Theorem infrpgernmpt 45482

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 1915	. . 3 ⊢ Ⅎ𝑤𝜑
2		infrpgernmpt.x	. . . 4 ⊢ Ⅎ𝑥𝜑
3		eqid 2730	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
4		infrpgernmpt.b	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ*)
5	2, 3, 4	rnmptssd 45212	. . 3 ⊢ (𝜑 → ran (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ ℝ*)
6		infrpgernmpt.a	. . . 4 ⊢ (𝜑 → 𝐴 ≠ ∅)
7	2, 4, 3, 6	rnmptn0 6188	. . 3 ⊢ (𝜑 → ran (𝑥 ∈ 𝐴 ↦ 𝐵) ≠ ∅)
8		infrpgernmpt.y	. . . . 5 ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 𝑦 ≤ 𝐵)
9		breq1 5092	. . . . . . 7 ⊢ (𝑦 = 𝑤 → (𝑦 ≤ 𝐵 ↔ 𝑤 ≤ 𝐵))
10	9	ralbidv 3153	. . . . . 6 ⊢ (𝑦 = 𝑤 → (∀𝑥 ∈ 𝐴 𝑦 ≤ 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑤 ≤ 𝐵))
11	10	cbvrexvw 3209	. . . . 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 45259	. . 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 45369	. 2 ⊢ (𝜑 → ∃𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑤 ≤ (inf(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ*, < ) +𝑒 𝐶))
16		simpll 766	. . . 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 3438	. . . . . . 7 ⊢ 𝑤 ∈ V
19	3	elrnmpt 5895	. . . . . . 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 2892	. . . . . . . 8 ⊢ Ⅎ𝑥𝑤
24		nfcv 2892	. . . . . . . 8 ⊢ Ⅎ𝑥 ≤
25		nfmpt1 5188	. . . . . . . . . . 11 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐵)
26	25	nfrn 5889	. . . . . . . . . 10 ⊢ Ⅎ𝑥ran (𝑥 ∈ 𝐴 ↦ 𝐵)
27		nfcv 2892	. . . . . . . . . 10 ⊢ Ⅎ𝑥ℝ*
28		nfcv 2892	. . . . . . . . . 10 ⊢ Ⅎ𝑥 <
29	26, 27, 28	nfinf 9362	. . . . . . . . 9 ⊢ Ⅎ𝑥inf(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ*, < )
30		nfcv 2892	. . . . . . . . 9 ⊢ Ⅎ𝑥 +𝑒
31		nfcv 2892	. . . . . . . . 9 ⊢ Ⅎ𝑥𝐶
32	29, 30, 31	nfov 7371	. . . . . . . 8 ⊢ Ⅎ𝑥(inf(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ*, < ) +𝑒 𝐶)
33	23, 24, 32	nfbr 5136	. . . . . . 7 ⊢ Ⅎ𝑥 𝑤 ≤ (inf(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ*, < ) +𝑒 𝐶)
34	2, 33	nfan 1900	. . . . . 6 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑤 ≤ (inf(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ*, < ) +𝑒 𝐶))
35		id 22	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝐵 → 𝑤 = 𝐵)
36	35	eqcomd 2736	. . . . . . . . . . 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 5111	. . . . . . . . 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 3232	. . . . 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 837	. . 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 3133	. 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 1541  Ⅎwnf 1784   ∈ wcel 2110   ≠ wne 2926  ∀wral 3045  ∃wrex 3054  Vcvv 3434  ∅c0 4281   class class class wbr 5089   ↦ cmpt 5170  ran crn 5615  (class class class)co 7341  infcinf 9320  ℝcr 10997  ℝ^*cxr 11137   < clt 11138   ≤ cle 11139  ℝ⁺crp 12882   +_𝑒 cxad 13001

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 1911  ax-6 1968  ax-7 2009  ax-8 2112  ax-9 2120  ax-10 2143  ax-11 2159  ax-12 2179  ax-ext 2702  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7663  ax-cnex 11054  ax-resscn 11055  ax-1cn 11056  ax-icn 11057  ax-addcl 11058  ax-addrcl 11059  ax-mulcl 11060  ax-mulrcl 11061  ax-mulcom 11062  ax-addass 11063  ax-mulass 11064  ax-distr 11065  ax-i2m1 11066  ax-1ne0 11067  ax-1rid 11068  ax-rnegex 11069  ax-rrecex 11070  ax-cnre 11071  ax-pre-lttri 11072  ax-pre-lttrn 11073  ax-pre-ltadd 11074  ax-pre-mulgt0 11075  ax-pre-sup 11076

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-rmo 3344  df-reu 3345  df-rab 3394  df-v 3436  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-nul 4282  df-if 4474  df-pw 4550  df-sn 4575  df-pr 4577  df-op 4581  df-uni 4858  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-po 5522  df-so 5523  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-iota 6433  df-fun 6479  df-fn 6480  df-f 6481  df-f1 6482  df-fo 6483  df-f1o 6484  df-fv 6485  df-riota 7298  df-ov 7344  df-oprab 7345  df-mpo 7346  df-1st 7916  df-2nd 7917  df-er 8617  df-en 8865  df-dom 8866  df-sdom 8867  df-sup 9321  df-inf 9322  df-pnf 11140  df-mnf 11141  df-xr 11142  df-ltxr 11143  df-le 11144  df-sub 11338  df-neg 11339  df-rp 12883  df-xadd 13004

This theorem is referenced by:  limsupgtlem  45794