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Theorem xrminadd 11283
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 9855 . . 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 9855 . . . 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 9855 . . . 4 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> -e C e. RR* )
7 xrmaxcl 11260 . . . 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 9868 . . 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 9884 . . . 4 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( A +e B ) e. RR* )
121, 5xaddcld 9884 . . . 4 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( A +e C ) e. RR* )
13 xrminmax 11273 . . . 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 9868 . . . . . . 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 9868 . . . . . . 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 3678 . . . . 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 6986 . . . 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 9827 . . . 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 11269 . . . . 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 9827 . . . 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 9833 . . . . 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 11273 . . . 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 5893 . 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 3594  (class class class)co 5875   supcsup 6981  infcinf 6982   RR*cxr 7991   < clt 7992   -ecxne 9769   +ecxad 9770
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 4119  ax-sep 4122  ax-nul 4130  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-iinf 4588  ax-cnex 7902  ax-resscn 7903  ax-1cn 7904  ax-1re 7905  ax-icn 7906  ax-addcl 7907  ax-addrcl 7908  ax-mulcl 7909  ax-mulrcl 7910  ax-addcom 7911  ax-mulcom 7912  ax-addass 7913  ax-mulass 7914  ax-distr 7915  ax-i2m1 7916  ax-0lt1 7917  ax-1rid 7918  ax-0id 7919  ax-rnegex 7920  ax-precex 7921  ax-cnre 7922  ax-pre-ltirr 7923  ax-pre-ltwlin 7924  ax-pre-lttrn 7925  ax-pre-apti 7926  ax-pre-ltadd 7927  ax-pre-mulgt0 7928  ax-pre-mulext 7929  ax-arch 7930  ax-caucvg 7931
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 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-if 3536  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-tr 4103  df-id 4294  df-po 4297  df-iso 4298  df-iord 4367  df-on 4369  df-ilim 4370  df-suc 4372  df-iom 4591  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-isom 5226  df-riota 5831  df-ov 5878  df-oprab 5879  df-mpo 5880  df-1st 6141  df-2nd 6142  df-recs 6306  df-frec 6392  df-sup 6983  df-inf 6984  df-pnf 7994  df-mnf 7995  df-xr 7996  df-ltxr 7997  df-le 7998  df-sub 8130  df-neg 8131  df-reap 8532  df-ap 8539  df-div 8630  df-inn 8920  df-2 8978  df-3 8979  df-4 8980  df-n0 9177  df-z 9254  df-uz 9529  df-rp 9654  df-xneg 9772  df-xadd 9773  df-seqfrec 10446  df-exp 10520  df-cj 10851  df-re 10852  df-im 10853  df-rsqrt 11007  df-abs 11008
This theorem is referenced by: (None)
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