) -> u = C )
50	48, 49	oveq12d 5915	. . 3 |- ( ( ( w = A /\ v = B ) /\ ( u = C /\ f = D ) ) -> ( w .N u ) = ( A .N C ) )
51		simplr 528	. . . 4 |- ( ( ( w = A /\ v = B ) /\ ( u = C /\ f = D ) ) -> v = B )
52		simprr 531	. . . 4 |- ( ( ( w = A /\ v = B ) /\ ( u = C /\ f = D ) ) -> f = D )
53	51, 52	oveq12d 5915	. . 3 |- ( ( ( w = A /\ v = B ) /\ ( u = C /\ f = D ) ) -> ( v .N f ) = ( B .N D ) )
54	50, 53	opeq12d 3801	. 2 |- ( ( ( w = A /\ v = B ) /\ ( u = C /\ f = D ) ) -> <. ( w .N u ) , ( v .N f ) >. = <. ( A .N C ) , ( B .N D ) >. )
55		df-mqqs 7380	. 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 7378	. 2 |- Q. = ( ( N. X. N. ) /. ~Q )
57		mulcmpblnq 7398	. 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 6668	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 104   = wceq 1364   e. wcel 2160   <.cop 3610   X. cxp 4642  (class class class)co 5897   [cec 6558   N.cnpi 7302   .N cmi 7304   .pQ cmpq 7307   ~Q ceq 7309   Q.cnq 7310   .Q cmq 7313

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-iinf 4605

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4311  df-iord 4384  df-on 4386  df-suc 4389  df-iom 4608  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-ov 5900  df-oprab 5901  df-mpo 5902  df-1st 6166  df-2nd 6167  df-recs 6331  df-irdg 6396  df-oadd 6446  df-omul 6447  df-er 6560  df-ec 6562  df-qs 6566  df-ni 7334  df-mi 7336  df-mpq 7375  df-enq 7377  df-nqqs 7378  df-mqqs 7380

This theorem is referenced by:  mulclnq  7406  mulcomnqg  7413  mulassnqg  7414  distrnqg  7417  mulidnq  7419  recexnq  7420  ltmnqg  7431  nqnq0m  7485