Theorem enqbreq2 7165

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

Assertion

Ref	Expression
enqbreq2	|- ( ( A e. ( N. X. N. ) /\ B e. ( N. X. N. ) ) -> ( A ~Q B <-> ( ( 1st ` A ) .N ( 2nd ` B ) ) = ( ( 1st ` B ) .N ( 2nd ` A ) ) ) )

Proof of Theorem enqbreq2

Step	Hyp	Ref	Expression
1		1st2nd2 6073	. . 3 |- ( A e. ( N. X. N. ) -> A = <. ( 1st ` A ) , ( 2nd ` A ) >. )
2		1st2nd2 6073	. . 3 |- ( B e. ( N. X. N. ) -> B = <. ( 1st ` B ) , ( 2nd ` B ) >. )
3	1, 2	breqan12d 3945	. 2 |- ( ( A e. ( N. X. N. ) /\ B e. ( N. X. N. ) ) -> ( A ~Q B <-> <. ( 1st ` A ) , ( 2nd ` A ) >. ~Q <. ( 1st ` B ) , ( 2nd ` B ) >. ) )
4		xp1st 6063	. . . 4 |- ( A e. ( N. X. N. ) -> ( 1st ` A ) e. N. )
5		xp2nd 6064	. . . 4 |- ( A e. ( N. X. N. ) -> ( 2nd ` A ) e. N. )
6	4, 5	jca 304	. . 3 |- ( A e. ( N. X. N. ) -> ( ( 1st ` A ) e. N. /\ ( 2nd ` A ) e. N. ) )
7		xp1st 6063	. . . 4 |- ( B e. ( N. X. N. ) -> ( 1st ` B ) e. N. )
8		xp2nd 6064	. . . 4 |- ( B e. ( N. X. N. ) -> ( 2nd ` B ) e. N. )
9	7, 8	jca 304	. . 3 |- ( B e. ( N. X. N. ) -> ( ( 1st ` B ) e. N. /\ ( 2nd ` B ) e. N. ) )
10		enqbreq 7164	. . 3 |- ( ( ( ( 1st ` A ) e. N. /\ ( 2nd ` A ) e. N. ) /\ ( ( 1st ` B ) e. N. /\ ( 2nd ` B ) e. N. ) ) -> ( <. ( 1st ` A ) , ( 2nd ` A ) >. ~Q <. ( 1st ` B ) , ( 2nd ` B ) >. <-> ( ( 1st ` A ) .N ( 2nd ` B ) ) = ( ( 2nd ` A ) .N ( 1st ` B ) ) ) )
11	6, 9, 10	syl2an 287	. 2 |- ( ( A e. ( N. X. N. ) /\ B e. ( N. X. N. ) ) -> ( <. ( 1st ` A ) , ( 2nd ` A ) >. ~Q <. ( 1st ` B ) , ( 2nd ` B ) >. <-> ( ( 1st ` A ) .N ( 2nd ` B ) ) = ( ( 2nd ` A ) .N ( 1st ` B ) ) ) )
12		mulcompig 7139	. . . 4 |- ( ( ( 2nd ` A ) e. N. /\ ( 1st ` B ) e. N. ) -> ( ( 2nd ` A ) .N ( 1st ` B ) ) = ( ( 1st ` B ) .N ( 2nd ` A ) ) )
13	5, 7, 12	syl2an 287	. . 3 |- ( ( A e. ( N. X. N. ) /\ B e. ( N. X. N. ) ) -> ( ( 2nd ` A ) .N ( 1st ` B ) ) = ( ( 1st ` B ) .N ( 2nd ` A ) ) )
14	13	eqeq2d 2151	. 2 |- ( ( A e. ( N. X. N. ) /\ B e. ( N. X. N. ) ) -> ( ( ( 1st ` A ) .N ( 2nd ` B ) ) = ( ( 2nd ` A ) .N ( 1st ` B ) ) <-> ( ( 1st ` A ) .N ( 2nd ` B ) ) = ( ( 1st ` B ) .N ( 2nd ` A ) ) ) )
15	3, 11, 14	3bitrd 213	1 |- ( ( A e. ( N. X. N. ) /\ B e. ( N. X. N. ) ) -> ( A ~Q B <-> ( ( 1st ` A ) .N ( 2nd ` B ) ) = ( ( 1st ` B ) .N ( 2nd ` A ) ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1331   e. wcel 1480   <.cop 3530   class class class wbr 3929   X. cxp 4537   ` cfv 5123  (class class class)co 5774   1stc1st 6036   2ndc2nd 6037   N.cnpi 7080   .N cmi 7082   ~Q ceq 7087

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-iord 4288  df-on 4290  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-irdg 6267  df-oadd 6317  df-omul 6318  df-ni 7112  df-mi 7114  df-enq 7155

This theorem is referenced by: (None)