Theorem infregelbex 9672

Description: Any lower bound of a set of real numbers with an infimum is less than or equal to the infimum. (Contributed by Jim Kingdon, 27-Sep-2024.)

Hypotheses

Ref	Expression
infregelbex.ex	⊢ (𝜑 → ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)))
infregelbex.ss	⊢ (𝜑 → 𝐴 ⊆ ℝ)
infregelbex.b	⊢ (𝜑 → 𝐵 ∈ ℝ)

Assertion

Ref	Expression
infregelbex	⊢ (𝜑 → (𝐵 ≤ inf(𝐴, ℝ, < ) ↔ ∀𝑧 ∈ 𝐴 𝐵 ≤ 𝑧))

Distinct variable groups:   𝑥,𝐴,𝑦,𝑧   𝑥,𝐵,𝑦,𝑧   𝜑,𝑥,𝑦,𝑧

Proof of Theorem infregelbex

Dummy variables 𝑎 𝑏 𝑐 𝑑 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		infregelbex.b	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ ℝ)
2	1	ad2antrr 488	. . . . 5 ⊢ (((𝜑 ∧ 𝐵 ≤ inf(𝐴, ℝ, < )) ∧ 𝑎 ∈ 𝐴) → 𝐵 ∈ ℝ)
3		lttri3 8106	. . . . . . . 8 ⊢ ((𝑏 ∈ ℝ ∧ 𝑐 ∈ ℝ) → (𝑏 = 𝑐 ↔ (¬ 𝑏 < 𝑐 ∧ ¬ 𝑐 < 𝑏)))
4	3	adantl 277	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑏 ∈ ℝ ∧ 𝑐 ∈ ℝ)) → (𝑏 = 𝑐 ↔ (¬ 𝑏 < 𝑐 ∧ ¬ 𝑐 < 𝑏)))
5		infregelbex.ex	. . . . . . 7 ⊢ (𝜑 → ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)))
6	4, 5	infclti 7089	. . . . . 6 ⊢ (𝜑 → inf(𝐴, ℝ, < ) ∈ ℝ)
7	6	ad2antrr 488	. . . . 5 ⊢ (((𝜑 ∧ 𝐵 ≤ inf(𝐴, ℝ, < )) ∧ 𝑎 ∈ 𝐴) → inf(𝐴, ℝ, < ) ∈ ℝ)
8		infregelbex.ss	. . . . . . 7 ⊢ (𝜑 → 𝐴 ⊆ ℝ)
9	8	sselda 3183	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → 𝑎 ∈ ℝ)
10	9	adantlr 477	. . . . 5 ⊢ (((𝜑 ∧ 𝐵 ≤ inf(𝐴, ℝ, < )) ∧ 𝑎 ∈ 𝐴) → 𝑎 ∈ ℝ)
11		simplr 528	. . . . 5 ⊢ (((𝜑 ∧ 𝐵 ≤ inf(𝐴, ℝ, < )) ∧ 𝑎 ∈ 𝐴) → 𝐵 ≤ inf(𝐴, ℝ, < ))
12	6	adantr 276	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → inf(𝐴, ℝ, < ) ∈ ℝ)
13	4, 5	inflbti 7090	. . . . . . . 8 ⊢ (𝜑 → (𝑎 ∈ 𝐴 → ¬ 𝑎 < inf(𝐴, ℝ, < )))
14	13	imp 124	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → ¬ 𝑎 < inf(𝐴, ℝ, < ))
15	12, 9, 14	nltled 8147	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → inf(𝐴, ℝ, < ) ≤ 𝑎)
16	15	adantlr 477	. . . . 5 ⊢ (((𝜑 ∧ 𝐵 ≤ inf(𝐴, ℝ, < )) ∧ 𝑎 ∈ 𝐴) → inf(𝐴, ℝ, < ) ≤ 𝑎)
17	2, 7, 10, 11, 16	letrd 8150	. . . 4 ⊢ (((𝜑 ∧ 𝐵 ≤ inf(𝐴, ℝ, < )) ∧ 𝑎 ∈ 𝐴) → 𝐵 ≤ 𝑎)
18	17	ralrimiva 2570	. . 3 ⊢ ((𝜑 ∧ 𝐵 ≤ inf(𝐴, ℝ, < )) → ∀𝑎 ∈ 𝐴 𝐵 ≤ 𝑎)
19		breq2 4037	. . . 4 ⊢ (𝑎 = 𝑧 → (𝐵 ≤ 𝑎 ↔ 𝐵 ≤ 𝑧))
20	19	cbvralv 2729	. . 3 ⊢ (∀𝑎 ∈ 𝐴 𝐵 ≤ 𝑎 ↔ ∀𝑧 ∈ 𝐴 𝐵 ≤ 𝑧)
21	18, 20	sylib 122	. 2 ⊢ ((𝜑 ∧ 𝐵 ≤ inf(𝐴, ℝ, < )) → ∀𝑧 ∈ 𝐴 𝐵 ≤ 𝑧)
22	1	adantr 276	. . 3 ⊢ ((𝜑 ∧ ∀𝑧 ∈ 𝐴 𝐵 ≤ 𝑧) → 𝐵 ∈ ℝ)
23	6	adantr 276	. . 3 ⊢ ((𝜑 ∧ ∀𝑧 ∈ 𝐴 𝐵 ≤ 𝑧) → inf(𝐴, ℝ, < ) ∈ ℝ)
24		simpl 109	. . . 4 ⊢ ((𝜑 ∧ ∀𝑧 ∈ 𝐴 𝐵 ≤ 𝑧) → 𝜑)
25		simpr 110	. . . . . . . 8 ⊢ ((𝜑 ∧ ∀𝑧 ∈ 𝐴 𝐵 ≤ 𝑧) → ∀𝑧 ∈ 𝐴 𝐵 ≤ 𝑧)
26		breq2 4037	. . . . . . . . 9 ⊢ (𝑧 = 𝑑 → (𝐵 ≤ 𝑧 ↔ 𝐵 ≤ 𝑑))
27	26	cbvralv 2729	. . . . . . . 8 ⊢ (∀𝑧 ∈ 𝐴 𝐵 ≤ 𝑧 ↔ ∀𝑑 ∈ 𝐴 𝐵 ≤ 𝑑)
28	25, 27	sylib 122	. . . . . . 7 ⊢ ((𝜑 ∧ ∀𝑧 ∈ 𝐴 𝐵 ≤ 𝑧) → ∀𝑑 ∈ 𝐴 𝐵 ≤ 𝑑)
29	1	ad2antrr 488	. . . . . . . . 9 ⊢ (((𝜑 ∧ ∀𝑧 ∈ 𝐴 𝐵 ≤ 𝑧) ∧ 𝑑 ∈ 𝐴) → 𝐵 ∈ ℝ)
30	8	ad2antrr 488	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ∀𝑧 ∈ 𝐴 𝐵 ≤ 𝑧) ∧ 𝑑 ∈ 𝐴) → 𝐴 ⊆ ℝ)
31		simpr 110	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ∀𝑧 ∈ 𝐴 𝐵 ≤ 𝑧) ∧ 𝑑 ∈ 𝐴) → 𝑑 ∈ 𝐴)
32	30, 31	sseldd 3184	. . . . . . . . 9 ⊢ (((𝜑 ∧ ∀𝑧 ∈ 𝐴 𝐵 ≤ 𝑧) ∧ 𝑑 ∈ 𝐴) → 𝑑 ∈ ℝ)
33	29, 32	lenltd 8144	. . . . . . . 8 ⊢ (((𝜑 ∧ ∀𝑧 ∈ 𝐴 𝐵 ≤ 𝑧) ∧ 𝑑 ∈ 𝐴) → (𝐵 ≤ 𝑑 ↔ ¬ 𝑑 < 𝐵))
34	33	ralbidva 2493	. . . . . . 7 ⊢ ((𝜑 ∧ ∀𝑧 ∈ 𝐴 𝐵 ≤ 𝑧) → (∀𝑑 ∈ 𝐴 𝐵 ≤ 𝑑 ↔ ∀𝑑 ∈ 𝐴 ¬ 𝑑 < 𝐵))
35	28, 34	mpbid 147	. . . . . 6 ⊢ ((𝜑 ∧ ∀𝑧 ∈ 𝐴 𝐵 ≤ 𝑧) → ∀𝑑 ∈ 𝐴 ¬ 𝑑 < 𝐵)
36		breq1 4036	. . . . . . . 8 ⊢ (𝑑 = 𝑧 → (𝑑 < 𝐵 ↔ 𝑧 < 𝐵))
37	36	notbid 668	. . . . . . 7 ⊢ (𝑑 = 𝑧 → (¬ 𝑑 < 𝐵 ↔ ¬ 𝑧 < 𝐵))
38	37	cbvralv 2729	. . . . . 6 ⊢ (∀𝑑 ∈ 𝐴 ¬ 𝑑 < 𝐵 ↔ ∀𝑧 ∈ 𝐴 ¬ 𝑧 < 𝐵)
39	35, 38	sylib 122	. . . . 5 ⊢ ((𝜑 ∧ ∀𝑧 ∈ 𝐴 𝐵 ≤ 𝑧) → ∀𝑧 ∈ 𝐴 ¬ 𝑧 < 𝐵)
40	22, 39	jca 306	. . . 4 ⊢ ((𝜑 ∧ ∀𝑧 ∈ 𝐴 𝐵 ≤ 𝑧) → (𝐵 ∈ ℝ ∧ ∀𝑧 ∈ 𝐴 ¬ 𝑧 < 𝐵))
41	4, 5	infnlbti 7092	. . . 4 ⊢ (𝜑 → ((𝐵 ∈ ℝ ∧ ∀𝑧 ∈ 𝐴 ¬ 𝑧 < 𝐵) → ¬ inf(𝐴, ℝ, < ) < 𝐵))
42	24, 40, 41	sylc 62	. . 3 ⊢ ((𝜑 ∧ ∀𝑧 ∈ 𝐴 𝐵 ≤ 𝑧) → ¬ inf(𝐴, ℝ, < ) < 𝐵)
43	22, 23, 42	nltled 8147	. 2 ⊢ ((𝜑 ∧ ∀𝑧 ∈ 𝐴 𝐵 ≤ 𝑧) → 𝐵 ≤ inf(𝐴, ℝ, < ))
44	21, 43	impbida 596	1 ⊢ (𝜑 → (𝐵 ≤ inf(𝐴, ℝ, < ) ↔ ∀𝑧 ∈ 𝐴 𝐵 ≤ 𝑧))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∈ wcel 2167  ∀wral 2475  ∃wrex 2476   ⊆ wss 3157   class class class wbr 4033  infcinf 7049  ℝcr 7878   < clt 8061   ≤ cle 8062

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-cnex 7970  ax-resscn 7971  ax-pre-ltirr 7991  ax-pre-ltwlin 7992  ax-pre-apti 7994

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-br 4034  df-opab 4095  df-xp 4669  df-cnv 4671  df-iota 5219  df-riota 5877  df-sup 7050  df-inf 7051  df-pnf 8063  df-mnf 8064  df-xr 8065  df-ltxr 8066  df-le 8067

This theorem is referenced by:  nninfdclemp1  12667