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Theorem xrminadd 11418
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 9921 . . 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 9921 . . . 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 9921 . . . 4 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> -e C e. RR* )
7 xrmaxcl 11395 . . . 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 9934 . . 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 9950 . . . 4 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( A +e B ) e. RR* )
121, 5xaddcld 9950 . . . 4 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( A +e C ) e. RR* )
13 xrminmax 11408 . . . 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 9934 . . . . . . 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 9934 . . . . . . 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 3703 . . . . 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 7046 . . . 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 9893 . . . 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 11404 . . . . 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 9893 . . . 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 2230 . 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 9899 . . . . 5 |- ( A e. RR* -> -e -e A = A )
2928eqcomd 2199 . . . 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 11408 . . . 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 5936 . 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 2236 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 2164   {cpr 3619  (class class class)co 5918   supcsup 7041  infcinf 7042   RR*cxr 8053   < clt 8054   -ecxne 9835   +ecxad 9836
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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-mulrcl 7971  ax-addcom 7972  ax-mulcom 7973  ax-addass 7974  ax-mulass 7975  ax-distr 7976  ax-i2m1 7977  ax-0lt1 7978  ax-1rid 7979  ax-0id 7980  ax-rnegex 7981  ax-precex 7982  ax-cnre 7983  ax-pre-ltirr 7984  ax-pre-ltwlin 7985  ax-pre-lttrn 7986  ax-pre-apti 7987  ax-pre-ltadd 7988  ax-pre-mulgt0 7989  ax-pre-mulext 7990  ax-arch 7991  ax-caucvg 7992
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 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-if 3558  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-id 4324  df-po 4327  df-iso 4328  df-iord 4397  df-on 4399  df-ilim 4400  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-isom 5263  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-recs 6358  df-frec 6444  df-sup 7043  df-inf 7044  df-pnf 8056  df-mnf 8057  df-xr 8058  df-ltxr 8059  df-le 8060  df-sub 8192  df-neg 8193  df-reap 8594  df-ap 8601  df-div 8692  df-inn 8983  df-2 9041  df-3 9042  df-4 9043  df-n0 9241  df-z 9318  df-uz 9593  df-rp 9720  df-xneg 9838  df-xadd 9839  df-seqfrec 10519  df-exp 10610  df-cj 10986  df-re 10987  df-im 10988  df-rsqrt 11142  df-abs 11143
This theorem is referenced by: (None)
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