Theorem infxrgelbrnmpt 46603

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 2763	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
3		infxrgelbrnmpt.b	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ*)
4	1, 2, 3	rnmptssd 7132	. . 3 ⊢ (𝜑 → ran (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ ℝ*)
5		infxrgelbrnmpt.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ ℝ*)
6		infxrgelb 13476	. . 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 5206	. . . . . . 7 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐵)
9	8	nfrn 5938	. . . . . 6 ⊢ Ⅎ𝑥ran (𝑥 ∈ 𝐴 ↦ 𝐵)
10		nfv 1948	. . . . . 6 ⊢ Ⅎ𝑥 𝐶 ≤ 𝑧
11	9, 10	nfralw 3312	. . . . 5 ⊢ Ⅎ𝑥∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝐶 ≤ 𝑧
12	1, 11	nfan 1933	. . . 4 ⊢ Ⅎ𝑥(𝜑 ∧ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝐶 ≤ 𝑧)
13		simpr 490	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
14	2	elrnmpt1 5946	. . . . . . . 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 5109	. . . . . . 7 ⊢ (𝑧 = 𝐵 → (𝐶 ≤ 𝑧 ↔ 𝐶 ≤ 𝐵))
19	18	rspcva 3577	. . . . . 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 3263	. . 3 ⊢ ((𝜑 ∧ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝐶 ≤ 𝑧) → ∀𝑥 ∈ 𝐴 𝐶 ≤ 𝐵)
23		vex 3457	. . . . . . . 8 ⊢ 𝑧 ∈ V
24	2	elrnmpt 5944	. . . . . . . 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 3289	. . . . . . . 8 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 𝐶 ≤ 𝐵
28		rspa 3254	. . . . . . . . . 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 3272	. . . . . . 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 3157	. . . 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 1569  Ⅎwnf 1815   ∈ wcel 2146  ∀wral 3079  ∃wrex 3089  Vcvv 3453   ⊆ wss 3899   class class class wbr 5105   ↦ cmpt 5188  ran crn 5655  infcinf 9443  ℝ^*cxr 11352   < clt 11353   ≤ cle 11354

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-sep 5251  ax-nul 5262  ax-pow 5330  ax-pr 5394  ax-un 7758  ax-cnex 11266  ax-resscn 11267  ax-1cn 11268  ax-icn 11269  ax-addcl 11270  ax-addrcl 11271  ax-mulcl 11272  ax-mulrcl 11273  ax-mulcom 11274  ax-addass 11275  ax-mulass 11276  ax-distr 11277  ax-i2m1 11278  ax-1ne0 11279  ax-1rid 11280  ax-rnegex 11281  ax-rrecex 11282  ax-cnre 11283  ax-pre-lttri 11284  ax-pre-lttrn 11285  ax-pre-ltadd 11286  ax-pre-mulgt0 11287  ax-pre-sup 11288

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3368  df-reu 3369  df-rab 3416  df-v 3455  df-sbc 3742  df-csb 3851  df-ss 3900  df-dif 4049  df-in 4093  df-un 4153  df-nul 4281  df-if 4485  df-pw 4561  df-sn 4588  df-pr 4590  df-op 4838  df-uni 4870  df-br 5106  df-opab 5170  df-mpt 5189  df-id 5549  df-po 5562  df-so 5563  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-iota 6500  df-fun 6548  df-fn 6549  df-f 6550  df-f1 6551  df-fo 6552  df-f1o 6553  df-fv 6554  df-riota 7384  df-ov 7430  df-oprab 7431  df-mpo 7432  df-er 8728  df-en 8985  df-dom 8986  df-sdom 8987  df-sup 9444  df-inf 9445  df-pnf 11355  df-mnf 11356  df-xr 11357  df-ltxr 11358  df-le 11359  df-sub 11553  df-neg 11554

This theorem is used by: (None)