Theorem supminfxrrnmpt 43011

Description: The indexed supremum of a set of reals is the negation of the indexed infimum of that set's image under negation. (Contributed by Glauco Siliprandi, 2-Jan-2022.)

Hypotheses

Ref	Expression
supminfxrrnmpt.x	⊢ Ⅎ𝑥𝜑
supminfxrrnmpt.b	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ*)

Assertion

Ref	Expression
supminfxrrnmpt	⊢ (𝜑 → sup(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ*, < ) = -𝑒inf(ran (𝑥 ∈ 𝐴 ↦ -𝑒𝐵), ℝ*, < ))

Distinct variable group:   𝑥,𝐴

Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)

Proof of Theorem supminfxrrnmpt

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		supminfxrrnmpt.x	. . . 4 ⊢ Ⅎ𝑥𝜑
2		eqid 2738	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
3		supminfxrrnmpt.b	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ*)
4	1, 2, 3	rnmptssd 42735	. . 3 ⊢ (𝜑 → ran (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ ℝ*)
5	4	supminfxr2 43009	. 2 ⊢ (𝜑 → sup(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ*, < ) = -𝑒inf({𝑦 ∈ ℝ* ∣ -𝑒𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)}, ℝ*, < ))
6		xnegex 12942	. . . . . . . . . . . 12 ⊢ -𝑒𝑦 ∈ V
7	2	elrnmpt 5865	. . . . . . . . . . . 12 ⊢ (-𝑒𝑦 ∈ V → (-𝑒𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ↔ ∃𝑥 ∈ 𝐴 -𝑒𝑦 = 𝐵))
8	6, 7	ax-mp 5	. . . . . . . . . . 11 ⊢ (-𝑒𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ↔ ∃𝑥 ∈ 𝐴 -𝑒𝑦 = 𝐵)
9	8	biimpi 215	. . . . . . . . . 10 ⊢ (-𝑒𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) → ∃𝑥 ∈ 𝐴 -𝑒𝑦 = 𝐵)
10		eqid 2738	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐴 ↦ -𝑒𝐵) = (𝑥 ∈ 𝐴 ↦ -𝑒𝐵)
11		xnegneg 12948	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ ℝ* → -𝑒-𝑒𝑦 = 𝑦)
12	11	eqcomd 2744	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ ℝ* → 𝑦 = -𝑒-𝑒𝑦)
13	12	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ ℝ* ∧ -𝑒𝑦 = 𝐵) → 𝑦 = -𝑒-𝑒𝑦)
14		xnegeq 12941	. . . . . . . . . . . . . . . 16 ⊢ (-𝑒𝑦 = 𝐵 → -𝑒-𝑒𝑦 = -𝑒𝐵)
15	14	adantl 482	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ ℝ* ∧ -𝑒𝑦 = 𝐵) → -𝑒-𝑒𝑦 = -𝑒𝐵)
16	13, 15	eqtrd 2778	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ ℝ* ∧ -𝑒𝑦 = 𝐵) → 𝑦 = -𝑒𝐵)
17	16	ex 413	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℝ* → (-𝑒𝑦 = 𝐵 → 𝑦 = -𝑒𝐵))
18	17	reximdv 3202	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℝ* → (∃𝑥 ∈ 𝐴 -𝑒𝑦 = 𝐵 → ∃𝑥 ∈ 𝐴 𝑦 = -𝑒𝐵))
19	18	imp 407	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℝ* ∧ ∃𝑥 ∈ 𝐴 -𝑒𝑦 = 𝐵) → ∃𝑥 ∈ 𝐴 𝑦 = -𝑒𝐵)
20		simpl 483	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℝ* ∧ ∃𝑥 ∈ 𝐴 -𝑒𝑦 = 𝐵) → 𝑦 ∈ ℝ*)
21	10, 19, 20	elrnmptd 5870	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ℝ* ∧ ∃𝑥 ∈ 𝐴 -𝑒𝑦 = 𝐵) → 𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ -𝑒𝐵))
22	9, 21	sylan2 593	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℝ* ∧ -𝑒𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)) → 𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ -𝑒𝐵))
23	22	ex 413	. . . . . . . 8 ⊢ (𝑦 ∈ ℝ* → (-𝑒𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) → 𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ -𝑒𝐵)))
24	23	rgen 3074	. . . . . . 7 ⊢ ∀𝑦 ∈ ℝ* (-𝑒𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) → 𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ -𝑒𝐵))
25		rabss 4005	. . . . . . . 8 ⊢ ({𝑦 ∈ ℝ* ∣ -𝑒𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)} ⊆ ran (𝑥 ∈ 𝐴 ↦ -𝑒𝐵) ↔ ∀𝑦 ∈ ℝ* (-𝑒𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) → 𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ -𝑒𝐵)))
26	25	biimpri 227	. . . . . . 7 ⊢ (∀𝑦 ∈ ℝ* (-𝑒𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) → 𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ -𝑒𝐵)) → {𝑦 ∈ ℝ* ∣ -𝑒𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)} ⊆ ran (𝑥 ∈ 𝐴 ↦ -𝑒𝐵))
27	24, 26	ax-mp 5	. . . . . 6 ⊢ {𝑦 ∈ ℝ* ∣ -𝑒𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)} ⊆ ran (𝑥 ∈ 𝐴 ↦ -𝑒𝐵)
28	27	a1i 11	. . . . 5 ⊢ (𝜑 → {𝑦 ∈ ℝ* ∣ -𝑒𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)} ⊆ ran (𝑥 ∈ 𝐴 ↦ -𝑒𝐵))
29		nfcv 2907	. . . . . . . 8 ⊢ Ⅎ𝑥-𝑒𝑦
30		nfmpt1 5182	. . . . . . . . 9 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐵)
31	30	nfrn 5861	. . . . . . . 8 ⊢ Ⅎ𝑥ran (𝑥 ∈ 𝐴 ↦ 𝐵)
32	29, 31	nfel 2921	. . . . . . 7 ⊢ Ⅎ𝑥-𝑒𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)
33		nfcv 2907	. . . . . . 7 ⊢ Ⅎ𝑥ℝ*
34	32, 33	nfrabw 3318	. . . . . 6 ⊢ Ⅎ𝑥{𝑦 ∈ ℝ* ∣ -𝑒𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)}
35		xnegeq 12941	. . . . . . . 8 ⊢ (𝑦 = -𝑒𝐵 → -𝑒𝑦 = -𝑒-𝑒𝐵)
36	35	eleq1d 2823	. . . . . . 7 ⊢ (𝑦 = -𝑒𝐵 → (-𝑒𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ↔ -𝑒-𝑒𝐵 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)))
37	3	xnegcld 13034	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → -𝑒𝐵 ∈ ℝ*)
38		xnegneg 12948	. . . . . . . . 9 ⊢ (𝐵 ∈ ℝ* → -𝑒-𝑒𝐵 = 𝐵)
39	3, 38	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → -𝑒-𝑒𝐵 = 𝐵)
40		simpr 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
41	2, 40, 3	elrnmpt1d 42773	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵))
42	39, 41	eqeltrd 2839	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → -𝑒-𝑒𝐵 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵))
43	36, 37, 42	elrabd 3626	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → -𝑒𝐵 ∈ {𝑦 ∈ ℝ* ∣ -𝑒𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)})
44	1, 34, 10, 43	rnmptssdf 42800	. . . . 5 ⊢ (𝜑 → ran (𝑥 ∈ 𝐴 ↦ -𝑒𝐵) ⊆ {𝑦 ∈ ℝ* ∣ -𝑒𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)})
45	28, 44	eqssd 3938	. . . 4 ⊢ (𝜑 → {𝑦 ∈ ℝ* ∣ -𝑒𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)} = ran (𝑥 ∈ 𝐴 ↦ -𝑒𝐵))
46	45	infeq1d 9236	. . 3 ⊢ (𝜑 → inf({𝑦 ∈ ℝ* ∣ -𝑒𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)}, ℝ*, < ) = inf(ran (𝑥 ∈ 𝐴 ↦ -𝑒𝐵), ℝ*, < ))
47	46	xnegeqd 42977	. 2 ⊢ (𝜑 → -𝑒inf({𝑦 ∈ ℝ* ∣ -𝑒𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)}, ℝ*, < ) = -𝑒inf(ran (𝑥 ∈ 𝐴 ↦ -𝑒𝐵), ℝ*, < ))
48	5, 47	eqtrd 2778	1 ⊢ (𝜑 → sup(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ*, < ) = -𝑒inf(ran (𝑥 ∈ 𝐴 ↦ -𝑒𝐵), ℝ*, < ))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539  Ⅎwnf 1786   ∈ wcel 2106  ∀wral 3064  ∃wrex 3065  {crab 3068  Vcvv 3432   ⊆ wss 3887   ↦ cmpt 5157  ran crn 5590  supcsup 9199  infcinf 9200  ℝ^*cxr 11008   < clt 11009  -_𝑒cxne 12845

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948  ax-pre-sup 10949

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-po 5503  df-so 5504  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-isom 6442  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-er 8498  df-en 8734  df-dom 8735  df-sdom 8736  df-sup 9201  df-inf 9202  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-xneg 12848

This theorem is referenced by:  liminfvalxr  43324