ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  xrminadd Unicode version
Theorem xrminadd 11835
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 1023 . . . 4 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> A e. RR* )
21xnegcld 10089 . . 3 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> -e A e. RR* )
3 simp2 1024 . . . . 5 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> B e. RR* )
43xnegcld 10089 . . . 4 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> -e B e. RR* )
5 simp3 1025 . . . . 5 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> C e. RR* )
65xnegcld 10089 . . . 4 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> -e C e. RR* )
7 xrmaxcl 11812 . . . 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 10102 . . 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 10118 . . . 4 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( A +e B ) e. RR* )
121, 5xaddcld 10118 . . . 4 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( A +e C ) e. RR* )
13 xrminmax 11825 . . . 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 10102 . . . . . . 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 10102 . . . . . . 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 3756 . . . . 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 7185 . . . 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 10061 . . . 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 11821 . . . . 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 1273 . . . 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 10061 . . . 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 2268 . 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 10067 . . . . 5 |- ( A e. RR* -> -e -e A = A )
2928eqcomd 2237 . . . 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 11825 . . . 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 6035 . 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 2274 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 1004   = wceq 1397   e. wcel 2202   {cpr 3670  (class class class)co 6017   supcsup 7180  infcinf 7181   RR*cxr 8212   < clt 8213   -ecxne 10003   +ecxad 10004
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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-mulrcl 8130  ax-addcom 8131  ax-mulcom 8132  ax-addass 8133  ax-mulass 8134  ax-distr 8135  ax-i2m1 8136  ax-0lt1 8137  ax-1rid 8138  ax-0id 8139  ax-rnegex 8140  ax-precex 8141  ax-cnre 8142  ax-pre-ltirr 8143  ax-pre-ltwlin 8144  ax-pre-lttrn 8145  ax-pre-apti 8146  ax-pre-ltadd 8147  ax-pre-mulgt0 8148  ax-pre-mulext 8149  ax-arch 8150  ax-caucvg 8151
This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-po 4393  df-iso 4394  df-iord 4463  df-on 4465  df-ilim 4466  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-isom 5335  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-recs 6470  df-frec 6556  df-sup 7182  df-inf 7183  df-pnf 8215  df-mnf 8216  df-xr 8217  df-ltxr 8218  df-le 8219  df-sub 8351  df-neg 8352  df-reap 8754  df-ap 8761  df-div 8852  df-inn 9143  df-2 9201  df-3 9202  df-4 9203  df-n0 9402  df-z 9479  df-uz 9755  df-rp 9888  df-xneg 10006  df-xadd 10007  df-seqfrec 10709  df-exp 10800  df-cj 11402  df-re 11403  df-im 11404  df-rsqrt 11558  df-abs 11559
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator