Theorem enqbreq2 7724

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 6409	. . 3 |- ( A e. ( N. X. N. ) -> A = <. ( 1st ` A ) , ( 2nd ` A ) >. )
2		1st2nd2 6409	. . 3 |- ( B e. ( N. X. N. ) -> B = <. ( 1st ` B ) , ( 2nd ` B ) >. )
3	1, 2	breqan12d 4146	. 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 6399	. . . 4 |- ( A e. ( N. X. N. ) -> ( 1st ` A ) e. N. )
5		xp2nd 6400	. . . 4 |- ( A e. ( N. X. N. ) -> ( 2nd ` A ) e. N. )
6	4, 5	jca 306	. . 3 |- ( A e. ( N. X. N. ) -> ( ( 1st ` A ) e. N. /\ ( 2nd ` A ) e. N. ) )
7		xp1st 6399	. . . 4 |- ( B e. ( N. X. N. ) -> ( 1st ` B ) e. N. )
8		xp2nd 6400	. . . 4 |- ( B e. ( N. X. N. ) -> ( 2nd ` B ) e. N. )
9	7, 8	jca 306	. . 3 |- ( B e. ( N. X. N. ) -> ( ( 1st ` B ) e. N. /\ ( 2nd ` B ) e. N. ) )
10		enqbreq 7723	. . 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 289	. 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 7698	. . . 4 |- ( ( ( 2nd ` A ) e. N. /\ ( 1st ` B ) e. N. ) -> ( ( 2nd ` A ) .N ( 1st ` B ) ) = ( ( 1st ` B ) .N ( 2nd ` A ) ) )
13	5, 7, 12	syl2an 289	. . 3 |- ( ( A e. ( N. X. N. ) /\ B e. ( N. X. N. ) ) -> ( ( 2nd ` A ) .N ( 1st ` B ) ) = ( ( 1st ` B ) .N ( 2nd ` A ) ) )
14	13	eqeq2d 2250	. 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 214	1 |- ( ( A e. ( N. X. N. ) /\ B e. ( N. X. N. ) ) -> ( A ~Q B <-> ( ( 1st ` A ) .N ( 2nd ` B ) ) = ( ( 1st ` B ) .N ( 2nd ` A ) ) ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   <.cop 3712   class class class wbr 4130   X. cxp 4772   ` cfv 5377  (class class class)co 6085   1stc1st 6372   2ndc2nd 6373   N.cnpi 7639   .N cmi 7641   ~Q ceq 7646

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-iord 4511  df-on 4513  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-irdg 6641  df-oadd 6691  df-omul 6692  df-ni 7671  df-mi 7673  df-enq 7714

This theorem is used by: (None)