Theorem mulpipq2 7554

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 6309	. . . 4 |- ( A e. ( N. X. N. ) -> ( 1st ` A ) e. N. )
2		xp1st 6309	. . . 4 |- ( B e. ( N. X. N. ) -> ( 1st ` B ) e. N. )
3		mulclpi 7511	. . . 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 6310	. . . 4 |- ( A e. ( N. X. N. ) -> ( 2nd ` A ) e. N. )
6		xp2nd 6310	. . . 4 |- ( B e. ( N. X. N. ) -> ( 2nd ` B ) e. N. )
7		mulclpi 7511	. . . 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 4313	. . 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 5626	. . . . 5 |- ( x = A -> ( 1st ` x ) = ( 1st ` A ) )
12	11	oveq1d 6015	. . . 4 |- ( x = A -> ( ( 1st ` x ) .N ( 1st ` y ) ) = ( ( 1st ` A ) .N ( 1st ` y ) ) )
13		fveq2 5626	. . . . 5 |- ( x = A -> ( 2nd ` x ) = ( 2nd ` A ) )
14	13	oveq1d 6015	. . . 4 |- ( x = A -> ( ( 2nd ` x ) .N ( 2nd ` y ) ) = ( ( 2nd ` A ) .N ( 2nd ` y ) ) )
15	12, 14	opeq12d 3864	. . 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 5626	. . . . 5 |- ( y = B -> ( 1st ` y ) = ( 1st ` B ) )
17	16	oveq2d 6016	. . . 4 |- ( y = B -> ( ( 1st ` A ) .N ( 1st ` y ) ) = ( ( 1st ` A ) .N ( 1st ` B ) ) )
18		fveq2 5626	. . . . 5 |- ( y = B -> ( 2nd ` y ) = ( 2nd ` B ) )
19	18	oveq2d 6016	. . . 4 |- ( y = B -> ( ( 2nd ` A ) .N ( 2nd ` y ) ) = ( ( 2nd ` A ) .N ( 2nd ` B ) ) )
20	17, 19	opeq12d 3864	. . 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 7528	. . 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 6138	. 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 1372	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 1395   e. wcel 2200   _Vcvv 2799   <.cop 3669   X. cxp 4716   ` cfv 5317  (class class class)co 6000   1stc1st 6282   2ndc2nd 6283   N.cnpi 7455   .N cmi 7457   .pQ cmpq 7460

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4198  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-iinf 4679

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-tr 4182  df-id 4383  df-iord 4456  df-on 4458  df-suc 4461  df-iom 4682  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325  df-ov 6003  df-oprab 6004  df-mpo 6005  df-1st 6284  df-2nd 6285  df-recs 6449  df-irdg 6514  df-oadd 6564  df-omul 6565  df-ni 7487  df-mi 7489  df-mpq 7528

This theorem is referenced by:  mulpipq  7555