Theorem supminfrnmpt 44616

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 2731	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
3		supminfrnmpt.b	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)
4	1, 2, 3	rnmptssd 44356	. . 3 ⊢ (𝜑 → ran (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ ℝ)
5		supminfrnmpt.a	. . . 4 ⊢ (𝜑 → 𝐴 ≠ ∅)
6	1, 3, 2, 5	rnmptn0 6243	. . 3 ⊢ (𝜑 → ran (𝑥 ∈ 𝐴 ↦ 𝐵) ≠ ∅)
7		supminfrnmpt.y	. . . 4 ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦)
8	1, 7	rnmptbdd 44410	. . 3 ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝑦)
9		supminf 12926	. . 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 1370	. 2 ⊢ (𝜑 → sup(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ, < ) = -inf({𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)}, ℝ, < ))
11		eqid 2731	. . . . . . . . 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 11530	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ ℝ → -𝑤 ∈ ℝ)
14	2	elrnmpt 5955	. . . . . . . . . . . . . 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 231	. . . . . . . . . . 11 ⊢ ((𝑤 ∈ ℝ ∧ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)) → ∃𝑥 ∈ 𝐴 -𝑤 = 𝐵)
18	17	adantll 711	. . . . . . . . . 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 11459	. . . . . . . . . . . . . . . . . . 19 ⊢ (-𝑤 = 𝐵 → --𝑤 = -𝐵)
22	21	eqcomd 2737	. . . . . . . . . . . . . . . . . 18 ⊢ (-𝑤 = 𝐵 → -𝐵 = --𝑤)
23	22	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑤 ∈ ℝ ∧ -𝑤 = 𝐵) → -𝐵 = --𝑤)
24		recn 11206	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑤 ∈ ℝ → 𝑤 ∈ ℂ)
25	24	negnegd 11569	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 ∈ ℝ → --𝑤 = 𝑤)
26	25	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑤 ∈ ℝ ∧ -𝑤 = 𝐵) → --𝑤 = 𝑤)
27	23, 26	eqtr2d 2772	. . . . . . . . . . . . . . . 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 11459	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 = -𝐵 → -𝑤 = --𝐵)
32	31	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑤 = -𝐵) → -𝑤 = --𝐵)
33	3	recnd 11249	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℂ)
34	33	negnegd 11569	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → --𝐵 = 𝐵)
35	34	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑤 = -𝐵) → --𝐵 = 𝐵)
36	32, 35	eqtrd 2771	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑤 = -𝐵) → -𝑤 = 𝐵)
37	36	ex 412	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑤 = -𝐵 → -𝑤 = 𝐵))
38	37	adantlr 712	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑤 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → (𝑤 = -𝐵 → -𝑤 = 𝐵))
39	30, 38	impbid 211	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑤 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → (-𝑤 = 𝐵 ↔ 𝑤 = -𝐵))
40	20, 39	rexbida 3268	. . . . . . . . . . 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 231	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑤 ∈ ℝ) ∧ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)) → ∃𝑥 ∈ 𝐴 𝑤 = -𝐵)
43		simplr 766	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑤 ∈ ℝ) ∧ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)) → 𝑤 ∈ ℝ)
44	11, 42, 43	elrnmptd 5960	. . . . . . . 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 3145	. . . . . 6 ⊢ (𝜑 → ∀𝑤 ∈ ℝ (-𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) → 𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ -𝐵)))
47		rabss 4069	. . . . . 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 233	. . . . 5 ⊢ (𝜑 → {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)} ⊆ ran (𝑥 ∈ 𝐴 ↦ -𝐵))
49		nfcv 2902	. . . . . . . 8 ⊢ Ⅎ𝑥-𝑤
50		nfmpt1 5256	. . . . . . . . 9 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐵)
51	50	nfrn 5951	. . . . . . . 8 ⊢ Ⅎ𝑥ran (𝑥 ∈ 𝐴 ↦ 𝐵)
52	49, 51	nfel 2916	. . . . . . 7 ⊢ Ⅎ𝑥-𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)
53		nfcv 2902	. . . . . . 7 ⊢ Ⅎ𝑥ℝ
54	52, 53	nfrabw 3467	. . . . . 6 ⊢ Ⅎ𝑥{𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)}
55	31	eleq1d 2817	. . . . . . 7 ⊢ (𝑤 = -𝐵 → (-𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ↔ --𝐵 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)))
56	3	renegcld 11648	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → -𝐵 ∈ ℝ)
57		simpr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
58	2	elrnmpt1 5957	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ ℝ) → 𝐵 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵))
59	57, 3, 58	syl2anc 583	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵))
60	34, 59	eqeltrd 2832	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → --𝐵 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵))
61	55, 56, 60	elrabd 3685	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → -𝐵 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)})
62	1, 54, 11, 61	rnmptssdf 44419	. . . . 5 ⊢ (𝜑 → ran (𝑥 ∈ 𝐴 ↦ -𝐵) ⊆ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)})
63	48, 62	eqssd 3999	. . . 4 ⊢ (𝜑 → {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)} = ran (𝑥 ∈ 𝐴 ↦ -𝐵))
64	63	infeq1d 9478	. . 3 ⊢ (𝜑 → inf({𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)}, ℝ, < ) = inf(ran (𝑥 ∈ 𝐴 ↦ -𝐵), ℝ, < ))
65	64	negeqd 11461	. 2 ⊢ (𝜑 → -inf({𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)}, ℝ, < ) = -inf(ran (𝑥 ∈ 𝐴 ↦ -𝐵), ℝ, < ))
66	10, 65	eqtrd 2771	1 ⊢ (𝜑 → sup(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ, < ) = -inf(ran (𝑥 ∈ 𝐴 ↦ -𝐵), ℝ, < ))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1540  Ⅎwnf 1784   ∈ wcel 2105   ≠ wne 2939  ∀wral 3060  ∃wrex 3069  {crab 3431   ⊆ wss 3948  ∅c0 4322   class class class wbr 5148   ↦ cmpt 5231  ran crn 5677  supcsup 9441  infcinf 9442  ℝcr 11115   < clt 11255   ≤ cle 11256  -cneg 11452

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729  ax-resscn 11173  ax-1cn 11174  ax-icn 11175  ax-addcl 11176  ax-addrcl 11177  ax-mulcl 11178  ax-mulrcl 11179  ax-mulcom 11180  ax-addass 11181  ax-mulass 11182  ax-distr 11183  ax-i2m1 11184  ax-1ne0 11185  ax-1rid 11186  ax-rnegex 11187  ax-rrecex 11188  ax-cnre 11189  ax-pre-lttri 11190  ax-pre-lttrn 11191  ax-pre-ltadd 11192  ax-pre-mulgt0 11193  ax-pre-sup 11194

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-po 5588  df-so 5589  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-er 8709  df-en 8946  df-dom 8947  df-sdom 8948  df-sup 9443  df-inf 9444  df-pnf 11257  df-mnf 11258  df-xr 11259  df-ltxr 11260  df-le 11261  df-sub 11453  df-neg 11454

This theorem is referenced by: (None)