Theorem infregelbex 9832

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 8259	. . . . . . . 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 7222	. . . . . 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 3227	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → 𝑎 ∈ ℝ)
10	9	adantlr 477	. . . . 5 ⊢ (((𝜑 ∧ 𝐵 ≤ inf(𝐴, ℝ, < )) ∧ 𝑎 ∈ 𝐴) → 𝑎 ∈ ℝ)
11		simplr 529	. . . . 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 7223	. . . . . . . 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 8300	. . . . . 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 8303	. . . 4 ⊢ (((𝜑 ∧ 𝐵 ≤ inf(𝐴, ℝ, < )) ∧ 𝑎 ∈ 𝐴) → 𝐵 ≤ 𝑎)
18	17	ralrimiva 2605	. . 3 ⊢ ((𝜑 ∧ 𝐵 ≤ inf(𝐴, ℝ, < )) → ∀𝑎 ∈ 𝐴 𝐵 ≤ 𝑎)
19		breq2 4092	. . . 4 ⊢ (𝑎 = 𝑧 → (𝐵 ≤ 𝑎 ↔ 𝐵 ≤ 𝑧))
20	19	cbvralv 2767	. . 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 4092	. . . . . . . . 9 ⊢ (𝑧 = 𝑑 → (𝐵 ≤ 𝑧 ↔ 𝐵 ≤ 𝑑))
27	26	cbvralv 2767	. . . . . . . 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 3228	. . . . . . . . 9 ⊢ (((𝜑 ∧ ∀𝑧 ∈ 𝐴 𝐵 ≤ 𝑧) ∧ 𝑑 ∈ 𝐴) → 𝑑 ∈ ℝ)
33	29, 32	lenltd 8297	. . . . . . . 8 ⊢ (((𝜑 ∧ ∀𝑧 ∈ 𝐴 𝐵 ≤ 𝑧) ∧ 𝑑 ∈ 𝐴) → (𝐵 ≤ 𝑑 ↔ ¬ 𝑑 < 𝐵))
34	33	ralbidva 2528	. . . . . . 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 4091	. . . . . . . 8 ⊢ (𝑑 = 𝑧 → (𝑑 < 𝐵 ↔ 𝑧 < 𝐵))
37	36	notbid 673	. . . . . . 7 ⊢ (𝑑 = 𝑧 → (¬ 𝑑 < 𝐵 ↔ ¬ 𝑧 < 𝐵))
38	37	cbvralv 2767	. . . . . 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 7225	. . . 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 8300	. 2 ⊢ ((𝜑 ∧ ∀𝑧 ∈ 𝐴 𝐵 ≤ 𝑧) → 𝐵 ≤ inf(𝐴, ℝ, < ))
44	21, 43	impbida 600	1 ⊢ (𝜑 → (𝐵 ≤ inf(𝐴, ℝ, < ) ↔ ∀𝑧 ∈ 𝐴 𝐵 ≤ 𝑧))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∈ wcel 2202  ∀wral 2510  ∃wrex 2511   ⊆ wss 3200   class class class wbr 4088  infcinf 7182  ℝcr 8031   < clt 8214   ≤ cle 8215

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8123  ax-resscn 8124  ax-pre-ltirr 8144  ax-pre-ltwlin 8145  ax-pre-apti 8147

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-xp 4731  df-cnv 4733  df-iota 5286  df-riota 5971  df-sup 7183  df-inf 7184  df-pnf 8216  df-mnf 8217  df-xr 8218  df-ltxr 8219  df-le 8220

This theorem is referenced by:  nninfdclemp1  13073