Theorem infxrgelbrnmpt 45443

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 2729	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
3		infxrgelbrnmpt.b	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ*)
4	1, 2, 3	rnmptssd 45184	. . 3 ⊢ (𝜑 → ran (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ ℝ*)
5		infxrgelbrnmpt.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ ℝ*)
6		infxrgelb 13238	. . 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 584	. 2 ⊢ (𝜑 → (𝐶 ≤ inf(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ*, < ) ↔ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝐶 ≤ 𝑧))
8		nfmpt1 5191	. . . . . . 7 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐵)
9	8	nfrn 5894	. . . . . 6 ⊢ Ⅎ𝑥ran (𝑥 ∈ 𝐴 ↦ 𝐵)
10		nfv 1914	. . . . . 6 ⊢ Ⅎ𝑥 𝐶 ≤ 𝑧
11	9, 10	nfralw 3276	. . . . 5 ⊢ Ⅎ𝑥∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝐶 ≤ 𝑧
12	1, 11	nfan 1899	. . . 4 ⊢ Ⅎ𝑥(𝜑 ∧ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝐶 ≤ 𝑧)
13		simpr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
14	2	elrnmpt1 5902	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ ℝ*) → 𝐵 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵))
15	13, 3, 14	syl2anc 584	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵))
16	15	adantlr 715	. . . . . 6 ⊢ (((𝜑 ∧ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝐶 ≤ 𝑧) ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵))
17		simplr 768	. . . . . 6 ⊢ (((𝜑 ∧ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝐶 ≤ 𝑧) ∧ 𝑥 ∈ 𝐴) → ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝐶 ≤ 𝑧)
18		breq2 5096	. . . . . . 7 ⊢ (𝑧 = 𝐵 → (𝐶 ≤ 𝑧 ↔ 𝐶 ≤ 𝐵))
19	18	rspcva 3575	. . . . . 6 ⊢ ((𝐵 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ∧ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝐶 ≤ 𝑧) → 𝐶 ≤ 𝐵)
20	16, 17, 19	syl2anc 584	. . . . 5 ⊢ (((𝜑 ∧ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝐶 ≤ 𝑧) ∧ 𝑥 ∈ 𝐴) → 𝐶 ≤ 𝐵)
21	20	ex 412	. . . 4 ⊢ ((𝜑 ∧ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝐶 ≤ 𝑧) → (𝑥 ∈ 𝐴 → 𝐶 ≤ 𝐵))
22	12, 21	ralrimi 3227	. . 3 ⊢ ((𝜑 ∧ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝐶 ≤ 𝑧) → ∀𝑥 ∈ 𝐴 𝐶 ≤ 𝐵)
23		vex 3440	. . . . . . . . 9 ⊢ 𝑧 ∈ V
24	2	elrnmpt 5900	. . . . . . . . 9 ⊢ (𝑧 ∈ V → (𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ↔ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵))
25	23, 24	ax-mp 5	. . . . . . . 8 ⊢ (𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ↔ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵)
26	25	biimpi 216	. . . . . . 7 ⊢ (𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) → ∃𝑥 ∈ 𝐴 𝑧 = 𝐵)
27	26	adantl 481	. . . . . 6 ⊢ ((∀𝑥 ∈ 𝐴 𝐶 ≤ 𝐵 ∧ 𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)) → ∃𝑥 ∈ 𝐴 𝑧 = 𝐵)
28		nfra1 3253	. . . . . . . 8 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 𝐶 ≤ 𝐵
29		rspa 3218	. . . . . . . . . 10 ⊢ ((∀𝑥 ∈ 𝐴 𝐶 ≤ 𝐵 ∧ 𝑥 ∈ 𝐴) → 𝐶 ≤ 𝐵)
30	18	biimprcd 250	. . . . . . . . . 10 ⊢ (𝐶 ≤ 𝐵 → (𝑧 = 𝐵 → 𝐶 ≤ 𝑧))
31	29, 30	syl 17	. . . . . . . . 9 ⊢ ((∀𝑥 ∈ 𝐴 𝐶 ≤ 𝐵 ∧ 𝑥 ∈ 𝐴) → (𝑧 = 𝐵 → 𝐶 ≤ 𝑧))
32	31	ex 412	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝐴 𝐶 ≤ 𝐵 → (𝑥 ∈ 𝐴 → (𝑧 = 𝐵 → 𝐶 ≤ 𝑧)))
33	28, 10, 32	rexlimd 3236	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 𝐶 ≤ 𝐵 → (∃𝑥 ∈ 𝐴 𝑧 = 𝐵 → 𝐶 ≤ 𝑧))
34	33	adantr 480	. . . . . 6 ⊢ ((∀𝑥 ∈ 𝐴 𝐶 ≤ 𝐵 ∧ 𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)) → (∃𝑥 ∈ 𝐴 𝑧 = 𝐵 → 𝐶 ≤ 𝑧))
35	27, 34	mpd 15	. . . . 5 ⊢ ((∀𝑥 ∈ 𝐴 𝐶 ≤ 𝐵 ∧ 𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)) → 𝐶 ≤ 𝑧)
36	35	ralrimiva 3121	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝐶 ≤ 𝐵 → ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝐶 ≤ 𝑧)
37	36	adantl 481	. . 3 ⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝐴 𝐶 ≤ 𝐵) → ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝐶 ≤ 𝑧)
38	22, 37	impbida 800	. 2 ⊢ (𝜑 → (∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝐶 ≤ 𝑧 ↔ ∀𝑥 ∈ 𝐴 𝐶 ≤ 𝐵))
39	7, 38	bitrd 279	1 ⊢ (𝜑 → (𝐶 ≤ inf(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ*, < ) ↔ ∀𝑥 ∈ 𝐴 𝐶 ≤ 𝐵))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540  Ⅎwnf 1783   ∈ wcel 2109  ∀wral 3044  ∃wrex 3053  Vcvv 3436   ⊆ wss 3903   class class class wbr 5092   ↦ cmpt 5173  ran crn 5620  infcinf 9331  ℝ^*cxr 11148   < clt 11149   ≤ cle 11150

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671  ax-cnex 11065  ax-resscn 11066  ax-1cn 11067  ax-icn 11068  ax-addcl 11069  ax-addrcl 11070  ax-mulcl 11071  ax-mulrcl 11072  ax-mulcom 11073  ax-addass 11074  ax-mulass 11075  ax-distr 11076  ax-i2m1 11077  ax-1ne0 11078  ax-1rid 11079  ax-rnegex 11080  ax-rrecex 11081  ax-cnre 11082  ax-pre-lttri 11083  ax-pre-lttrn 11084  ax-pre-ltadd 11085  ax-pre-mulgt0 11086  ax-pre-sup 11087

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3343  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-br 5093  df-opab 5155  df-mpt 5174  df-id 5514  df-po 5527  df-so 5528  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-riota 7306  df-ov 7352  df-oprab 7353  df-mpo 7354  df-er 8625  df-en 8873  df-dom 8874  df-sdom 8875  df-sup 9332  df-inf 9333  df-pnf 11151  df-mnf 11152  df-xr 11153  df-ltxr 11154  df-le 11155  df-sub 11349  df-neg 11350

This theorem is referenced by: (None)