Theorem infrenegsupex 9976

Description: The infimum of a set of reals 𝐴 is the negative of the supremum of the negatives of its elements. (Contributed by Jim Kingdon, 14-Jan-2022.)

Hypotheses

Ref	Expression
infrenegsupex.ex	⊢ (𝜑 → ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)))
infrenegsupex.ss	⊢ (𝜑 → 𝐴 ⊆ ℝ)

Assertion

Ref	Expression
infrenegsupex	⊢ (𝜑 → inf(𝐴, ℝ, < ) = -sup({𝑧 ∈ ℝ ∣ -𝑧 ∈ 𝐴}, ℝ, < ))

Distinct variable groups:   𝑥,𝐴,𝑦,𝑧   𝜑,𝑥,𝑦,𝑧

Proof of Theorem infrenegsupex

Dummy variables 𝑓 𝑔 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		lttri3 8398	. . . . . 6 ⊢ ((𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ) → (𝑓 = 𝑔 ↔ (¬ 𝑓 < 𝑔 ∧ ¬ 𝑔 < 𝑓)))
2	1	adantl 277	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ)) → (𝑓 = 𝑔 ↔ (¬ 𝑓 < 𝑔 ∧ ¬ 𝑔 < 𝑓)))
3		infrenegsupex.ex	. . . . 5 ⊢ (𝜑 → ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)))
4	2, 3	infclti 7356	. . . 4 ⊢ (𝜑 → inf(𝐴, ℝ, < ) ∈ ℝ)
5	4	recnd 8347	. . 3 ⊢ (𝜑 → inf(𝐴, ℝ, < ) ∈ ℂ)
6	5	negnegd 8621	. 2 ⊢ (𝜑 → --inf(𝐴, ℝ, < ) = inf(𝐴, ℝ, < ))
7		negeq 8512	. . . . . . . . 9 ⊢ (𝑤 = 𝑧 → -𝑤 = -𝑧)
8	7	cbvmptv 4225	. . . . . . . 8 ⊢ (𝑤 ∈ ℝ ↦ -𝑤) = (𝑧 ∈ ℝ ↦ -𝑧)
9	8	mptpreima 5279	. . . . . . 7 ⊢ (◡(𝑤 ∈ ℝ ↦ -𝑤) “ 𝐴) = {𝑧 ∈ ℝ ∣ -𝑧 ∈ 𝐴}
10		eqid 2238	. . . . . . . . . 10 ⊢ (𝑤 ∈ ℝ ↦ -𝑤) = (𝑤 ∈ ℝ ↦ -𝑤)
11	10	negiso 9278	. . . . . . . . 9 ⊢ ((𝑤 ∈ ℝ ↦ -𝑤) Isom < , ◡ < (ℝ, ℝ) ∧ ◡(𝑤 ∈ ℝ ↦ -𝑤) = (𝑤 ∈ ℝ ↦ -𝑤))
12	11	simpri 113	. . . . . . . 8 ⊢ ◡(𝑤 ∈ ℝ ↦ -𝑤) = (𝑤 ∈ ℝ ↦ -𝑤)
13	12	imaeq1i 5121	. . . . . . 7 ⊢ (◡(𝑤 ∈ ℝ ↦ -𝑤) “ 𝐴) = ((𝑤 ∈ ℝ ↦ -𝑤) “ 𝐴)
14	9, 13	eqtr3i 2261	. . . . . 6 ⊢ {𝑧 ∈ ℝ ∣ -𝑧 ∈ 𝐴} = ((𝑤 ∈ ℝ ↦ -𝑤) “ 𝐴)
15	14	supeq1i 7321	. . . . 5 ⊢ sup({𝑧 ∈ ℝ ∣ -𝑧 ∈ 𝐴}, ℝ, < ) = sup(((𝑤 ∈ ℝ ↦ -𝑤) “ 𝐴), ℝ, < )
16	11	simpli 111	. . . . . . . . 9 ⊢ (𝑤 ∈ ℝ ↦ -𝑤) Isom < , ◡ < (ℝ, ℝ)
17		isocnv 6010	. . . . . . . . 9 ⊢ ((𝑤 ∈ ℝ ↦ -𝑤) Isom < , ◡ < (ℝ, ℝ) → ◡(𝑤 ∈ ℝ ↦ -𝑤) Isom ◡ < , < (ℝ, ℝ))
18	16, 17	ax-mp 5	. . . . . . . 8 ⊢ ◡(𝑤 ∈ ℝ ↦ -𝑤) Isom ◡ < , < (ℝ, ℝ)
19		isoeq1 6000	. . . . . . . . 9 ⊢ (◡(𝑤 ∈ ℝ ↦ -𝑤) = (𝑤 ∈ ℝ ↦ -𝑤) → (◡(𝑤 ∈ ℝ ↦ -𝑤) Isom ◡ < , < (ℝ, ℝ) ↔ (𝑤 ∈ ℝ ↦ -𝑤) Isom ◡ < , < (ℝ, ℝ)))
20	12, 19	ax-mp 5	. . . . . . . 8 ⊢ (◡(𝑤 ∈ ℝ ↦ -𝑤) Isom ◡ < , < (ℝ, ℝ) ↔ (𝑤 ∈ ℝ ↦ -𝑤) Isom ◡ < , < (ℝ, ℝ))
21	18, 20	mpbi 145	. . . . . . 7 ⊢ (𝑤 ∈ ℝ ↦ -𝑤) Isom ◡ < , < (ℝ, ℝ)
22	21	a1i 9	. . . . . 6 ⊢ (𝜑 → (𝑤 ∈ ℝ ↦ -𝑤) Isom ◡ < , < (ℝ, ℝ))
23		infrenegsupex.ss	. . . . . 6 ⊢ (𝜑 → 𝐴 ⊆ ℝ)
24	3	cnvinfex 7351	. . . . . 6 ⊢ (𝜑 → ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝐴 ¬ 𝑥◡ < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦◡ < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦◡ < 𝑧)))
25	2	cnvti 7352	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ)) → (𝑓 = 𝑔 ↔ (¬ 𝑓◡ < 𝑔 ∧ ¬ 𝑔◡ < 𝑓)))
26	22, 23, 24, 25	supisoti 7343	. . . . 5 ⊢ (𝜑 → sup(((𝑤 ∈ ℝ ↦ -𝑤) “ 𝐴), ℝ, < ) = ((𝑤 ∈ ℝ ↦ -𝑤)‘sup(𝐴, ℝ, ◡ < )))
27	15, 26	eqtrid 2283	. . . 4 ⊢ (𝜑 → sup({𝑧 ∈ ℝ ∣ -𝑧 ∈ 𝐴}, ℝ, < ) = ((𝑤 ∈ ℝ ↦ -𝑤)‘sup(𝐴, ℝ, ◡ < )))
28		df-inf 7318	. . . . . . 7 ⊢ inf(𝐴, ℝ, < ) = sup(𝐴, ℝ, ◡ < )
29	28	eqcomi 2242	. . . . . 6 ⊢ sup(𝐴, ℝ, ◡ < ) = inf(𝐴, ℝ, < )
30	29	fveq2i 5696	. . . . 5 ⊢ ((𝑤 ∈ ℝ ↦ -𝑤)‘sup(𝐴, ℝ, ◡ < )) = ((𝑤 ∈ ℝ ↦ -𝑤)‘inf(𝐴, ℝ, < ))
31		eqidd 2239	. . . . . 6 ⊢ (𝜑 → (𝑤 ∈ ℝ ↦ -𝑤) = (𝑤 ∈ ℝ ↦ -𝑤))
32		negeq 8512	. . . . . . 7 ⊢ (𝑤 = inf(𝐴, ℝ, < ) → -𝑤 = -inf(𝐴, ℝ, < ))
33	32	adantl 277	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 = inf(𝐴, ℝ, < )) → -𝑤 = -inf(𝐴, ℝ, < ))
34	5	negcld 8617	. . . . . 6 ⊢ (𝜑 → -inf(𝐴, ℝ, < ) ∈ ℂ)
35	31, 33, 4, 34	fvmptd 5783	. . . . 5 ⊢ (𝜑 → ((𝑤 ∈ ℝ ↦ -𝑤)‘inf(𝐴, ℝ, < )) = -inf(𝐴, ℝ, < ))
36	30, 35	eqtrid 2283	. . . 4 ⊢ (𝜑 → ((𝑤 ∈ ℝ ↦ -𝑤)‘sup(𝐴, ℝ, ◡ < )) = -inf(𝐴, ℝ, < ))
37	27, 36	eqtr2d 2272	. . 3 ⊢ (𝜑 → -inf(𝐴, ℝ, < ) = sup({𝑧 ∈ ℝ ∣ -𝑧 ∈ 𝐴}, ℝ, < ))
38	37	negeqd 8514	. 2 ⊢ (𝜑 → --inf(𝐴, ℝ, < ) = -sup({𝑧 ∈ ℝ ∣ -𝑧 ∈ 𝐴}, ℝ, < ))
39	6, 38	eqtr3d 2273	1 ⊢ (𝜑 → inf(𝐴, ℝ, < ) = -sup({𝑧 ∈ ℝ ∣ -𝑧 ∈ 𝐴}, ℝ, < ))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1402   ∈ wcel 2209  ∀wral 2528  ∃wrex 2529  {crab 2532   ⊆ wss 3220   class class class wbr 4128   ↦ cmpt 4190  ^◡ccnv 4771   “ cima 4775  ‘cfv 5375   Isom wiso 5376  supcsup 7315  infcinf 7316  ℂcc 8170  ℝcr 8171   < clt 8353  -cneg 8491

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-sep 4247  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-cnex 8263  ax-resscn 8264  ax-1cn 8265  ax-1re 8266  ax-icn 8267  ax-addcl 8268  ax-addrcl 8269  ax-mulcl 8270  ax-addcom 8272  ax-addass 8274  ax-distr 8276  ax-i2m1 8277  ax-0id 8280  ax-rnegex 8281  ax-cnre 8283  ax-pre-ltirr 8284  ax-pre-apti 8287  ax-pre-ltadd 8288

This theorem depends on definitions:  df-bi 117  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-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-br 4129  df-opab 4191  df-mpt 4192  df-id 4436  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 6031  df-ov 6081  df-oprab 6082  df-mpo 6083  df-sup 7317  df-inf 7318  df-pnf 8355  df-mnf 8356  df-ltxr 8358  df-sub 8492  df-neg 8493

This theorem is referenced by:  supminfex  9979  infssuzcldc  10649  minmax  11977