Theorem supminfrnmpt 39672

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 2622	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
3		supminfrnmpt.b	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)
4	1, 2, 3	rnmptssd 39385	. . 3 ⊢ (𝜑 → ran (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ ℝ)
5		supminfrnmpt.a	. . . 4 ⊢ (𝜑 → 𝐴 ≠ ∅)
6	1, 3, 2, 5	rnmptn0 39413	. . 3 ⊢ (𝜑 → ran (𝑥 ∈ 𝐴 ↦ 𝐵) ≠ ∅)
7		supminfrnmpt.y	. . . 4 ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦)
8	1, 3	rnmptbd 39471	. . . 4 ⊢ (𝜑 → (∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 ↔ ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝑦))
9	7, 8	mpbid 222	. . 3 ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝑦)
10		supminf 11775	. . 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, < ))
11	4, 6, 9, 10	syl3anc 1326	. 2 ⊢ (𝜑 → sup(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ, < ) = -inf({𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)}, ℝ, < ))
12		eqid 2622	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐴 ↦ -𝐵) = (𝑥 ∈ 𝐴 ↦ -𝐵)
13		simpr 477	. . . . . . . . . . . 12 ⊢ ((𝑤 ∈ ℝ ∧ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)) → -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵))
14		renegcl 10344	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ ℝ → -𝑤 ∈ ℝ)
15	2	elrnmpt 5372	. . . . . . . . . . . . . 14 ⊢ (-𝑤 ∈ ℝ → (-𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ↔ ∃𝑥 ∈ 𝐴 -𝑤 = 𝐵))
16	14, 15	syl 17	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ ℝ → (-𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ↔ ∃𝑥 ∈ 𝐴 -𝑤 = 𝐵))
17	16	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝑤 ∈ ℝ ∧ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)) → (-𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ↔ ∃𝑥 ∈ 𝐴 -𝑤 = 𝐵))
18	13, 17	mpbid 222	. . . . . . . . . . 11 ⊢ ((𝑤 ∈ ℝ ∧ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)) → ∃𝑥 ∈ 𝐴 -𝑤 = 𝐵)
19	18	adantll 750	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑤 ∈ ℝ) ∧ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)) → ∃𝑥 ∈ 𝐴 -𝑤 = 𝐵)
20		nfv 1843	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥 𝑤 ∈ ℝ
21	1, 20	nfan 1828	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑤 ∈ ℝ)
22		negeq 10273	. . . . . . . . . . . . . . . . . . 19 ⊢ (-𝑤 = 𝐵 → --𝑤 = -𝐵)
23	22	eqcomd 2628	. . . . . . . . . . . . . . . . . 18 ⊢ (-𝑤 = 𝐵 → -𝐵 = --𝑤)
24	23	adantl 482	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑤 ∈ ℝ ∧ -𝑤 = 𝐵) → -𝐵 = --𝑤)
25		recn 10026	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑤 ∈ ℝ → 𝑤 ∈ ℂ)
26	25	negnegd 10383	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 ∈ ℝ → --𝑤 = 𝑤)
27	26	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑤 ∈ ℝ ∧ -𝑤 = 𝐵) → --𝑤 = 𝑤)
28	24, 27	eqtr2d 2657	. . . . . . . . . . . . . . . 16 ⊢ ((𝑤 ∈ ℝ ∧ -𝑤 = 𝐵) → 𝑤 = -𝐵)
29	28	ex 450	. . . . . . . . . . . . . . 15 ⊢ (𝑤 ∈ ℝ → (-𝑤 = 𝐵 → 𝑤 = -𝐵))
30	29	adantl 482	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑤 ∈ ℝ) → (-𝑤 = 𝐵 → 𝑤 = -𝐵))
31	30	adantr 481	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑤 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → (-𝑤 = 𝐵 → 𝑤 = -𝐵))
32		negeq 10273	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 = -𝐵 → -𝑤 = --𝐵)
33	32	adantl 482	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑤 = -𝐵) → -𝑤 = --𝐵)
34	3	recnd 10068	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℂ)
35	34	negnegd 10383	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → --𝐵 = 𝐵)
36	35	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑤 = -𝐵) → --𝐵 = 𝐵)
37	33, 36	eqtrd 2656	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑤 = -𝐵) → -𝑤 = 𝐵)
38	37	ex 450	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑤 = -𝐵 → -𝑤 = 𝐵))
39	38	adantlr 751	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑤 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → (𝑤 = -𝐵 → -𝑤 = 𝐵))
40	31, 39	impbid 202	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑤 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → (-𝑤 = 𝐵 ↔ 𝑤 = -𝐵))
41	21, 40	rexbida 3047	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑤 ∈ ℝ) → (∃𝑥 ∈ 𝐴 -𝑤 = 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑤 = -𝐵))
42	41	adantr 481	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑤 ∈ ℝ) ∧ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)) → (∃𝑥 ∈ 𝐴 -𝑤 = 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑤 = -𝐵))
43	19, 42	mpbid 222	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑤 ∈ ℝ) ∧ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)) → ∃𝑥 ∈ 𝐴 𝑤 = -𝐵)
44		simplr 792	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑤 ∈ ℝ) ∧ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)) → 𝑤 ∈ ℝ)
45	12, 43, 44	elrnmptd 39366	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑤 ∈ ℝ) ∧ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)) → 𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ -𝐵))
46	45	ex 450	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ ℝ) → (-𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) → 𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ -𝐵)))
47	46	ralrimiva 2966	. . . . . 6 ⊢ (𝜑 → ∀𝑤 ∈ ℝ (-𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) → 𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ -𝐵)))
48		rabss 3679	. . . . . 6 ⊢ ({𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)} ⊆ ran (𝑥 ∈ 𝐴 ↦ -𝐵) ↔ ∀𝑤 ∈ ℝ (-𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) → 𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ -𝐵)))
49	47, 48	sylibr 224	. . . . 5 ⊢ (𝜑 → {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)} ⊆ ran (𝑥 ∈ 𝐴 ↦ -𝐵))
50		nfcv 2764	. . . . . . . 8 ⊢ Ⅎ𝑥-𝑤
51		nfmpt1 4747	. . . . . . . . 9 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐵)
52	51	nfrn 5368	. . . . . . . 8 ⊢ Ⅎ𝑥ran (𝑥 ∈ 𝐴 ↦ 𝐵)
53	50, 52	nfel 2777	. . . . . . 7 ⊢ Ⅎ𝑥-𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)
54		nfcv 2764	. . . . . . 7 ⊢ Ⅎ𝑥ℝ
55	53, 54	nfrab 3123	. . . . . 6 ⊢ Ⅎ𝑥{𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)}
56	32	eleq1d 2686	. . . . . . 7 ⊢ (𝑤 = -𝐵 → (-𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ↔ --𝐵 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)))
57	3	renegcld 10457	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → -𝐵 ∈ ℝ)
58		simpr 477	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
59	2	elrnmpt1 5374	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ ℝ) → 𝐵 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵))
60	58, 3, 59	syl2anc 693	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵))
61	35, 60	eqeltrd 2701	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → --𝐵 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵))
62	56, 57, 61	elrabd 3365	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → -𝐵 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)})
63	1, 55, 12, 62	rnmptssdf 39469	. . . . 5 ⊢ (𝜑 → ran (𝑥 ∈ 𝐴 ↦ -𝐵) ⊆ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)})
64	49, 63	eqssd 3620	. . . 4 ⊢ (𝜑 → {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)} = ran (𝑥 ∈ 𝐴 ↦ -𝐵))
65	64	infeq1d 8383	. . 3 ⊢ (𝜑 → inf({𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)}, ℝ, < ) = inf(ran (𝑥 ∈ 𝐴 ↦ -𝐵), ℝ, < ))
66	65	negeqd 10275	. 2 ⊢ (𝜑 → -inf({𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)}, ℝ, < ) = -inf(ran (𝑥 ∈ 𝐴 ↦ -𝐵), ℝ, < ))
67	11, 66	eqtrd 2656	1 ⊢ (𝜑 → sup(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ, < ) = -inf(ran (𝑥 ∈ 𝐴 ↦ -𝐵), ℝ, < ))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483  Ⅎwnf 1708   ∈ wcel 1990   ≠ wne 2794  ∀wral 2912  ∃wrex 2913  {crab 2916   ⊆ wss 3574  ∅c0 3915   class class class wbr 4653   ↦ cmpt 4729  ran crn 5115  supcsup 8346  infcinf 8347  ℝcr 9935   < clt 10074   ≤ cle 10075  -cneg 10267

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-inf 8349  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269

This theorem is referenced by: (None)