Theorem infrpgernmpt 45915

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 1921	. . 3 ⊢ Ⅎ𝑤𝜑
2		infrpgernmpt.x	. . . 4 ⊢ Ⅎ𝑥𝜑
3		eqid 2740	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
4		infrpgernmpt.b	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ*)
5	2, 3, 4	rnmptssd 7072	. . 3 ⊢ (𝜑 → ran (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ ℝ*)
6		infrpgernmpt.a	. . . 4 ⊢ (𝜑 → 𝐴 ≠ ∅)
7	2, 4, 3, 6	rnmptn0 6202	. . 3 ⊢ (𝜑 → ran (𝑥 ∈ 𝐴 ↦ 𝐵) ≠ ∅)
8		infrpgernmpt.y	. . . . 5 ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 𝑦 ≤ 𝐵)
9		breq1 5082	. . . . . . 7 ⊢ (𝑦 = 𝑤 → (𝑦 ≤ 𝐵 ↔ 𝑤 ≤ 𝐵))
10	9	ralbidv 3163	. . . . . 6 ⊢ (𝑦 = 𝑤 → (∀𝑥 ∈ 𝐴 𝑦 ≤ 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑤 ≤ 𝐵))
11	10	cbvrexvw 3219	. . . . 5 ⊢ (∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 𝑦 ≤ 𝐵 ↔ ∃𝑤 ∈ ℝ ∀𝑥 ∈ 𝐴 𝑤 ≤ 𝐵)
12	8, 11	sylib 219	. . . 4 ⊢ (𝜑 → ∃𝑤 ∈ ℝ ∀𝑥 ∈ 𝐴 𝑤 ≤ 𝐵)
13	12	rnmptlb 45694	. . 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 45803	. 2 ⊢ (𝜑 → ∃𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑤 ≤ (inf(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ*, < ) +𝑒 𝐶))
16		simpll 772	. . . 4 ⊢ (((𝜑 ∧ 𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)) ∧ 𝑤 ≤ (inf(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ*, < ) +𝑒 𝐶)) → 𝜑)
17		simpr 485	. . . 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 3436	. . . . . . 7 ⊢ 𝑤 ∈ V
19	3	elrnmpt 5907	. . . . . . 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 217	. . . . 5 ⊢ (𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) → ∃𝑥 ∈ 𝐴 𝑤 = 𝐵)
22	21	ad2antlr 733	. . . 4 ⊢ (((𝜑 ∧ 𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)) ∧ 𝑤 ≤ (inf(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ*, < ) +𝑒 𝐶)) → ∃𝑥 ∈ 𝐴 𝑤 = 𝐵)
23		nfcv 2902	. . . . . . . 8 ⊢ Ⅎ𝑥𝑤
24		nfcv 2902	. . . . . . . 8 ⊢ Ⅎ𝑥 ≤
25		nfmpt1 5178	. . . . . . . . . . 11 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐵)
26	25	nfrn 5901	. . . . . . . . . 10 ⊢ Ⅎ𝑥ran (𝑥 ∈ 𝐴 ↦ 𝐵)
27		nfcv 2902	. . . . . . . . . 10 ⊢ Ⅎ𝑥ℝ*
28		nfcv 2902	. . . . . . . . . 10 ⊢ Ⅎ𝑥 <
29	26, 27, 28	nfinf 9393	. . . . . . . . 9 ⊢ Ⅎ𝑥inf(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ*, < )
30		nfcv 2902	. . . . . . . . 9 ⊢ Ⅎ𝑥 +𝑒
31		nfcv 2902	. . . . . . . . 9 ⊢ Ⅎ𝑥𝐶
32	29, 30, 31	nfov 7393	. . . . . . . 8 ⊢ Ⅎ𝑥(inf(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ*, < ) +𝑒 𝐶)
33	23, 24, 32	nfbr 5126	. . . . . . 7 ⊢ Ⅎ𝑥 𝑤 ≤ (inf(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ*, < ) +𝑒 𝐶)
34	2, 33	nfan 1906	. . . . . 6 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑤 ≤ (inf(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ*, < ) +𝑒 𝐶))
35		id 22	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝐵 → 𝑤 = 𝐵)
36	35	eqcomd 2746	. . . . . . . . . . 11 ⊢ (𝑤 = 𝐵 → 𝐵 = 𝑤)
37	36	adantl 482	. . . . . . . . . 10 ⊢ ((𝑤 ≤ (inf(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ*, < ) +𝑒 𝐶) ∧ 𝑤 = 𝐵) → 𝐵 = 𝑤)
38		simpl 483	. . . . . . . . . 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 5101	. . . . . . . . 9 ⊢ ((𝑤 ≤ (inf(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ*, < ) +𝑒 𝐶) ∧ 𝑤 = 𝐵) → 𝐵 ≤ (inf(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ*, < ) +𝑒 𝐶))
40	39	ex 413	. . . . . . . 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 482	. . . . . 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 3242	. . . . 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 407	. . . 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 843	. . 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 3143	. 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 207   ∧ wa 396   = wceq 1547  Ⅎwnf 1790   ∈ wcel 2119   ≠ wne 2935  ∀wral 3054  ∃wrex 3064  Vcvv 3432  ∅c0 4268   class class class wbr 5079   ↦ cmpt 5160  ran crn 5626  (class class class)co 7363  infcinf 9351  ℝcr 11035  ℝ^*cxr 11176   < clt 11177   ≤ cle 11178  ℝ⁺crp 12940   +_𝑒 cxad 13059

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685  ax-cnex 11092  ax-resscn 11093  ax-1cn 11094  ax-icn 11095  ax-addcl 11096  ax-addrcl 11097  ax-mulcl 11098  ax-mulrcl 11099  ax-mulcom 11100  ax-addass 11101  ax-mulass 11102  ax-distr 11103  ax-i2m1 11104  ax-1ne0 11105  ax-1rid 11106  ax-rnegex 11107  ax-rrecex 11108  ax-cnre 11109  ax-pre-lttri 11110  ax-pre-lttrn 11111  ax-pre-ltadd 11112  ax-pre-mulgt0 11113  ax-pre-sup 11114

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-nel 3040  df-ral 3055  df-rex 3065  df-rmo 3345  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-iun 4930  df-br 5080  df-opab 5142  df-mpt 5161  df-id 5520  df-po 5533  df-so 5534  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7320  df-ov 7366  df-oprab 7367  df-mpo 7368  df-1st 7938  df-2nd 7939  df-er 8640  df-en 8891  df-dom 8892  df-sdom 8893  df-sup 9352  df-inf 9353  df-pnf 11179  df-mnf 11180  df-xr 11181  df-ltxr 11182  df-le 11183  df-sub 11377  df-neg 11378  df-rp 12941  df-xadd 13062

This theorem is referenced by:  limsupgtlem  46227