xrminadd - Intuitionistic Logic Explorer

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Theorem xrminadd 11044

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 981	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → 𝐴 ∈ ℝ^)
2	1	xnegcld 9638	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → -_𝑒𝐴 ∈ ℝ^)
3		simp2 982	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → 𝐵 ∈ ℝ^)
4	3	xnegcld 9638	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → -_𝑒𝐵 ∈ ℝ^)
5		simp3 983	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → 𝐶 ∈ ℝ^)
6	5	xnegcld 9638	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → -_𝑒𝐶 ∈ ℝ^)
7		xrmaxcl 11021	. . . 4 ⊢ ((-_𝑒𝐵 ∈ ℝ^* ∧ -_𝑒𝐶 ∈ ℝ^) → sup({-_𝑒𝐵, -_𝑒𝐶}, ℝ^, < ) ∈ ℝ^*)
8	4, 6, 7	syl2anc 408	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → sup({-_𝑒𝐵, -_𝑒𝐶}, ℝ^, < ) ∈ ℝ^*)
9		xnegdi 9651	. . 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 408	. 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 9667	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → (𝐴 +_𝑒 𝐵) ∈ ℝ^)
12	1, 5	xaddcld 9667	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → (𝐴 +_𝑒 𝐶) ∈ ℝ^)
13		xrminmax 11034	. . . 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 408	. . 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 9651	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → -_𝑒(𝐴 +_𝑒 𝐵) = (-_𝑒𝐴 +_𝑒 -_𝑒𝐵))
16	1, 3, 15	syl2anc 408	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → -_𝑒(𝐴 +_𝑒 𝐵) = (-_𝑒𝐴 +_𝑒 -_𝑒𝐵))
17		xnegdi 9651	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → -_𝑒(𝐴 +_𝑒 𝐶) = (-_𝑒𝐴 +_𝑒 -_𝑒𝐶))
18	1, 5, 17	syl2anc 408	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → -_𝑒(𝐴 +_𝑒 𝐶) = (-_𝑒𝐴 +_𝑒 -_𝑒𝐶))
19	16, 18	preq12d 3608	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → {-_𝑒(𝐴 +_𝑒 𝐵), -_𝑒(𝐴 +_𝑒 𝐶)} = {(-_𝑒𝐴 +_𝑒 -_𝑒𝐵), (-_𝑒𝐴 +_𝑒 -_𝑒𝐶)})
20	19	supeq1d 6874	. . . 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 9610	. . . 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 11030	. . . . 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 1216	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → sup({(-_𝑒𝐴 +_𝑒 -_𝑒𝐵), (-_𝑒𝐴 +_𝑒 -_𝑒𝐶)}, ℝ^, < ) = (-_𝑒𝐴 +_𝑒 sup({-_𝑒𝐵, -_𝑒𝐶}, ℝ^*, < )))
25		xnegeq 9610	. . . 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 2176	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → inf({(𝐴 +_𝑒 𝐵), (𝐴 +_𝑒 𝐶)}, ℝ^, < ) = -_𝑒(-_𝑒𝐴 +_𝑒 sup({-_𝑒𝐵, -_𝑒𝐶}, ℝ^*, < )))
28		xnegneg 9616	. . . . 5 ⊢ (𝐴 ∈ ℝ^* → -_𝑒-_𝑒𝐴 = 𝐴)
29	28	eqcomd 2145	. . . 4 ⊢ (𝐴 ∈ ℝ^* → 𝐴 = -_𝑒-_𝑒𝐴)
30	1, 29	syl 14	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → 𝐴 = -_𝑒-_𝑒𝐴)
31		xrminmax 11034	. . . 4 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → inf({𝐵, 𝐶}, ℝ^, < ) = -_𝑒sup({-_𝑒𝐵, -_𝑒𝐶}, ℝ^*, < ))
32	3, 5, 31	syl2anc 408	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → inf({𝐵, 𝐶}, ℝ^, < ) = -_𝑒sup({-_𝑒𝐵, -_𝑒𝐶}, ℝ^*, < ))
33	30, 32	oveq12d 5792	. 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 2182	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 962 = wceq 1331 ∈ wcel 1480 {cpr 3528 (class class class)co 5774 supcsup 6869 infcinf 6870 ℝ^*cxr 7799 < clt 7800 -_𝑒cxne 9556 +_𝑒 cxad 9557

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-mulrcl 7719 ax-addcom 7720 ax-mulcom 7721 ax-addass 7722 ax-mulass 7723 ax-distr 7724 ax-i2m1 7725 ax-0lt1 7726 ax-1rid 7727 ax-0id 7728 ax-rnegex 7729 ax-precex 7730 ax-cnre 7731 ax-pre-ltirr 7732 ax-pre-ltwlin 7733 ax-pre-lttrn 7734 ax-pre-apti 7735 ax-pre-ltadd 7736 ax-pre-mulgt0 7737 ax-pre-mulext 7738 ax-arch 7739 ax-caucvg 7740

This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rmo 2424 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-if 3475 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-id 4215 df-po 4218 df-iso 4219 df-iord 4288 df-on 4290 df-ilim 4291 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-isom 5132 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-recs 6202 df-frec 6288 df-sup 6871 df-inf 6872 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-sub 7935 df-neg 7936 df-reap 8337 df-ap 8344 df-div 8433 df-inn 8721 df-2 8779 df-3 8780 df-4 8781 df-n0 8978 df-z 9055 df-uz 9327 df-rp 9442 df-xneg 9559 df-xadd 9560 df-seqfrec 10219 df-exp 10293 df-cj 10614 df-re 10615 df-im 10616 df-rsqrt 10770 df-abs 10771

This theorem is referenced by: (None)

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