ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  xrminadd Unicode version

Theorem xrminadd 11278
Description: Distributing addition over minimum. (Contributed by Jim Kingdon, 10-May-2023.)
Assertion
Ref Expression
xrminadd  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  -> inf ( { ( A +e
B ) ,  ( A +e C ) } ,  RR* ,  <  )  =  ( A +einf ( { B ,  C } ,  RR* ,  <  ) ) )

Proof of Theorem xrminadd
StepHypRef Expression
1 simp1 997 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  A  e.  RR* )
21xnegcld 9853 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  -e
A  e.  RR* )
3 simp2 998 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  B  e.  RR* )
43xnegcld 9853 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  -e
B  e.  RR* )
5 simp3 999 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  C  e.  RR* )
65xnegcld 9853 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  -e
C  e.  RR* )
7 xrmaxcl 11255 . . . 4  |-  ( ( 
-e B  e. 
RR*  /\  -e C  e.  RR* )  ->  sup ( {  -e B ,  -e C } ,  RR* ,  <  )  e.  RR* )
84, 6, 7syl2anc 411 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  sup ( {  -e B ,  -e C } ,  RR* ,  <  )  e.  RR* )
9 xnegdi 9866 . . 3  |-  ( ( 
-e A  e. 
RR*  /\  sup ( {  -e B ,  -e C } ,  RR* ,  <  )  e. 
RR* )  ->  -e
(  -e A +e sup ( {  -e
B ,  -e
C } ,  RR* ,  <  ) )  =  (  -e  -e A +e  -e sup ( { 
-e B ,  -e C } ,  RR* ,  <  ) ) )
102, 8, 9syl2anc 411 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  -e
(  -e A +e sup ( {  -e
B ,  -e
C } ,  RR* ,  <  ) )  =  (  -e  -e A +e  -e sup ( { 
-e B ,  -e C } ,  RR* ,  <  ) ) )
111, 3xaddcld 9882 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  ( A +e B )  e.  RR* )
121, 5xaddcld 9882 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  ( A +e C )  e.  RR* )
13 xrminmax 11268 . . . 4  |-  ( ( ( A +e
B )  e.  RR*  /\  ( A +e
C )  e.  RR* )  -> inf ( { ( A +e B ) ,  ( A +e C ) } ,  RR* ,  <  )  =  -e sup ( {  -e
( A +e
B ) ,  -e ( A +e C ) } ,  RR* ,  <  )
)
1411, 12, 13syl2anc 411 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  -> inf ( { ( A +e
B ) ,  ( A +e C ) } ,  RR* ,  <  )  =  -e sup ( {  -e
( A +e
B ) ,  -e ( A +e C ) } ,  RR* ,  <  )
)
15 xnegdi 9866 . . . . . . 7  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  -e
( A +e
B )  =  ( 
-e A +e  -e B ) )
161, 3, 15syl2anc 411 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  -e
( A +e
B )  =  ( 
-e A +e  -e B ) )
17 xnegdi 9866 . . . . . . 7  |-  ( ( A  e.  RR*  /\  C  e.  RR* )  ->  -e
( A +e
C )  =  ( 
-e A +e  -e C ) )
181, 5, 17syl2anc 411 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  -e
( A +e
C )  =  ( 
-e A +e  -e C ) )
1916, 18preq12d 3677 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  {  -e
( A +e
B ) ,  -e ( A +e C ) }  =  { (  -e A +e  -e B ) ,  (  -e A +e  -e
C ) } )
2019supeq1d 6985 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  sup ( {  -e ( A +e B ) ,  -e
( A +e
C ) } ,  RR* ,  <  )  =  sup ( { ( 
-e A +e  -e B ) ,  (  -e
A +e  -e C ) } ,  RR* ,  <  )
)
21 xnegeq 9825 . . . 4  |-  ( sup ( {  -e
( A +e
B ) ,  -e ( A +e C ) } ,  RR* ,  <  )  =  sup ( { ( 
-e A +e  -e B ) ,  (  -e
A +e  -e C ) } ,  RR* ,  <  )  -> 
-e sup ( {  -e ( A +e B ) ,  -e ( A +e C ) } ,  RR* ,  <  )  =  -e sup ( { ( 
-e A +e  -e B ) ,  (  -e
A +e  -e C ) } ,  RR* ,  <  )
)
2220, 21syl 14 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  -e sup ( {  -e
( A +e
B ) ,  -e ( A +e C ) } ,  RR* ,  <  )  =  -e sup ( { (  -e
A +e  -e B ) ,  (  -e A +e  -e
C ) } ,  RR* ,  <  ) )
23 xrmaxadd 11264 . . . . 5  |-  ( ( 
-e A  e. 
RR*  /\  -e B  e.  RR*  /\  -e
C  e.  RR* )  ->  sup ( { ( 
-e A +e  -e B ) ,  (  -e
A +e  -e C ) } ,  RR* ,  <  )  =  (  -e A +e sup ( {  -e B ,  -e C } ,  RR* ,  <  ) ) )
242, 4, 6, 23syl3anc 1238 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  sup ( { (  -e
A +e  -e B ) ,  (  -e A +e  -e
C ) } ,  RR* ,  <  )  =  (  -e A +e sup ( {  -e B ,  -e C } ,  RR* ,  <  ) ) )
25 xnegeq 9825 . . . 4  |-  ( sup ( { (  -e A +e  -e B ) ,  (  -e A +e  -e
C ) } ,  RR* ,  <  )  =  (  -e A +e sup ( {  -e B ,  -e C } ,  RR* ,  <  ) )  ->  -e sup ( { (  -e
A +e  -e B ) ,  (  -e A +e  -e
C ) } ,  RR* ,  <  )  = 
-e (  -e A +e sup ( {  -e
B ,  -e
C } ,  RR* ,  <  ) ) )
2624, 25syl 14 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  -e sup ( { (  -e A +e  -e B ) ,  (  -e A +e  -e
C ) } ,  RR* ,  <  )  = 
-e (  -e A +e sup ( {  -e
B ,  -e
C } ,  RR* ,  <  ) ) )
2714, 22, 263eqtrd 2214 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  -> inf ( { ( A +e
B ) ,  ( A +e C ) } ,  RR* ,  <  )  =  -e (  -e
A +e sup ( {  -e
B ,  -e
C } ,  RR* ,  <  ) ) )
28 xnegneg 9831 . . . . 5  |-  ( A  e.  RR*  ->  -e  -e A  =  A )
2928eqcomd 2183 . . . 4  |-  ( A  e.  RR*  ->  A  = 
-e  -e
A )
301, 29syl 14 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  A  =  -e  -e
A )
31 xrminmax 11268 . . . 4  |-  ( ( B  e.  RR*  /\  C  e.  RR* )  -> inf ( { B ,  C } ,  RR* ,  <  )  =  -e sup ( {  -e B ,  -e C } ,  RR* ,  <  ) )
323, 5, 31syl2anc 411 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  -> inf ( { B ,  C } ,  RR* ,  <  )  =  -e sup ( {  -e B ,  -e C } ,  RR* ,  <  ) )
3330, 32oveq12d 5892 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  ( A +einf ( { B ,  C } ,  RR* ,  <  )
)  =  (  -e  -e A +e  -e sup ( {  -e B ,  -e C } ,  RR* ,  <  ) ) )
3410, 27, 333eqtr4d 2220 1  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  -> inf ( { ( A +e
B ) ,  ( A +e C ) } ,  RR* ,  <  )  =  ( A +einf ( { B ,  C } ,  RR* ,  <  ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 978    = wceq 1353    e. wcel 2148   {cpr 3593  (class class class)co 5874   supcsup 6980  infcinf 6981   RR*cxr 7989    < clt 7990    -ecxne 9767   +ecxad 9768
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-1re 7904  ax-icn 7905  ax-addcl 7906  ax-addrcl 7907  ax-mulcl 7908  ax-mulrcl 7909  ax-addcom 7910  ax-mulcom 7911  ax-addass 7912  ax-mulass 7913  ax-distr 7914  ax-i2m1 7915  ax-0lt1 7916  ax-1rid 7917  ax-0id 7918  ax-rnegex 7919  ax-precex 7920  ax-cnre 7921  ax-pre-ltirr 7922  ax-pre-ltwlin 7923  ax-pre-lttrn 7924  ax-pre-apti 7925  ax-pre-ltadd 7926  ax-pre-mulgt0 7927  ax-pre-mulext 7928  ax-arch 7929  ax-caucvg 7930
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-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  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-po 4296  df-iso 4297  df-iord 4366  df-on 4368  df-ilim 4369  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-isom 5225  df-riota 5830  df-ov 5877  df-oprab 5878  df-mpo 5879  df-1st 6140  df-2nd 6141  df-recs 6305  df-frec 6391  df-sup 6982  df-inf 6983  df-pnf 7992  df-mnf 7993  df-xr 7994  df-ltxr 7995  df-le 7996  df-sub 8128  df-neg 8129  df-reap 8530  df-ap 8537  df-div 8628  df-inn 8918  df-2 8976  df-3 8977  df-4 8978  df-n0 9175  df-z 9252  df-uz 9527  df-rp 9652  df-xneg 9770  df-xadd 9771  df-seqfrec 10443  df-exp 10517  df-cj 10846  df-re 10847  df-im 10848  df-rsqrt 11002  df-abs 11003
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator