xrminadd - Intuitionistic Logic Explorer

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Theorem xrminadd 11988

Description: Distributing addition over minimum. (Contributed by Jim Kingdon, 10-May-2023.)

**Assertion**
Ref	Expression
xrminadd	⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → inf({(𝐴 +_𝑒 𝐵), (𝐴 +_𝑒 𝐶)}, ℝ^, < ) = (𝐴 +_𝑒 inf({𝐵, 𝐶}, ℝ^*, < )))

**Proof of Theorem xrminadd**
Step	Hyp	Ref	Expression
1		simp1 1024	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → 𝐴 ∈ ℝ^)
2	1	xnegcld 10210	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → -_𝑒𝐴 ∈ ℝ^)
3		simp2 1025	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → 𝐵 ∈ ℝ^)
4	3	xnegcld 10210	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → -_𝑒𝐵 ∈ ℝ^)
5		simp3 1026	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → 𝐶 ∈ ℝ^)
6	5	xnegcld 10210	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → -_𝑒𝐶 ∈ ℝ^)
7		xrmaxcl 11965	. . . 4 ⊢ ((-_𝑒𝐵 ∈ ℝ^* ∧ -_𝑒𝐶 ∈ ℝ^) → sup({-_𝑒𝐵, -_𝑒𝐶}, ℝ^, < ) ∈ ℝ^*)
8	4, 6, 7	syl2anc 411	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → sup({-_𝑒𝐵, -_𝑒𝐶}, ℝ^, < ) ∈ ℝ^*)
9		xnegdi 10223	. . 3 ⊢ ((-_𝑒𝐴 ∈ ℝ^* ∧ sup({-_𝑒𝐵, -_𝑒𝐶}, ℝ^, < ) ∈ ℝ^) → -_𝑒(-_𝑒𝐴 +_𝑒 sup({-_𝑒𝐵, -_𝑒𝐶}, ℝ^, < )) = (-_𝑒-_𝑒𝐴 +_𝑒 -_𝑒sup({-_𝑒𝐵, -_𝑒𝐶}, ℝ^, < )))
10	2, 8, 9	syl2anc 411	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → -_𝑒(-_𝑒𝐴 +_𝑒 sup({-_𝑒𝐵, -_𝑒𝐶}, ℝ^, < )) = (-_𝑒-_𝑒𝐴 +_𝑒 -_𝑒sup({-_𝑒𝐵, -_𝑒𝐶}, ℝ^*, < )))
11	1, 3	xaddcld 10239	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → (𝐴 +_𝑒 𝐵) ∈ ℝ^)
12	1, 5	xaddcld 10239	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → (𝐴 +_𝑒 𝐶) ∈ ℝ^)
13		xrminmax 11978	. . . 4 ⊢ (((𝐴 +_𝑒 𝐵) ∈ ℝ^* ∧ (𝐴 +_𝑒 𝐶) ∈ ℝ^) → inf({(𝐴 +_𝑒 𝐵), (𝐴 +_𝑒 𝐶)}, ℝ^, < ) = -_𝑒sup({-_𝑒(𝐴 +_𝑒 𝐵), -_𝑒(𝐴 +_𝑒 𝐶)}, ℝ^*, < ))
14	11, 12, 13	syl2anc 411	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → inf({(𝐴 +_𝑒 𝐵), (𝐴 +_𝑒 𝐶)}, ℝ^, < ) = -_𝑒sup({-_𝑒(𝐴 +_𝑒 𝐵), -_𝑒(𝐴 +_𝑒 𝐶)}, ℝ^*, < ))
15		xnegdi 10223	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → -_𝑒(𝐴 +_𝑒 𝐵) = (-_𝑒𝐴 +_𝑒 -_𝑒𝐵))
16	1, 3, 15	syl2anc 411	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → -_𝑒(𝐴 +_𝑒 𝐵) = (-_𝑒𝐴 +_𝑒 -_𝑒𝐵))
17		xnegdi 10223	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → -_𝑒(𝐴 +_𝑒 𝐶) = (-_𝑒𝐴 +_𝑒 -_𝑒𝐶))
18	1, 5, 17	syl2anc 411	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → -_𝑒(𝐴 +_𝑒 𝐶) = (-_𝑒𝐴 +_𝑒 -_𝑒𝐶))
19	16, 18	preq12d 3781	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → {-_𝑒(𝐴 +_𝑒 𝐵), -_𝑒(𝐴 +_𝑒 𝐶)} = {(-_𝑒𝐴 +_𝑒 -_𝑒𝐵), (-_𝑒𝐴 +_𝑒 -_𝑒𝐶)})
20	19	supeq1d 7291	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → sup({-_𝑒(𝐴 +_𝑒 𝐵), -_𝑒(𝐴 +_𝑒 𝐶)}, ℝ^, < ) = sup({(-_𝑒𝐴 +_𝑒 -_𝑒𝐵), (-_𝑒𝐴 +_𝑒 -_𝑒𝐶)}, ℝ^*, < ))
21		xnegeq 10182	. . . 4 ⊢ (sup({-_𝑒(𝐴 +_𝑒 𝐵), -_𝑒(𝐴 +_𝑒 𝐶)}, ℝ^, < ) = sup({(-_𝑒𝐴 +_𝑒 -_𝑒𝐵), (-_𝑒𝐴 +_𝑒 -_𝑒𝐶)}, ℝ^, < ) → -_𝑒sup({-_𝑒(𝐴 +_𝑒 𝐵), -_𝑒(𝐴 +_𝑒 𝐶)}, ℝ^, < ) = -_𝑒sup({(-_𝑒𝐴 +_𝑒 -_𝑒𝐵), (-_𝑒𝐴 +_𝑒 -_𝑒𝐶)}, ℝ^, < ))
22	20, 21	syl 14	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → -_𝑒sup({-_𝑒(𝐴 +_𝑒 𝐵), -_𝑒(𝐴 +_𝑒 𝐶)}, ℝ^, < ) = -_𝑒sup({(-_𝑒𝐴 +_𝑒 -_𝑒𝐵), (-_𝑒𝐴 +_𝑒 -_𝑒𝐶)}, ℝ^*, < ))
23		xrmaxadd 11974	. . . . 5 ⊢ ((-_𝑒𝐴 ∈ ℝ^* ∧ -_𝑒𝐵 ∈ ℝ^* ∧ -_𝑒𝐶 ∈ ℝ^) → sup({(-_𝑒𝐴 +_𝑒 -_𝑒𝐵), (-_𝑒𝐴 +_𝑒 -_𝑒𝐶)}, ℝ^, < ) = (-_𝑒𝐴 +_𝑒 sup({-_𝑒𝐵, -_𝑒𝐶}, ℝ^*, < )))
24	2, 4, 6, 23	syl3anc 1274	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → sup({(-_𝑒𝐴 +_𝑒 -_𝑒𝐵), (-_𝑒𝐴 +_𝑒 -_𝑒𝐶)}, ℝ^, < ) = (-_𝑒𝐴 +_𝑒 sup({-_𝑒𝐵, -_𝑒𝐶}, ℝ^*, < )))
25		xnegeq 10182	. . . 4 ⊢ (sup({(-_𝑒𝐴 +_𝑒 -_𝑒𝐵), (-_𝑒𝐴 +_𝑒 -_𝑒𝐶)}, ℝ^, < ) = (-_𝑒𝐴 +_𝑒 sup({-_𝑒𝐵, -_𝑒𝐶}, ℝ^, < )) → -_𝑒sup({(-_𝑒𝐴 +_𝑒 -_𝑒𝐵), (-_𝑒𝐴 +_𝑒 -_𝑒𝐶)}, ℝ^, < ) = -_𝑒(-_𝑒𝐴 +_𝑒 sup({-_𝑒𝐵, -_𝑒𝐶}, ℝ^, < )))
26	24, 25	syl 14	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → -_𝑒sup({(-_𝑒𝐴 +_𝑒 -_𝑒𝐵), (-_𝑒𝐴 +_𝑒 -_𝑒𝐶)}, ℝ^, < ) = -_𝑒(-_𝑒𝐴 +_𝑒 sup({-_𝑒𝐵, -_𝑒𝐶}, ℝ^*, < )))
27	14, 22, 26	3eqtrd 2271	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → inf({(𝐴 +_𝑒 𝐵), (𝐴 +_𝑒 𝐶)}, ℝ^, < ) = -_𝑒(-_𝑒𝐴 +_𝑒 sup({-_𝑒𝐵, -_𝑒𝐶}, ℝ^*, < )))
28		xnegneg 10188	. . . . 5 ⊢ (𝐴 ∈ ℝ^* → -_𝑒-_𝑒𝐴 = 𝐴)
29	28	eqcomd 2240	. . . 4 ⊢ (𝐴 ∈ ℝ^* → 𝐴 = -_𝑒-_𝑒𝐴)
30	1, 29	syl 14	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → 𝐴 = -_𝑒-_𝑒𝐴)
31		xrminmax 11978	. . . 4 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → inf({𝐵, 𝐶}, ℝ^, < ) = -_𝑒sup({-_𝑒𝐵, -_𝑒𝐶}, ℝ^*, < ))
32	3, 5, 31	syl2anc 411	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → inf({𝐵, 𝐶}, ℝ^, < ) = -_𝑒sup({-_𝑒𝐵, -_𝑒𝐶}, ℝ^*, < ))
33	30, 32	oveq12d 6076	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → (𝐴 +_𝑒 inf({𝐵, 𝐶}, ℝ^, < )) = (-_𝑒-_𝑒𝐴 +_𝑒 -_𝑒sup({-_𝑒𝐵, -_𝑒𝐶}, ℝ^*, < )))
34	10, 27, 33	3eqtr4d 2277	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → inf({(𝐴 +_𝑒 𝐵), (𝐴 +_𝑒 𝐶)}, ℝ^, < ) = (𝐴 +_𝑒 inf({𝐵, 𝐶}, ℝ^*, < )))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ w3a 1005 = wceq 1398 ∈ wcel 2205 {cpr 3695 (class class class)co 6058 supcsup 7286 infcinf 7287 ℝ^*cxr 8323 < clt 8324 -_𝑒cxne 10124 +_𝑒 cxad 10125

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 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2207 ax-14 2208 ax-ext 2216 ax-coll 4230 ax-sep 4233 ax-nul 4241 ax-pow 4292 ax-pr 4327 ax-un 4559 ax-setind 4664 ax-iinf 4715 ax-cnex 8234 ax-resscn 8235 ax-1cn 8236 ax-1re 8237 ax-icn 8238 ax-addcl 8239 ax-addrcl 8240 ax-mulcl 8241 ax-mulrcl 8242 ax-addcom 8243 ax-mulcom 8244 ax-addass 8245 ax-mulass 8246 ax-distr 8247 ax-i2m1 8248 ax-0lt1 8249 ax-1rid 8250 ax-0id 8251 ax-rnegex 8252 ax-precex 8253 ax-cnre 8254 ax-pre-ltirr 8255 ax-pre-ltwlin 8256 ax-pre-lttrn 8257 ax-pre-apti 8258 ax-pre-ltadd 8259 ax-pre-mulgt0 8260 ax-pre-mulext 8261 ax-arch 8262 ax-caucvg 8263

This theorem depends on definitions: df-bi 117 df-dc 843 df-3or 1006 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1812 df-eu 2085 df-mo 2086 df-clab 2221 df-cleq 2227 df-clel 2230 df-nfc 2375 df-ne 2415 df-nel 2510 df-ral 2527 df-rex 2528 df-reu 2529 df-rmo 2530 df-rab 2531 df-v 2817 df-sbc 3046 df-csb 3142 df-dif 3216 df-un 3218 df-in 3220 df-ss 3227 df-nul 3513 df-if 3625 df-pw 3676 df-sn 3700 df-pr 3701 df-op 3703 df-uni 3920 df-int 3955 df-iun 3998 df-br 4115 df-opab 4177 df-mpt 4178 df-tr 4214 df-id 4419 df-po 4422 df-iso 4423 df-iord 4492 df-on 4494 df-ilim 4495 df-suc 4497 df-iom 4718 df-xp 4760 df-rel 4761 df-cnv 4762 df-co 4763 df-dm 4764 df-rn 4765 df-res 4766 df-ima 4767 df-iota 5317 df-fun 5359 df-fn 5360 df-f 5361 df-f1 5362 df-fo 5363 df-f1o 5364 df-fv 5365 df-isom 5366 df-riota 6011 df-ov 6061 df-oprab 6062 df-mpo 6063 df-1st 6347 df-2nd 6348 df-recs 6549 df-frec 6635 df-sup 7288 df-inf 7289 df-pnf 8326 df-mnf 8327 df-xr 8328 df-ltxr 8329 df-le 8330 df-sub 8463 df-neg 8464 df-reap 8867 df-ap 8874 df-div 8967 df-inn 9258 df-2 9316 df-3 9317 df-4 9318 df-n0 9517 df-z 9598 df-uz 9875 df-rp 10008 df-xneg 10127 df-xadd 10128 df-seqfrec 10837 df-exp 10928 df-cj 11555 df-re 11556 df-im 11557 df-rsqrt 11711 df-abs 11712

This theorem is referenced by: (None)

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