xrminadd - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > xrminadd		GIF version

Theorem xrminadd 12024

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 1028	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → 𝐴 ∈ ℝ^)
2	1	xnegcld 10240	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → -_𝑒𝐴 ∈ ℝ^)
3		simp2 1029	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → 𝐵 ∈ ℝ^)
4	3	xnegcld 10240	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → -_𝑒𝐵 ∈ ℝ^)
5		simp3 1030	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → 𝐶 ∈ ℝ^)
6	5	xnegcld 10240	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → -_𝑒𝐶 ∈ ℝ^)
7		xrmaxcl 12001	. . . 4 ⊢ ((-_𝑒𝐵 ∈ ℝ^* ∧ -_𝑒𝐶 ∈ ℝ^) → sup({-_𝑒𝐵, -_𝑒𝐶}, ℝ^, < ) ∈ ℝ^*)
8	4, 6, 7	syl2anc 415	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → sup({-_𝑒𝐵, -_𝑒𝐶}, ℝ^, < ) ∈ ℝ^*)
9		xnegdi 10253	. . 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 415	. 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 10269	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → (𝐴 +_𝑒 𝐵) ∈ ℝ^)
12	1, 5	xaddcld 10269	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → (𝐴 +_𝑒 𝐶) ∈ ℝ^)
13		xrminmax 12014	. . . 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 415	. . 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 10253	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → -_𝑒(𝐴 +_𝑒 𝐵) = (-_𝑒𝐴 +_𝑒 -_𝑒𝐵))
16	1, 3, 15	syl2anc 415	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → -_𝑒(𝐴 +_𝑒 𝐵) = (-_𝑒𝐴 +_𝑒 -_𝑒𝐵))
17		xnegdi 10253	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → -_𝑒(𝐴 +_𝑒 𝐶) = (-_𝑒𝐴 +_𝑒 -_𝑒𝐶))
18	1, 5, 17	syl2anc 415	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → -_𝑒(𝐴 +_𝑒 𝐶) = (-_𝑒𝐴 +_𝑒 -_𝑒𝐶))
19	16, 18	preq12d 3795	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → {-_𝑒(𝐴 +_𝑒 𝐵), -_𝑒(𝐴 +_𝑒 𝐶)} = {(-_𝑒𝐴 +_𝑒 -_𝑒𝐵), (-_𝑒𝐴 +_𝑒 -_𝑒𝐶)})
20	19	supeq1d 7321	. . . 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 10212	. . . 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 12010	. . . . 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 1278	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → sup({(-_𝑒𝐴 +_𝑒 -_𝑒𝐵), (-_𝑒𝐴 +_𝑒 -_𝑒𝐶)}, ℝ^, < ) = (-_𝑒𝐴 +_𝑒 sup({-_𝑒𝐵, -_𝑒𝐶}, ℝ^*, < )))
25		xnegeq 10212	. . . 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 2275	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → inf({(𝐴 +_𝑒 𝐵), (𝐴 +_𝑒 𝐶)}, ℝ^, < ) = -_𝑒(-_𝑒𝐴 +_𝑒 sup({-_𝑒𝐵, -_𝑒𝐶}, ℝ^*, < )))
28		xnegneg 10218	. . . . 5 ⊢ (𝐴 ∈ ℝ^* → -_𝑒-_𝑒𝐴 = 𝐴)
29	28	eqcomd 2244	. . . 4 ⊢ (𝐴 ∈ ℝ^* → 𝐴 = -_𝑒-_𝑒𝐴)
30	1, 29	syl 14	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → 𝐴 = -_𝑒-_𝑒𝐴)
31		xrminmax 12014	. . . 4 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → inf({𝐵, 𝐶}, ℝ^, < ) = -_𝑒sup({-_𝑒𝐵, -_𝑒𝐶}, ℝ^*, < ))
32	3, 5, 31	syl2anc 415	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → inf({𝐵, 𝐶}, ℝ^, < ) = -_𝑒sup({-_𝑒𝐵, -_𝑒𝐶}, ℝ^*, < ))
33	30, 32	oveq12d 6097	. 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 2281	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 1009 = wceq 1402 ∈ wcel 2209 {cpr 3709 (class class class)co 6079 supcsup 7316 infcinf 7317 ℝ^*cxr 8353 < clt 8354 -_𝑒cxne 10154 +_𝑒 cxad 10155

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 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-coll 4244 ax-sep 4247 ax-nul 4257 ax-pow 4309 ax-pr 4344 ax-un 4576 ax-setind 4682 ax-iinf 4733 ax-cnex 8264 ax-resscn 8265 ax-1cn 8266 ax-1re 8267 ax-icn 8268 ax-addcl 8269 ax-addrcl 8270 ax-mulcl 8271 ax-mulrcl 8272 ax-addcom 8273 ax-mulcom 8274 ax-addass 8275 ax-mulass 8276 ax-distr 8277 ax-i2m1 8278 ax-0lt1 8279 ax-1rid 8280 ax-0id 8281 ax-rnegex 8282 ax-precex 8283 ax-cnre 8284 ax-pre-ltirr 8285 ax-pre-ltwlin 8286 ax-pre-lttrn 8287 ax-pre-apti 8288 ax-pre-ltadd 8289 ax-pre-mulgt0 8290 ax-pre-mulext 8291 ax-arch 8292 ax-caucvg 8293

This theorem depends on definitions: df-bi 117 df-dc 847 df-3or 1010 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-nel 2516 df-ral 2533 df-rex 2534 df-reu 2535 df-rmo 2536 df-rab 2537 df-v 2823 df-sbc 3052 df-csb 3148 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-nul 3521 df-if 3639 df-pw 3690 df-sn 3714 df-pr 3715 df-op 3717 df-uni 3934 df-int 3969 df-iun 4012 df-br 4129 df-opab 4191 df-mpt 4192 df-tr 4228 df-id 4436 df-po 4439 df-iso 4440 df-iord 4509 df-on 4511 df-ilim 4512 df-suc 4514 df-iom 4736 df-xp 4778 df-rel 4779 df-cnv 4780 df-co 4781 df-dm 4782 df-rn 4783 df-res 4784 df-ima 4785 df-iota 5335 df-fun 5377 df-fn 5378 df-f 5379 df-f1 5380 df-fo 5381 df-f1o 5382 df-fv 5383 df-isom 5384 df-riota 6032 df-ov 6082 df-oprab 6083 df-mpo 6084 df-1st 6368 df-2nd 6369 df-recs 6570 df-frec 6656 df-sup 7318 df-inf 7319 df-pnf 8356 df-mnf 8357 df-xr 8358 df-ltxr 8359 df-le 8360 df-sub 8493 df-neg 8494 df-reap 8897 df-ap 8904 df-div 8997 df-inn 9288 df-2 9346 df-3 9347 df-4 9348 df-n0 9547 df-z 9628 df-uz 9905 df-rp 10038 df-xneg 10157 df-xadd 10158 df-seqfrec 10868 df-exp 10959 df-cj 11590 df-re 11591 df-im 11592 df-rsqrt 11747 df-abs 11748

This theorem is referenced by: (None)

W3C validator