Theorem mulpipq 7334

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 4643	. . 3 |- ( ( A e. N. /\ B e. N. ) -> <. A , B >. e. ( N. X. N. ) )
2		opelxpi 4643	. . 3 |- ( ( C e. N. /\ D e. N. ) -> <. C , D >. e. ( N. X. N. ) )
3		mulpipq2 7333	. . 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 287	. 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 6129	. . . 4 |- ( ( A e. N. /\ B e. N. ) -> ( 1st ` <. A , B >. ) = A )
6		op1stg 6129	. . . 4 |- ( ( C e. N. /\ D e. N. ) -> ( 1st ` <. C , D >. ) = C )
7	5, 6	oveqan12d 5872	. . 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 6130	. . . 4 |- ( ( A e. N. /\ B e. N. ) -> ( 2nd ` <. A , B >. ) = B )
9		op2ndg 6130	. . . 4 |- ( ( C e. N. /\ D e. N. ) -> ( 2nd ` <. C , D >. ) = D )
10	8, 9	oveqan12d 5872	. . 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 3773	. 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 2203	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 103   = wceq 1348   e. wcel 2141   <.cop 3586   X. cxp 4609   ` cfv 5198  (class class class)co 5853   1stc1st 6117   2ndc2nd 6118   N.cnpi 7234   .N cmi 7236   .pQ cmpq 7239

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-iord 4351  df-on 4353  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-irdg 6349  df-oadd 6399  df-omul 6400  df-ni 7266  df-mi 7268  df-mpq 7307

This theorem is referenced by: (None)