Theorem enqbreq2 6547

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 5821	. . 3 |- ( A e. ( N. X. N. ) -> A = <. ( 1st ` A ) , ( 2nd ` A ) >. )
2		1st2nd2 5821	. . 3 |- ( B e. ( N. X. N. ) -> B = <. ( 1st ` B ) , ( 2nd ` B ) >. )
3	1, 2	breqan12d 3800	. 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 5812	. . . 4 |- ( A e. ( N. X. N. ) -> ( 1st ` A ) e. N. )
5		xp2nd 5813	. . . 4 |- ( A e. ( N. X. N. ) -> ( 2nd ` A ) e. N. )
6	4, 5	jca 300	. . 3 |- ( A e. ( N. X. N. ) -> ( ( 1st ` A ) e. N. /\ ( 2nd ` A ) e. N. ) )
7		xp1st 5812	. . . 4 |- ( B e. ( N. X. N. ) -> ( 1st ` B ) e. N. )
8		xp2nd 5813	. . . 4 |- ( B e. ( N. X. N. ) -> ( 2nd ` B ) e. N. )
9	7, 8	jca 300	. . 3 |- ( B e. ( N. X. N. ) -> ( ( 1st ` B ) e. N. /\ ( 2nd ` B ) e. N. ) )
10		enqbreq 6546	. . 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 283	. 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 6521	. . . 4 |- ( ( ( 2nd ` A ) e. N. /\ ( 1st ` B ) e. N. ) -> ( ( 2nd ` A ) .N ( 1st ` B ) ) = ( ( 1st ` B ) .N ( 2nd ` A ) ) )
13	5, 7, 12	syl2an 283	. . 3 |- ( ( A e. ( N. X. N. ) /\ B e. ( N. X. N. ) ) -> ( ( 2nd ` A ) .N ( 1st ` B ) ) = ( ( 1st ` B ) .N ( 2nd ` A ) ) )
14	13	eqeq2d 2092	. 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 212	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 102   <-> wb 103   = wceq 1284   e. wcel 1433   <.cop 3401   class class class wbr 3785   X. cxp 4361   ` cfv 4922  (class class class)co 5532   1stc1st 5785   2ndc2nd 5786   N.cnpi 6462   .N cmi 6464   ~Q ceq 6469

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-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-enq 6537

This theorem is referenced by: (None)