Theorem mulpipq2 7455

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

Assertion

Ref	Expression
mulpipq2	|- ( ( A e. ( N. X. N. ) /\ B e. ( N. X. N. ) ) -> ( A .pQ B ) = <. ( ( 1st ` A ) .N ( 1st ` B ) ) , ( ( 2nd ` A ) .N ( 2nd ` B ) ) >. )

Proof of Theorem mulpipq2

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

Step	Hyp	Ref	Expression
1		xp1st 6232	. . . 4 |- ( A e. ( N. X. N. ) -> ( 1st ` A ) e. N. )
2		xp1st 6232	. . . 4 |- ( B e. ( N. X. N. ) -> ( 1st ` B ) e. N. )
3		mulclpi 7412	. . . 4 |- ( ( ( 1st ` A ) e. N. /\ ( 1st ` B ) e. N. ) -> ( ( 1st ` A ) .N ( 1st ` B ) ) e. N. )
4	1, 2, 3	syl2an 289	. . 3 |- ( ( A e. ( N. X. N. ) /\ B e. ( N. X. N. ) ) -> ( ( 1st ` A ) .N ( 1st ` B ) ) e. N. )
5		xp2nd 6233	. . . 4 |- ( A e. ( N. X. N. ) -> ( 2nd ` A ) e. N. )
6		xp2nd 6233	. . . 4 |- ( B e. ( N. X. N. ) -> ( 2nd ` B ) e. N. )
7		mulclpi 7412	. . . 4 |- ( ( ( 2nd ` A ) e. N. /\ ( 2nd ` B ) e. N. ) -> ( ( 2nd ` A ) .N ( 2nd ` B ) ) e. N. )
8	5, 6, 7	syl2an 289	. . 3 |- ( ( A e. ( N. X. N. ) /\ B e. ( N. X. N. ) ) -> ( ( 2nd ` A ) .N ( 2nd ` B ) ) e. N. )
9		opexg 4262	. . 3 |- ( ( ( ( 1st ` A ) .N ( 1st ` B ) ) e. N. /\ ( ( 2nd ` A ) .N ( 2nd ` B ) ) e. N. ) -> <. ( ( 1st ` A ) .N ( 1st ` B ) ) , ( ( 2nd ` A ) .N ( 2nd ` B ) ) >. e. _V )
10	4, 8, 9	syl2anc 411	. 2 |- ( ( A e. ( N. X. N. ) /\ B e. ( N. X. N. ) ) -> <. ( ( 1st ` A ) .N ( 1st ` B ) ) , ( ( 2nd ` A ) .N ( 2nd ` B ) ) >. e. _V )
11		fveq2 5561	. . . . 5 |- ( x = A -> ( 1st ` x ) = ( 1st ` A ) )
12	11	oveq1d 5940	. . . 4 |- ( x = A -> ( ( 1st ` x ) .N ( 1st ` y ) ) = ( ( 1st ` A ) .N ( 1st ` y ) ) )
13		fveq2 5561	. . . . 5 |- ( x = A -> ( 2nd ` x ) = ( 2nd ` A ) )
14	13	oveq1d 5940	. . . 4 |- ( x = A -> ( ( 2nd ` x ) .N ( 2nd ` y ) ) = ( ( 2nd ` A ) .N ( 2nd ` y ) ) )
15	12, 14	opeq12d 3817	. . 3 |- ( x = A -> <. ( ( 1st ` x ) .N ( 1st ` y ) ) , ( ( 2nd ` x ) .N ( 2nd ` y ) ) >. = <. ( ( 1st ` A ) .N ( 1st ` y ) ) , ( ( 2nd ` A ) .N ( 2nd ` y ) ) >. )
16		fveq2 5561	. . . . 5 |- ( y = B -> ( 1st ` y ) = ( 1st ` B ) )
17	16	oveq2d 5941	. . . 4 |- ( y = B -> ( ( 1st ` A ) .N ( 1st ` y ) ) = ( ( 1st ` A ) .N ( 1st ` B ) ) )
18		fveq2 5561	. . . . 5 |- ( y = B -> ( 2nd ` y ) = ( 2nd ` B ) )
19	18	oveq2d 5941	. . . 4 |- ( y = B -> ( ( 2nd ` A ) .N ( 2nd ` y ) ) = ( ( 2nd ` A ) .N ( 2nd ` B ) ) )
20	17, 19	opeq12d 3817	. . 3 |- ( y = B -> <. ( ( 1st ` A ) .N ( 1st ` y ) ) , ( ( 2nd ` A ) .N ( 2nd ` y ) ) >. = <. ( ( 1st ` A ) .N ( 1st ` B ) ) , ( ( 2nd ` A ) .N ( 2nd ` B ) ) >. )
21		df-mpq 7429	. . 3 |- .pQ = ( x e. ( N. X. N. ) , y e. ( N. X. N. ) |-> <. ( ( 1st ` x ) .N ( 1st ` y ) ) , ( ( 2nd ` x ) .N ( 2nd ` y ) ) >. )
22	15, 20, 21	ovmpog 6061	. 2 |- ( ( A e. ( N. X. N. ) /\ B e. ( N. X. N. ) /\ <. ( ( 1st ` A ) .N ( 1st ` B ) ) , ( ( 2nd ` A ) .N ( 2nd ` B ) ) >. e. _V ) -> ( A .pQ B ) = <. ( ( 1st ` A ) .N ( 1st ` B ) ) , ( ( 2nd ` A ) .N ( 2nd ` B ) ) >. )
23	10, 22	mpd3an3 1349	1 |- ( ( A e. ( N. X. N. ) /\ B e. ( N. X. N. ) ) -> ( A .pQ B ) = <. ( ( 1st ` A ) .N ( 1st ` B ) ) , ( ( 2nd ` A ) .N ( 2nd ` B ) ) >. )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2167   _Vcvv 2763   <.cop 3626   X. cxp 4662   ` cfv 5259  (class class class)co 5925   1stc1st 6205   2ndc2nd 6206   N.cnpi 7356   .N cmi 7358   .pQ cmpq 7361

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-id 4329  df-iord 4402  df-on 4404  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-recs 6372  df-irdg 6437  df-oadd 6487  df-omul 6488  df-ni 7388  df-mi 7390  df-mpq 7429

This theorem is referenced by:  mulpipq  7456