Theorem mulpipq2 7345

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 6156	. . . 4 |- ( A e. ( N. X. N. ) -> ( 1st ` A ) e. N. )
2		xp1st 6156	. . . 4 |- ( B e. ( N. X. N. ) -> ( 1st ` B ) e. N. )
3		mulclpi 7302	. . . 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 6157	. . . 4 |- ( A e. ( N. X. N. ) -> ( 2nd ` A ) e. N. )
6		xp2nd 6157	. . . 4 |- ( B e. ( N. X. N. ) -> ( 2nd ` B ) e. N. )
7		mulclpi 7302	. . . 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 4222	. . 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 5507	. . . . 5 |- ( x = A -> ( 1st ` x ) = ( 1st ` A ) )
12	11	oveq1d 5880	. . . 4 |- ( x = A -> ( ( 1st ` x ) .N ( 1st ` y ) ) = ( ( 1st ` A ) .N ( 1st ` y ) ) )
13		fveq2 5507	. . . . 5 |- ( x = A -> ( 2nd ` x ) = ( 2nd ` A ) )
14	13	oveq1d 5880	. . . 4 |- ( x = A -> ( ( 2nd ` x ) .N ( 2nd ` y ) ) = ( ( 2nd ` A ) .N ( 2nd ` y ) ) )
15	12, 14	opeq12d 3782	. . 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 5507	. . . . 5 |- ( y = B -> ( 1st ` y ) = ( 1st ` B ) )
17	16	oveq2d 5881	. . . 4 |- ( y = B -> ( ( 1st ` A ) .N ( 1st ` y ) ) = ( ( 1st ` A ) .N ( 1st ` B ) ) )
18		fveq2 5507	. . . . 5 |- ( y = B -> ( 2nd ` y ) = ( 2nd ` B ) )
19	18	oveq2d 5881	. . . 4 |- ( y = B -> ( ( 2nd ` A ) .N ( 2nd ` y ) ) = ( ( 2nd ` A ) .N ( 2nd ` B ) ) )
20	17, 19	opeq12d 3782	. . 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 7319	. . 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 5999	. 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 1338	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 1353   e. wcel 2146   _Vcvv 2735   <.cop 3592   X. cxp 4618   ` cfv 5208  (class class class)co 5865   1stc1st 6129   2ndc2nd 6130   N.cnpi 7246   .N cmi 7248   .pQ cmpq 7251

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 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-coll 4113  ax-sep 4116  ax-nul 4124  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-setind 4530  ax-iinf 4581

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-ral 2458  df-rex 2459  df-reu 2460  df-rab 2462  df-v 2737  df-sbc 2961  df-csb 3056  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-nul 3421  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-int 3841  df-iun 3884  df-br 3999  df-opab 4060  df-mpt 4061  df-tr 4097  df-id 4287  df-iord 4360  df-on 4362  df-suc 4365  df-iom 4584  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-f1 5213  df-fo 5214  df-f1o 5215  df-fv 5216  df-ov 5868  df-oprab 5869  df-mpo 5870  df-1st 6131  df-2nd 6132  df-recs 6296  df-irdg 6361  df-oadd 6411  df-omul 6412  df-ni 7278  df-mi 7280  df-mpq 7319

This theorem is referenced by:  mulpipq  7346