Theorem addpipqqs 7702

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

Assertion

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

Proof of Theorem addpipqqs

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		addpipqqslem 7701	. 2 |- ( ( ( A e. N. /\ B e. N. ) /\ ( C e. N. /\ D e. N. ) ) -> <. ( ( A .N D ) +N ( B .N C ) ) , ( B .N D ) >. e. ( N. X. N. ) )
2		addpipqqslem 7701	. 2 |- ( ( ( a e. N. /\ b e. N. ) /\ ( g e. N. /\ h e. N. ) ) -> <. ( ( a .N h ) +N ( b .N g ) ) , ( b .N h ) >. e. ( N. X. N. ) )
3		addpipqqslem 7701	. 2 |- ( ( ( c e. N. /\ d e. N. ) /\ ( t e. N. /\ s e. N. ) ) -> <. ( ( c .N s ) +N ( d .N t ) ) , ( d .N s ) >. e. ( N. X. N. ) )
4		enqex 7692	. 2 |- ~Q e. _V
5		enqer 7690	. 2 |- ~Q Er ( N. X. N. )
6		df-enq 7679	. 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 ) ) ) }
7		oveq12 6068	. . . 4 |- ( ( z = a /\ u = d ) -> ( z .N u ) = ( a .N d ) )
8		oveq12 6068	. . . 4 |- ( ( w = b /\ v = c ) -> ( w .N v ) = ( b .N c ) )
9	7, 8	eqeqan12d 2250	. . 3 |- ( ( ( z = a /\ u = d ) /\ ( w = b /\ v = c ) ) -> ( ( z .N u ) = ( w .N v ) <-> ( a .N d ) = ( b .N c ) ) )
10	9	an42s 593	. 2 |- ( ( ( z = a /\ w = b ) /\ ( v = c /\ u = d ) ) -> ( ( z .N u ) = ( w .N v ) <-> ( a .N d ) = ( b .N c ) ) )
11		oveq12 6068	. . . 4 |- ( ( z = g /\ u = s ) -> ( z .N u ) = ( g .N s ) )
12		oveq12 6068	. . . 4 |- ( ( w = h /\ v = t ) -> ( w .N v ) = ( h .N t ) )
13	11, 12	eqeqan12d 2250	. . 3 |- ( ( ( z = g /\ u = s ) /\ ( w = h /\ v = t ) ) -> ( ( z .N u ) = ( w .N v ) <-> ( g .N s ) = ( h .N t ) ) )
14	13	an42s 593	. 2 |- ( ( ( z = g /\ w = h ) /\ ( v = t /\ u = s ) ) -> ( ( z .N u ) = ( w .N v ) <-> ( g .N s ) = ( h .N t ) ) )
15		dfplpq2 7686	. 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 f ) +N ( v .N u ) ) , ( v .N f ) >. ) ) }
16		oveq12 6068	. . . . 5 |- ( ( w = a /\ f = h ) -> ( w .N f ) = ( a .N h ) )
17		oveq12 6068	. . . . 5 |- ( ( v = b /\ u = g ) -> ( v .N u ) = ( b .N g ) )
18	16, 17	oveqan12d 6078	. . . 4 |- ( ( ( w = a /\ f = h ) /\ ( v = b /\ u = g ) ) -> ( ( w .N f ) +N ( v .N u ) ) = ( ( a .N h ) +N ( b .N g ) ) )
19	18	an42s 593	. . 3 |- ( ( ( w = a /\ v = b ) /\ ( u = g /\ f = h ) ) -> ( ( w .N f ) +N ( v .N u ) ) = ( ( a .N h ) +N ( b .N g ) ) )
20		oveq12 6068	. . . 4 |- ( ( v = b /\ f = h ) -> ( v .N f ) = ( b .N h ) )
21	20	ad2ant2l 508	. . 3 |- ( ( ( w = a /\ v = b ) /\ ( u = g /\ f = h ) ) -> ( v .N f ) = ( b .N h ) )
22	19, 21	opeq12d 3897	. 2 |- ( ( ( w = a /\ v = b ) /\ ( u = g /\ f = h ) ) -> <. ( ( w .N f ) +N ( v .N u ) ) , ( v .N f ) >. = <. ( ( a .N h ) +N ( b .N g ) ) , ( b .N h ) >. )
23		oveq12 6068	. . . . 5 |- ( ( w = c /\ f = s ) -> ( w .N f ) = ( c .N s ) )
24		oveq12 6068	. . . . 5 |- ( ( v = d /\ u = t ) -> ( v .N u ) = ( d .N t ) )
25	23, 24	oveqan12d 6078	. . . 4 |- ( ( ( w = c /\ f = s ) /\ ( v = d /\ u = t ) ) -> ( ( w .N f ) +N ( v .N u ) ) = ( ( c .N s ) +N ( d .N t ) ) )
26	25	an42s 593	. . 3 |- ( ( ( w = c /\ v = d ) /\ ( u = t /\ f = s ) ) -> ( ( w .N f ) +N ( v .N u ) ) = ( ( c .N s ) +N ( d .N t ) ) )
27		oveq12 6068	. . . 4 |- ( ( v = d /\ f = s ) -> ( v .N f ) = ( d .N s ) )
28	27	ad2ant2l 508	. . 3 |- ( ( ( w = c /\ v = d ) /\ ( u = t /\ f = s ) ) -> ( v .N f ) = ( d .N s ) )
29	26, 28	opeq12d 3897	. 2 |- ( ( ( w = c /\ v = d ) /\ ( u = t /\ f = s ) ) -> <. ( ( w .N f ) +N ( v .N u ) ) , ( v .N f ) >. = <. ( ( c .N s ) +N ( d .N t ) ) , ( d .N s ) >. )
30		oveq12 6068	. . . . 5 |- ( ( w = A /\ f = D ) -> ( w .N f ) = ( A .N D ) )
31		oveq12 6068	. . . . 5 |- ( ( v = B /\ u = C ) -> ( v .N u ) = ( B .N C ) )
32	30, 31	oveqan12d 6078	. . . 4 |- ( ( ( w = A /\ f = D ) /\ ( v = B /\ u = C ) ) -> ( ( w .N f ) +N ( v .N u ) ) = ( ( A .N D ) +N ( B .N C ) ) )
33	32	an42s 593	. . 3 |- ( ( ( w = A /\ v = B ) /\ ( u = C /\ f = D ) ) -> ( ( w .N f ) +N ( v .N u ) ) = ( ( A .N D ) +N ( B .N C ) ) )
34		oveq12 6068	. . . 4 |- ( ( v = B /\ f = D ) -> ( v .N f ) = ( B .N D ) )
35	34	ad2ant2l 508	. . 3 |- ( ( ( w = A /\ v = B ) /\ ( u = C /\ f = D ) ) -> ( v .N f ) = ( B .N D ) )
36	33, 35	opeq12d 3897	. 2 |- ( ( ( w = A /\ v = B ) /\ ( u = C /\ f = D ) ) -> <. ( ( w .N f ) +N ( v .N u ) ) , ( v .N f ) >. = <. ( ( A .N D ) +N ( B .N C ) ) , ( B .N D ) >. )
37		df-plqqs 7681	. 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 ) ) }
38		df-nqqs 7680	. 2 |- Q. = ( ( N. X. N. ) /. ~Q )
39		addcmpblnq 7699	. 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 h ) +N ( b .N g ) ) , ( b .N h ) >. ~Q <. ( ( c .N s ) +N ( d .N t ) ) , ( d .N s ) >. ) )
40	1, 2, 3, 4, 5, 6, 10, 14, 15, 22, 29, 36, 37, 38, 39	oviec 6889	1 |- ( ( ( A e. N. /\ B e. N. ) /\ ( C e. N. /\ D e. N. ) ) -> ( [ <. A , B >. ] ~Q +Q [ <. C , D >. ] ~Q ) = [ <. ( ( A .N D ) +N ( B .N C ) ) , ( B .N D ) >. ] ~Q )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2205   <.cop 3698  (class class class)co 6059   [cec 6779   N.cnpi 7604   +N cpli 7605   .N cmi 7606   +pQ cplpq 7608   ~Q ceq 7611   Q.cnq 7612   +Q cplq 7614

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 2207  ax-14 2208  ax-ext 2216  ax-coll 4231  ax-sep 4234  ax-nul 4242  ax-pow 4293  ax-pr 4328  ax-un 4560  ax-setind 4665  ax-iinf 4716

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 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3677  df-sn 3701  df-pr 3702  df-op 3704  df-uni 3921  df-int 3956  df-iun 3999  df-br 4116  df-opab 4178  df-mpt 4179  df-tr 4215  df-id 4420  df-iord 4493  df-on 4495  df-suc 4498  df-iom 4719  df-xp 4761  df-rel 4762  df-cnv 4763  df-co 4764  df-dm 4765  df-rn 4766  df-res 4767  df-ima 4768  df-iota 5318  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 6062  df-oprab 6063  df-mpo 6064  df-1st 6348  df-2nd 6349  df-recs 6550  df-irdg 6615  df-oadd 6665  df-omul 6666  df-er 6781  df-ec 6783  df-qs 6787  df-ni 7636  df-pli 7637  df-mi 7638  df-plpq 7676  df-enq 7679  df-nqqs 7680  df-plqqs 7681

This theorem is referenced by:  addclnq  7707  addcomnqg  7713  addassnqg  7714  distrnqg  7719  ltanqg  7732  1lt2nq  7738  ltexnqq  7740  nqnq0a  7786  addpinq1  7796