Theorem infregelbex 9557

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 485	. . . . 5 ⊢ (((𝜑 ∧ 𝐵 ≤ inf(𝐴, ℝ, < )) ∧ 𝑎 ∈ 𝐴) → 𝐵 ∈ ℝ)
3		lttri3 7999	. . . . . . . 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 7000	. . . . . 6 ⊢ (𝜑 → inf(𝐴, ℝ, < ) ∈ ℝ)
7	6	ad2antrr 485	. . . . 5 ⊢ (((𝜑 ∧ 𝐵 ≤ inf(𝐴, ℝ, < )) ∧ 𝑎 ∈ 𝐴) → inf(𝐴, ℝ, < ) ∈ ℝ)
8		infregelbex.ss	. . . . . . 7 ⊢ (𝜑 → 𝐴 ⊆ ℝ)
9	8	sselda 3147	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → 𝑎 ∈ ℝ)
10	9	adantlr 474	. . . . 5 ⊢ (((𝜑 ∧ 𝐵 ≤ inf(𝐴, ℝ, < )) ∧ 𝑎 ∈ 𝐴) → 𝑎 ∈ ℝ)
11		simplr 525	. . . . 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 7001	. . . . . . . 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 8040	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → inf(𝐴, ℝ, < ) ≤ 𝑎)
16	15	adantlr 474	. . . . 5 ⊢ (((𝜑 ∧ 𝐵 ≤ inf(𝐴, ℝ, < )) ∧ 𝑎 ∈ 𝐴) → inf(𝐴, ℝ, < ) ≤ 𝑎)
17	2, 7, 10, 11, 16	letrd 8043	. . . 4 ⊢ (((𝜑 ∧ 𝐵 ≤ inf(𝐴, ℝ, < )) ∧ 𝑎 ∈ 𝐴) → 𝐵 ≤ 𝑎)
18	17	ralrimiva 2543	. . 3 ⊢ ((𝜑 ∧ 𝐵 ≤ inf(𝐴, ℝ, < )) → ∀𝑎 ∈ 𝐴 𝐵 ≤ 𝑎)
19		breq2 3993	. . . 4 ⊢ (𝑎 = 𝑧 → (𝐵 ≤ 𝑎 ↔ 𝐵 ≤ 𝑧))
20	19	cbvralv 2696	. . 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 3993	. . . . . . . . 9 ⊢ (𝑧 = 𝑑 → (𝐵 ≤ 𝑧 ↔ 𝐵 ≤ 𝑑))
27	26	cbvralv 2696	. . . . . . . 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 485	. . . . . . . . 9 ⊢ (((𝜑 ∧ ∀𝑧 ∈ 𝐴 𝐵 ≤ 𝑧) ∧ 𝑑 ∈ 𝐴) → 𝐵 ∈ ℝ)
30	8	ad2antrr 485	. . . . . . . . . 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 3148	. . . . . . . . 9 ⊢ (((𝜑 ∧ ∀𝑧 ∈ 𝐴 𝐵 ≤ 𝑧) ∧ 𝑑 ∈ 𝐴) → 𝑑 ∈ ℝ)
33	29, 32	lenltd 8037	. . . . . . . 8 ⊢ (((𝜑 ∧ ∀𝑧 ∈ 𝐴 𝐵 ≤ 𝑧) ∧ 𝑑 ∈ 𝐴) → (𝐵 ≤ 𝑑 ↔ ¬ 𝑑 < 𝐵))
34	33	ralbidva 2466	. . . . . . 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 3992	. . . . . . . 8 ⊢ (𝑑 = 𝑧 → (𝑑 < 𝐵 ↔ 𝑧 < 𝐵))
37	36	notbid 662	. . . . . . 7 ⊢ (𝑑 = 𝑧 → (¬ 𝑑 < 𝐵 ↔ ¬ 𝑧 < 𝐵))
38	37	cbvralv 2696	. . . . . 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 7003	. . . 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 8040	. 2 ⊢ ((𝜑 ∧ ∀𝑧 ∈ 𝐴 𝐵 ≤ 𝑧) → 𝐵 ≤ inf(𝐴, ℝ, < ))
44	21, 43	impbida 591	1 ⊢ (𝜑 → (𝐵 ≤ inf(𝐴, ℝ, < ) ↔ ∀𝑧 ∈ 𝐴 𝐵 ≤ 𝑧))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104   ∈ wcel 2141  ∀wral 2448  ∃wrex 2449   ⊆ wss 3121   class class class wbr 3989  infcinf 6960  ℝcr 7773   < clt 7954   ≤ cle 7955

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-cnex 7865  ax-resscn 7866  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-apti 7889

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-xp 4617  df-cnv 4619  df-iota 5160  df-riota 5809  df-sup 6961  df-inf 6962  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960

This theorem is referenced by:  nninfdclemp1  12405