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Theorem negfi 11231
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 3149 . . . . . . . . . 10  |-  ( A 
C_  RR  ->  ( a  e.  A  ->  a  e.  RR ) )
2 renegcl 8216 . . . . . . . . . 10  |-  ( a  e.  RR  ->  -u a  e.  RR )
31, 2syl6 33 . . . . . . . . 9  |-  ( A 
C_  RR  ->  ( a  e.  A  ->  -u a  e.  RR ) )
43imp 124 . . . . . . . 8  |-  ( ( A  C_  RR  /\  a  e.  A )  ->  -u a  e.  RR )
54ralrimiva 2550 . . . . . . 7  |-  ( A 
C_  RR  ->  A. a  e.  A  -u a  e.  RR )
6 dmmptg 5126 . . . . . . 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 2183 . . . . 5  |-  ( A 
C_  RR  ->  A  =  dom  ( a  e.  A  |->  -u a ) )
98eleq1d 2246 . . . 4  |-  ( A 
C_  RR  ->  ( A  e.  Fin  <->  dom  ( a  e.  A  |->  -u a
)  e.  Fin )
)
10 funmpt 5254 . . . . 5  |-  Fun  (
a  e.  A  |->  -u a )
11 fundmfibi 6937 . . . . 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 191 . . 3  |-  ( A 
C_  RR  ->  ( A  e.  Fin  <->  ( a  e.  A  |->  -u a
)  e.  Fin )
)
14 reex 7944 . . . . . 6  |-  RR  e.  _V
1514ssex 4140 . . . . 5  |-  ( A 
C_  RR  ->  A  e. 
_V )
16 mptexg 5741 . . . . 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 2177 . . . . . 6  |-  ( a  e.  A  |->  -u a
)  =  ( a  e.  A  |->  -u a
)
1918negf1o 8337 . . . . 5  |-  ( A 
C_  RR  ->  ( a  e.  A  |->  -u a
) : A -1-1-onto-> { x  e.  RR  |  -u x  e.  A } )
20 f1of1 5460 . . . . 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 6943 . . . 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 411 . . 3  |-  ( A 
C_  RR  ->  ( ( a  e.  A  |->  -u a )  e.  Fin  <->  ran  ( a  e.  A  |-> 
-u a )  e. 
Fin ) )
241imp 124 . . . . . . . . . 10  |-  ( ( A  C_  RR  /\  a  e.  A )  ->  a  e.  RR )
252adantl 277 . . . . . . . . . . 11  |-  ( ( ( A  C_  RR  /\  a  e.  A )  /\  a  e.  RR )  ->  -u a  e.  RR )
26 recn 7943 . . . . . . . . . . . . . . . . 17  |-  ( a  e.  RR  ->  a  e.  CC )
2726negnegd 8257 . . . . . . . . . . . . . . . 16  |-  ( a  e.  RR  ->  -u -u a  =  a )
2827eqcomd 2183 . . . . . . . . . . . . . . 15  |-  ( a  e.  RR  ->  a  =  -u -u a )
2928eleq1d 2246 . . . . . . . . . . . . . 14  |-  ( a  e.  RR  ->  (
a  e.  A  <->  -u -u a  e.  A ) )
3029biimpcd 159 . . . . . . . . . . . . 13  |-  ( a  e.  A  ->  (
a  e.  RR  ->  -u -u a  e.  A ) )
3130adantl 277 . . . . . . . . . . . 12  |-  ( ( A  C_  RR  /\  a  e.  A )  ->  (
a  e.  RR  ->  -u -u a  e.  A ) )
3231imp 124 . . . . . . . . . . 11  |-  ( ( ( A  C_  RR  /\  a  e.  A )  /\  a  e.  RR )  ->  -u -u a  e.  A
)
3325, 32jca 306 . . . . . . . . . 10  |-  ( ( ( A  C_  RR  /\  a  e.  A )  /\  a  e.  RR )  ->  ( -u a  e.  RR  /\  -u -u a  e.  A ) )
3424, 33mpdan 421 . . . . . . . . 9  |-  ( ( A  C_  RR  /\  a  e.  A )  ->  ( -u a  e.  RR  /\  -u -u a  e.  A
) )
35 eleq1 2240 . . . . . . . . . 10  |-  ( n  =  -u a  ->  (
n  e.  RR  <->  -u a  e.  RR ) )
36 negeq 8148 . . . . . . . . . . 11  |-  ( n  =  -u a  ->  -u n  =  -u -u a )
3736eleq1d 2246 . . . . . . . . . 10  |-  ( n  =  -u a  ->  ( -u n  e.  A  <->  -u -u a  e.  A ) )
3835, 37anbi12d 473 . . . . . . . . 9  |-  ( n  =  -u a  ->  (
( n  e.  RR  /\  -u n  e.  A
)  <->  ( -u a  e.  RR  /\  -u -u a  e.  A ) ) )
3934, 38syl5ibrcom 157 . . . . . . . 8  |-  ( ( A  C_  RR  /\  a  e.  A )  ->  (
n  =  -u a  ->  ( n  e.  RR  /\  -u n  e.  A
) ) )
4039rexlimdva 2594 . . . . . . 7  |-  ( A 
C_  RR  ->  ( E. a  e.  A  n  =  -u a  ->  (
n  e.  RR  /\  -u n  e.  A ) ) )
41 simprr 531 . . . . . . . . 9  |-  ( ( A  C_  RR  /\  (
n  e.  RR  /\  -u n  e.  A ) )  ->  -u n  e.  A )
42 negeq 8148 . . . . . . . . . . 11  |-  ( a  =  -u n  ->  -u a  =  -u -u n )
4342eqeq2d 2189 . . . . . . . . . 10  |-  ( a  =  -u n  ->  (
n  =  -u a  <->  n  =  -u -u n ) )
4443adantl 277 . . . . . . . . 9  |-  ( ( ( A  C_  RR  /\  ( n  e.  RR  /\  -u n  e.  A
) )  /\  a  =  -u n )  -> 
( n  =  -u a 
<->  n  =  -u -u n
) )
45 recn 7943 . . . . . . . . . . 11  |-  ( n  e.  RR  ->  n  e.  CC )
46 negneg 8205 . . . . . . . . . . . 12  |-  ( n  e.  CC  ->  -u -u n  =  n )
4746eqcomd 2183 . . . . . . . . . . 11  |-  ( n  e.  CC  ->  n  =  -u -u n )
4845, 47syl 14 . . . . . . . . . 10  |-  ( n  e.  RR  ->  n  =  -u -u n )
4948ad2antrl 490 . . . . . . . . 9  |-  ( ( A  C_  RR  /\  (
n  e.  RR  /\  -u n  e.  A ) )  ->  n  =  -u -u n )
5041, 44, 49rspcedvd 2847 . . . . . . . 8  |-  ( ( A  C_  RR  /\  (
n  e.  RR  /\  -u n  e.  A ) )  ->  E. a  e.  A  n  =  -u a )
5150ex 115 . . . . . . 7  |-  ( A 
C_  RR  ->  ( ( n  e.  RR  /\  -u n  e.  A )  ->  E. a  e.  A  n  =  -u a ) )
5240, 51impbid 129 . . . . . 6  |-  ( A 
C_  RR  ->  ( E. a  e.  A  n  =  -u a  <->  ( n  e.  RR  /\  -u n  e.  A ) ) )
5352abbidv 2295 . . . . 5  |-  ( A 
C_  RR  ->  { n  |  E. a  e.  A  n  =  -u a }  =  { n  |  ( n  e.  RR  /\  -u n  e.  A
) } )
5418rnmpt 4875 . . . . 5  |-  ran  (
a  e.  A  |->  -u a )  =  {
n  |  E. a  e.  A  n  =  -u a }
55 df-rab 2464 . . . . 5  |-  { n  e.  RR  |  -u n  e.  A }  =  {
n  |  ( n  e.  RR  /\  -u n  e.  A ) }
5653, 54, 553eqtr4g 2235 . . . 4  |-  ( A 
C_  RR  ->  ran  (
a  e.  A  |->  -u a )  =  {
n  e.  RR  |  -u n  e.  A }
)
5756eleq1d 2246 . . 3  |-  ( A 
C_  RR  ->  ( ran  ( a  e.  A  |-> 
-u a )  e. 
Fin 
<->  { n  e.  RR  |  -u n  e.  A }  e.  Fin )
)
5813, 23, 573bitrd 214 . 2  |-  ( A 
C_  RR  ->  ( A  e.  Fin  <->  { n  e.  RR  |  -u n  e.  A }  e.  Fin ) )
5958biimpa 296 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 104    <-> wb 105    = wceq 1353    e. wcel 2148   {cab 2163   A.wral 2455   E.wrex 2456   {crab 2459   _Vcvv 2737    C_ wss 3129    |-> cmpt 4064   dom cdm 4626   ran crn 4627   Fun wfun 5210   -1-1->wf1 5213   -1-1-onto->wf1o 5215   Fincfn 6739   CCcc 7808   RRcr 7809   -ucneg 8127
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4118  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-iinf 4587  ax-cnex 7901  ax-resscn 7902  ax-1cn 7903  ax-icn 7905  ax-addcl 7906  ax-addrcl 7907  ax-mulcl 7908  ax-addcom 7910  ax-addass 7912  ax-distr 7914  ax-i2m1 7915  ax-0id 7918  ax-rnegex 7919  ax-cnre 7921
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-tr 4102  df-id 4293  df-iord 4366  df-on 4368  df-suc 4371  df-iom 4590  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-riota 5830  df-ov 5877  df-oprab 5878  df-mpo 5879  df-1st 6140  df-2nd 6141  df-1o 6416  df-er 6534  df-en 6740  df-fin 6742  df-sub 8128  df-neg 8129
This theorem is referenced by: (None)
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