Theorem mulpqnq 9763

Description: Multiplication of positive fractions in terms of positive integers. (Contributed by NM, 28-Aug-1995.) (Revised by Mario Carneiro, 26-Dec-2014.) (New usage is discouraged.)

Assertion

Ref	Expression
mulpqnq	|- ( ( A e. Q. /\ B e. Q. ) -> ( A .Q B ) = ( /Q ` ( A .pQ B ) ) )

Proof of Theorem mulpqnq

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-mq 9737	. . . . 5 |- .Q = ( ( /Q o. .pQ ) |` ( Q. X. Q. ) )
2	1	fveq1i 6192	. . . 4 |- ( .Q ` <. A , B >. ) = ( ( ( /Q o. .pQ ) |` ( Q. X. Q. ) ) ` <. A , B >. )
3	2	a1i 11	. . 3 |- ( ( A e. Q. /\ B e. Q. ) -> ( .Q ` <. A , B >. ) = ( ( ( /Q o. .pQ ) |` ( Q. X. Q. ) ) ` <. A , B >. ) )
4		opelxpi 5148	. . . 4 |- ( ( A e. Q. /\ B e. Q. ) -> <. A , B >. e. ( Q. X. Q. ) )
5		fvres 6207	. . . 4 |- ( <. A , B >. e. ( Q. X. Q. ) -> ( ( ( /Q o. .pQ ) |` ( Q. X. Q. ) ) ` <. A , B >. ) = ( ( /Q o. .pQ ) ` <. A , B >. ) )
6	4, 5	syl 17	. . 3 |- ( ( A e. Q. /\ B e. Q. ) -> ( ( ( /Q o. .pQ ) |` ( Q. X. Q. ) ) ` <. A , B >. ) = ( ( /Q o. .pQ ) ` <. A , B >. ) )
7		df-mpq 9731	. . . . 5 |- .pQ = ( x e. ( N. X. N. ) , y e. ( N. X. N. ) |-> <. ( ( 1st ` x ) .N ( 1st ` y ) ) , ( ( 2nd ` x ) .N ( 2nd ` y ) ) >. )
8		opex 4932	. . . . 5 |- <. ( ( 1st ` x ) .N ( 1st ` y ) ) , ( ( 2nd ` x ) .N ( 2nd ` y ) ) >. e. _V
9	7, 8	fnmpt2i 7239	. . . 4 |- .pQ Fn ( ( N. X. N. ) X. ( N. X. N. ) )
10		elpqn 9747	. . . . 5 |- ( A e. Q. -> A e. ( N. X. N. ) )
11		elpqn 9747	. . . . 5 |- ( B e. Q. -> B e. ( N. X. N. ) )
12		opelxpi 5148	. . . . 5 |- ( ( A e. ( N. X. N. ) /\ B e. ( N. X. N. ) ) -> <. A , B >. e. ( ( N. X. N. ) X. ( N. X. N. ) ) )
13	10, 11, 12	syl2an 494	. . . 4 |- ( ( A e. Q. /\ B e. Q. ) -> <. A , B >. e. ( ( N. X. N. ) X. ( N. X. N. ) ) )
14		fvco2 6273	. . . 4 |- ( ( .pQ Fn ( ( N. X. N. ) X. ( N. X. N. ) ) /\ <. A , B >. e. ( ( N. X. N. ) X. ( N. X. N. ) ) ) -> ( ( /Q o. .pQ ) ` <. A , B >. ) = ( /Q ` ( .pQ ` <. A , B >. ) ) )
15	9, 13, 14	sylancr 695	. . 3 |- ( ( A e. Q. /\ B e. Q. ) -> ( ( /Q o. .pQ ) ` <. A , B >. ) = ( /Q ` ( .pQ ` <. A , B >. ) ) )
16	3, 6, 15	3eqtrd 2660	. 2 |- ( ( A e. Q. /\ B e. Q. ) -> ( .Q ` <. A , B >. ) = ( /Q ` ( .pQ ` <. A , B >. ) ) )
17		df-ov 6653	. 2 |- ( A .Q B ) = ( .Q ` <. A , B >. )
18		df-ov 6653	. . 3 |- ( A .pQ B ) = ( .pQ ` <. A , B >. )
19	18	fveq2i 6194	. 2 |- ( /Q ` ( A .pQ B ) ) = ( /Q ` ( .pQ ` <. A , B >. ) )
20	16, 17, 19	3eqtr4g 2681	1 |- ( ( A e. Q. /\ B e. Q. ) -> ( A .Q B ) = ( /Q ` ( A .pQ B ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   <.cop 4183   X. cxp 5112   |` cres 5116   o. ccom 5118   Fn wfn 5883   ` cfv 5888  (class class class)co 6650   1stc1st 7166   2ndc2nd 7167   N.cnpi 9666   .N cmi 9668   .pQ cmpq 9671   Q.cnq 9674   /Qcerq 9676   .Q cmq 9678

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-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  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-iun 4522  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-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-mpq 9731  df-nq 9734  df-mq 9737

This theorem is referenced by:  mulclnq  9769  mulcomnq  9775  mulerpq  9779  mulassnq  9781  distrnq  9783  mulidnq  9785  ltmnq  9794