Theorem addpipq 9759

Description: Addition of positive fractions in terms of positive integers. (Contributed by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)

Assertion

Ref	Expression
addpipq	|- ( ( ( A e. N. /\ B e. N. ) /\ ( C e. N. /\ D e. N. ) ) -> ( <. A , B >. +pQ <. C , D >. ) = <. ( ( A .N D ) +N ( C .N B ) ) , ( B .N D ) >. )

Proof of Theorem addpipq

Step	Hyp	Ref	Expression
1		opelxpi 5148	. . 3 |- ( ( A e. N. /\ B e. N. ) -> <. A , B >. e. ( N. X. N. ) )
2		opelxpi 5148	. . 3 |- ( ( C e. N. /\ D e. N. ) -> <. C , D >. e. ( N. X. N. ) )
3		addpipq2 9758	. . 3 |- ( ( <. A , B >. e. ( N. X. N. ) /\ <. C , D >. e. ( N. X. N. ) ) -> ( <. A , B >. +pQ <. C , D >. ) = <. ( ( ( 1st ` <. A , B >. ) .N ( 2nd ` <. C , D >. ) ) +N ( ( 1st ` <. C , D >. ) .N ( 2nd ` <. A , B >. ) ) ) , ( ( 2nd ` <. A , B >. ) .N ( 2nd ` <. C , D >. ) ) >. )
4	1, 2, 3	syl2an 494	. 2 |- ( ( ( A e. N. /\ B e. N. ) /\ ( C e. N. /\ D e. N. ) ) -> ( <. A , B >. +pQ <. C , D >. ) = <. ( ( ( 1st ` <. A , B >. ) .N ( 2nd ` <. C , D >. ) ) +N ( ( 1st ` <. C , D >. ) .N ( 2nd ` <. A , B >. ) ) ) , ( ( 2nd ` <. A , B >. ) .N ( 2nd ` <. C , D >. ) ) >. )
5		op1stg 7180	. . . . 5 |- ( ( A e. N. /\ B e. N. ) -> ( 1st ` <. A , B >. ) = A )
6		op2ndg 7181	. . . . 5 |- ( ( C e. N. /\ D e. N. ) -> ( 2nd ` <. C , D >. ) = D )
7	5, 6	oveqan12d 6669	. . . 4 |- ( ( ( A e. N. /\ B e. N. ) /\ ( C e. N. /\ D e. N. ) ) -> ( ( 1st ` <. A , B >. ) .N ( 2nd ` <. C , D >. ) ) = ( A .N D ) )
8		op1stg 7180	. . . . 5 |- ( ( C e. N. /\ D e. N. ) -> ( 1st ` <. C , D >. ) = C )
9		op2ndg 7181	. . . . 5 |- ( ( A e. N. /\ B e. N. ) -> ( 2nd ` <. A , B >. ) = B )
10	8, 9	oveqan12rd 6670	. . . 4 |- ( ( ( A e. N. /\ B e. N. ) /\ ( C e. N. /\ D e. N. ) ) -> ( ( 1st ` <. C , D >. ) .N ( 2nd ` <. A , B >. ) ) = ( C .N B ) )
11	7, 10	oveq12d 6668	. . 3 |- ( ( ( A e. N. /\ B e. N. ) /\ ( C e. N. /\ D e. N. ) ) -> ( ( ( 1st ` <. A , B >. ) .N ( 2nd ` <. C , D >. ) ) +N ( ( 1st ` <. C , D >. ) .N ( 2nd ` <. A , B >. ) ) ) = ( ( A .N D ) +N ( C .N B ) ) )
12	9, 6	oveqan12d 6669	. . 3 |- ( ( ( A e. N. /\ B e. N. ) /\ ( C e. N. /\ D e. N. ) ) -> ( ( 2nd ` <. A , B >. ) .N ( 2nd ` <. C , D >. ) ) = ( B .N D ) )
13	11, 12	opeq12d 4410	. 2 |- ( ( ( A e. N. /\ B e. N. ) /\ ( C e. N. /\ D e. N. ) ) -> <. ( ( ( 1st ` <. A , B >. ) .N ( 2nd ` <. C , D >. ) ) +N ( ( 1st ` <. C , D >. ) .N ( 2nd ` <. A , B >. ) ) ) , ( ( 2nd ` <. A , B >. ) .N ( 2nd ` <. C , D >. ) ) >. = <. ( ( A .N D ) +N ( C .N B ) ) , ( B .N D ) >. )
14	4, 13	eqtrd 2656	1 |- ( ( ( A e. N. /\ B e. N. ) /\ ( C e. N. /\ D e. N. ) ) -> ( <. A , B >. +pQ <. C , D >. ) = <. ( ( A .N D ) +N ( C .N B ) ) , ( B .N D ) >. )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   <.cop 4183   X. cxp 5112   ` cfv 5888  (class class class)co 6650   1stc1st 7166   2ndc2nd 7167   N.cnpi 9666   +N cpli 9667   .N cmi 9668   +pQ cplpq 9670

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-plpq 9730

This theorem is referenced by:  addassnq  9780  distrnq  9783  1lt2nq  9795  ltexnq  9797  prlem934  9855