Theorem mulpipqqs 7688

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 7643	. . . 4 |- ( ( A e. N. /\ C e. N. ) -> ( A .N C ) e. N. )
2		mulclpi 7643	. . . 4 |- ( ( B e. N. /\ D e. N. ) -> ( B .N D ) e. N. )
3		opelxpi 4781	. . . 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 592	. 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 7643	. . . 4 |- ( ( a e. N. /\ g e. N. ) -> ( a .N g ) e. N. )
7		mulclpi 7643	. . . 4 |- ( ( b e. N. /\ h e. N. ) -> ( b .N h ) e. N. )
8		opelxpi 4781	. . . 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 592	. 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 7643	. . . 4 |- ( ( c e. N. /\ t e. N. ) -> ( c .N t ) e. N. )
12		mulclpi 7643	. . . 4 |- ( ( d e. N. /\ s e. N. ) -> ( d .N s ) e. N. )
13		opelxpi 4781	. . . 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 592	. 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 7675	. 2 |- ~Q e. _V
17		enqer 7673	. 2 |- ~Q Er ( N. X. N. )
18		df-enq 7662	. 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 533	. . . 4 |- ( ( ( z = a /\ w = b ) /\ ( v = c /\ u = d ) ) -> u = d )
21	19, 20	oveq12d 6068	. . 3 |- ( ( ( z = a /\ w = b ) /\ ( v = c /\ u = d ) ) -> ( z .N u ) = ( a .N d ) )
22		simplr 529	. . . 4 |- ( ( ( z = a /\ w = b ) /\ ( v = c /\ u = d ) ) -> w = b )
23		simprl 531	. . . 4 |- ( ( ( z = a /\ w = b ) /\ ( v = c /\ u = d ) ) -> v = c )
24	22, 23	oveq12d 6068	. . 3 |- ( ( ( z = a /\ w = b ) /\ ( v = c /\ u = d ) ) -> ( w .N v ) = ( b .N c ) )
25	21, 24	eqeq12d 2247	. 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 533	. . . 4 |- ( ( ( z = g /\ w = h ) /\ ( v = t /\ u = s ) ) -> u = s )
28	26, 27	oveq12d 6068	. . 3 |- ( ( ( z = g /\ w = h ) /\ ( v = t /\ u = s ) ) -> ( z .N u ) = ( g .N s ) )
29		simplr 529	. . . 4 |- ( ( ( z = g /\ w = h ) /\ ( v = t /\ u = s ) ) -> w = h )
30		simprl 531	. . . 4 |- ( ( ( z = g /\ w = h ) /\ ( v = t /\ u = s ) ) -> v = t )
31	29, 30	oveq12d 6068	. . 3 |- ( ( ( z = g /\ w = h ) /\ ( v = t /\ u = s ) ) -> ( w .N v ) = ( h .N t ) )
32	28, 31	eqeq12d 2247	. 2 |- ( ( ( z = g /\ w = h ) /\ ( v = t /\ u = s ) ) -> ( ( z .N u ) = ( w .N v ) <-> ( g .N s ) = ( h .N t ) ) )
33		dfmpq2 7670	. 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 531	. . . 4 |- ( ( ( w = a /\ v = b ) /\ ( u = g /\ f = h ) ) -> u = g )
36	34, 35	oveq12d 6068	. . 3 |- ( ( ( w = a /\ v = b ) /\ ( u = g /\ f = h ) ) -> ( w .N u ) = ( a .N g ) )
37		simplr 529	. . . 4 |- ( ( ( w = a /\ v = b ) /\ ( u = g /\ f = h ) ) -> v = b )
38		simprr 533	. . . 4 |- ( ( ( w = a /\ v = b ) /\ ( u = g /\ f = h ) ) -> f = h )
39	37, 38	oveq12d 6068	. . 3 |- ( ( ( w = a /\ v = b ) /\ ( u = g /\ f = h ) ) -> ( v .N f ) = ( b .N h ) )
40	36, 39	opeq12d 3891	. 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 531	. . . 4 |- ( ( ( w = c /\ v = d ) /\ ( u = t /\ f = s ) ) -> u = t )
43	41, 42	oveq12d 6068	. . 3 |- ( ( ( w = c /\ v = d ) /\ ( u = t /\ f = s ) ) -> ( w .N u ) = ( c .N t ) )
44		simplr 529	. . . 4 |- ( ( ( w = c /\ v = d ) /\ ( u = t /\ f = s ) ) -> v = d )
45		simprr 533	. . . 4 |- ( ( ( w = c /\ v = d ) /\ ( u = t /\ f = s ) ) -> f = s )
46	44, 45	oveq12d 6068	. . 3 |- ( ( ( w = c /\ v = d ) /\ ( u = t /\ f = s ) ) -> ( v .N f ) = ( d .N s ) )
47	43, 46	opeq12d 3891	. 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 531	. . . 4 |- ( ( ( w = A /\ v = B ) /\ ( u = C /\ f = D ) ) -> u = C )
50	48, 49	oveq12d 6068	. . 3 |- ( ( ( w = A /\ v = B ) /\ ( u = C /\ f = D ) ) -> ( w .N u ) = ( A .N C ) )
51		simplr 529	. . . 4 |- ( ( ( w = A /\ v = B ) /\ ( u = C /\ f = D ) ) -> v = B )
52		simprr 533	. . . 4 |- ( ( ( w = A /\ v = B ) /\ ( u = C /\ f = D ) ) -> f = D )
53	51, 52	oveq12d 6068	. . 3 |- ( ( ( w = A /\ v = B ) /\ ( u = C /\ f = D ) ) -> ( v .N f ) = ( B .N D ) )
54	50, 53	opeq12d 3891	. 2 |- ( ( ( w = A /\ v = B ) /\ ( u = C /\ f = D ) ) -> <. ( w .N u ) , ( v .N f ) >. = <. ( A .N C ) , ( B .N D ) >. )
55		df-mqqs 7665	. 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 7663	. 2 |- Q. = ( ( N. X. N. ) /. ~Q )
57		mulcmpblnq 7683	. 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 6875	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 1398   e. wcel 2203   <.cop 3692   X. cxp 4747  (class class class)co 6050   [cec 6765   N.cnpi 7587   .N cmi 7589   .pQ cmpq 7592   ~Q ceq 7594   Q.cnq 7595   .Q cmq 7598

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-iinf 4710

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-tr 4209  df-id 4414  df-iord 4487  df-on 4489  df-suc 4492  df-iom 4713  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-recs 6536  df-irdg 6601  df-oadd 6651  df-omul 6652  df-er 6767  df-ec 6769  df-qs 6773  df-ni 7619  df-mi 7621  df-mpq 7660  df-enq 7662  df-nqqs 7663  df-mqqs 7665

This theorem is referenced by:  mulclnq  7691  mulcomnqg  7698  mulassnqg  7699  distrnqg  7702  mulidnq  7704  recexnq  7705  ltmnqg  7716  nqnq0m  7770