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Theorem nqereq 9202
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 9198 . . . . 5  |-  ( A  e.  ( N.  X.  N. )  ->  ( /Q
`  A )  e. 
Q. )
213ad2ant1 1009 . . . 4  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. )  /\  A  ~Q  B
)  ->  ( /Q `  A )  e.  Q. )
3 nqercl 9198 . . . . 5  |-  ( B  e.  ( N.  X.  N. )  ->  ( /Q
`  B )  e. 
Q. )
433ad2ant2 1010 . . . 4  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. )  /\  A  ~Q  B
)  ->  ( /Q `  B )  e.  Q. )
5 enqer 9188 . . . . . 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 9199 . . . . . . 7  |-  ( A  e.  ( N.  X.  N. )  ->  A  ~Q  ( /Q `  A ) )
873ad2ant1 1009 . . . . . 6  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. )  /\  A  ~Q  B
)  ->  A  ~Q  ( /Q `  A ) )
9 simp3 990 . . . . . 6  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. )  /\  A  ~Q  B
)  ->  A  ~Q  B )
106, 8, 9ertr3d 7216 . . . . 5  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. )  /\  A  ~Q  B
)  ->  ( /Q `  A )  ~Q  B
)
11 nqerrel 9199 . . . . . 6  |-  ( B  e.  ( N.  X.  N. )  ->  B  ~Q  ( /Q `  B ) )
12113ad2ant2 1010 . . . . 5  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. )  /\  A  ~Q  B
)  ->  B  ~Q  ( /Q `  B ) )
136, 10, 12ertrd 7214 . . . 4  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. )  /\  A  ~Q  B
)  ->  ( /Q `  A )  ~Q  ( /Q `  B ) )
14 enqeq 9201 . . . 4  |-  ( ( ( /Q `  A
)  e.  Q.  /\  ( /Q `  B )  e.  Q.  /\  ( /Q `  A )  ~Q  ( /Q `  B ) )  ->  ( /Q `  A )  =  ( /Q `  B ) )
152, 4, 13, 14syl3anc 1219 . . 3  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. )  /\  A  ~Q  B
)  ->  ( /Q `  A )  =  ( /Q `  B ) )
16153expia 1190 . 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 465 . . . . 5  |-  ( ( A  e.  ( N. 
X.  N. )  /\  ( B  e.  ( N.  X.  N. )  /\  ( /Q `  A )  =  ( /Q `  B
) ) )  ->  A  ~Q  ( /Q `  A ) )
19 simprr 756 . . . . 5  |-  ( ( A  e.  ( N. 
X.  N. )  /\  ( B  e.  ( N.  X.  N. )  /\  ( /Q `  A )  =  ( /Q `  B
) ) )  -> 
( /Q `  A
)  =  ( /Q
`  B ) )
2018, 19breqtrd 4411 . . . 4  |-  ( ( A  e.  ( N. 
X.  N. )  /\  ( B  e.  ( N.  X.  N. )  /\  ( /Q `  A )  =  ( /Q `  B
) ) )  ->  A  ~Q  ( /Q `  B ) )
2111ad2antrl 727 . . . 4  |-  ( ( A  e.  ( N. 
X.  N. )  /\  ( B  e.  ( N.  X.  N. )  /\  ( /Q `  A )  =  ( /Q `  B
) ) )  ->  B  ~Q  ( /Q `  B ) )
2217, 20, 21ertr4d 7217 . . 3  |-  ( ( A  e.  ( N. 
X.  N. )  /\  ( B  e.  ( N.  X.  N. )  /\  ( /Q `  A )  =  ( /Q `  B
) ) )  ->  A  ~Q  B )
2322expr 615 . 2  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. ) )  ->  (
( /Q `  A
)  =  ( /Q
`  B )  ->  A  ~Q  B ) )
2416, 23impbid 191 1  |-  ( ( A  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. ) )  ->  ( A  ~Q  B  <->  ( /Q `  A )  =  ( /Q `  B ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   class class class wbr 4387    X. cxp 4933   ` cfv 5513    Er wer 7195   N.cnpi 9109    ~Q ceq 9116   Q.cnq 9117   /Qcerq 9119
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4508  ax-nul 4516  ax-pow 4565  ax-pr 4626  ax-un 6469
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2599  df-ne 2644  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3067  df-sbc 3282  df-csb 3384  df-dif 3426  df-un 3428  df-in 3430  df-ss 3437  df-pss 3439  df-nul 3733  df-if 3887  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4187  df-iun 4268  df-br 4388  df-opab 4446  df-mpt 4447  df-tr 4481  df-eprel 4727  df-id 4731  df-po 4736  df-so 4737  df-fr 4774  df-we 4776  df-ord 4817  df-on 4818  df-lim 4819  df-suc 4820  df-xp 4941  df-rel 4942  df-cnv 4943  df-co 4944  df-dm 4945  df-rn 4946  df-res 4947  df-ima 4948  df-iota 5476  df-fun 5515  df-fn 5516  df-f 5517  df-f1 5518  df-fo 5519  df-f1o 5520  df-fv 5521  df-ov 6190  df-oprab 6191  df-mpt2 6192  df-om 6574  df-1st 6674  df-2nd 6675  df-recs 6929  df-rdg 6963  df-1o 7017  df-oadd 7021  df-omul 7022  df-er 7198  df-ni 9139  df-mi 9141  df-lti 9142  df-enq 9178  df-nq 9179  df-erq 9180  df-1nq 9183
This theorem is referenced by:  adderpq  9223  mulerpq  9224  distrnq  9228  recmulnq  9231  ltexnq  9242
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