Theorem supminfxrrnmpt 46623

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 2763	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
3		supminfxrrnmpt.b	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ*)
4	1, 2, 3	rnmptssd 7135	. . 3 ⊢ (𝜑 → ran (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ ℝ*)
5	4	supminfxr2 46621	. 2 ⊢ (𝜑 → sup(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ*, < ) = -einf({𝑦 ∈ ℝ* ∣ -e𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)}, ℝ*, < ))
6		xnegex 13351	. . . . . . . . . . . 12 ⊢ -e𝑦 ∈ V
7	2	elrnmpt 5945	. . . . . . . . . . . 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 219	. . . . . . . . . 10 ⊢ ( -e𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) → ∃𝑥 ∈ 𝐴 -e𝑦 = 𝐵)
10		eqid 2763	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐴 ↦ -e𝐵) = (𝑥 ∈ 𝐴 ↦ -e𝐵)
11		xnegneg 13357	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ ℝ* → -e -e𝑦 = 𝑦)
12	11	eqcomd 2769	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ ℝ* → 𝑦 = -e -e𝑦)
13	12	adantr 486	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ ℝ* ∧ -e𝑦 = 𝐵) → 𝑦 = -e -e𝑦)
14		xnegeq 13350	. . . . . . . . . . . . . . . 16 ⊢ ( -e𝑦 = 𝐵 → -e -e𝑦 = -e𝐵)
15	14	adantl 487	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ ℝ* ∧ -e𝑦 = 𝐵) → -e -e𝑦 = -e𝐵)
16	13, 15	eqtrd 2798	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ ℝ* ∧ -e𝑦 = 𝐵) → 𝑦 = -e𝐵)
17	16	ex 418	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℝ* → ( -e𝑦 = 𝐵 → 𝑦 = -e𝐵))
18	17	reximdv 3181	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℝ* → (∃𝑥 ∈ 𝐴 -e𝑦 = 𝐵 → ∃𝑥 ∈ 𝐴 𝑦 = -e𝐵))
19	18	imp 412	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℝ* ∧ ∃𝑥 ∈ 𝐴 -e𝑦 = 𝐵) → ∃𝑥 ∈ 𝐴 𝑦 = -e𝐵)
20		simpl 488	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℝ* ∧ ∃𝑥 ∈ 𝐴 -e𝑦 = 𝐵) → 𝑦 ∈ ℝ*)
21	10, 19, 20	elrnmptd 5950	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ℝ* ∧ ∃𝑥 ∈ 𝐴 -e𝑦 = 𝐵) → 𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ -e𝐵))
22	9, 21	sylan2 605	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℝ* ∧ -e𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)) → 𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ -e𝐵))
23	22	ex 418	. . . . . . . 8 ⊢ (𝑦 ∈ ℝ* → ( -e𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) → 𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ -e𝐵)))
24	23	rgen 3082	. . . . . . 7 ⊢ ∀𝑦 ∈ ℝ* ( -e𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) → 𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ -e𝐵))
25		rabss 3998	. . . . . . . 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 231	. . . . . . 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 2926	. . . . . . . 8 ⊢ Ⅎ𝑥 -e𝑦
30		nfmpt1 5207	. . . . . . . . 9 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐵)
31	30	nfrn 5939	. . . . . . . 8 ⊢ Ⅎ𝑥ran (𝑥 ∈ 𝐴 ↦ 𝐵)
32	29, 31	nfel 2940	. . . . . . 7 ⊢ Ⅎ𝑥 -e𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)
33		nfcv 2926	. . . . . . 7 ⊢ Ⅎ𝑥ℝ*
34	32, 33	nfrabw 3451	. . . . . 6 ⊢ Ⅎ𝑥{𝑦 ∈ ℝ* ∣ -e𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)}
35		xnegeq 13350	. . . . . . . 8 ⊢ (𝑦 = -e𝐵 → -e𝑦 = -e -e𝐵)
36	35	eleq1d 2849	. . . . . . 7 ⊢ (𝑦 = -e𝐵 → ( -e𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ↔ -e -e𝐵 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)))
37	3	xnegcld 13443	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → -e𝐵 ∈ ℝ*)
38		xnegneg 13357	. . . . . . . . 9 ⊢ (𝐵 ∈ ℝ* → -e -e𝐵 = 𝐵)
39	3, 38	syl 18	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → -e -e𝐵 = 𝐵)
40		simpr 490	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
41	2, 40, 3	elrnmpt1d 5951	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵))
42	39, 41	eqeltrd 2864	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → -e -e𝐵 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵))
43	36, 37, 42	elrabd 3650	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → -e𝐵 ∈ {𝑦 ∈ ℝ* ∣ -e𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)})
44	1, 34, 10, 43	rnmptssdf 46408	. . . . 5 ⊢ (𝜑 → ran (𝑥 ∈ 𝐴 ↦ -e𝐵) ⊆ {𝑦 ∈ ℝ* ∣ -e𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)})
45	28, 44	eqssd 3930	. . . 4 ⊢ (𝜑 → {𝑦 ∈ ℝ* ∣ -e𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)} = ran (𝑥 ∈ 𝐴 ↦ -e𝐵))
46	45	infeq1d 9483	. . 3 ⊢ (𝜑 → inf({𝑦 ∈ ℝ* ∣ -e𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)}, ℝ*, < ) = inf(ran (𝑥 ∈ 𝐴 ↦ -e𝐵), ℝ*, < ))
47	46	xnegeqd 46589	. 2 ⊢ (𝜑 → -einf({𝑦 ∈ ℝ* ∣ -e𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)}, ℝ*, < ) = -einf(ran (𝑥 ∈ 𝐴 ↦ -e𝐵), ℝ*, < ))
48	5, 47	eqtrd 2798	1 ⊢ (𝜑 → sup(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ*, < ) = -einf(ran (𝑥 ∈ 𝐴 ↦ -e𝐵), ℝ*, < ))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569  Ⅎwnf 1815   ∈ wcel 2146  ∀wral 3080  ∃wrex 3090  {crab 3416  Vcvv 3454   ⊆ wss 3900   ↦ cmpt 5189  ran crn 5656  supcsup 9445  infcinf 9446  ℝ^*cxr 11355   < clt 11356   -_ecxne 13251

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7761  ax-cnex 11269  ax-resscn 11270  ax-1cn 11271  ax-icn 11272  ax-addcl 11273  ax-addrcl 11274  ax-mulcl 11275  ax-mulrcl 11276  ax-mulcom 11277  ax-addass 11278  ax-mulass 11279  ax-distr 11280  ax-i2m1 11281  ax-1ne0 11282  ax-1rid 11283  ax-rnegex 11284  ax-rrecex 11285  ax-cnre 11286  ax-pre-lttri 11287  ax-pre-lttrn 11288  ax-pre-ltadd 11289  ax-pre-mulgt0 11290  ax-pre-sup 11291

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-po 5563  df-so 5564  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6501  df-fun 6549  df-fn 6550  df-f 6551  df-f1 6552  df-fo 6553  df-f1o 6554  df-fv 6555  df-isom 6556  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-er 8731  df-en 8988  df-dom 8989  df-sdom 8990  df-sup 9447  df-inf 9448  df-pnf 11358  df-mnf 11359  df-xr 11360  df-ltxr 11361  df-le 11362  df-sub 11556  df-neg 11557  df-xneg 13254

This theorem is used by:  liminfvalxr  46935