Theorem mulpipq2 6561

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 5812	. . . 4 |- ( A e. ( N. X. N. ) -> ( 1st ` A ) e. N. )
2		xp1st 5812	. . . 4 |- ( B e. ( N. X. N. ) -> ( 1st ` B ) e. N. )
3		mulclpi 6518	. . . 4 |- ( ( ( 1st ` A ) e. N. /\ ( 1st ` B ) e. N. ) -> ( ( 1st ` A ) .N ( 1st ` B ) ) e. N. )
4	1, 2, 3	syl2an 283	. . 3 |- ( ( A e. ( N. X. N. ) /\ B e. ( N. X. N. ) ) -> ( ( 1st ` A ) .N ( 1st ` B ) ) e. N. )
5		xp2nd 5813	. . . 4 |- ( A e. ( N. X. N. ) -> ( 2nd ` A ) e. N. )
6		xp2nd 5813	. . . 4 |- ( B e. ( N. X. N. ) -> ( 2nd ` B ) e. N. )
7		mulclpi 6518	. . . 4 |- ( ( ( 2nd ` A ) e. N. /\ ( 2nd ` B ) e. N. ) -> ( ( 2nd ` A ) .N ( 2nd ` B ) ) e. N. )
8	5, 6, 7	syl2an 283	. . 3 |- ( ( A e. ( N. X. N. ) /\ B e. ( N. X. N. ) ) -> ( ( 2nd ` A ) .N ( 2nd ` B ) ) e. N. )
9		opexg 3983	. . 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 403	. 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 5198	. . . . 5 |- ( x = A -> ( 1st ` x ) = ( 1st ` A ) )
12	11	oveq1d 5547	. . . 4 |- ( x = A -> ( ( 1st ` x ) .N ( 1st ` y ) ) = ( ( 1st ` A ) .N ( 1st ` y ) ) )
13		fveq2 5198	. . . . 5 |- ( x = A -> ( 2nd ` x ) = ( 2nd ` A ) )
14	13	oveq1d 5547	. . . 4 |- ( x = A -> ( ( 2nd ` x ) .N ( 2nd ` y ) ) = ( ( 2nd ` A ) .N ( 2nd ` y ) ) )
15	12, 14	opeq12d 3578	. . 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 5198	. . . . 5 |- ( y = B -> ( 1st ` y ) = ( 1st ` B ) )
17	16	oveq2d 5548	. . . 4 |- ( y = B -> ( ( 1st ` A ) .N ( 1st ` y ) ) = ( ( 1st ` A ) .N ( 1st ` B ) ) )
18		fveq2 5198	. . . . 5 |- ( y = B -> ( 2nd ` y ) = ( 2nd ` B ) )
19	18	oveq2d 5548	. . . 4 |- ( y = B -> ( ( 2nd ` A ) .N ( 2nd ` y ) ) = ( ( 2nd ` A ) .N ( 2nd ` B ) ) )
20	17, 19	opeq12d 3578	. . 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 6535	. . 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	ovmpt2g 5655	. 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 1269	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 102   = wceq 1284   e. wcel 1433   _Vcvv 2601   <.cop 3401   X. cxp 4361   ` cfv 4922  (class class class)co 5532   1stc1st 5785   2ndc2nd 5786   N.cnpi 6462   .N cmi 6464   .pQ cmpq 6467

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-coll 3893  ax-sep 3896  ax-nul 3904  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-iinf 4329

This theorem depends on definitions:  df-bi 115  df-dc 776  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-ral 2353  df-rex 2354  df-reu 2355  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-int 3637  df-iun 3680  df-br 3786  df-opab 3840  df-mpt 3841  df-tr 3876  df-id 4048  df-iord 4121  df-on 4123  df-suc 4126  df-iom 4332  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-1st 5787  df-2nd 5788  df-recs 5943  df-irdg 5980  df-oadd 6028  df-omul 6029  df-ni 6494  df-mi 6496  df-mpq 6535

This theorem is referenced by:  mulpipq  6562