Theorem infxrunb3rnmpt 45877

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 5185	. . . . . . 7 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐵)
4	3	nfrn 5902	. . . . . 6 ⊢ Ⅎ𝑥ran (𝑥 ∈ 𝐴 ↦ 𝐵)
5		nfv 1916	. . . . . 6 ⊢ Ⅎ𝑥 𝑧 ≤ 𝑦
6	4, 5	nfrexw 3286	. . . . 5 ⊢ Ⅎ𝑥∃𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝑦
7		simpr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
8		infxrunb3rnmpt.3	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ*)
9		eqid 2737	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
10	9	elrnmpt1 5910	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ ℝ*) → 𝐵 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵))
11	7, 8, 10	syl2anc 585	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵))
12	11	3adant3 1133	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝐵 ≤ 𝑦) → 𝐵 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵))
13		simp3 1139	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝐵 ≤ 𝑦) → 𝐵 ≤ 𝑦)
14		breq1 5089	. . . . . . . 8 ⊢ (𝑧 = 𝐵 → (𝑧 ≤ 𝑦 ↔ 𝐵 ≤ 𝑦))
15	14	rspcev 3565	. . . . . . 7 ⊢ ((𝐵 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ∧ 𝐵 ≤ 𝑦) → ∃𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝑦)
16	12, 13, 15	syl2anc 585	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝐵 ≤ 𝑦) → ∃𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝑦)
17	16	3exp 1120	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → (𝐵 ≤ 𝑦 → ∃𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝑦)))
18	2, 6, 17	rexlimd 3245	. . . 4 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 → ∃𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝑦))
19		nfv 1916	. . . . . 6 ⊢ Ⅎ𝑧∃𝑥 ∈ 𝐴 𝐵 ≤ 𝑦
20		vex 3434	. . . . . . . . 9 ⊢ 𝑧 ∈ V
21	9	elrnmpt 5908	. . . . . . . . 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 216	. . . . . . 7 ⊢ (𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) → ∃𝑥 ∈ 𝐴 𝑧 = 𝐵)
24	14	biimpcd 249	. . . . . . . . . 10 ⊢ (𝑧 ≤ 𝑦 → (𝑧 = 𝐵 → 𝐵 ≤ 𝑦))
25	24	a1d 25	. . . . . . . . 9 ⊢ (𝑧 ≤ 𝑦 → (𝑥 ∈ 𝐴 → (𝑧 = 𝐵 → 𝐵 ≤ 𝑦)))
26	5, 25	reximdai 3240	. . . . . . . 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 3238	. . . . 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 212	. . 3 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 ↔ ∃𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝑦))
32	1, 31	ralbid 3251	. 2 ⊢ (𝜑 → (∀𝑦 ∈ ℝ ∃𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 ↔ ∀𝑦 ∈ ℝ ∃𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝑦))
33	2, 9, 8	rnmptssd 7071	. . 3 ⊢ (𝜑 → ran (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ ℝ*)
34		infxrunb3 45873	. . 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 279	1 ⊢ (𝜑 → (∀𝑦 ∈ ℝ ∃𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 ↔ inf(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ*, < ) = -∞))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542  Ⅎwnf 1785   ∈ wcel 2114  ∀wral 3052  ∃wrex 3062  Vcvv 3430   ⊆ wss 3890   class class class wbr 5086   ↦ cmpt 5167  ran crn 5626  infcinf 9348  ℝcr 11031  -∞cmnf 11171  ℝ^*cxr 11172   < clt 11173   ≤ cle 11174

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5232  ax-nul 5242  ax-pow 5303  ax-pr 5371  ax-un 7683  ax-cnex 11088  ax-resscn 11089  ax-1cn 11090  ax-icn 11091  ax-addcl 11092  ax-addrcl 11093  ax-mulcl 11094  ax-mulrcl 11095  ax-mulcom 11096  ax-addass 11097  ax-mulass 11098  ax-distr 11099  ax-i2m1 11100  ax-1ne0 11101  ax-1rid 11102  ax-rnegex 11103  ax-rrecex 11104  ax-cnre 11105  ax-pre-lttri 11106  ax-pre-lttrn 11107  ax-pre-ltadd 11108  ax-pre-mulgt0 11109  ax-pre-sup 11110

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5520  df-po 5533  df-so 5534  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-riota 7318  df-ov 7364  df-oprab 7365  df-mpo 7366  df-er 8637  df-en 8888  df-dom 8889  df-sdom 8890  df-sup 9349  df-inf 9350  df-pnf 11175  df-mnf 11176  df-xr 11177  df-ltxr 11178  df-le 11179  df-sub 11373  df-neg 11374

This theorem is referenced by:  limsupmnflem  46169