Theorem mulpipq 7385

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 4670	. . 3 |- ( ( A e. N. /\ B e. N. ) -> <. A , B >. e. ( N. X. N. ) )
2		opelxpi 4670	. . 3 |- ( ( C e. N. /\ D e. N. ) -> <. C , D >. e. ( N. X. N. ) )
3		mulpipq2 7384	. . 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 6165	. . . 4 |- ( ( A e. N. /\ B e. N. ) -> ( 1st ` <. A , B >. ) = A )
6		op1stg 6165	. . . 4 |- ( ( C e. N. /\ D e. N. ) -> ( 1st ` <. C , D >. ) = C )
7	5, 6	oveqan12d 5907	. . 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 6166	. . . 4 |- ( ( A e. N. /\ B e. N. ) -> ( 2nd ` <. A , B >. ) = B )
9		op2ndg 6166	. . . 4 |- ( ( C e. N. /\ D e. N. ) -> ( 2nd ` <. C , D >. ) = D )
10	8, 9	oveqan12d 5907	. . 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 3798	. 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 2220	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 1363   e. wcel 2158   <.cop 3607   X. cxp 4636   ` cfv 5228  (class class class)co 5888   1stc1st 6153   2ndc2nd 6154   N.cnpi 7285   .N cmi 7287   .pQ cmpq 7290

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 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-coll 4130  ax-sep 4133  ax-nul 4141  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-iinf 4599

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-ral 2470  df-rex 2471  df-reu 2472  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-tr 4114  df-id 4305  df-iord 4378  df-on 4380  df-suc 4383  df-iom 4602  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-ov 5891  df-oprab 5892  df-mpo 5893  df-1st 6155  df-2nd 6156  df-recs 6320  df-irdg 6385  df-oadd 6435  df-omul 6436  df-ni 7317  df-mi 7319  df-mpq 7358

This theorem is referenced by: (None)