Theorem enqbreq2 7472

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 6263	. . 3 |- ( A e. ( N. X. N. ) -> A = <. ( 1st ` A ) , ( 2nd ` A ) >. )
2		1st2nd2 6263	. . 3 |- ( B e. ( N. X. N. ) -> B = <. ( 1st ` B ) , ( 2nd ` B ) >. )
3	1, 2	breqan12d 4061	. 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 6253	. . . 4 |- ( A e. ( N. X. N. ) -> ( 1st ` A ) e. N. )
5		xp2nd 6254	. . . 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 6253	. . . 4 |- ( B e. ( N. X. N. ) -> ( 1st ` B ) e. N. )
8		xp2nd 6254	. . . 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 7471	. . 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 7446	. . . 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 2217	. 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 ) ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1373   e. wcel 2176   <.cop 3636   class class class wbr 4045   X. cxp 4674   ` cfv 5272  (class class class)co 5946   1stc1st 6226   2ndc2nd 6227   N.cnpi 7387   .N cmi 7389   ~Q ceq 7394

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 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-coll 4160  ax-sep 4163  ax-nul 4171  ax-pow 4219  ax-pr 4254  ax-un 4481  ax-setind 4586  ax-iinf 4637

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-ral 2489  df-rex 2490  df-reu 2491  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-br 4046  df-opab 4107  df-mpt 4108  df-tr 4144  df-id 4341  df-iord 4414  df-on 4416  df-suc 4419  df-iom 4640  df-xp 4682  df-rel 4683  df-cnv 4684  df-co 4685  df-dm 4686  df-rn 4687  df-res 4688  df-ima 4689  df-iota 5233  df-fun 5274  df-fn 5275  df-f 5276  df-f1 5277  df-fo 5278  df-f1o 5279  df-fv 5280  df-ov 5949  df-oprab 5950  df-mpo 5951  df-1st 6228  df-2nd 6229  df-recs 6393  df-irdg 6458  df-oadd 6508  df-omul 6509  df-ni 7419  df-mi 7421  df-enq 7462

This theorem is referenced by: (None)