xrminadd - Intuitionistic Logic Explorer

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Theorem xrminadd 11238

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 992	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → 𝐴 ∈ ℝ^)
2	1	xnegcld 9812	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → -_𝑒𝐴 ∈ ℝ^)
3		simp2 993	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → 𝐵 ∈ ℝ^)
4	3	xnegcld 9812	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → -_𝑒𝐵 ∈ ℝ^)
5		simp3 994	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → 𝐶 ∈ ℝ^)
6	5	xnegcld 9812	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → -_𝑒𝐶 ∈ ℝ^)
7		xrmaxcl 11215	. . . 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 9825	. . 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 9841	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → (𝐴 +_𝑒 𝐵) ∈ ℝ^)
12	1, 5	xaddcld 9841	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → (𝐴 +_𝑒 𝐶) ∈ ℝ^)
13		xrminmax 11228	. . . 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 9825	. . . . . . 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 9825	. . . . . . 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 3668	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → {-_𝑒(𝐴 +_𝑒 𝐵), -_𝑒(𝐴 +_𝑒 𝐶)} = {(-_𝑒𝐴 +_𝑒 -_𝑒𝐵), (-_𝑒𝐴 +_𝑒 -_𝑒𝐶)})
20	19	supeq1d 6964	. . . 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 9784	. . . 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 11224	. . . . 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 1233	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → sup({(-_𝑒𝐴 +_𝑒 -_𝑒𝐵), (-_𝑒𝐴 +_𝑒 -_𝑒𝐶)}, ℝ^, < ) = (-_𝑒𝐴 +_𝑒 sup({-_𝑒𝐵, -_𝑒𝐶}, ℝ^*, < )))
25		xnegeq 9784	. . . 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 2207	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → inf({(𝐴 +_𝑒 𝐵), (𝐴 +_𝑒 𝐶)}, ℝ^, < ) = -_𝑒(-_𝑒𝐴 +_𝑒 sup({-_𝑒𝐵, -_𝑒𝐶}, ℝ^*, < )))
28		xnegneg 9790	. . . . 5 ⊢ (𝐴 ∈ ℝ^* → -_𝑒-_𝑒𝐴 = 𝐴)
29	28	eqcomd 2176	. . . 4 ⊢ (𝐴 ∈ ℝ^* → 𝐴 = -_𝑒-_𝑒𝐴)
30	1, 29	syl 14	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → 𝐴 = -_𝑒-_𝑒𝐴)
31		xrminmax 11228	. . . 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 5871	. 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 2213	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 973 = wceq 1348 ∈ wcel 2141 {cpr 3584 (class class class)co 5853 supcsup 6959 infcinf 6960 ℝ^*cxr 7953 < clt 7954 -_𝑒cxne 9726 +_𝑒 cxad 9727

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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4104 ax-sep 4107 ax-nul 4115 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-iinf 4572 ax-cnex 7865 ax-resscn 7866 ax-1cn 7867 ax-1re 7868 ax-icn 7869 ax-addcl 7870 ax-addrcl 7871 ax-mulcl 7872 ax-mulrcl 7873 ax-addcom 7874 ax-mulcom 7875 ax-addass 7876 ax-mulass 7877 ax-distr 7878 ax-i2m1 7879 ax-0lt1 7880 ax-1rid 7881 ax-0id 7882 ax-rnegex 7883 ax-precex 7884 ax-cnre 7885 ax-pre-ltirr 7886 ax-pre-ltwlin 7887 ax-pre-lttrn 7888 ax-pre-apti 7889 ax-pre-ltadd 7890 ax-pre-mulgt0 7891 ax-pre-mulext 7892 ax-arch 7893 ax-caucvg 7894

This theorem depends on definitions: df-bi 116 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rmo 2456 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-if 3527 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-iun 3875 df-br 3990 df-opab 4051 df-mpt 4052 df-tr 4088 df-id 4278 df-po 4281 df-iso 4282 df-iord 4351 df-on 4353 df-ilim 4354 df-suc 4356 df-iom 4575 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-f1 5203 df-fo 5204 df-f1o 5205 df-fv 5206 df-isom 5207 df-riota 5809 df-ov 5856 df-oprab 5857 df-mpo 5858 df-1st 6119 df-2nd 6120 df-recs 6284 df-frec 6370 df-sup 6961 df-inf 6962 df-pnf 7956 df-mnf 7957 df-xr 7958 df-ltxr 7959 df-le 7960 df-sub 8092 df-neg 8093 df-reap 8494 df-ap 8501 df-div 8590 df-inn 8879 df-2 8937 df-3 8938 df-4 8939 df-n0 9136 df-z 9213 df-uz 9488 df-rp 9611 df-xneg 9729 df-xadd 9730 df-seqfrec 10402 df-exp 10476 df-cj 10806 df-re 10807 df-im 10808 df-rsqrt 10962 df-abs 10963

This theorem is referenced by: (None)

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