Theorem infrnmptle 41717

Description: An indexed infimum of extended reals is smaller than another indexed infimum of extended reals, when every indexed element is smaller than the corresponding one. (Contributed by Glauco Siliprandi, 23-Oct-2021.)

Hypotheses

Ref	Expression
infrnmptle.x	⊢ Ⅎ𝑥𝜑
infrnmptle.b	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ*)
infrnmptle.c	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ ℝ*)
infrnmptle.l	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ≤ 𝐶)

Assertion

Ref	Expression
infrnmptle	⊢ (𝜑 → inf(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ*, < ) ≤ inf(ran (𝑥 ∈ 𝐴 ↦ 𝐶), ℝ*, < ))

Distinct variable group:   𝑥,𝐴

Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)   𝐶(𝑥)

Proof of Theorem infrnmptle

Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1915	. 2 ⊢ Ⅎ𝑦𝜑
2		nfv 1915	. 2 ⊢ Ⅎ𝑧𝜑
3		infrnmptle.x	. . 3 ⊢ Ⅎ𝑥𝜑
4		eqid 2821	. . 3 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
5		infrnmptle.b	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ*)
6	3, 4, 5	rnmptssd 41478	. 2 ⊢ (𝜑 → ran (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ ℝ*)
7		eqid 2821	. . 3 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐴 ↦ 𝐶)
8		infrnmptle.c	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ ℝ*)
9	3, 7, 8	rnmptssd 41478	. 2 ⊢ (𝜑 → ran (𝑥 ∈ 𝐴 ↦ 𝐶) ⊆ ℝ*)
10		vex 3497	. . . . . 6 ⊢ 𝑦 ∈ V
11	7	elrnmpt 5828	. . . . . 6 ⊢ (𝑦 ∈ V → (𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐶) ↔ ∃𝑥 ∈ 𝐴 𝑦 = 𝐶))
12	10, 11	ax-mp 5	. . . . 5 ⊢ (𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐶) ↔ ∃𝑥 ∈ 𝐴 𝑦 = 𝐶)
13	12	biimpi 218	. . . 4 ⊢ (𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐶) → ∃𝑥 ∈ 𝐴 𝑦 = 𝐶)
14	13	adantl 484	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐶)) → ∃𝑥 ∈ 𝐴 𝑦 = 𝐶)
15		nfmpt1 5164	. . . . . . 7 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐵)
16	15	nfrn 5824	. . . . . 6 ⊢ Ⅎ𝑥ran (𝑥 ∈ 𝐴 ↦ 𝐵)
17		nfv 1915	. . . . . 6 ⊢ Ⅎ𝑥 𝑧 ≤ 𝑦
18	16, 17	nfrex 3309	. . . . 5 ⊢ Ⅎ𝑥∃𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝑦
19		simpr 487	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
20	4	elrnmpt1 5830	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ ℝ*) → 𝐵 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵))
21	19, 5, 20	syl2anc 586	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵))
22	21	3adant3 1128	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶) → 𝐵 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵))
23		infrnmptle.l	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ≤ 𝐶)
24	23	3adant3 1128	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶) → 𝐵 ≤ 𝐶)
25		id 22	. . . . . . . . . 10 ⊢ (𝑦 = 𝐶 → 𝑦 = 𝐶)
26	25	eqcomd 2827	. . . . . . . . 9 ⊢ (𝑦 = 𝐶 → 𝐶 = 𝑦)
27	26	3ad2ant3 1131	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶) → 𝐶 = 𝑦)
28	24, 27	breqtrd 5092	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶) → 𝐵 ≤ 𝑦)
29		breq1 5069	. . . . . . . 8 ⊢ (𝑧 = 𝐵 → (𝑧 ≤ 𝑦 ↔ 𝐵 ≤ 𝑦))
30	29	rspcev 3623	. . . . . . 7 ⊢ ((𝐵 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ∧ 𝐵 ≤ 𝑦) → ∃𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝑦)
31	22, 28, 30	syl2anc 586	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶) → ∃𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝑦)
32	31	3exp 1115	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → (𝑦 = 𝐶 → ∃𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝑦)))
33	3, 18, 32	rexlimd 3317	. . . 4 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝑦 = 𝐶 → ∃𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝑦))
34	33	adantr 483	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐶)) → (∃𝑥 ∈ 𝐴 𝑦 = 𝐶 → ∃𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝑦))
35	14, 34	mpd 15	. 2 ⊢ ((𝜑 ∧ 𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐶)) → ∃𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝑦)
36	1, 2, 6, 9, 35	infleinf2 41708	1 ⊢ (𝜑 → inf(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ*, < ) ≤ inf(ran (𝑥 ∈ 𝐴 ↦ 𝐶), ℝ*, < ))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1537  Ⅎwnf 1784   ∈ wcel 2114  ∃wrex 3139  Vcvv 3494   class class class wbr 5066   ↦ cmpt 5146  ran crn 5556  infcinf 8905  ℝ^*cxr 10674   < clt 10675   ≤ cle 10676

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461  ax-cnex 10593  ax-resscn 10594  ax-1cn 10595  ax-icn 10596  ax-addcl 10597  ax-addrcl 10598  ax-mulcl 10599  ax-mulrcl 10600  ax-mulcom 10601  ax-addass 10602  ax-mulass 10603  ax-distr 10604  ax-i2m1 10605  ax-1ne0 10606  ax-1rid 10607  ax-rnegex 10608  ax-rrecex 10609  ax-cnre 10610  ax-pre-lttri 10611  ax-pre-lttrn 10612  ax-pre-ltadd 10613  ax-pre-mulgt0 10614  ax-pre-sup 10615

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-br 5067  df-opab 5129  df-mpt 5147  df-id 5460  df-po 5474  df-so 5475  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-riota 7114  df-ov 7159  df-oprab 7160  df-mpo 7161  df-er 8289  df-en 8510  df-dom 8511  df-sdom 8512  df-sup 8906  df-inf 8907  df-pnf 10677  df-mnf 10678  df-xr 10679  df-ltxr 10680  df-le 10681  df-sub 10872  df-neg 10873

This theorem is referenced by:  limsupres  42006