Theorem infrenegsupex 9944

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 8369	. . . . . 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 7327	. . . 4 ⊢ (𝜑 → inf(𝐴, ℝ, < ) ∈ ℝ)
5	4	recnd 8318	. . 3 ⊢ (𝜑 → inf(𝐴, ℝ, < ) ∈ ℂ)
6	5	negnegd 8591	. 2 ⊢ (𝜑 → --inf(𝐴, ℝ, < ) = inf(𝐴, ℝ, < ))
7		negeq 8482	. . . . . . . . 9 ⊢ (𝑤 = 𝑧 → -𝑤 = -𝑧)
8	7	cbvmptv 4211	. . . . . . . 8 ⊢ (𝑤 ∈ ℝ ↦ -𝑤) = (𝑧 ∈ ℝ ↦ -𝑧)
9	8	mptpreima 5261	. . . . . . 7 ⊢ (◡(𝑤 ∈ ℝ ↦ -𝑤) “ 𝐴) = {𝑧 ∈ ℝ ∣ -𝑧 ∈ 𝐴}
10		eqid 2234	. . . . . . . . . 10 ⊢ (𝑤 ∈ ℝ ↦ -𝑤) = (𝑤 ∈ ℝ ↦ -𝑤)
11	10	negiso 9246	. . . . . . . . 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 5103	. . . . . . 7 ⊢ (◡(𝑤 ∈ ℝ ↦ -𝑤) “ 𝐴) = ((𝑤 ∈ ℝ ↦ -𝑤) “ 𝐴)
14	9, 13	eqtr3i 2257	. . . . . 6 ⊢ {𝑧 ∈ ℝ ∣ -𝑧 ∈ 𝐴} = ((𝑤 ∈ ℝ ↦ -𝑤) “ 𝐴)
15	14	supeq1i 7292	. . . . 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 5990	. . . . . . . . 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 5980	. . . . . . . . 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 7322	. . . . . 6 ⊢ (𝜑 → ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝐴 ¬ 𝑥◡ < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦◡ < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦◡ < 𝑧)))
25	2	cnvti 7323	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ)) → (𝑓 = 𝑔 ↔ (¬ 𝑓◡ < 𝑔 ∧ ¬ 𝑔◡ < 𝑓)))
26	22, 23, 24, 25	supisoti 7314	. . . . 5 ⊢ (𝜑 → sup(((𝑤 ∈ ℝ ↦ -𝑤) “ 𝐴), ℝ, < ) = ((𝑤 ∈ ℝ ↦ -𝑤)‘sup(𝐴, ℝ, ◡ < )))
27	15, 26	eqtrid 2279	. . . 4 ⊢ (𝜑 → sup({𝑧 ∈ ℝ ∣ -𝑧 ∈ 𝐴}, ℝ, < ) = ((𝑤 ∈ ℝ ↦ -𝑤)‘sup(𝐴, ℝ, ◡ < )))
28		df-inf 7289	. . . . . . 7 ⊢ inf(𝐴, ℝ, < ) = sup(𝐴, ℝ, ◡ < )
29	28	eqcomi 2238	. . . . . 6 ⊢ sup(𝐴, ℝ, ◡ < ) = inf(𝐴, ℝ, < )
30	29	fveq2i 5678	. . . . 5 ⊢ ((𝑤 ∈ ℝ ↦ -𝑤)‘sup(𝐴, ℝ, ◡ < )) = ((𝑤 ∈ ℝ ↦ -𝑤)‘inf(𝐴, ℝ, < ))
31		eqidd 2235	. . . . . 6 ⊢ (𝜑 → (𝑤 ∈ ℝ ↦ -𝑤) = (𝑤 ∈ ℝ ↦ -𝑤))
32		negeq 8482	. . . . . . 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 8587	. . . . . 6 ⊢ (𝜑 → -inf(𝐴, ℝ, < ) ∈ ℂ)
35	31, 33, 4, 34	fvmptd 5763	. . . . 5 ⊢ (𝜑 → ((𝑤 ∈ ℝ ↦ -𝑤)‘inf(𝐴, ℝ, < )) = -inf(𝐴, ℝ, < ))
36	30, 35	eqtrid 2279	. . . 4 ⊢ (𝜑 → ((𝑤 ∈ ℝ ↦ -𝑤)‘sup(𝐴, ℝ, ◡ < )) = -inf(𝐴, ℝ, < ))
37	27, 36	eqtr2d 2268	. . 3 ⊢ (𝜑 → -inf(𝐴, ℝ, < ) = sup({𝑧 ∈ ℝ ∣ -𝑧 ∈ 𝐴}, ℝ, < ))
38	37	negeqd 8484	. 2 ⊢ (𝜑 → --inf(𝐴, ℝ, < ) = -sup({𝑧 ∈ ℝ ∣ -𝑧 ∈ 𝐴}, ℝ, < ))
39	6, 38	eqtr3d 2269	1 ⊢ (𝜑 → inf(𝐴, ℝ, < ) = -sup({𝑧 ∈ ℝ ∣ -𝑧 ∈ 𝐴}, ℝ, < ))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1398   ∈ wcel 2205  ∀wral 2522  ∃wrex 2523  {crab 2526   ⊆ wss 3214   class class class wbr 4114   ↦ cmpt 4176  ^◡ccnv 4753   “ cima 4757  ‘cfv 5357   Isom wiso 5358  supcsup 7286  infcinf 7287  ℂcc 8141  ℝcr 8142   < clt 8324  -cneg 8461

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-addass 8245  ax-distr 8247  ax-i2m1 8248  ax-0id 8251  ax-rnegex 8252  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-apti 8258  ax-pre-ltadd 8259

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-isom 5366  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-sup 7288  df-inf 7289  df-pnf 8326  df-mnf 8327  df-ltxr 8329  df-sub 8462  df-neg 8463

This theorem is referenced by:  supminfex  9947  infssuzcldc  10617  minmax  11940