Theorem supminfrnmpt 45428

Description: The indexed supremum of a bounded-above 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
supminfrnmpt.x	⊢ Ⅎ𝑥𝜑
supminfrnmpt.a	⊢ (𝜑 → 𝐴 ≠ ∅)
supminfrnmpt.b	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)
supminfrnmpt.y	⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦)

Assertion

Ref	Expression
supminfrnmpt	⊢ (𝜑 → sup(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ, < ) = -inf(ran (𝑥 ∈ 𝐴 ↦ -𝐵), ℝ, < ))

Distinct variable groups:   𝑥,𝐴,𝑦   𝑦,𝐵

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

Proof of Theorem supminfrnmpt

Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		supminfrnmpt.x	. . . 4 ⊢ Ⅎ𝑥𝜑
2		eqid 2729	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
3		supminfrnmpt.b	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)
4	1, 2, 3	rnmptssd 45178	. . 3 ⊢ (𝜑 → ran (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ ℝ)
5		supminfrnmpt.a	. . . 4 ⊢ (𝜑 → 𝐴 ≠ ∅)
6	1, 3, 2, 5	rnmptn0 6193	. . 3 ⊢ (𝜑 → ran (𝑥 ∈ 𝐴 ↦ 𝐵) ≠ ∅)
7		supminfrnmpt.y	. . . 4 ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦)
8	1, 7	rnmptbdd 45227	. . 3 ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝑦)
9		supminf 12836	. . 3 ⊢ ((ran (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ ℝ ∧ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ≠ ∅ ∧ ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝑦) → sup(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ, < ) = -inf({𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)}, ℝ, < ))
10	4, 6, 8, 9	syl3anc 1373	. 2 ⊢ (𝜑 → sup(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ, < ) = -inf({𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)}, ℝ, < ))
11		eqid 2729	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐴 ↦ -𝐵) = (𝑥 ∈ 𝐴 ↦ -𝐵)
12		simpr 484	. . . . . . . . . . . 12 ⊢ ((𝑤 ∈ ℝ ∧ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)) → -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵))
13		renegcl 11427	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ ℝ → -𝑤 ∈ ℝ)
14	2	elrnmpt 5900	. . . . . . . . . . . . . 14 ⊢ (-𝑤 ∈ ℝ → (-𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ↔ ∃𝑥 ∈ 𝐴 -𝑤 = 𝐵))
15	13, 14	syl 17	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ ℝ → (-𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ↔ ∃𝑥 ∈ 𝐴 -𝑤 = 𝐵))
16	15	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝑤 ∈ ℝ ∧ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)) → (-𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ↔ ∃𝑥 ∈ 𝐴 -𝑤 = 𝐵))
17	12, 16	mpbid 232	. . . . . . . . . . 11 ⊢ ((𝑤 ∈ ℝ ∧ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)) → ∃𝑥 ∈ 𝐴 -𝑤 = 𝐵)
18	17	adantll 714	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑤 ∈ ℝ) ∧ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)) → ∃𝑥 ∈ 𝐴 -𝑤 = 𝐵)
19		nfv 1914	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥 𝑤 ∈ ℝ
20	1, 19	nfan 1899	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑤 ∈ ℝ)
21		negeq 11355	. . . . . . . . . . . . . . . . . . 19 ⊢ (-𝑤 = 𝐵 → --𝑤 = -𝐵)
22	21	eqcomd 2735	. . . . . . . . . . . . . . . . . 18 ⊢ (-𝑤 = 𝐵 → -𝐵 = --𝑤)
23	22	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑤 ∈ ℝ ∧ -𝑤 = 𝐵) → -𝐵 = --𝑤)
24		recn 11099	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑤 ∈ ℝ → 𝑤 ∈ ℂ)
25	24	negnegd 11466	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 ∈ ℝ → --𝑤 = 𝑤)
26	25	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑤 ∈ ℝ ∧ -𝑤 = 𝐵) → --𝑤 = 𝑤)
27	23, 26	eqtr2d 2765	. . . . . . . . . . . . . . . 16 ⊢ ((𝑤 ∈ ℝ ∧ -𝑤 = 𝐵) → 𝑤 = -𝐵)
28	27	ex 412	. . . . . . . . . . . . . . 15 ⊢ (𝑤 ∈ ℝ → (-𝑤 = 𝐵 → 𝑤 = -𝐵))
29	28	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑤 ∈ ℝ) → (-𝑤 = 𝐵 → 𝑤 = -𝐵))
30	29	adantr 480	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑤 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → (-𝑤 = 𝐵 → 𝑤 = -𝐵))
31		negeq 11355	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 = -𝐵 → -𝑤 = --𝐵)
32	31	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑤 = -𝐵) → -𝑤 = --𝐵)
33	3	recnd 11143	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℂ)
34	33	negnegd 11466	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → --𝐵 = 𝐵)
35	34	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑤 = -𝐵) → --𝐵 = 𝐵)
36	32, 35	eqtrd 2764	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑤 = -𝐵) → -𝑤 = 𝐵)
37	36	ex 412	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑤 = -𝐵 → -𝑤 = 𝐵))
38	37	adantlr 715	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑤 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → (𝑤 = -𝐵 → -𝑤 = 𝐵))
39	30, 38	impbid 212	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑤 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → (-𝑤 = 𝐵 ↔ 𝑤 = -𝐵))
40	20, 39	rexbida 3241	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑤 ∈ ℝ) → (∃𝑥 ∈ 𝐴 -𝑤 = 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑤 = -𝐵))
41	40	adantr 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑤 ∈ ℝ) ∧ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)) → (∃𝑥 ∈ 𝐴 -𝑤 = 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑤 = -𝐵))
42	18, 41	mpbid 232	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑤 ∈ ℝ) ∧ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)) → ∃𝑥 ∈ 𝐴 𝑤 = -𝐵)
43		simplr 768	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑤 ∈ ℝ) ∧ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)) → 𝑤 ∈ ℝ)
44	11, 42, 43	elrnmptd 5905	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑤 ∈ ℝ) ∧ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)) → 𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ -𝐵))
45	44	ex 412	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ ℝ) → (-𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) → 𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ -𝐵)))
46	45	ralrimiva 3121	. . . . . 6 ⊢ (𝜑 → ∀𝑤 ∈ ℝ (-𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) → 𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ -𝐵)))
47		rabss 4023	. . . . . 6 ⊢ ({𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)} ⊆ ran (𝑥 ∈ 𝐴 ↦ -𝐵) ↔ ∀𝑤 ∈ ℝ (-𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) → 𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ -𝐵)))
48	46, 47	sylibr 234	. . . . 5 ⊢ (𝜑 → {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)} ⊆ ran (𝑥 ∈ 𝐴 ↦ -𝐵))
49		nfcv 2891	. . . . . . . 8 ⊢ Ⅎ𝑥-𝑤
50		nfmpt1 5191	. . . . . . . . 9 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐵)
51	50	nfrn 5894	. . . . . . . 8 ⊢ Ⅎ𝑥ran (𝑥 ∈ 𝐴 ↦ 𝐵)
52	49, 51	nfel 2906	. . . . . . 7 ⊢ Ⅎ𝑥-𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)
53		nfcv 2891	. . . . . . 7 ⊢ Ⅎ𝑥ℝ
54	52, 53	nfrabw 3432	. . . . . 6 ⊢ Ⅎ𝑥{𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)}
55	31	eleq1d 2813	. . . . . . 7 ⊢ (𝑤 = -𝐵 → (-𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ↔ --𝐵 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)))
56	3	renegcld 11547	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → -𝐵 ∈ ℝ)
57		simpr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
58	2	elrnmpt1 5902	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ ℝ) → 𝐵 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵))
59	57, 3, 58	syl2anc 584	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵))
60	34, 59	eqeltrd 2828	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → --𝐵 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵))
61	55, 56, 60	elrabd 3650	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → -𝐵 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)})
62	1, 54, 11, 61	rnmptssdf 45236	. . . . 5 ⊢ (𝜑 → ran (𝑥 ∈ 𝐴 ↦ -𝐵) ⊆ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)})
63	48, 62	eqssd 3953	. . . 4 ⊢ (𝜑 → {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)} = ran (𝑥 ∈ 𝐴 ↦ -𝐵))
64	63	infeq1d 9368	. . 3 ⊢ (𝜑 → inf({𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)}, ℝ, < ) = inf(ran (𝑥 ∈ 𝐴 ↦ -𝐵), ℝ, < ))
65	64	negeqd 11357	. 2 ⊢ (𝜑 → -inf({𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)}, ℝ, < ) = -inf(ran (𝑥 ∈ 𝐴 ↦ -𝐵), ℝ, < ))
66	10, 65	eqtrd 2764	1 ⊢ (𝜑 → sup(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ, < ) = -inf(ran (𝑥 ∈ 𝐴 ↦ -𝐵), ℝ, < ))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540  Ⅎwnf 1783   ∈ wcel 2109   ≠ wne 2925  ∀wral 3044  ∃wrex 3053  {crab 3394   ⊆ wss 3903  ∅c0 4284   class class class wbr 5092   ↦ cmpt 5173  ran crn 5620  supcsup 9330  infcinf 9331  ℝcr 11008   < clt 11149   ≤ cle 11150  -cneg 11348

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671  ax-resscn 11066  ax-1cn 11067  ax-icn 11068  ax-addcl 11069  ax-addrcl 11070  ax-mulcl 11071  ax-mulrcl 11072  ax-mulcom 11073  ax-addass 11074  ax-mulass 11075  ax-distr 11076  ax-i2m1 11077  ax-1ne0 11078  ax-1rid 11079  ax-rnegex 11080  ax-rrecex 11081  ax-cnre 11082  ax-pre-lttri 11083  ax-pre-lttrn 11084  ax-pre-ltadd 11085  ax-pre-mulgt0 11086  ax-pre-sup 11087

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3343  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-br 5093  df-opab 5155  df-mpt 5174  df-id 5514  df-po 5527  df-so 5528  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-isom 6491  df-riota 7306  df-ov 7352  df-oprab 7353  df-mpo 7354  df-er 8625  df-en 8873  df-dom 8874  df-sdom 8875  df-sup 9332  df-inf 9333  df-pnf 11151  df-mnf 11152  df-xr 11153  df-ltxr 11154  df-le 11155  df-sub 11349  df-neg 11350

This theorem is referenced by: (None)