Theorem infrpgernmpt 45461

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 2729	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
4		infrpgernmpt.b	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ*)
5	2, 3, 4	rnmptssd 45190	. . 3 ⊢ (𝜑 → ran (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ ℝ*)
6		infrpgernmpt.a	. . . 4 ⊢ (𝜑 → 𝐴 ≠ ∅)
7	2, 4, 3, 6	rnmptn0 6217	. . 3 ⊢ (𝜑 → ran (𝑥 ∈ 𝐴 ↦ 𝐵) ≠ ∅)
8		infrpgernmpt.y	. . . . 5 ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 𝑦 ≤ 𝐵)
9		breq1 5110	. . . . . . 7 ⊢ (𝑦 = 𝑤 → (𝑦 ≤ 𝐵 ↔ 𝑤 ≤ 𝐵))
10	9	ralbidv 3156	. . . . . 6 ⊢ (𝑦 = 𝑤 → (∀𝑥 ∈ 𝐴 𝑦 ≤ 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑤 ≤ 𝐵))
11	10	cbvrexvw 3216	. . . . 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 45237	. . 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 45347	. 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 3451	. . . . . . 7 ⊢ 𝑤 ∈ V
19	3	elrnmpt 5922	. . . . . . 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 2891	. . . . . . . 8 ⊢ Ⅎ𝑥𝑤
24		nfcv 2891	. . . . . . . 8 ⊢ Ⅎ𝑥 ≤
25		nfmpt1 5206	. . . . . . . . . . 11 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐵)
26	25	nfrn 5916	. . . . . . . . . 10 ⊢ Ⅎ𝑥ran (𝑥 ∈ 𝐴 ↦ 𝐵)
27		nfcv 2891	. . . . . . . . . 10 ⊢ Ⅎ𝑥ℝ*
28		nfcv 2891	. . . . . . . . . 10 ⊢ Ⅎ𝑥 <
29	26, 27, 28	nfinf 9434	. . . . . . . . 9 ⊢ Ⅎ𝑥inf(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ*, < )
30		nfcv 2891	. . . . . . . . 9 ⊢ Ⅎ𝑥 +𝑒
31		nfcv 2891	. . . . . . . . 9 ⊢ Ⅎ𝑥𝐶
32	29, 30, 31	nfov 7417	. . . . . . . 8 ⊢ Ⅎ𝑥(inf(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ*, < ) +𝑒 𝐶)
33	23, 24, 32	nfbr 5154	. . . . . . 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 2735	. . . . . . . . . . 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 5129	. . . . . . . . 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 3239	. . . . 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 3136	. 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 2109   ≠ wne 2925  ∀wral 3044  ∃wrex 3053  Vcvv 3447  ∅c0 4296   class class class wbr 5107   ↦ cmpt 5188  ran crn 5639  (class class class)co 7387  infcinf 9392  ℝcr 11067  ℝ^*cxr 11207   < clt 11208   ≤ cle 11209  ℝ⁺crp 12951   +_𝑒 cxad 13070

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-cnex 11124  ax-resscn 11125  ax-1cn 11126  ax-icn 11127  ax-addcl 11128  ax-addrcl 11129  ax-mulcl 11130  ax-mulrcl 11131  ax-mulcom 11132  ax-addass 11133  ax-mulass 11134  ax-distr 11135  ax-i2m1 11136  ax-1ne0 11137  ax-1rid 11138  ax-rnegex 11139  ax-rrecex 11140  ax-cnre 11141  ax-pre-lttri 11142  ax-pre-lttrn 11143  ax-pre-ltadd 11144  ax-pre-mulgt0 11145  ax-pre-sup 11146

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-po 5546  df-so 5547  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-1st 7968  df-2nd 7969  df-er 8671  df-en 8919  df-dom 8920  df-sdom 8921  df-sup 9393  df-inf 9394  df-pnf 11210  df-mnf 11211  df-xr 11212  df-ltxr 11213  df-le 11214  df-sub 11407  df-neg 11408  df-rp 12952  df-xadd 13073

This theorem is referenced by:  limsupgtlem  45775