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Theorem nqereq 8513
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 8509 . . . . 5  |-  ( A  e.  ( N.  X.  N. )  ->  ( /Q
`  A )  e. 
Q. )
213ad2ant1 981 . . . 4  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. )  /\  A  ~Q  B
)  ->  ( /Q `  A )  e.  Q. )
3 nqercl 8509 . . . . 5  |-  ( B  e.  ( N.  X.  N. )  ->  ( /Q
`  B )  e. 
Q. )
433ad2ant2 982 . . . 4  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. )  /\  A  ~Q  B
)  ->  ( /Q `  B )  e.  Q. )
5 enqer 8499 . . . . . 6  |-  ~Q  Er  ( N.  X.  N. )
65a1i 12 . . . . 5  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. )  /\  A  ~Q  B
)  ->  ~Q  Er  ( N.  X.  N. ) )
7 nqerrel 8510 . . . . . . 7  |-  ( A  e.  ( N.  X.  N. )  ->  A  ~Q  ( /Q `  A ) )
873ad2ant1 981 . . . . . 6  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. )  /\  A  ~Q  B
)  ->  A  ~Q  ( /Q `  A ) )
9 simp3 962 . . . . . 6  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. )  /\  A  ~Q  B
)  ->  A  ~Q  B )
106, 8, 9ertr3d 6632 . . . . 5  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. )  /\  A  ~Q  B
)  ->  ( /Q `  A )  ~Q  B
)
11 nqerrel 8510 . . . . . 6  |-  ( B  e.  ( N.  X.  N. )  ->  B  ~Q  ( /Q `  B ) )
12113ad2ant2 982 . . . . 5  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. )  /\  A  ~Q  B
)  ->  B  ~Q  ( /Q `  B ) )
136, 10, 12ertrd 6630 . . . 4  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. )  /\  A  ~Q  B
)  ->  ( /Q `  A )  ~Q  ( /Q `  B ) )
14 enqeq 8512 . . . 4  |-  ( ( ( /Q `  A
)  e.  Q.  /\  ( /Q `  B )  e.  Q.  /\  ( /Q `  A )  ~Q  ( /Q `  B ) )  ->  ( /Q `  A )  =  ( /Q `  B ) )
152, 4, 13, 14syl3anc 1187 . . 3  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. )  /\  A  ~Q  B
)  ->  ( /Q `  A )  =  ( /Q `  B ) )
16153expia 1158 . 2  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. ) )  ->  ( A  ~Q  B  ->  ( /Q `  A )  =  ( /Q `  B
) ) )
175a1i 12 . . . 4  |-  ( ( A  e.  ( N. 
X.  N. )  /\  ( B  e.  ( N.  X.  N. )  /\  ( /Q `  A )  =  ( /Q `  B
) ) )  ->  ~Q  Er  ( N.  X.  N. ) )
187adantr 453 . . . . 5  |-  ( ( A  e.  ( N. 
X.  N. )  /\  ( B  e.  ( N.  X.  N. )  /\  ( /Q `  A )  =  ( /Q `  B
) ) )  ->  A  ~Q  ( /Q `  A ) )
19 simprr 736 . . . . 5  |-  ( ( A  e.  ( N. 
X.  N. )  /\  ( B  e.  ( N.  X.  N. )  /\  ( /Q `  A )  =  ( /Q `  B
) ) )  -> 
( /Q `  A
)  =  ( /Q
`  B ) )
2018, 19breqtrd 4007 . . . 4  |-  ( ( A  e.  ( N. 
X.  N. )  /\  ( B  e.  ( N.  X.  N. )  /\  ( /Q `  A )  =  ( /Q `  B
) ) )  ->  A  ~Q  ( /Q `  B ) )
2111ad2antrl 711 . . . 4  |-  ( ( A  e.  ( N. 
X.  N. )  /\  ( B  e.  ( N.  X.  N. )  /\  ( /Q `  A )  =  ( /Q `  B
) ) )  ->  B  ~Q  ( /Q `  B ) )
2217, 20, 21ertr4d 6633 . . 3  |-  ( ( A  e.  ( N. 
X.  N. )  /\  ( B  e.  ( N.  X.  N. )  /\  ( /Q `  A )  =  ( /Q `  B
) ) )  ->  A  ~Q  B )
2322expr 601 . 2  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. ) )  ->  (
( /Q `  A
)  =  ( /Q
`  B )  ->  A  ~Q  B ) )
2416, 23impbid 185 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 6    <-> wb 178    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621   class class class wbr 3983    X. cxp 4645   ` cfv 4659    Er wer 6611   N.cnpi 8420    ~Q ceq 8427   Q.cnq 8428   /Qcerq 8430
This theorem is referenced by:  adderpq  8534  mulerpq  8535  distrnq  8539  recmulnq  8542  ltexnq  8553
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-sep 4101  ax-nul 4109  ax-pr 4172  ax-un 4470
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2521  df-rex 2522  df-reu 2523  df-rmo 2524  df-rab 2525  df-v 2759  df-sbc 2953  df-csb 3043  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-pss 3129  df-nul 3417  df-if 3526  df-pw 3587  df-sn 3606  df-pr 3607  df-tp 3608  df-op 3609  df-uni 3788  df-iun 3867  df-br 3984  df-opab 4038  df-mpt 4039  df-tr 4074  df-eprel 4263  df-id 4267  df-po 4272  df-so 4273  df-fr 4310  df-we 4312  df-ord 4353  df-on 4354  df-lim 4355  df-suc 4356  df-om 4615  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-fun 4669  df-fn 4670  df-f 4671  df-f1 4672  df-fo 4673  df-f1o 4674  df-fv 4675  df-ov 5781  df-oprab 5782  df-mpt2 5783  df-1st 6042  df-2nd 6043  df-recs 6342  df-rdg 6377  df-1o 6433  df-oadd 6437  df-omul 6438  df-er 6614  df-ni 8450  df-mi 8452  df-lti 8453  df-enq 8489  df-nq 8490  df-erq 8491  df-1nq 8494
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