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Theorem negfi 10311
Description: The negation of a finite set of real numbers is finite. (Contributed by AV, 9-Aug-2020.)
Assertion
Ref Expression
negfi  |-  ( ( A  C_  RR  /\  A  e.  Fin )  ->  { n  e.  RR  |  -u n  e.  A }  e.  Fin )
Distinct variable group:    A, n

Proof of Theorem negfi
Dummy variables  a  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssel 3002 . . . . . . . . . 10  |-  ( A 
C_  RR  ->  ( a  e.  A  ->  a  e.  RR ) )
2 renegcl 7488 . . . . . . . . . 10  |-  ( a  e.  RR  ->  -u a  e.  RR )
31, 2syl6 33 . . . . . . . . 9  |-  ( A 
C_  RR  ->  ( a  e.  A  ->  -u a  e.  RR ) )
43imp 122 . . . . . . . 8  |-  ( ( A  C_  RR  /\  a  e.  A )  ->  -u a  e.  RR )
54ralrimiva 2439 . . . . . . 7  |-  ( A 
C_  RR  ->  A. a  e.  A  -u a  e.  RR )
6 dmmptg 4868 . . . . . . 7  |-  ( A. a  e.  A  -u a  e.  RR  ->  dom  ( a  e.  A  |->  -u a
)  =  A )
75, 6syl 14 . . . . . 6  |-  ( A 
C_  RR  ->  dom  (
a  e.  A  |->  -u a )  =  A )
87eqcomd 2088 . . . . 5  |-  ( A 
C_  RR  ->  A  =  dom  ( a  e.  A  |->  -u a ) )
98eleq1d 2151 . . . 4  |-  ( A 
C_  RR  ->  ( A  e.  Fin  <->  dom  ( a  e.  A  |->  -u a
)  e.  Fin )
)
10 funmpt 4988 . . . . 5  |-  Fun  (
a  e.  A  |->  -u a )
11 fundmfibi 6480 . . . . 5  |-  ( Fun  ( a  e.  A  |-> 
-u a )  -> 
( ( a  e.  A  |->  -u a )  e. 
Fin 
<->  dom  ( a  e.  A  |->  -u a )  e. 
Fin ) )
1210, 11mp1i 10 . . . 4  |-  ( A 
C_  RR  ->  ( ( a  e.  A  |->  -u a )  e.  Fin  <->  dom  ( a  e.  A  |-> 
-u a )  e. 
Fin ) )
139, 12bitr4d 189 . . 3  |-  ( A 
C_  RR  ->  ( A  e.  Fin  <->  ( a  e.  A  |->  -u a
)  e.  Fin )
)
14 reex 7221 . . . . . 6  |-  RR  e.  _V
1514ssex 3935 . . . . 5  |-  ( A 
C_  RR  ->  A  e. 
_V )
16 mptexg 5438 . . . . 5  |-  ( A  e.  _V  ->  (
a  e.  A  |->  -u a )  e.  _V )
1715, 16syl 14 . . . 4  |-  ( A 
C_  RR  ->  ( a  e.  A  |->  -u a
)  e.  _V )
18 eqid 2083 . . . . . 6  |-  ( a  e.  A  |->  -u a
)  =  ( a  e.  A  |->  -u a
)
1918negf1o 7605 . . . . 5  |-  ( A 
C_  RR  ->  ( a  e.  A  |->  -u a
) : A -1-1-onto-> { x  e.  RR  |  -u x  e.  A } )
20 f1of1 5176 . . . . 5  |-  ( ( a  e.  A  |->  -u a ) : A -1-1-onto-> {
x  e.  RR  |  -u x  e.  A }  ->  ( a  e.  A  |-> 
-u a ) : A -1-1-> { x  e.  RR  |  -u x  e.  A } )
2119, 20syl 14 . . . 4  |-  ( A 
C_  RR  ->  ( a  e.  A  |->  -u a
) : A -1-1-> {
x  e.  RR  |  -u x  e.  A }
)
22 f1vrnfibi 6485 . . . 4  |-  ( ( ( a  e.  A  |-> 
-u a )  e. 
_V  /\  ( a  e.  A  |->  -u a
) : A -1-1-> {
x  e.  RR  |  -u x  e.  A }
)  ->  ( (
a  e.  A  |->  -u a )  e.  Fin  <->  ran  ( a  e.  A  |-> 
-u a )  e. 
Fin ) )
2317, 21, 22syl2anc 403 . . 3  |-  ( A 
C_  RR  ->  ( ( a  e.  A  |->  -u a )  e.  Fin  <->  ran  ( a  e.  A  |-> 
-u a )  e. 
Fin ) )
241imp 122 . . . . . . . . . 10  |-  ( ( A  C_  RR  /\  a  e.  A )  ->  a  e.  RR )
252adantl 271 . . . . . . . . . . 11  |-  ( ( ( A  C_  RR  /\  a  e.  A )  /\  a  e.  RR )  ->  -u a  e.  RR )
26 recn 7220 . . . . . . . . . . . . . . . . 17  |-  ( a  e.  RR  ->  a  e.  CC )
2726negnegd 7529 . . . . . . . . . . . . . . . 16  |-  ( a  e.  RR  ->  -u -u a  =  a )
2827eqcomd 2088 . . . . . . . . . . . . . . 15  |-  ( a  e.  RR  ->  a  =  -u -u a )
2928eleq1d 2151 . . . . . . . . . . . . . 14  |-  ( a  e.  RR  ->  (
a  e.  A  <->  -u -u a  e.  A ) )
3029biimpcd 157 . . . . . . . . . . . . 13  |-  ( a  e.  A  ->  (
a  e.  RR  ->  -u -u a  e.  A ) )
3130adantl 271 . . . . . . . . . . . 12  |-  ( ( A  C_  RR  /\  a  e.  A )  ->  (
a  e.  RR  ->  -u -u a  e.  A ) )
3231imp 122 . . . . . . . . . . 11  |-  ( ( ( A  C_  RR  /\  a  e.  A )  /\  a  e.  RR )  ->  -u -u a  e.  A
)
3325, 32jca 300 . . . . . . . . . 10  |-  ( ( ( A  C_  RR  /\  a  e.  A )  /\  a  e.  RR )  ->  ( -u a  e.  RR  /\  -u -u a  e.  A ) )
3424, 33mpdan 412 . . . . . . . . 9  |-  ( ( A  C_  RR  /\  a  e.  A )  ->  ( -u a  e.  RR  /\  -u -u a  e.  A
) )
35 eleq1 2145 . . . . . . . . . 10  |-  ( n  =  -u a  ->  (
n  e.  RR  <->  -u a  e.  RR ) )
36 negeq 7420 . . . . . . . . . . 11  |-  ( n  =  -u a  ->  -u n  =  -u -u a )
3736eleq1d 2151 . . . . . . . . . 10  |-  ( n  =  -u a  ->  ( -u n  e.  A  <->  -u -u a  e.  A ) )
3835, 37anbi12d 457 . . . . . . . . 9  |-  ( n  =  -u a  ->  (
( n  e.  RR  /\  -u n  e.  A
)  <->  ( -u a  e.  RR  /\  -u -u a  e.  A ) ) )
3934, 38syl5ibrcom 155 . . . . . . . 8  |-  ( ( A  C_  RR  /\  a  e.  A )  ->  (
n  =  -u a  ->  ( n  e.  RR  /\  -u n  e.  A
) ) )
4039rexlimdva 2482 . . . . . . 7  |-  ( A 
C_  RR  ->  ( E. a  e.  A  n  =  -u a  ->  (
n  e.  RR  /\  -u n  e.  A ) ) )
41 simprr 499 . . . . . . . . 9  |-  ( ( A  C_  RR  /\  (
n  e.  RR  /\  -u n  e.  A ) )  ->  -u n  e.  A )
42 negeq 7420 . . . . . . . . . . 11  |-  ( a  =  -u n  ->  -u a  =  -u -u n )
4342eqeq2d 2094 . . . . . . . . . 10  |-  ( a  =  -u n  ->  (
n  =  -u a  <->  n  =  -u -u n ) )
4443adantl 271 . . . . . . . . 9  |-  ( ( ( A  C_  RR  /\  ( n  e.  RR  /\  -u n  e.  A
) )  /\  a  =  -u n )  -> 
( n  =  -u a 
<->  n  =  -u -u n
) )
45 recn 7220 . . . . . . . . . . 11  |-  ( n  e.  RR  ->  n  e.  CC )
46 negneg 7477 . . . . . . . . . . . 12  |-  ( n  e.  CC  ->  -u -u n  =  n )
4746eqcomd 2088 . . . . . . . . . . 11  |-  ( n  e.  CC  ->  n  =  -u -u n )
4845, 47syl 14 . . . . . . . . . 10  |-  ( n  e.  RR  ->  n  =  -u -u n )
4948ad2antrl 474 . . . . . . . . 9  |-  ( ( A  C_  RR  /\  (
n  e.  RR  /\  -u n  e.  A ) )  ->  n  =  -u -u n )
5041, 44, 49rspcedvd 2716 . . . . . . . 8  |-  ( ( A  C_  RR  /\  (
n  e.  RR  /\  -u n  e.  A ) )  ->  E. a  e.  A  n  =  -u a )
5150ex 113 . . . . . . 7  |-  ( A 
C_  RR  ->  ( ( n  e.  RR  /\  -u n  e.  A )  ->  E. a  e.  A  n  =  -u a ) )
5240, 51impbid 127 . . . . . 6  |-  ( A 
C_  RR  ->  ( E. a  e.  A  n  =  -u a  <->  ( n  e.  RR  /\  -u n  e.  A ) ) )
5352abbidv 2200 . . . . 5  |-  ( A 
C_  RR  ->  { n  |  E. a  e.  A  n  =  -u a }  =  { n  |  ( n  e.  RR  /\  -u n  e.  A
) } )
5418rnmpt 4630 . . . . 5  |-  ran  (
a  e.  A  |->  -u a )  =  {
n  |  E. a  e.  A  n  =  -u a }
55 df-rab 2362 . . . . 5  |-  { n  e.  RR  |  -u n  e.  A }  =  {
n  |  ( n  e.  RR  /\  -u n  e.  A ) }
5653, 54, 553eqtr4g 2140 . . . 4  |-  ( A 
C_  RR  ->  ran  (
a  e.  A  |->  -u a )  =  {
n  e.  RR  |  -u n  e.  A }
)
5756eleq1d 2151 . . 3  |-  ( A 
C_  RR  ->  ( ran  ( a  e.  A  |-> 
-u a )  e. 
Fin 
<->  { n  e.  RR  |  -u n  e.  A }  e.  Fin )
)
5813, 23, 573bitrd 212 . 2  |-  ( A 
C_  RR  ->  ( A  e.  Fin  <->  { n  e.  RR  |  -u n  e.  A }  e.  Fin ) )
5958biimpa 290 1  |-  ( ( A  C_  RR  /\  A  e.  Fin )  ->  { n  e.  RR  |  -u n  e.  A }  e.  Fin )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1285    e. wcel 1434   {cab 2069   A.wral 2353   E.wrex 2354   {crab 2357   _Vcvv 2610    C_ wss 2982    |-> cmpt 3859   dom cdm 4391   ran crn 4392   Fun wfun 4946   -1-1->wf1 4949   -1-1-onto->wf1o 4951   Fincfn 6308   CCcc 7093   RRcr 7094   -ucneg 7399
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3913  ax-sep 3916  ax-nul 3924  ax-pow 3968  ax-pr 3992  ax-un 4216  ax-setind 4308  ax-iinf 4357  ax-cnex 7181  ax-resscn 7182  ax-1cn 7183  ax-icn 7185  ax-addcl 7186  ax-addrcl 7187  ax-mulcl 7188  ax-addcom 7190  ax-addass 7192  ax-distr 7194  ax-i2m1 7195  ax-0id 7198  ax-rnegex 7199  ax-cnre 7201
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2612  df-sbc 2825  df-csb 2918  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-nul 3268  df-if 3369  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-int 3657  df-iun 3700  df-br 3806  df-opab 3860  df-mpt 3861  df-tr 3896  df-id 4076  df-iord 4149  df-on 4151  df-suc 4154  df-iom 4360  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402  df-res 4403  df-ima 4404  df-iota 4917  df-fun 4954  df-fn 4955  df-f 4956  df-f1 4957  df-fo 4958  df-f1o 4959  df-fv 4960  df-riota 5519  df-ov 5566  df-oprab 5567  df-mpt2 5568  df-1st 5818  df-2nd 5819  df-1o 6085  df-er 6193  df-en 6309  df-fin 6311  df-sub 7400  df-neg 7401
This theorem is referenced by: (None)
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