Theorem supminfxrrnmpt 46480

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 2761	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
3		supminfxrrnmpt.b	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ*)
4	1, 2, 3	rnmptssd 7124	. . 3 ⊢ (𝜑 → ran (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ ℝ*)
5	4	supminfxr2 46478	. 2 ⊢ (𝜑 → sup(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ*, < ) = -einf({𝑦 ∈ ℝ* ∣ -e𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)}, ℝ*, < ))
6		xnegex 13338	. . . . . . . . . . . 12 ⊢ -e𝑦 ∈ V
7	2	elrnmpt 5940	. . . . . . . . . . . 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 2761	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐴 ↦ -e𝐵) = (𝑥 ∈ 𝐴 ↦ -e𝐵)
11		xnegneg 13344	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ ℝ* → -e -e𝑦 = 𝑦)
12	11	eqcomd 2767	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ ℝ* → 𝑦 = -e -e𝑦)
13	12	adantr 486	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ ℝ* ∧ -e𝑦 = 𝐵) → 𝑦 = -e -e𝑦)
14		xnegeq 13337	. . . . . . . . . . . . . . . 16 ⊢ ( -e𝑦 = 𝐵 → -e -e𝑦 = -e𝐵)
15	14	adantl 487	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ ℝ* ∧ -e𝑦 = 𝐵) → -e -e𝑦 = -e𝐵)
16	13, 15	eqtrd 2796	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ ℝ* ∧ -e𝑦 = 𝐵) → 𝑦 = -e𝐵)
17	16	ex 418	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℝ* → ( -e𝑦 = 𝐵 → 𝑦 = -e𝐵))
18	17	reximdv 3178	. . . . . . . . . . . 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 5945	. . . . . . . . . 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 3079	. . . . . . 7 ⊢ ∀𝑦 ∈ ℝ* ( -e𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) → 𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ -e𝐵))
25		rabss 4018	. . . . . . . 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 2923	. . . . . . . 8 ⊢ Ⅎ𝑥 -e𝑦
30		nfmpt1 5204	. . . . . . . . 9 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐵)
31	30	nfrn 5934	. . . . . . . 8 ⊢ Ⅎ𝑥ran (𝑥 ∈ 𝐴 ↦ 𝐵)
32	29, 31	nfel 2937	. . . . . . 7 ⊢ Ⅎ𝑥 -e𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)
33		nfcv 2923	. . . . . . 7 ⊢ Ⅎ𝑥ℝ*
34	32, 33	nfrabw 3448	. . . . . 6 ⊢ Ⅎ𝑥{𝑦 ∈ ℝ* ∣ -e𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)}
35		xnegeq 13337	. . . . . . . 8 ⊢ (𝑦 = -e𝐵 → -e𝑦 = -e -e𝐵)
36	35	eleq1d 2846	. . . . . . 7 ⊢ (𝑦 = -e𝐵 → ( -e𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ↔ -e -e𝐵 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)))
37	3	xnegcld 13430	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → -e𝐵 ∈ ℝ*)
38		xnegneg 13344	. . . . . . . . 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 5946	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵))
42	39, 41	eqeltrd 2861	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → -e -e𝐵 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵))
43	36, 37, 42	elrabd 3647	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → -e𝐵 ∈ {𝑦 ∈ ℝ* ∣ -e𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)})
44	1, 34, 10, 43	rnmptssdf 46265	. . . . 5 ⊢ (𝜑 → ran (𝑥 ∈ 𝐴 ↦ -e𝐵) ⊆ {𝑦 ∈ ℝ* ∣ -e𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)})
45	28, 44	eqssd 3948	. . . 4 ⊢ (𝜑 → {𝑦 ∈ ℝ* ∣ -e𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)} = ran (𝑥 ∈ 𝐴 ↦ -e𝐵))
46	45	infeq1d 9470	. . 3 ⊢ (𝜑 → inf({𝑦 ∈ ℝ* ∣ -e𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)}, ℝ*, < ) = inf(ran (𝑥 ∈ 𝐴 ↦ -e𝐵), ℝ*, < ))
47	46	xnegeqd 46446	. 2 ⊢ (𝜑 → -einf({𝑦 ∈ ℝ* ∣ -e𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)}, ℝ*, < ) = -einf(ran (𝑥 ∈ 𝐴 ↦ -e𝐵), ℝ*, < ))
48	5, 47	eqtrd 2796	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 1570  Ⅎwnf 1816   ∈ wcel 2145  ∀wral 3077  ∃wrex 3087  {crab 3413  Vcvv 3451   ⊆ wss 3899   ↦ cmpt 5186  ran crn 5652  supcsup 9432  infcinf 9433  ℝ^*cxr 11342   < clt 11343   -_ecxne 13238

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7751  ax-cnex 11256  ax-resscn 11257  ax-1cn 11258  ax-icn 11259  ax-addcl 11260  ax-addrcl 11261  ax-mulcl 11262  ax-mulrcl 11263  ax-mulcom 11264  ax-addass 11265  ax-mulass 11266  ax-distr 11267  ax-i2m1 11268  ax-1ne0 11269  ax-1rid 11270  ax-rnegex 11271  ax-rrecex 11272  ax-cnre 11273  ax-pre-lttri 11274  ax-pre-lttrn 11275  ax-pre-ltadd 11276  ax-pre-mulgt0 11277  ax-pre-sup 11278

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-po 5559  df-so 5560  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-isom 6547  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-er 8717  df-en 8974  df-dom 8975  df-sdom 8976  df-sup 9434  df-inf 9435  df-pnf 11345  df-mnf 11346  df-xr 11347  df-ltxr 11348  df-le 11349  df-sub 11543  df-neg 11544  df-xneg 13241

This theorem is used by:  liminfvalxr  46792