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Theorem nqereq 8812
Description: The function  /Q acts as a substitute for equivalence classes, and it satisfies the fundamental requirement for equivalence representatives: the representatives are equal iff the members are equivalent. (Contributed by Mario Carneiro, 6-May-2013.) (Revised by Mario Carneiro, 12-Aug-2015.) (New usage is discouraged.)
Assertion
Ref Expression
nqereq  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. ) )  ->  ( A  ~Q  B  <->  ( /Q `  A )  =  ( /Q `  B ) ) )

Proof of Theorem nqereq
StepHypRef Expression
1 nqercl 8808 . . . . 5  |-  ( A  e.  ( N.  X.  N. )  ->  ( /Q
`  A )  e. 
Q. )
213ad2ant1 978 . . . 4  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. )  /\  A  ~Q  B
)  ->  ( /Q `  A )  e.  Q. )
3 nqercl 8808 . . . . 5  |-  ( B  e.  ( N.  X.  N. )  ->  ( /Q
`  B )  e. 
Q. )
433ad2ant2 979 . . . 4  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. )  /\  A  ~Q  B
)  ->  ( /Q `  B )  e.  Q. )
5 enqer 8798 . . . . . 6  |-  ~Q  Er  ( N.  X.  N. )
65a1i 11 . . . . 5  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. )  /\  A  ~Q  B
)  ->  ~Q  Er  ( N.  X.  N. ) )
7 nqerrel 8809 . . . . . . 7  |-  ( A  e.  ( N.  X.  N. )  ->  A  ~Q  ( /Q `  A ) )
873ad2ant1 978 . . . . . 6  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. )  /\  A  ~Q  B
)  ->  A  ~Q  ( /Q `  A ) )
9 simp3 959 . . . . . 6  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. )  /\  A  ~Q  B
)  ->  A  ~Q  B )
106, 8, 9ertr3d 6923 . . . . 5  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. )  /\  A  ~Q  B
)  ->  ( /Q `  A )  ~Q  B
)
11 nqerrel 8809 . . . . . 6  |-  ( B  e.  ( N.  X.  N. )  ->  B  ~Q  ( /Q `  B ) )
12113ad2ant2 979 . . . . 5  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. )  /\  A  ~Q  B
)  ->  B  ~Q  ( /Q `  B ) )
136, 10, 12ertrd 6921 . . . 4  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. )  /\  A  ~Q  B
)  ->  ( /Q `  A )  ~Q  ( /Q `  B ) )
14 enqeq 8811 . . . 4  |-  ( ( ( /Q `  A
)  e.  Q.  /\  ( /Q `  B )  e.  Q.  /\  ( /Q `  A )  ~Q  ( /Q `  B ) )  ->  ( /Q `  A )  =  ( /Q `  B ) )
152, 4, 13, 14syl3anc 1184 . . 3  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. )  /\  A  ~Q  B
)  ->  ( /Q `  A )  =  ( /Q `  B ) )
16153expia 1155 . 2  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. ) )  ->  ( A  ~Q  B  ->  ( /Q `  A )  =  ( /Q `  B
) ) )
175a1i 11 . . . 4  |-  ( ( A  e.  ( N. 
X.  N. )  /\  ( B  e.  ( N.  X.  N. )  /\  ( /Q `  A )  =  ( /Q `  B
) ) )  ->  ~Q  Er  ( N.  X.  N. ) )
187adantr 452 . . . . 5  |-  ( ( A  e.  ( N. 
X.  N. )  /\  ( B  e.  ( N.  X.  N. )  /\  ( /Q `  A )  =  ( /Q `  B
) ) )  ->  A  ~Q  ( /Q `  A ) )
19 simprr 734 . . . . 5  |-  ( ( A  e.  ( N. 
X.  N. )  /\  ( B  e.  ( N.  X.  N. )  /\  ( /Q `  A )  =  ( /Q `  B
) ) )  -> 
( /Q `  A
)  =  ( /Q
`  B ) )
2018, 19breqtrd 4236 . . . 4  |-  ( ( A  e.  ( N. 
X.  N. )  /\  ( B  e.  ( N.  X.  N. )  /\  ( /Q `  A )  =  ( /Q `  B
) ) )  ->  A  ~Q  ( /Q `  B ) )
2111ad2antrl 709 . . . 4  |-  ( ( A  e.  ( N. 
X.  N. )  /\  ( B  e.  ( N.  X.  N. )  /\  ( /Q `  A )  =  ( /Q `  B
) ) )  ->  B  ~Q  ( /Q `  B ) )
2217, 20, 21ertr4d 6924 . . 3  |-  ( ( A  e.  ( N. 
X.  N. )  /\  ( B  e.  ( N.  X.  N. )  /\  ( /Q `  A )  =  ( /Q `  B
) ) )  ->  A  ~Q  B )
2322expr 599 . 2  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. ) )  ->  (
( /Q `  A
)  =  ( /Q
`  B )  ->  A  ~Q  B ) )
2416, 23impbid 184 1  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. ) )  ->  ( A  ~Q  B  <->  ( /Q `  A )  =  ( /Q `  B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   class class class wbr 4212    X. cxp 4876   ` cfv 5454    Er wer 6902   N.cnpi 8719    ~Q ceq 8726   Q.cnq 8727   /Qcerq 8729
This theorem is referenced by:  adderpq  8833  mulerpq  8834  distrnq  8838  recmulnq  8841  ltexnq  8852
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-recs 6633  df-rdg 6668  df-1o 6724  df-oadd 6728  df-omul 6729  df-er 6905  df-ni 8749  df-mi 8751  df-lti 8752  df-enq 8788  df-nq 8789  df-erq 8790  df-1nq 8793
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