Theorem supminfxrrnmpt 42901

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 42624	. . 3 ⊢ (𝜑 → ran (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ ℝ*)
5	4	supminfxr2 42899	. 2 ⊢ (𝜑 → sup(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ*, < ) = -𝑒inf({𝑦 ∈ ℝ* ∣ -𝑒𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)}, ℝ*, < ))
6		xnegex 12871	. . . . . . . . . . . 12 ⊢ -𝑒𝑦 ∈ V
7	2	elrnmpt 5854	. . . . . . . . . . . 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 12877	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ ℝ* → -𝑒-𝑒𝑦 = 𝑦)
12	11	eqcomd 2744	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ ℝ* → 𝑦 = -𝑒-𝑒𝑦)
13	12	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ ℝ* ∧ -𝑒𝑦 = 𝐵) → 𝑦 = -𝑒-𝑒𝑦)
14		xnegeq 12870	. . . . . . . . . . . . . . . 16 ⊢ (-𝑒𝑦 = 𝐵 → -𝑒-𝑒𝑦 = -𝑒𝐵)
15	14	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ ℝ* ∧ -𝑒𝑦 = 𝐵) → -𝑒-𝑒𝑦 = -𝑒𝐵)
16	13, 15	eqtrd 2778	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ ℝ* ∧ -𝑒𝑦 = 𝐵) → 𝑦 = -𝑒𝐵)
17	16	ex 412	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℝ* → (-𝑒𝑦 = 𝐵 → 𝑦 = -𝑒𝐵))
18	17	reximdv 3201	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℝ* → (∃𝑥 ∈ 𝐴 -𝑒𝑦 = 𝐵 → ∃𝑥 ∈ 𝐴 𝑦 = -𝑒𝐵))
19	18	imp 406	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℝ* ∧ ∃𝑥 ∈ 𝐴 -𝑒𝑦 = 𝐵) → ∃𝑥 ∈ 𝐴 𝑦 = -𝑒𝐵)
20		simpl 482	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℝ* ∧ ∃𝑥 ∈ 𝐴 -𝑒𝑦 = 𝐵) → 𝑦 ∈ ℝ*)
21	10, 19, 20	elrnmptd 5859	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ℝ* ∧ ∃𝑥 ∈ 𝐴 -𝑒𝑦 = 𝐵) → 𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ -𝑒𝐵))
22	9, 21	sylan2 592	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℝ* ∧ -𝑒𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)) → 𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ -𝑒𝐵))
23	22	ex 412	. . . . . . . 8 ⊢ (𝑦 ∈ ℝ* → (-𝑒𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) → 𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ -𝑒𝐵)))
24	23	rgen 3073	. . . . . . 7 ⊢ ∀𝑦 ∈ ℝ* (-𝑒𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) → 𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ -𝑒𝐵))
25		rabss 4001	. . . . . . . 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 2906	. . . . . . . 8 ⊢ Ⅎ𝑥-𝑒𝑦
30		nfmpt1 5178	. . . . . . . . 9 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐵)
31	30	nfrn 5850	. . . . . . . 8 ⊢ Ⅎ𝑥ran (𝑥 ∈ 𝐴 ↦ 𝐵)
32	29, 31	nfel 2920	. . . . . . 7 ⊢ Ⅎ𝑥-𝑒𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)
33		nfcv 2906	. . . . . . 7 ⊢ Ⅎ𝑥ℝ*
34	32, 33	nfrabw 3311	. . . . . 6 ⊢ Ⅎ𝑥{𝑦 ∈ ℝ* ∣ -𝑒𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)}
35		xnegeq 12870	. . . . . . . 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 12963	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → -𝑒𝐵 ∈ ℝ*)
38		xnegneg 12877	. . . . . . . . 9 ⊢ (𝐵 ∈ ℝ* → -𝑒-𝑒𝐵 = 𝐵)
39	3, 38	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → -𝑒-𝑒𝐵 = 𝐵)
40		simpr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
41	2, 40, 3	elrnmpt1d 42662	. . . . . . . 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 3619	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → -𝑒𝐵 ∈ {𝑦 ∈ ℝ* ∣ -𝑒𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)})
44	1, 34, 10, 43	rnmptssdf 42689	. . . . 5 ⊢ (𝜑 → ran (𝑥 ∈ 𝐴 ↦ -𝑒𝐵) ⊆ {𝑦 ∈ ℝ* ∣ -𝑒𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)})
45	28, 44	eqssd 3934	. . . 4 ⊢ (𝜑 → {𝑦 ∈ ℝ* ∣ -𝑒𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)} = ran (𝑥 ∈ 𝐴 ↦ -𝑒𝐵))
46	45	infeq1d 9166	. . 3 ⊢ (𝜑 → inf({𝑦 ∈ ℝ* ∣ -𝑒𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)}, ℝ*, < ) = inf(ran (𝑥 ∈ 𝐴 ↦ -𝑒𝐵), ℝ*, < ))
47	46	xnegeqd 42867	. 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 395   = wceq 1539  Ⅎwnf 1787   ∈ wcel 2108  ∀wral 3063  ∃wrex 3064  {crab 3067  Vcvv 3422   ⊆ wss 3883   ↦ cmpt 5153  ran crn 5581  supcsup 9129  infcinf 9130  ℝ^*cxr 10939   < clt 10940  -_𝑒cxne 12774

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879  ax-pre-sup 10880

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-po 5494  df-so 5495  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-isom 6427  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-er 8456  df-en 8692  df-dom 8693  df-sdom 8694  df-sup 9131  df-inf 9132  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-xneg 12777

This theorem is referenced by:  liminfvalxr  43214