Theorem infxrunb3rnmpt 45885

Description: The infimum of an unbounded-below set of extended reals is minus infinity. (Contributed by Glauco Siliprandi, 23-Oct-2021.)

Hypotheses

Ref	Expression
infxrunb3rnmpt.1	⊢ Ⅎ𝑥𝜑
infxrunb3rnmpt.2	⊢ Ⅎ𝑦𝜑
infxrunb3rnmpt.3	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ*)

Assertion

Ref	Expression
infxrunb3rnmpt	⊢ (𝜑 → (∀𝑦 ∈ ℝ ∃𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 ↔ inf(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ*, < ) = -∞))

Distinct variable groups:   𝑥,𝐴,𝑦   𝑦,𝐵

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

Proof of Theorem infxrunb3rnmpt

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		infxrunb3rnmpt.2	. . 3 ⊢ Ⅎ𝑦𝜑
2		infxrunb3rnmpt.1	. . . . 5 ⊢ Ⅎ𝑥𝜑
3		nfmpt1 5174	. . . . . . 7 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐵)
4	3	nfrn 5901	. . . . . 6 ⊢ Ⅎ𝑥ran (𝑥 ∈ 𝐴 ↦ 𝐵)
5		nfv 1922	. . . . . 6 ⊢ Ⅎ𝑥 𝑧 ≤ 𝑦
6	4, 5	nfrexw 3289	. . . . 5 ⊢ Ⅎ𝑥∃𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝑦
7		simpr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
8		infxrunb3rnmpt.3	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ*)
9		eqid 2741	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
10	9	elrnmpt1 5909	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ ℝ*) → 𝐵 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵))
11	7, 8, 10	syl2anc 591	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵))
12	11	3adant3 1139	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝐵 ≤ 𝑦) → 𝐵 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵))
13		simp3 1145	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝐵 ≤ 𝑦) → 𝐵 ≤ 𝑦)
14		breq1 5078	. . . . . . . 8 ⊢ (𝑧 = 𝐵 → (𝑧 ≤ 𝑦 ↔ 𝐵 ≤ 𝑦))
15	14	rspcev 3562	. . . . . . 7 ⊢ ((𝐵 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ∧ 𝐵 ≤ 𝑦) → ∃𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝑦)
16	12, 13, 15	syl2anc 591	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝐵 ≤ 𝑦) → ∃𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝑦)
17	16	3exp 1126	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → (𝐵 ≤ 𝑦 → ∃𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝑦)))
18	2, 6, 17	rexlimd 3248	. . . 4 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 → ∃𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝑦))
19		nfv 1922	. . . . . 6 ⊢ Ⅎ𝑧∃𝑥 ∈ 𝐴 𝐵 ≤ 𝑦
20		vex 3437	. . . . . . . . 9 ⊢ 𝑧 ∈ V
21	9	elrnmpt 5907	. . . . . . . . 9 ⊢ (𝑧 ∈ V → (𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ↔ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵))
22	20, 21	ax-mp 5	. . . . . . . 8 ⊢ (𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ↔ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵)
23	22	biimpi 218	. . . . . . 7 ⊢ (𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) → ∃𝑥 ∈ 𝐴 𝑧 = 𝐵)
24	14	biimpcd 251	. . . . . . . . . 10 ⊢ (𝑧 ≤ 𝑦 → (𝑧 = 𝐵 → 𝐵 ≤ 𝑦))
25	24	a1d 25	. . . . . . . . 9 ⊢ (𝑧 ≤ 𝑦 → (𝑥 ∈ 𝐴 → (𝑧 = 𝐵 → 𝐵 ≤ 𝑦)))
26	5, 25	reximdai 3243	. . . . . . . 8 ⊢ (𝑧 ≤ 𝑦 → (∃𝑥 ∈ 𝐴 𝑧 = 𝐵 → ∃𝑥 ∈ 𝐴 𝐵 ≤ 𝑦))
27	26	com12 32	. . . . . . 7 ⊢ (∃𝑥 ∈ 𝐴 𝑧 = 𝐵 → (𝑧 ≤ 𝑦 → ∃𝑥 ∈ 𝐴 𝐵 ≤ 𝑦))
28	23, 27	syl 17	. . . . . 6 ⊢ (𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) → (𝑧 ≤ 𝑦 → ∃𝑥 ∈ 𝐴 𝐵 ≤ 𝑦))
29	19, 28	rexlimi 3241	. . . . 5 ⊢ (∃𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝑦 → ∃𝑥 ∈ 𝐴 𝐵 ≤ 𝑦)
30	29	a1i 11	. . . 4 ⊢ (𝜑 → (∃𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝑦 → ∃𝑥 ∈ 𝐴 𝐵 ≤ 𝑦))
31	18, 30	impbid 214	. . 3 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 ↔ ∃𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝑦))
32	1, 31	ralbid 3254	. 2 ⊢ (𝜑 → (∀𝑦 ∈ ℝ ∃𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 ↔ ∀𝑦 ∈ ℝ ∃𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝑦))
33	2, 9, 8	rnmptssd 7069	. . 3 ⊢ (𝜑 → ran (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ ℝ*)
34		infxrunb3 45881	. . 3 ⊢ (ran (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ ℝ* → (∀𝑦 ∈ ℝ ∃𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝑦 ↔ inf(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ*, < ) = -∞))
35	33, 34	syl 17	. 2 ⊢ (𝜑 → (∀𝑦 ∈ ℝ ∃𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝑦 ↔ inf(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ*, < ) = -∞))
36	32, 35	bitrd 281	1 ⊢ (𝜑 → (∀𝑦 ∈ ℝ ∃𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 ↔ inf(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ*, < ) = -∞))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 397   ∧ w3a 1093   = wceq 1548  Ⅎwnf 1791   ∈ wcel 2121  ∀wral 3055  ∃wrex 3065  Vcvv 3433   ⊆ wss 3885   class class class wbr 5075   ↦ cmpt 5156  ran crn 5622  infcinf 9348  ℝcr 11032  -∞cmnf 11172  ℝ^*cxr 11173   < clt 11174   ≤ cle 11175

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-sep 5221  ax-nul 5231  ax-pow 5297  ax-pr 5365  ax-un 7682  ax-cnex 11089  ax-resscn 11090  ax-1cn 11091  ax-icn 11092  ax-addcl 11093  ax-addrcl 11094  ax-mulcl 11095  ax-mulrcl 11096  ax-mulcom 11097  ax-addass 11098  ax-mulass 11099  ax-distr 11100  ax-i2m1 11101  ax-1ne0 11102  ax-1rid 11103  ax-rnegex 11104  ax-rrecex 11105  ax-cnre 11106  ax-pre-lttri 11107  ax-pre-lttrn 11108  ax-pre-ltadd 11109  ax-pre-mulgt0 11110  ax-pre-sup 11111

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3or 1094  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-nel 3041  df-ral 3056  df-rex 3066  df-rmo 3346  df-reu 3347  df-rab 3394  df-v 3435  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4265  df-if 4458  df-pw 4534  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-br 5076  df-opab 5138  df-mpt 5157  df-id 5516  df-po 5529  df-so 5530  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-er 8637  df-en 8888  df-dom 8889  df-sdom 8890  df-sup 9349  df-inf 9350  df-pnf 11176  df-mnf 11177  df-xr 11178  df-ltxr 11179  df-le 11180  df-sub 11374  df-neg 11375

This theorem is referenced by:  limsupmnflem  46177