Theorem infrenegsupex 9714

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 8151	. . . . . 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 7124	. . . 4 ⊢ (𝜑 → inf(𝐴, ℝ, < ) ∈ ℝ)
5	4	recnd 8100	. . 3 ⊢ (𝜑 → inf(𝐴, ℝ, < ) ∈ ℂ)
6	5	negnegd 8373	. 2 ⊢ (𝜑 → --inf(𝐴, ℝ, < ) = inf(𝐴, ℝ, < ))
7		negeq 8264	. . . . . . . . 9 ⊢ (𝑤 = 𝑧 → -𝑤 = -𝑧)
8	7	cbvmptv 4139	. . . . . . . 8 ⊢ (𝑤 ∈ ℝ ↦ -𝑤) = (𝑧 ∈ ℝ ↦ -𝑧)
9	8	mptpreima 5175	. . . . . . 7 ⊢ (◡(𝑤 ∈ ℝ ↦ -𝑤) “ 𝐴) = {𝑧 ∈ ℝ ∣ -𝑧 ∈ 𝐴}
10		eqid 2204	. . . . . . . . . 10 ⊢ (𝑤 ∈ ℝ ↦ -𝑤) = (𝑤 ∈ ℝ ↦ -𝑤)
11	10	negiso 9027	. . . . . . . . 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 5018	. . . . . . 7 ⊢ (◡(𝑤 ∈ ℝ ↦ -𝑤) “ 𝐴) = ((𝑤 ∈ ℝ ↦ -𝑤) “ 𝐴)
14	9, 13	eqtr3i 2227	. . . . . 6 ⊢ {𝑧 ∈ ℝ ∣ -𝑧 ∈ 𝐴} = ((𝑤 ∈ ℝ ↦ -𝑤) “ 𝐴)
15	14	supeq1i 7089	. . . . 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 5879	. . . . . . . . 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 5869	. . . . . . . . 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 7119	. . . . . 6 ⊢ (𝜑 → ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝐴 ¬ 𝑥◡ < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦◡ < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦◡ < 𝑧)))
25	2	cnvti 7120	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ)) → (𝑓 = 𝑔 ↔ (¬ 𝑓◡ < 𝑔 ∧ ¬ 𝑔◡ < 𝑓)))
26	22, 23, 24, 25	supisoti 7111	. . . . 5 ⊢ (𝜑 → sup(((𝑤 ∈ ℝ ↦ -𝑤) “ 𝐴), ℝ, < ) = ((𝑤 ∈ ℝ ↦ -𝑤)‘sup(𝐴, ℝ, ◡ < )))
27	15, 26	eqtrid 2249	. . . 4 ⊢ (𝜑 → sup({𝑧 ∈ ℝ ∣ -𝑧 ∈ 𝐴}, ℝ, < ) = ((𝑤 ∈ ℝ ↦ -𝑤)‘sup(𝐴, ℝ, ◡ < )))
28		df-inf 7086	. . . . . . 7 ⊢ inf(𝐴, ℝ, < ) = sup(𝐴, ℝ, ◡ < )
29	28	eqcomi 2208	. . . . . 6 ⊢ sup(𝐴, ℝ, ◡ < ) = inf(𝐴, ℝ, < )
30	29	fveq2i 5578	. . . . 5 ⊢ ((𝑤 ∈ ℝ ↦ -𝑤)‘sup(𝐴, ℝ, ◡ < )) = ((𝑤 ∈ ℝ ↦ -𝑤)‘inf(𝐴, ℝ, < ))
31		eqidd 2205	. . . . . 6 ⊢ (𝜑 → (𝑤 ∈ ℝ ↦ -𝑤) = (𝑤 ∈ ℝ ↦ -𝑤))
32		negeq 8264	. . . . . . 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 8369	. . . . . 6 ⊢ (𝜑 → -inf(𝐴, ℝ, < ) ∈ ℂ)
35	31, 33, 4, 34	fvmptd 5659	. . . . 5 ⊢ (𝜑 → ((𝑤 ∈ ℝ ↦ -𝑤)‘inf(𝐴, ℝ, < )) = -inf(𝐴, ℝ, < ))
36	30, 35	eqtrid 2249	. . . 4 ⊢ (𝜑 → ((𝑤 ∈ ℝ ↦ -𝑤)‘sup(𝐴, ℝ, ◡ < )) = -inf(𝐴, ℝ, < ))
37	27, 36	eqtr2d 2238	. . 3 ⊢ (𝜑 → -inf(𝐴, ℝ, < ) = sup({𝑧 ∈ ℝ ∣ -𝑧 ∈ 𝐴}, ℝ, < ))
38	37	negeqd 8266	. 2 ⊢ (𝜑 → --inf(𝐴, ℝ, < ) = -sup({𝑧 ∈ ℝ ∣ -𝑧 ∈ 𝐴}, ℝ, < ))
39	6, 38	eqtr3d 2239	1 ⊢ (𝜑 → inf(𝐴, ℝ, < ) = -sup({𝑧 ∈ ℝ ∣ -𝑧 ∈ 𝐴}, ℝ, < ))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1372   ∈ wcel 2175  ∀wral 2483  ∃wrex 2484  {crab 2487   ⊆ wss 3165   class class class wbr 4043   ↦ cmpt 4104  ^◡ccnv 4673   “ cima 4677  ‘cfv 5270   Isom wiso 5271  supcsup 7083  infcinf 7084  ℂcc 7922  ℝcr 7923   < clt 8106  -cneg 8243

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 615  ax-in2 616  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252  ax-un 4479  ax-setind 4584  ax-cnex 8015  ax-resscn 8016  ax-1cn 8017  ax-1re 8018  ax-icn 8019  ax-addcl 8020  ax-addrcl 8021  ax-mulcl 8022  ax-addcom 8024  ax-addass 8026  ax-distr 8028  ax-i2m1 8029  ax-0id 8032  ax-rnegex 8033  ax-cnre 8035  ax-pre-ltirr 8036  ax-pre-apti 8039  ax-pre-ltadd 8040

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-nel 2471  df-ral 2488  df-rex 2489  df-reu 2490  df-rmo 2491  df-rab 2492  df-v 2773  df-sbc 2998  df-csb 3093  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-br 4044  df-opab 4105  df-mpt 4106  df-id 4339  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-res 4686  df-ima 4687  df-iota 5231  df-fun 5272  df-fn 5273  df-f 5274  df-f1 5275  df-fo 5276  df-f1o 5277  df-fv 5278  df-isom 5279  df-riota 5898  df-ov 5946  df-oprab 5947  df-mpo 5948  df-sup 7085  df-inf 7086  df-pnf 8108  df-mnf 8109  df-ltxr 8111  df-sub 8244  df-neg 8245

This theorem is referenced by:  supminfex  9717  infssuzcldc  10376  minmax  11483