Theorem infxrunb3rnmpt 45693

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 5197	. . . . . . 7 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐵)
4	3	nfrn 5901	. . . . . 6 ⊢ Ⅎ𝑥ran (𝑥 ∈ 𝐴 ↦ 𝐵)
5		nfv 1915	. . . . . 6 ⊢ Ⅎ𝑥 𝑧 ≤ 𝑦
6	4, 5	nfrexw 3284	. . . . 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 2736	. . . . . . . . . 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 584	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵))
12	11	3adant3 1132	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝐵 ≤ 𝑦) → 𝐵 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵))
13		simp3 1138	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝐵 ≤ 𝑦) → 𝐵 ≤ 𝑦)
14		breq1 5101	. . . . . . . 8 ⊢ (𝑧 = 𝐵 → (𝑧 ≤ 𝑦 ↔ 𝐵 ≤ 𝑦))
15	14	rspcev 3576	. . . . . . 7 ⊢ ((𝐵 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ∧ 𝐵 ≤ 𝑦) → ∃𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝑦)
16	12, 13, 15	syl2anc 584	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝐵 ≤ 𝑦) → ∃𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝑦)
17	16	3exp 1119	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → (𝐵 ≤ 𝑦 → ∃𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝑦)))
18	2, 6, 17	rexlimd 3243	. . . 4 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 → ∃𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝑦))
19		nfv 1915	. . . . . 6 ⊢ Ⅎ𝑧∃𝑥 ∈ 𝐴 𝐵 ≤ 𝑦
20		vex 3444	. . . . . . . . 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 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 3238	. . . . . . . 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 3236	. . . . 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 3249	. 2 ⊢ (𝜑 → (∀𝑦 ∈ ℝ ∃𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 ↔ ∀𝑦 ∈ ℝ ∃𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝑦))
33	2, 9, 8	rnmptssd 45461	. . 3 ⊢ (𝜑 → ran (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ ℝ*)
34		infxrunb3 45689	. . 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 1086   = wceq 1541  Ⅎwnf 1784   ∈ wcel 2113  ∀wral 3051  ∃wrex 3060  Vcvv 3440   ⊆ wss 3901   class class class wbr 5098   ↦ cmpt 5179  ran crn 5625  infcinf 9346  ℝcr 11027  -∞cmnf 11166  ℝ^*cxr 11167   < clt 11168   ≤ cle 11169

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-cnex 11084  ax-resscn 11085  ax-1cn 11086  ax-icn 11087  ax-addcl 11088  ax-addrcl 11089  ax-mulcl 11090  ax-mulrcl 11091  ax-mulcom 11092  ax-addass 11093  ax-mulass 11094  ax-distr 11095  ax-i2m1 11096  ax-1ne0 11097  ax-1rid 11098  ax-rnegex 11099  ax-rrecex 11100  ax-cnre 11101  ax-pre-lttri 11102  ax-pre-lttrn 11103  ax-pre-ltadd 11104  ax-pre-mulgt0 11105  ax-pre-sup 11106

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-po 5532  df-so 5533  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-er 8635  df-en 8886  df-dom 8887  df-sdom 8888  df-sup 9347  df-inf 9348  df-pnf 11170  df-mnf 11171  df-xr 11172  df-ltxr 11173  df-le 11174  df-sub 11368  df-neg 11369

This theorem is referenced by:  limsupmnflem  45985