Theorem supminfrnmpt 45894

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 2737	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
3		supminfrnmpt.b	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)
4	1, 2, 3	rnmptssd 7071	. . 3 ⊢ (𝜑 → ran (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ ℝ)
5		supminfrnmpt.a	. . . 4 ⊢ (𝜑 → 𝐴 ≠ ∅)
6	1, 3, 2, 5	rnmptn0 6203	. . 3 ⊢ (𝜑 → ran (𝑥 ∈ 𝐴 ↦ 𝐵) ≠ ∅)
7		supminfrnmpt.y	. . . 4 ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦)
8	1, 7	rnmptbdd 45695	. . 3 ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝑦)
9		supminf 12879	. . 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 1374	. 2 ⊢ (𝜑 → sup(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ, < ) = -inf({𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)}, ℝ, < ))
11		eqid 2737	. . . . . . . . 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 11451	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ ℝ → -𝑤 ∈ ℝ)
14	2	elrnmpt 5908	. . . . . . . . . . . . . 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 715	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑤 ∈ ℝ) ∧ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)) → ∃𝑥 ∈ 𝐴 -𝑤 = 𝐵)
19		nfv 1916	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥 𝑤 ∈ ℝ
20	1, 19	nfan 1901	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑤 ∈ ℝ)
21		negeq 11379	. . . . . . . . . . . . . . . . . . 19 ⊢ (-𝑤 = 𝐵 → --𝑤 = -𝐵)
22	21	eqcomd 2743	. . . . . . . . . . . . . . . . . 18 ⊢ (-𝑤 = 𝐵 → -𝐵 = --𝑤)
23	22	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑤 ∈ ℝ ∧ -𝑤 = 𝐵) → -𝐵 = --𝑤)
24		recn 11122	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑤 ∈ ℝ → 𝑤 ∈ ℂ)
25	24	negnegd 11490	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 ∈ ℝ → --𝑤 = 𝑤)
26	25	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑤 ∈ ℝ ∧ -𝑤 = 𝐵) → --𝑤 = 𝑤)
27	23, 26	eqtr2d 2773	. . . . . . . . . . . . . . . 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 11379	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 = -𝐵 → -𝑤 = --𝐵)
32	31	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑤 = -𝐵) → -𝑤 = --𝐵)
33	3	recnd 11167	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℂ)
34	33	negnegd 11490	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → --𝐵 = 𝐵)
35	34	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑤 = -𝐵) → --𝐵 = 𝐵)
36	32, 35	eqtrd 2772	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑤 = -𝐵) → -𝑤 = 𝐵)
37	36	ex 412	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑤 = -𝐵 → -𝑤 = 𝐵))
38	37	adantlr 716	. . . . . . . . . . . . 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 3250	. . . . . . . . . . 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 769	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑤 ∈ ℝ) ∧ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)) → 𝑤 ∈ ℝ)
44	11, 42, 43	elrnmptd 5913	. . . . . . . 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 3130	. . . . . 6 ⊢ (𝜑 → ∀𝑤 ∈ ℝ (-𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) → 𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ -𝐵)))
47		rabss 4011	. . . . . 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 2899	. . . . . . . 8 ⊢ Ⅎ𝑥-𝑤
50		nfmpt1 5185	. . . . . . . . 9 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐵)
51	50	nfrn 5902	. . . . . . . 8 ⊢ Ⅎ𝑥ran (𝑥 ∈ 𝐴 ↦ 𝐵)
52	49, 51	nfel 2914	. . . . . . 7 ⊢ Ⅎ𝑥-𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)
53		nfcv 2899	. . . . . . 7 ⊢ Ⅎ𝑥ℝ
54	52, 53	nfrabw 3427	. . . . . 6 ⊢ Ⅎ𝑥{𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)}
55	31	eleq1d 2822	. . . . . . 7 ⊢ (𝑤 = -𝐵 → (-𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ↔ --𝐵 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)))
56	3	renegcld 11571	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → -𝐵 ∈ ℝ)
57		simpr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
58	2	elrnmpt1 5910	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ ℝ) → 𝐵 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵))
59	57, 3, 58	syl2anc 585	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵))
60	34, 59	eqeltrd 2837	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → --𝐵 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵))
61	55, 56, 60	elrabd 3637	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → -𝐵 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)})
62	1, 54, 11, 61	rnmptssdf 45704	. . . . 5 ⊢ (𝜑 → ran (𝑥 ∈ 𝐴 ↦ -𝐵) ⊆ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)})
63	48, 62	eqssd 3940	. . . 4 ⊢ (𝜑 → {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)} = ran (𝑥 ∈ 𝐴 ↦ -𝐵))
64	63	infeq1d 9385	. . 3 ⊢ (𝜑 → inf({𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)}, ℝ, < ) = inf(ran (𝑥 ∈ 𝐴 ↦ -𝐵), ℝ, < ))
65	64	negeqd 11381	. 2 ⊢ (𝜑 → -inf({𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)}, ℝ, < ) = -inf(ran (𝑥 ∈ 𝐴 ↦ -𝐵), ℝ, < ))
66	10, 65	eqtrd 2772	1 ⊢ (𝜑 → sup(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ, < ) = -inf(ran (𝑥 ∈ 𝐴 ↦ -𝐵), ℝ, < ))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542  Ⅎwnf 1785   ∈ wcel 2114   ≠ wne 2933  ∀wral 3052  ∃wrex 3062  {crab 3390   ⊆ wss 3890  ∅c0 4274   class class class wbr 5086   ↦ cmpt 5167  ran crn 5626  supcsup 9347  infcinf 9348  ℝcr 11031   < clt 11173   ≤ cle 11174  -cneg 11372

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5232  ax-nul 5242  ax-pow 5303  ax-pr 5371  ax-un 7683  ax-resscn 11089  ax-1cn 11090  ax-icn 11091  ax-addcl 11092  ax-addrcl 11093  ax-mulcl 11094  ax-mulrcl 11095  ax-mulcom 11096  ax-addass 11097  ax-mulass 11098  ax-distr 11099  ax-i2m1 11100  ax-1ne0 11101  ax-1rid 11102  ax-rnegex 11103  ax-rrecex 11104  ax-cnre 11105  ax-pre-lttri 11106  ax-pre-lttrn 11107  ax-pre-ltadd 11108  ax-pre-mulgt0 11109  ax-pre-sup 11110

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5520  df-po 5533  df-so 5534  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-isom 6502  df-riota 7318  df-ov 7364  df-oprab 7365  df-mpo 7366  df-er 8637  df-en 8888  df-dom 8889  df-sdom 8890  df-sup 9349  df-inf 9350  df-pnf 11175  df-mnf 11176  df-xr 11177  df-ltxr 11178  df-le 11179  df-sub 11373  df-neg 11374

This theorem is referenced by: (None)