Theorem infxrgelbrnmpt 46285

Description: The infimum of an indexed set of extended reals is greater than or equal to a lower bound. (Contributed by Glauco Siliprandi, 2-Jan-2022.)

Hypotheses

Ref	Expression
infxrgelbrnmpt.x	⊢ Ⅎ𝑥𝜑
infxrgelbrnmpt.b	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ*)
infxrgelbrnmpt.c	⊢ (𝜑 → 𝐶 ∈ ℝ*)

Assertion

Ref	Expression
infxrgelbrnmpt	⊢ (𝜑 → (𝐶 ≤ inf(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ*, < ) ↔ ∀𝑥 ∈ 𝐴 𝐶 ≤ 𝐵))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐶

Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)

Proof of Theorem infxrgelbrnmpt

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		infxrgelbrnmpt.x	. . . 4 ⊢ Ⅎ𝑥𝜑
2		eqid 2760	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
3		infxrgelbrnmpt.b	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ*)
4	1, 2, 3	rnmptssd 7118	. . 3 ⊢ (𝜑 → ran (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ ℝ*)
5		infxrgelbrnmpt.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ ℝ*)
6		infxrgelb 13391	. . 3 ⊢ ((ran (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ ℝ* ∧ 𝐶 ∈ ℝ*) → (𝐶 ≤ inf(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ*, < ) ↔ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝐶 ≤ 𝑧))
7	4, 5, 6	syl2anc 596	. 2 ⊢ (𝜑 → (𝐶 ≤ inf(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ*, < ) ↔ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝐶 ≤ 𝑧))
8		nfmpt1 5204	. . . . . . 7 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐵)
9	8	nfrn 5936	. . . . . 6 ⊢ Ⅎ𝑥ran (𝑥 ∈ 𝐴 ↦ 𝐵)
10		nfv 1947	. . . . . 6 ⊢ Ⅎ𝑥 𝐶 ≤ 𝑧
11	9, 10	nfralw 3309	. . . . 5 ⊢ Ⅎ𝑥∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝐶 ≤ 𝑧
12	1, 11	nfan 1932	. . . 4 ⊢ Ⅎ𝑥(𝜑 ∧ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝐶 ≤ 𝑧)
13		simpr 490	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
14	2	elrnmpt1 5944	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ ℝ*) → 𝐵 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵))
15	13, 3, 14	syl2anc 596	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵))
16	15	adantlr 728	. . . . . 6 ⊢ (((𝜑 ∧ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝐶 ≤ 𝑧) ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵))
17		simplr 781	. . . . . 6 ⊢ (((𝜑 ∧ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝐶 ≤ 𝑧) ∧ 𝑥 ∈ 𝐴) → ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝐶 ≤ 𝑧)
18		breq2 5107	. . . . . . 7 ⊢ (𝑧 = 𝐵 → (𝐶 ≤ 𝑧 ↔ 𝐶 ≤ 𝐵))
19	18	rspcva 3574	. . . . . 6 ⊢ ((𝐵 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ∧ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝐶 ≤ 𝑧) → 𝐶 ≤ 𝐵)
20	16, 17, 19	syl2anc 596	. . . . 5 ⊢ (((𝜑 ∧ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝐶 ≤ 𝑧) ∧ 𝑥 ∈ 𝐴) → 𝐶 ≤ 𝐵)
21	20	ex 418	. . . 4 ⊢ ((𝜑 ∧ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝐶 ≤ 𝑧) → (𝑥 ∈ 𝐴 → 𝐶 ≤ 𝐵))
22	12, 21	ralrimi 3260	. . 3 ⊢ ((𝜑 ∧ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝐶 ≤ 𝑧) → ∀𝑥 ∈ 𝐴 𝐶 ≤ 𝐵)
23		vex 3454	. . . . . . . 8 ⊢ 𝑧 ∈ V
24	2	elrnmpt 5942	. . . . . . . 8 ⊢ (𝑧 ∈ V → (𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ↔ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵))
25	23, 24	ax-mp 5	. . . . . . 7 ⊢ (𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ↔ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵)
26	25	bilani 510	. . . . . 6 ⊢ ((∀𝑥 ∈ 𝐴 𝐶 ≤ 𝐵 ∧ 𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)) → ∃𝑥 ∈ 𝐴 𝑧 = 𝐵)
27		nfra1 3286	. . . . . . . 8 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 𝐶 ≤ 𝐵
28		rspa 3251	. . . . . . . . . 10 ⊢ ((∀𝑥 ∈ 𝐴 𝐶 ≤ 𝐵 ∧ 𝑥 ∈ 𝐴) → 𝐶 ≤ 𝐵)
29	18	biimprcd 253	. . . . . . . . . 10 ⊢ (𝐶 ≤ 𝐵 → (𝑧 = 𝐵 → 𝐶 ≤ 𝑧))
30	28, 29	syl 18	. . . . . . . . 9 ⊢ ((∀𝑥 ∈ 𝐴 𝐶 ≤ 𝐵 ∧ 𝑥 ∈ 𝐴) → (𝑧 = 𝐵 → 𝐶 ≤ 𝑧))
31	30	ex 418	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝐴 𝐶 ≤ 𝐵 → (𝑥 ∈ 𝐴 → (𝑧 = 𝐵 → 𝐶 ≤ 𝑧)))
32	27, 10, 31	rexlimd 3269	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 𝐶 ≤ 𝐵 → (∃𝑥 ∈ 𝐴 𝑧 = 𝐵 → 𝐶 ≤ 𝑧))
33	32	adantr 486	. . . . . 6 ⊢ ((∀𝑥 ∈ 𝐴 𝐶 ≤ 𝐵 ∧ 𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)) → (∃𝑥 ∈ 𝐴 𝑧 = 𝐵 → 𝐶 ≤ 𝑧))
34	26, 33	mpd 16	. . . . 5 ⊢ ((∀𝑥 ∈ 𝐴 𝐶 ≤ 𝐵 ∧ 𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)) → 𝐶 ≤ 𝑧)
35	34	ralrimiva 3154	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝐶 ≤ 𝐵 → ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝐶 ≤ 𝑧)
36	35	adantl 487	. . 3 ⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝐴 𝐶 ≤ 𝐵) → ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝐶 ≤ 𝑧)
37	22, 36	impbida 813	. 2 ⊢ (𝜑 → (∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝐶 ≤ 𝑧 ↔ ∀𝑥 ∈ 𝐴 𝐶 ≤ 𝐵))
38	7, 37	bitrd 282	1 ⊢ (𝜑 → (𝐶 ≤ inf(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ*, < ) ↔ ∀𝑥 ∈ 𝐴 𝐶 ≤ 𝐵))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570  Ⅎwnf 1816   ∈ wcel 2145  ∀wral 3076  ∃wrex 3086  Vcvv 3450   ⊆ wss 3899   class class class wbr 5103   ↦ cmpt 5186  ran crn 5656  infcinf 9414  ℝ^*cxr 11269   < clt 11270   ≤ cle 11271

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7737  ax-cnex 11183  ax-resscn 11184  ax-1cn 11185  ax-icn 11186  ax-addcl 11187  ax-addrcl 11188  ax-mulcl 11189  ax-mulrcl 11190  ax-mulcom 11191  ax-addass 11192  ax-mulass 11193  ax-distr 11194  ax-i2m1 11195  ax-1ne0 11196  ax-1rid 11197  ax-rnegex 11198  ax-rrecex 11199  ax-cnre 11200  ax-pre-lttri 11201  ax-pre-lttrn 11202  ax-pre-ltadd 11203  ax-pre-mulgt0 11204  ax-pre-sup 11205

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-po 5563  df-so 5564  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7371  df-ov 7417  df-oprab 7418  df-mpo 7419  df-er 8699  df-en 8956  df-dom 8957  df-sdom 8958  df-sup 9415  df-inf 9416  df-pnf 11272  df-mnf 11273  df-xr 11274  df-ltxr 11275  df-le 11276  df-sub 11470  df-neg 11471

This theorem is used by: (None)