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Theorem xrminadd 11440
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 999 . . . 4 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> A e. RR* )
21xnegcld 9930 . . 3 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> -e A e. RR* )
3 simp2 1000 . . . . 5 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> B e. RR* )
43xnegcld 9930 . . . 4 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> -e B e. RR* )
5 simp3 1001 . . . . 5 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> C e. RR* )
65xnegcld 9930 . . . 4 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> -e C e. RR* )
7 xrmaxcl 11417 . . . 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 9943 . . 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 9959 . . . 4 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( A +e B ) e. RR* )
121, 5xaddcld 9959 . . . 4 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( A +e C ) e. RR* )
13 xrminmax 11430 . . . 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 9943 . . . . . . 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 9943 . . . . . . 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 3707 . . . . 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 7053 . . . 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 9902 . . . 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 11426 . . . . 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 1249 . . . 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 9902 . . . 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 2233 . 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 9908 . . . . 5 |- ( A e. RR* -> -e -e A = A )
2928eqcomd 2202 . . . 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 11430 . . . 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 5940 . 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 2239 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 980   = wceq 1364   e. wcel 2167   {cpr 3623  (class class class)co 5922   supcsup 7048  infcinf 7049   RR*cxr 8060   < clt 8061   -ecxne 9844   +ecxad 9845
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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-iinf 4624  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-mulrcl 7978  ax-addcom 7979  ax-mulcom 7980  ax-addass 7981  ax-mulass 7982  ax-distr 7983  ax-i2m1 7984  ax-0lt1 7985  ax-1rid 7986  ax-0id 7987  ax-rnegex 7988  ax-precex 7989  ax-cnre 7990  ax-pre-ltirr 7991  ax-pre-ltwlin 7992  ax-pre-lttrn 7993  ax-pre-apti 7994  ax-pre-ltadd 7995  ax-pre-mulgt0 7996  ax-pre-mulext 7997  ax-arch 7998  ax-caucvg 7999
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-if 3562  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-id 4328  df-po 4331  df-iso 4332  df-iord 4401  df-on 4403  df-ilim 4404  df-suc 4406  df-iom 4627  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-isom 5267  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-recs 6363  df-frec 6449  df-sup 7050  df-inf 7051  df-pnf 8063  df-mnf 8064  df-xr 8065  df-ltxr 8066  df-le 8067  df-sub 8199  df-neg 8200  df-reap 8602  df-ap 8609  df-div 8700  df-inn 8991  df-2 9049  df-3 9050  df-4 9051  df-n0 9250  df-z 9327  df-uz 9602  df-rp 9729  df-xneg 9847  df-xadd 9848  df-seqfrec 10540  df-exp 10631  df-cj 11007  df-re 11008  df-im 11009  df-rsqrt 11163  df-abs 11164
This theorem is referenced by: (None)
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