Theorem infregelbex 9532

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 480	. . . . 5 ⊢ (((𝜑 ∧ 𝐵 ≤ inf(𝐴, ℝ, < )) ∧ 𝑎 ∈ 𝐴) → 𝐵 ∈ ℝ)
3		lttri3 7974	. . . . . . . 8 ⊢ ((𝑏 ∈ ℝ ∧ 𝑐 ∈ ℝ) → (𝑏 = 𝑐 ↔ (¬ 𝑏 < 𝑐 ∧ ¬ 𝑐 < 𝑏)))
4	3	adantl 275	. . . . . . 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 6984	. . . . . 6 ⊢ (𝜑 → inf(𝐴, ℝ, < ) ∈ ℝ)
7	6	ad2antrr 480	. . . . 5 ⊢ (((𝜑 ∧ 𝐵 ≤ inf(𝐴, ℝ, < )) ∧ 𝑎 ∈ 𝐴) → inf(𝐴, ℝ, < ) ∈ ℝ)
8		infregelbex.ss	. . . . . . 7 ⊢ (𝜑 → 𝐴 ⊆ ℝ)
9	8	sselda 3141	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → 𝑎 ∈ ℝ)
10	9	adantlr 469	. . . . 5 ⊢ (((𝜑 ∧ 𝐵 ≤ inf(𝐴, ℝ, < )) ∧ 𝑎 ∈ 𝐴) → 𝑎 ∈ ℝ)
11		simplr 520	. . . . 5 ⊢ (((𝜑 ∧ 𝐵 ≤ inf(𝐴, ℝ, < )) ∧ 𝑎 ∈ 𝐴) → 𝐵 ≤ inf(𝐴, ℝ, < ))
12	6	adantr 274	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → inf(𝐴, ℝ, < ) ∈ ℝ)
13	4, 5	inflbti 6985	. . . . . . . 8 ⊢ (𝜑 → (𝑎 ∈ 𝐴 → ¬ 𝑎 < inf(𝐴, ℝ, < )))
14	13	imp 123	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → ¬ 𝑎 < inf(𝐴, ℝ, < ))
15	12, 9, 14	nltled 8015	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → inf(𝐴, ℝ, < ) ≤ 𝑎)
16	15	adantlr 469	. . . . 5 ⊢ (((𝜑 ∧ 𝐵 ≤ inf(𝐴, ℝ, < )) ∧ 𝑎 ∈ 𝐴) → inf(𝐴, ℝ, < ) ≤ 𝑎)
17	2, 7, 10, 11, 16	letrd 8018	. . . 4 ⊢ (((𝜑 ∧ 𝐵 ≤ inf(𝐴, ℝ, < )) ∧ 𝑎 ∈ 𝐴) → 𝐵 ≤ 𝑎)
18	17	ralrimiva 2538	. . 3 ⊢ ((𝜑 ∧ 𝐵 ≤ inf(𝐴, ℝ, < )) → ∀𝑎 ∈ 𝐴 𝐵 ≤ 𝑎)
19		breq2 3985	. . . 4 ⊢ (𝑎 = 𝑧 → (𝐵 ≤ 𝑎 ↔ 𝐵 ≤ 𝑧))
20	19	cbvralv 2691	. . 3 ⊢ (∀𝑎 ∈ 𝐴 𝐵 ≤ 𝑎 ↔ ∀𝑧 ∈ 𝐴 𝐵 ≤ 𝑧)
21	18, 20	sylib 121	. 2 ⊢ ((𝜑 ∧ 𝐵 ≤ inf(𝐴, ℝ, < )) → ∀𝑧 ∈ 𝐴 𝐵 ≤ 𝑧)
22	1	adantr 274	. . 3 ⊢ ((𝜑 ∧ ∀𝑧 ∈ 𝐴 𝐵 ≤ 𝑧) → 𝐵 ∈ ℝ)
23	6	adantr 274	. . 3 ⊢ ((𝜑 ∧ ∀𝑧 ∈ 𝐴 𝐵 ≤ 𝑧) → inf(𝐴, ℝ, < ) ∈ ℝ)
24		simpl 108	. . . 4 ⊢ ((𝜑 ∧ ∀𝑧 ∈ 𝐴 𝐵 ≤ 𝑧) → 𝜑)
25		simpr 109	. . . . . . . 8 ⊢ ((𝜑 ∧ ∀𝑧 ∈ 𝐴 𝐵 ≤ 𝑧) → ∀𝑧 ∈ 𝐴 𝐵 ≤ 𝑧)
26		breq2 3985	. . . . . . . . 9 ⊢ (𝑧 = 𝑑 → (𝐵 ≤ 𝑧 ↔ 𝐵 ≤ 𝑑))
27	26	cbvralv 2691	. . . . . . . 8 ⊢ (∀𝑧 ∈ 𝐴 𝐵 ≤ 𝑧 ↔ ∀𝑑 ∈ 𝐴 𝐵 ≤ 𝑑)
28	25, 27	sylib 121	. . . . . . 7 ⊢ ((𝜑 ∧ ∀𝑧 ∈ 𝐴 𝐵 ≤ 𝑧) → ∀𝑑 ∈ 𝐴 𝐵 ≤ 𝑑)
29	1	ad2antrr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ ∀𝑧 ∈ 𝐴 𝐵 ≤ 𝑧) ∧ 𝑑 ∈ 𝐴) → 𝐵 ∈ ℝ)
30	8	ad2antrr 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ∀𝑧 ∈ 𝐴 𝐵 ≤ 𝑧) ∧ 𝑑 ∈ 𝐴) → 𝐴 ⊆ ℝ)
31		simpr 109	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ∀𝑧 ∈ 𝐴 𝐵 ≤ 𝑧) ∧ 𝑑 ∈ 𝐴) → 𝑑 ∈ 𝐴)
32	30, 31	sseldd 3142	. . . . . . . . 9 ⊢ (((𝜑 ∧ ∀𝑧 ∈ 𝐴 𝐵 ≤ 𝑧) ∧ 𝑑 ∈ 𝐴) → 𝑑 ∈ ℝ)
33	29, 32	lenltd 8012	. . . . . . . 8 ⊢ (((𝜑 ∧ ∀𝑧 ∈ 𝐴 𝐵 ≤ 𝑧) ∧ 𝑑 ∈ 𝐴) → (𝐵 ≤ 𝑑 ↔ ¬ 𝑑 < 𝐵))
34	33	ralbidva 2461	. . . . . . 7 ⊢ ((𝜑 ∧ ∀𝑧 ∈ 𝐴 𝐵 ≤ 𝑧) → (∀𝑑 ∈ 𝐴 𝐵 ≤ 𝑑 ↔ ∀𝑑 ∈ 𝐴 ¬ 𝑑 < 𝐵))
35	28, 34	mpbid 146	. . . . . 6 ⊢ ((𝜑 ∧ ∀𝑧 ∈ 𝐴 𝐵 ≤ 𝑧) → ∀𝑑 ∈ 𝐴 ¬ 𝑑 < 𝐵)
36		breq1 3984	. . . . . . . 8 ⊢ (𝑑 = 𝑧 → (𝑑 < 𝐵 ↔ 𝑧 < 𝐵))
37	36	notbid 657	. . . . . . 7 ⊢ (𝑑 = 𝑧 → (¬ 𝑑 < 𝐵 ↔ ¬ 𝑧 < 𝐵))
38	37	cbvralv 2691	. . . . . 6 ⊢ (∀𝑑 ∈ 𝐴 ¬ 𝑑 < 𝐵 ↔ ∀𝑧 ∈ 𝐴 ¬ 𝑧 < 𝐵)
39	35, 38	sylib 121	. . . . 5 ⊢ ((𝜑 ∧ ∀𝑧 ∈ 𝐴 𝐵 ≤ 𝑧) → ∀𝑧 ∈ 𝐴 ¬ 𝑧 < 𝐵)
40	22, 39	jca 304	. . . 4 ⊢ ((𝜑 ∧ ∀𝑧 ∈ 𝐴 𝐵 ≤ 𝑧) → (𝐵 ∈ ℝ ∧ ∀𝑧 ∈ 𝐴 ¬ 𝑧 < 𝐵))
41	4, 5	infnlbti 6987	. . . 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 8015	. 2 ⊢ ((𝜑 ∧ ∀𝑧 ∈ 𝐴 𝐵 ≤ 𝑧) → 𝐵 ≤ inf(𝐴, ℝ, < ))
44	21, 43	impbida 586	1 ⊢ (𝜑 → (𝐵 ≤ inf(𝐴, ℝ, < ) ↔ ∀𝑧 ∈ 𝐴 𝐵 ≤ 𝑧))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104   ∈ wcel 2136  ∀wral 2443  ∃wrex 2444   ⊆ wss 3115   class class class wbr 3981  infcinf 6944  ℝcr 7748   < clt 7929   ≤ cle 7930

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4099  ax-pow 4152  ax-pr 4186  ax-un 4410  ax-setind 4513  ax-cnex 7840  ax-resscn 7841  ax-pre-ltirr 7861  ax-pre-ltwlin 7862  ax-pre-apti 7864

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2296  df-ne 2336  df-nel 2431  df-ral 2448  df-rex 2449  df-reu 2450  df-rmo 2451  df-rab 2452  df-v 2727  df-sbc 2951  df-dif 3117  df-un 3119  df-in 3121  df-ss 3128  df-pw 3560  df-sn 3581  df-pr 3582  df-op 3584  df-uni 3789  df-br 3982  df-opab 4043  df-xp 4609  df-cnv 4611  df-iota 5152  df-riota 5797  df-sup 6945  df-inf 6946  df-pnf 7931  df-mnf 7932  df-xr 7933  df-ltxr 7934  df-le 7935

This theorem is referenced by:  nninfdclemp1  12379