Theorem infrpgernmpt 39695

Description: The infimum of a non empty, 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 1843	. . 3 ⊢ Ⅎ𝑤𝜑
2		infrpgernmpt.x	. . . 4 ⊢ Ⅎ𝑥𝜑
3		eqid 2622	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
4		infrpgernmpt.b	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ*)
5	2, 3, 4	rnmptssd 39385	. . 3 ⊢ (𝜑 → ran (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ ℝ*)
6		infrpgernmpt.a	. . . 4 ⊢ (𝜑 → 𝐴 ≠ ∅)
7	2, 4, 3, 6	rnmptn0 39413	. . 3 ⊢ (𝜑 → ran (𝑥 ∈ 𝐴 ↦ 𝐵) ≠ ∅)
8		infrpgernmpt.y	. . . . 5 ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 𝑦 ≤ 𝐵)
9		breq1 4656	. . . . . . 7 ⊢ (𝑦 = 𝑤 → (𝑦 ≤ 𝐵 ↔ 𝑤 ≤ 𝐵))
10	9	ralbidv 2986	. . . . . 6 ⊢ (𝑦 = 𝑤 → (∀𝑥 ∈ 𝐴 𝑦 ≤ 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑤 ≤ 𝐵))
11	10	cbvrexv 3172	. . . . 5 ⊢ (∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 𝑦 ≤ 𝐵 ↔ ∃𝑤 ∈ ℝ ∀𝑥 ∈ 𝐴 𝑤 ≤ 𝐵)
12	8, 11	sylib 208	. . . 4 ⊢ (𝜑 → ∃𝑤 ∈ ℝ ∀𝑥 ∈ 𝐴 𝑤 ≤ 𝐵)
13	12	rnmptlb 39453	. . 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 39567	. 2 ⊢ (𝜑 → ∃𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑤 ≤ (inf(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ*, < ) +𝑒 𝐶))
16		simpll 790	. . . 4 ⊢ (((𝜑 ∧ 𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)) ∧ 𝑤 ≤ (inf(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ*, < ) +𝑒 𝐶)) → 𝜑)
17		simpr 477	. . . 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 3203	. . . . . . 7 ⊢ 𝑤 ∈ V
19	3	elrnmpt 5372	. . . . . . 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 206	. . . . 5 ⊢ (𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) → ∃𝑥 ∈ 𝐴 𝑤 = 𝐵)
22	21	ad2antlr 763	. . . 4 ⊢ (((𝜑 ∧ 𝑤 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)) ∧ 𝑤 ≤ (inf(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ*, < ) +𝑒 𝐶)) → ∃𝑥 ∈ 𝐴 𝑤 = 𝐵)
23		nfcv 2764	. . . . . . . 8 ⊢ Ⅎ𝑥𝑤
24		nfcv 2764	. . . . . . . 8 ⊢ Ⅎ𝑥 ≤
25		nfmpt1 4747	. . . . . . . . . . 11 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐵)
26	25	nfrn 5368	. . . . . . . . . 10 ⊢ Ⅎ𝑥ran (𝑥 ∈ 𝐴 ↦ 𝐵)
27		nfcv 2764	. . . . . . . . . 10 ⊢ Ⅎ𝑥ℝ*
28		nfcv 2764	. . . . . . . . . 10 ⊢ Ⅎ𝑥 <
29	26, 27, 28	nfinf 8388	. . . . . . . . 9 ⊢ Ⅎ𝑥inf(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ*, < )
30		nfcv 2764	. . . . . . . . 9 ⊢ Ⅎ𝑥 +𝑒
31		nfcv 2764	. . . . . . . . 9 ⊢ Ⅎ𝑥𝐶
32	29, 30, 31	nfov 6676	. . . . . . . 8 ⊢ Ⅎ𝑥(inf(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ*, < ) +𝑒 𝐶)
33	23, 24, 32	nfbr 4699	. . . . . . 7 ⊢ Ⅎ𝑥 𝑤 ≤ (inf(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ*, < ) +𝑒 𝐶)
34	2, 33	nfan 1828	. . . . . 6 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑤 ≤ (inf(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ*, < ) +𝑒 𝐶))
35		id 22	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝐵 → 𝑤 = 𝐵)
36	35	eqcomd 2628	. . . . . . . . . . 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 473	. . . . . . . . . 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 4675	. . . . . . . . 9 ⊢ ((𝑤 ≤ (inf(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ*, < ) +𝑒 𝐶) ∧ 𝑤 = 𝐵) → 𝐵 ≤ (inf(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ*, < ) +𝑒 𝐶))
40	39	ex 450	. . . . . . . 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 3012	. . . . 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 445	. . . 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 1325	. . 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 39339	. 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 196   ∧ wa 384   = wceq 1483  Ⅎwnf 1708   ∈ wcel 1990   ≠ wne 2794  ∀wral 2912  ∃wrex 2913  Vcvv 3200  ∅c0 3915   class class class wbr 4653   ↦ cmpt 4729  ran crn 5115  (class class class)co 6650  infcinf 8347  ℝcr 9935  ℝ^*cxr 10073   < clt 10074   ≤ cle 10075  ℝ⁺crp 11832   +_𝑒 cxad 11944

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-inf 8349  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-rp 11833  df-xadd 11947

This theorem is referenced by:  limsupgtlem  40009