Theorem infrpgernmpt 45476

Description: The infimum of a nonempty, bounded below, indexed subset of extended reals can be approximated from above by an element of the set. (Contributed by Glauco Siliprandi, 2-Jan-2022.)

Hypotheses

Ref	Expression
infrpgernmpt.x	⊢ Ⅎ𝑥𝜑
infrpgernmpt.a	⊢ (𝜑 → 𝐴 ≠ ∅)
infrpgernmpt.b	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ*)
infrpgernmpt.y	⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 𝑦 ≤ 𝐵)
infrpgernmpt.c	⊢ (𝜑 → 𝐶 ∈ ℝ+)

Assertion

Ref	Expression
infrpgernmpt	⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝐵 ≤ (inf(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ*, < ) +𝑒 𝐶))

Distinct variable groups:   𝑥,𝐴,𝑦   𝑦,𝐵   𝑥,𝐶

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐵(𝑥)   𝐶(𝑦)

Proof of Theorem infrpgernmpt

Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1914	. . 3 ⊢ Ⅎ𝑤𝜑
2		infrpgernmpt.x	. . . 4 ⊢ Ⅎ𝑥𝜑
3		eqid 2737	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
4		infrpgernmpt.b	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ*)
5	2, 3, 4	rnmptssd 45201	. . 3 ⊢ (𝜑 → ran (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ ℝ*)
6		infrpgernmpt.a	. . . 4 ⊢ (𝜑 → 𝐴 ≠ ∅)
7	2, 4, 3, 6	rnmptn0 6264	. . 3 ⊢ (𝜑 → ran (𝑥 ∈ 𝐴 ↦ 𝐵) ≠ ∅)
8		infrpgernmpt.y	. . . . 5 ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 𝑦 ≤ 𝐵)
9		breq1 5146	. . . . . . 7 ⊢ (𝑦 = 𝑤 → (𝑦 ≤ 𝐵 ↔ 𝑤 ≤ 𝐵))
10	9	ralbidv 3178	. . . . . 6 ⊢ (𝑦 = 𝑤 → (∀𝑥 ∈ 𝐴 𝑦 ≤ 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑤 ≤ 𝐵))
11	10	cbvrexvw 3238	. . . . 5 ⊢ (∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 𝑦 ≤ 𝐵 ↔ ∃𝑤 ∈ ℝ ∀𝑥 ∈ 𝐴 𝑤 ≤ 𝐵)
12	8, 11	sylib 218	. . . 4 ⊢ (𝜑 → ∃𝑤 ∈ ℝ ∀𝑥 ∈ 𝐴 𝑤 ≤ 𝐵)
13	12	rnmptlb 45250	. . 3 ⊢ (𝜑 → ∃𝑤 ∈ ℝ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑤 ≤ 𝑧)
14		infrpgernmpt.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ ℝ+)
15	1, 5, 7, 13, 14	infrpge 45362	. 2 ⊢ (𝜑 → ∃𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑤 ≤ (inf(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ*, < ) +𝑒 𝐶))
16		simpll 767	. . . 4 ⊢ (((𝜑 ∧ 𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)) ∧ 𝑤 ≤ (inf(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ*, < ) +𝑒 𝐶)) → 𝜑)
17		simpr 484	. . . 4 ⊢ (((𝜑 ∧ 𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)) ∧ 𝑤 ≤ (inf(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ*, < ) +𝑒 𝐶)) → 𝑤 ≤ (inf(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ*, < ) +𝑒 𝐶))
18		vex 3484	. . . . . . 7 ⊢ 𝑤 ∈ V
19	3	elrnmpt 5969	. . . . . . 7 ⊢ (𝑤 ∈ V → (𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ↔ ∃𝑥 ∈ 𝐴 𝑤 = 𝐵))
20	18, 19	ax-mp 5	. . . . . 6 ⊢ (𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ↔ ∃𝑥 ∈ 𝐴 𝑤 = 𝐵)
21	20	biimpi 216	. . . . 5 ⊢ (𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) → ∃𝑥 ∈ 𝐴 𝑤 = 𝐵)
22	21	ad2antlr 727	. . . 4 ⊢ (((𝜑 ∧ 𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)) ∧ 𝑤 ≤ (inf(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ*, < ) +𝑒 𝐶)) → ∃𝑥 ∈ 𝐴 𝑤 = 𝐵)
23		nfcv 2905	. . . . . . . 8 ⊢ Ⅎ𝑥𝑤
24		nfcv 2905	. . . . . . . 8 ⊢ Ⅎ𝑥 ≤
25		nfmpt1 5250	. . . . . . . . . . 11 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐵)
26	25	nfrn 5963	. . . . . . . . . 10 ⊢ Ⅎ𝑥ran (𝑥 ∈ 𝐴 ↦ 𝐵)
27		nfcv 2905	. . . . . . . . . 10 ⊢ Ⅎ𝑥ℝ*
28		nfcv 2905	. . . . . . . . . 10 ⊢ Ⅎ𝑥 <
29	26, 27, 28	nfinf 9522	. . . . . . . . 9 ⊢ Ⅎ𝑥inf(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ*, < )
30		nfcv 2905	. . . . . . . . 9 ⊢ Ⅎ𝑥 +𝑒
31		nfcv 2905	. . . . . . . . 9 ⊢ Ⅎ𝑥𝐶
32	29, 30, 31	nfov 7461	. . . . . . . 8 ⊢ Ⅎ𝑥(inf(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ*, < ) +𝑒 𝐶)
33	23, 24, 32	nfbr 5190	. . . . . . 7 ⊢ Ⅎ𝑥 𝑤 ≤ (inf(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ*, < ) +𝑒 𝐶)
34	2, 33	nfan 1899	. . . . . 6 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑤 ≤ (inf(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ*, < ) +𝑒 𝐶))
35		id 22	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝐵 → 𝑤 = 𝐵)
36	35	eqcomd 2743	. . . . . . . . . . 11 ⊢ (𝑤 = 𝐵 → 𝐵 = 𝑤)
37	36	adantl 481	. . . . . . . . . 10 ⊢ ((𝑤 ≤ (inf(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ*, < ) +𝑒 𝐶) ∧ 𝑤 = 𝐵) → 𝐵 = 𝑤)
38		simpl 482	. . . . . . . . . 10 ⊢ ((𝑤 ≤ (inf(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ*, < ) +𝑒 𝐶) ∧ 𝑤 = 𝐵) → 𝑤 ≤ (inf(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ*, < ) +𝑒 𝐶))
39	37, 38	eqbrtrd 5165	. . . . . . . . 9 ⊢ ((𝑤 ≤ (inf(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ*, < ) +𝑒 𝐶) ∧ 𝑤 = 𝐵) → 𝐵 ≤ (inf(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ*, < ) +𝑒 𝐶))
40	39	ex 412	. . . . . . . 8 ⊢ (𝑤 ≤ (inf(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ*, < ) +𝑒 𝐶) → (𝑤 = 𝐵 → 𝐵 ≤ (inf(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ*, < ) +𝑒 𝐶)))
41	40	a1d 25	. . . . . . 7 ⊢ (𝑤 ≤ (inf(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ*, < ) +𝑒 𝐶) → (𝑥 ∈ 𝐴 → (𝑤 = 𝐵 → 𝐵 ≤ (inf(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ*, < ) +𝑒 𝐶))))
42	41	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ≤ (inf(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ*, < ) +𝑒 𝐶)) → (𝑥 ∈ 𝐴 → (𝑤 = 𝐵 → 𝐵 ≤ (inf(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ*, < ) +𝑒 𝐶))))
43	34, 42	reximdai 3261	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ≤ (inf(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ*, < ) +𝑒 𝐶)) → (∃𝑥 ∈ 𝐴 𝑤 = 𝐵 → ∃𝑥 ∈ 𝐴 𝐵 ≤ (inf(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ*, < ) +𝑒 𝐶)))
44	43	imp 406	. . . 4 ⊢ (((𝜑 ∧ 𝑤 ≤ (inf(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ*, < ) +𝑒 𝐶)) ∧ ∃𝑥 ∈ 𝐴 𝑤 = 𝐵) → ∃𝑥 ∈ 𝐴 𝐵 ≤ (inf(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ*, < ) +𝑒 𝐶))
45	16, 17, 22, 44	syl21anc 838	. . 3 ⊢ (((𝜑 ∧ 𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)) ∧ 𝑤 ≤ (inf(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ*, < ) +𝑒 𝐶)) → ∃𝑥 ∈ 𝐴 𝐵 ≤ (inf(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ*, < ) +𝑒 𝐶))
46	45	rexlimdva2 3157	. 2 ⊢ (𝜑 → (∃𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑤 ≤ (inf(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ*, < ) +𝑒 𝐶) → ∃𝑥 ∈ 𝐴 𝐵 ≤ (inf(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ*, < ) +𝑒 𝐶)))
47	15, 46	mpd 15	1 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝐵 ≤ (inf(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ*, < ) +𝑒 𝐶))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540  Ⅎwnf 1783   ∈ wcel 2108   ≠ wne 2940  ∀wral 3061  ∃wrex 3070  Vcvv 3480  ∅c0 4333   class class class wbr 5143   ↦ cmpt 5225  ran crn 5686  (class class class)co 7431  infcinf 9481  ℝcr 11154  ℝ^*cxr 11294   < clt 11295   ≤ cle 11296  ℝ⁺crp 13034   +_𝑒 cxad 13152

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-po 5592  df-so 5593  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-sup 9482  df-inf 9483  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-rp 13035  df-xadd 13155

This theorem is referenced by:  limsupgtlem  45792