Theorem mulpipq 7591

Description: Multiplication of positive fractions in terms of positive integers. (Contributed by NM, 28-Aug-1995.) (Revised by Mario Carneiro, 8-May-2013.)

Assertion

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

Proof of Theorem mulpipq

Step	Hyp	Ref	Expression
1		opelxpi 4757	. . 3 |- ( ( A e. N. /\ B e. N. ) -> <. A , B >. e. ( N. X. N. ) )
2		opelxpi 4757	. . 3 |- ( ( C e. N. /\ D e. N. ) -> <. C , D >. e. ( N. X. N. ) )
3		mulpipq2 7590	. . 3 |- ( ( <. A , B >. e. ( N. X. N. ) /\ <. C , D >. e. ( N. X. N. ) ) -> ( <. A , B >. .pQ <. C , D >. ) = <. ( ( 1st ` <. A , B >. ) .N ( 1st ` <. C , D >. ) ) , ( ( 2nd ` <. A , B >. ) .N ( 2nd ` <. C , D >. ) ) >. )
4	1, 2, 3	syl2an 289	. 2 |- ( ( ( A e. N. /\ B e. N. ) /\ ( C e. N. /\ D e. N. ) ) -> ( <. A , B >. .pQ <. C , D >. ) = <. ( ( 1st ` <. A , B >. ) .N ( 1st ` <. C , D >. ) ) , ( ( 2nd ` <. A , B >. ) .N ( 2nd ` <. C , D >. ) ) >. )
5		op1stg 6312	. . . 4 |- ( ( A e. N. /\ B e. N. ) -> ( 1st ` <. A , B >. ) = A )
6		op1stg 6312	. . . 4 |- ( ( C e. N. /\ D e. N. ) -> ( 1st ` <. C , D >. ) = C )
7	5, 6	oveqan12d 6036	. . 3 |- ( ( ( A e. N. /\ B e. N. ) /\ ( C e. N. /\ D e. N. ) ) -> ( ( 1st ` <. A , B >. ) .N ( 1st ` <. C , D >. ) ) = ( A .N C ) )
8		op2ndg 6313	. . . 4 |- ( ( A e. N. /\ B e. N. ) -> ( 2nd ` <. A , B >. ) = B )
9		op2ndg 6313	. . . 4 |- ( ( C e. N. /\ D e. N. ) -> ( 2nd ` <. C , D >. ) = D )
10	8, 9	oveqan12d 6036	. . 3 |- ( ( ( A e. N. /\ B e. N. ) /\ ( C e. N. /\ D e. N. ) ) -> ( ( 2nd ` <. A , B >. ) .N ( 2nd ` <. C , D >. ) ) = ( B .N D ) )
11	7, 10	opeq12d 3870	. 2 |- ( ( ( A e. N. /\ B e. N. ) /\ ( C e. N. /\ D e. N. ) ) -> <. ( ( 1st ` <. A , B >. ) .N ( 1st ` <. C , D >. ) ) , ( ( 2nd ` <. A , B >. ) .N ( 2nd ` <. C , D >. ) ) >. = <. ( A .N C ) , ( B .N D ) >. )
12	4, 11	eqtrd 2264	1 |- ( ( ( A e. N. /\ B e. N. ) /\ ( C e. N. /\ D e. N. ) ) -> ( <. A , B >. .pQ <. C , D >. ) = <. ( A .N C ) , ( B .N D ) >. )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1397   e. wcel 2202   <.cop 3672   X. cxp 4723   ` cfv 5326  (class class class)co 6017   1stc1st 6300   2ndc2nd 6301   N.cnpi 7491   .N cmi 7493   .pQ cmpq 7496

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-iord 4463  df-on 4465  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-recs 6470  df-irdg 6535  df-oadd 6585  df-omul 6586  df-ni 7523  df-mi 7525  df-mpq 7564

This theorem is referenced by: (None)