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Theorem negfi 10623
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 3017 . . . . . . . . . 10  |-  ( A 
C_  RR  ->  ( a  e.  A  ->  a  e.  RR ) )
2 renegcl 7722 . . . . . . . . . 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 2446 . . . . . . 7  |-  ( A 
C_  RR  ->  A. a  e.  A  -u a  e.  RR )
6 dmmptg 4915 . . . . . . 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 2093 . . . . 5  |-  ( A 
C_  RR  ->  A  =  dom  ( a  e.  A  |->  -u a ) )
98eleq1d 2156 . . . 4  |-  ( A 
C_  RR  ->  ( A  e.  Fin  <->  dom  ( a  e.  A  |->  -u a
)  e.  Fin )
)
10 funmpt 5038 . . . . 5  |-  Fun  (
a  e.  A  |->  -u a )
11 fundmfibi 6627 . . . . 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 7455 . . . . . 6  |-  RR  e.  _V
1514ssex 3968 . . . . 5  |-  ( A 
C_  RR  ->  A  e. 
_V )
16 mptexg 5504 . . . . 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 2088 . . . . . 6  |-  ( a  e.  A  |->  -u a
)  =  ( a  e.  A  |->  -u a
)
1918negf1o 7839 . . . . 5  |-  ( A 
C_  RR  ->  ( a  e.  A  |->  -u a
) : A -1-1-onto-> { x  e.  RR  |  -u x  e.  A } )
20 f1of1 5236 . . . . 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 6633 . . . 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 7454 . . . . . . . . . . . . . . . . 17  |-  ( a  e.  RR  ->  a  e.  CC )
2726negnegd 7763 . . . . . . . . . . . . . . . 16  |-  ( a  e.  RR  ->  -u -u a  =  a )
2827eqcomd 2093 . . . . . . . . . . . . . . 15  |-  ( a  e.  RR  ->  a  =  -u -u a )
2928eleq1d 2156 . . . . . . . . . . . . . 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 2150 . . . . . . . . . 10  |-  ( n  =  -u a  ->  (
n  e.  RR  <->  -u a  e.  RR ) )
36 negeq 7654 . . . . . . . . . . 11  |-  ( n  =  -u a  ->  -u n  =  -u -u a )
3736eleq1d 2156 . . . . . . . . . 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 2489 . . . . . . 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 7654 . . . . . . . . . . 11  |-  ( a  =  -u n  ->  -u a  =  -u -u n )
4342eqeq2d 2099 . . . . . . . . . 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 7454 . . . . . . . . . . 11  |-  ( n  e.  RR  ->  n  e.  CC )
46 negneg 7711 . . . . . . . . . . . 12  |-  ( n  e.  CC  ->  -u -u n  =  n )
4746eqcomd 2093 . . . . . . . . . . 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 2728 . . . . . . . 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 2205 . . . . 5  |-  ( A 
C_  RR  ->  { n  |  E. a  e.  A  n  =  -u a }  =  { n  |  ( n  e.  RR  /\  -u n  e.  A
) } )
5418rnmpt 4671 . . . . 5  |-  ran  (
a  e.  A  |->  -u a )  =  {
n  |  E. a  e.  A  n  =  -u a }
55 df-rab 2368 . . . . 5  |-  { n  e.  RR  |  -u n  e.  A }  =  {
n  |  ( n  e.  RR  /\  -u n  e.  A ) }
5653, 54, 553eqtr4g 2145 . . . 4  |-  ( A 
C_  RR  ->  ran  (
a  e.  A  |->  -u a )  =  {
n  e.  RR  |  -u n  e.  A }
)
5756eleq1d 2156 . . 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 1289    e. wcel 1438   {cab 2074   A.wral 2359   E.wrex 2360   {crab 2363   _Vcvv 2619    C_ wss 2997    |-> cmpt 3891   dom cdm 4428   ran crn 4429   Fun wfun 4996   -1-1->wf1 4999   -1-1-onto->wf1o 5001   Fincfn 6437   CCcc 7327   RRcr 7328   -ucneg 7633
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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3946  ax-sep 3949  ax-nul 3957  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343  ax-iinf 4393  ax-cnex 7415  ax-resscn 7416  ax-1cn 7417  ax-icn 7419  ax-addcl 7420  ax-addrcl 7421  ax-mulcl 7422  ax-addcom 7424  ax-addass 7426  ax-distr 7428  ax-i2m1 7429  ax-0id 7432  ax-rnegex 7433  ax-cnre 7435
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2839  df-csb 2932  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-if 3390  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-int 3684  df-iun 3727  df-br 3838  df-opab 3892  df-mpt 3893  df-tr 3929  df-id 4111  df-iord 4184  df-on 4186  df-suc 4189  df-iom 4396  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-f1 5007  df-fo 5008  df-f1o 5009  df-fv 5010  df-riota 5590  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-1st 5893  df-2nd 5894  df-1o 6163  df-er 6272  df-en 6438  df-fin 6440  df-sub 7634  df-neg 7635
This theorem is referenced by: (None)
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