Theorem addpipqqs 7290

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 7289	. 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 7289	. 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 7289	. 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 7280	. 2 |- ~Q e. _V
5		enqer 7278	. 2 |- ~Q Er ( N. X. N. )
6		df-enq 7267	. 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 5833	. . . 4 |- ( ( z = a /\ u = d ) -> ( z .N u ) = ( a .N d ) )
8		oveq12 5833	. . . 4 |- ( ( w = b /\ v = c ) -> ( w .N v ) = ( b .N c ) )
9	7, 8	eqeqan12d 2173	. . 3 |- ( ( ( z = a /\ u = d ) /\ ( w = b /\ v = c ) ) -> ( ( z .N u ) = ( w .N v ) <-> ( a .N d ) = ( b .N c ) ) )
10	9	an42s 579	. 2 |- ( ( ( z = a /\ w = b ) /\ ( v = c /\ u = d ) ) -> ( ( z .N u ) = ( w .N v ) <-> ( a .N d ) = ( b .N c ) ) )
11		oveq12 5833	. . . 4 |- ( ( z = g /\ u = s ) -> ( z .N u ) = ( g .N s ) )
12		oveq12 5833	. . . 4 |- ( ( w = h /\ v = t ) -> ( w .N v ) = ( h .N t ) )
13	11, 12	eqeqan12d 2173	. . 3 |- ( ( ( z = g /\ u = s ) /\ ( w = h /\ v = t ) ) -> ( ( z .N u ) = ( w .N v ) <-> ( g .N s ) = ( h .N t ) ) )
14	13	an42s 579	. 2 |- ( ( ( z = g /\ w = h ) /\ ( v = t /\ u = s ) ) -> ( ( z .N u ) = ( w .N v ) <-> ( g .N s ) = ( h .N t ) ) )
15		dfplpq2 7274	. 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 5833	. . . . 5 |- ( ( w = a /\ f = h ) -> ( w .N f ) = ( a .N h ) )
17		oveq12 5833	. . . . 5 |- ( ( v = b /\ u = g ) -> ( v .N u ) = ( b .N g ) )
18	16, 17	oveqan12d 5843	. . . 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 579	. . 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 5833	. . . 4 |- ( ( v = b /\ f = h ) -> ( v .N f ) = ( b .N h ) )
21	20	ad2ant2l 500	. . 3 |- ( ( ( w = a /\ v = b ) /\ ( u = g /\ f = h ) ) -> ( v .N f ) = ( b .N h ) )
22	19, 21	opeq12d 3749	. 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 5833	. . . . 5 |- ( ( w = c /\ f = s ) -> ( w .N f ) = ( c .N s ) )
24		oveq12 5833	. . . . 5 |- ( ( v = d /\ u = t ) -> ( v .N u ) = ( d .N t ) )
25	23, 24	oveqan12d 5843	. . . 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 579	. . 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 5833	. . . 4 |- ( ( v = d /\ f = s ) -> ( v .N f ) = ( d .N s ) )
28	27	ad2ant2l 500	. . 3 |- ( ( ( w = c /\ v = d ) /\ ( u = t /\ f = s ) ) -> ( v .N f ) = ( d .N s ) )
29	26, 28	opeq12d 3749	. 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 5833	. . . . 5 |- ( ( w = A /\ f = D ) -> ( w .N f ) = ( A .N D ) )
31		oveq12 5833	. . . . 5 |- ( ( v = B /\ u = C ) -> ( v .N u ) = ( B .N C ) )
32	30, 31	oveqan12d 5843	. . . 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 579	. . 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 5833	. . . 4 |- ( ( v = B /\ f = D ) -> ( v .N f ) = ( B .N D ) )
35	34	ad2ant2l 500	. . 3 |- ( ( ( w = A /\ v = B ) /\ ( u = C /\ f = D ) ) -> ( v .N f ) = ( B .N D ) )
36	33, 35	opeq12d 3749	. 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 7269	. 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 7268	. 2 |- Q. = ( ( N. X. N. ) /. ~Q )
39		addcmpblnq 7287	. 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 6586	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 103   <-> wb 104   = wceq 1335   e. wcel 2128   <.cop 3563  (class class class)co 5824   [cec 6478   N.cnpi 7192   +N cpli 7193   .N cmi 7194   +pQ cplpq 7196   ~Q ceq 7199   Q.cnq 7200   +Q cplq 7202

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4079  ax-sep 4082  ax-nul 4090  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4496  ax-iinf 4547

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-int 3808  df-iun 3851  df-br 3966  df-opab 4026  df-mpt 4027  df-tr 4063  df-id 4253  df-iord 4326  df-on 4328  df-suc 4331  df-iom 4550  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-rn 4597  df-res 4598  df-ima 4599  df-iota 5135  df-fun 5172  df-fn 5173  df-f 5174  df-f1 5175  df-fo 5176  df-f1o 5177  df-fv 5178  df-ov 5827  df-oprab 5828  df-mpo 5829  df-1st 6088  df-2nd 6089  df-recs 6252  df-irdg 6317  df-oadd 6367  df-omul 6368  df-er 6480  df-ec 6482  df-qs 6486  df-ni 7224  df-pli 7225  df-mi 7226  df-plpq 7264  df-enq 7267  df-nqqs 7268  df-plqqs 7269

This theorem is referenced by:  addclnq  7295  addcomnqg  7301  addassnqg  7302  distrnqg  7307  ltanqg  7320  1lt2nq  7326  ltexnqq  7328  nqnq0a  7374  addpinq1  7384