xrminadd - Intuitionistic Logic Explorer

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Theorem xrminadd 11216

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 987	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → 𝐴 ∈ ℝ^)
2	1	xnegcld 9791	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → -_𝑒𝐴 ∈ ℝ^)
3		simp2 988	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → 𝐵 ∈ ℝ^)
4	3	xnegcld 9791	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → -_𝑒𝐵 ∈ ℝ^)
5		simp3 989	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → 𝐶 ∈ ℝ^)
6	5	xnegcld 9791	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → -_𝑒𝐶 ∈ ℝ^)
7		xrmaxcl 11193	. . . 4 ⊢ ((-_𝑒𝐵 ∈ ℝ^* ∧ -_𝑒𝐶 ∈ ℝ^) → sup({-_𝑒𝐵, -_𝑒𝐶}, ℝ^, < ) ∈ ℝ^*)
8	4, 6, 7	syl2anc 409	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → sup({-_𝑒𝐵, -_𝑒𝐶}, ℝ^, < ) ∈ ℝ^*)
9		xnegdi 9804	. . 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 409	. 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 9820	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → (𝐴 +_𝑒 𝐵) ∈ ℝ^)
12	1, 5	xaddcld 9820	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → (𝐴 +_𝑒 𝐶) ∈ ℝ^)
13		xrminmax 11206	. . . 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 409	. . 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 9804	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → -_𝑒(𝐴 +_𝑒 𝐵) = (-_𝑒𝐴 +_𝑒 -_𝑒𝐵))
16	1, 3, 15	syl2anc 409	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → -_𝑒(𝐴 +_𝑒 𝐵) = (-_𝑒𝐴 +_𝑒 -_𝑒𝐵))
17		xnegdi 9804	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → -_𝑒(𝐴 +_𝑒 𝐶) = (-_𝑒𝐴 +_𝑒 -_𝑒𝐶))
18	1, 5, 17	syl2anc 409	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → -_𝑒(𝐴 +_𝑒 𝐶) = (-_𝑒𝐴 +_𝑒 -_𝑒𝐶))
19	16, 18	preq12d 3661	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → {-_𝑒(𝐴 +_𝑒 𝐵), -_𝑒(𝐴 +_𝑒 𝐶)} = {(-_𝑒𝐴 +_𝑒 -_𝑒𝐵), (-_𝑒𝐴 +_𝑒 -_𝑒𝐶)})
20	19	supeq1d 6952	. . . 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 9763	. . . 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 11202	. . . . 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 1228	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → sup({(-_𝑒𝐴 +_𝑒 -_𝑒𝐵), (-_𝑒𝐴 +_𝑒 -_𝑒𝐶)}, ℝ^, < ) = (-_𝑒𝐴 +_𝑒 sup({-_𝑒𝐵, -_𝑒𝐶}, ℝ^*, < )))
25		xnegeq 9763	. . . 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 2202	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → inf({(𝐴 +_𝑒 𝐵), (𝐴 +_𝑒 𝐶)}, ℝ^, < ) = -_𝑒(-_𝑒𝐴 +_𝑒 sup({-_𝑒𝐵, -_𝑒𝐶}, ℝ^*, < )))
28		xnegneg 9769	. . . . 5 ⊢ (𝐴 ∈ ℝ^* → -_𝑒-_𝑒𝐴 = 𝐴)
29	28	eqcomd 2171	. . . 4 ⊢ (𝐴 ∈ ℝ^* → 𝐴 = -_𝑒-_𝑒𝐴)
30	1, 29	syl 14	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → 𝐴 = -_𝑒-_𝑒𝐴)
31		xrminmax 11206	. . . 4 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → inf({𝐵, 𝐶}, ℝ^, < ) = -_𝑒sup({-_𝑒𝐵, -_𝑒𝐶}, ℝ^*, < ))
32	3, 5, 31	syl2anc 409	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → inf({𝐵, 𝐶}, ℝ^, < ) = -_𝑒sup({-_𝑒𝐵, -_𝑒𝐶}, ℝ^*, < ))
33	30, 32	oveq12d 5860	. 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 2208	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 968 = wceq 1343 ∈ wcel 2136 {cpr 3577 (class class class)co 5842 supcsup 6947 infcinf 6948 ℝ^*cxr 7932 < clt 7933 -_𝑒cxne 9705 +_𝑒 cxad 9706

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 604 ax-in2 605 ax-io 699 ax-5 1435 ax-7 1436 ax-gen 1437 ax-ie1 1481 ax-ie2 1482 ax-8 1492 ax-10 1493 ax-11 1494 ax-i12 1495 ax-bndl 1497 ax-4 1498 ax-17 1514 ax-i9 1518 ax-ial 1522 ax-i5r 1523 ax-13 2138 ax-14 2139 ax-ext 2147 ax-coll 4097 ax-sep 4100 ax-nul 4108 ax-pow 4153 ax-pr 4187 ax-un 4411 ax-setind 4514 ax-iinf 4565 ax-cnex 7844 ax-resscn 7845 ax-1cn 7846 ax-1re 7847 ax-icn 7848 ax-addcl 7849 ax-addrcl 7850 ax-mulcl 7851 ax-mulrcl 7852 ax-addcom 7853 ax-mulcom 7854 ax-addass 7855 ax-mulass 7856 ax-distr 7857 ax-i2m1 7858 ax-0lt1 7859 ax-1rid 7860 ax-0id 7861 ax-rnegex 7862 ax-precex 7863 ax-cnre 7864 ax-pre-ltirr 7865 ax-pre-ltwlin 7866 ax-pre-lttrn 7867 ax-pre-apti 7868 ax-pre-ltadd 7869 ax-pre-mulgt0 7870 ax-pre-mulext 7871 ax-arch 7872 ax-caucvg 7873

This theorem depends on definitions: df-bi 116 df-dc 825 df-3or 969 df-3an 970 df-tru 1346 df-fal 1349 df-nf 1449 df-sb 1751 df-eu 2017 df-mo 2018 df-clab 2152 df-cleq 2158 df-clel 2161 df-nfc 2297 df-ne 2337 df-nel 2432 df-ral 2449 df-rex 2450 df-reu 2451 df-rmo 2452 df-rab 2453 df-v 2728 df-sbc 2952 df-csb 3046 df-dif 3118 df-un 3120 df-in 3122 df-ss 3129 df-nul 3410 df-if 3521 df-pw 3561 df-sn 3582 df-pr 3583 df-op 3585 df-uni 3790 df-int 3825 df-iun 3868 df-br 3983 df-opab 4044 df-mpt 4045 df-tr 4081 df-id 4271 df-po 4274 df-iso 4275 df-iord 4344 df-on 4346 df-ilim 4347 df-suc 4349 df-iom 4568 df-xp 4610 df-rel 4611 df-cnv 4612 df-co 4613 df-dm 4614 df-rn 4615 df-res 4616 df-ima 4617 df-iota 5153 df-fun 5190 df-fn 5191 df-f 5192 df-f1 5193 df-fo 5194 df-f1o 5195 df-fv 5196 df-isom 5197 df-riota 5798 df-ov 5845 df-oprab 5846 df-mpo 5847 df-1st 6108 df-2nd 6109 df-recs 6273 df-frec 6359 df-sup 6949 df-inf 6950 df-pnf 7935 df-mnf 7936 df-xr 7937 df-ltxr 7938 df-le 7939 df-sub 8071 df-neg 8072 df-reap 8473 df-ap 8480 df-div 8569 df-inn 8858 df-2 8916 df-3 8917 df-4 8918 df-n0 9115 df-z 9192 df-uz 9467 df-rp 9590 df-xneg 9708 df-xadd 9709 df-seqfrec 10381 df-exp 10455 df-cj 10784 df-re 10785 df-im 10786 df-rsqrt 10940 df-abs 10941

This theorem is referenced by: (None)

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