Theorem infrnmptle 45434

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 1914	. 2 ⊢ Ⅎ𝑦𝜑
2		nfv 1914	. 2 ⊢ Ⅎ𝑧𝜑
3		infrnmptle.x	. . 3 ⊢ Ⅎ𝑥𝜑
4		eqid 2737	. . 3 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
5		infrnmptle.b	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ*)
6	3, 4, 5	rnmptssd 45201	. 2 ⊢ (𝜑 → ran (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ ℝ*)
7		eqid 2737	. . 3 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐴 ↦ 𝐶)
8		infrnmptle.c	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ ℝ*)
9	3, 7, 8	rnmptssd 45201	. 2 ⊢ (𝜑 → ran (𝑥 ∈ 𝐴 ↦ 𝐶) ⊆ ℝ*)
10		vex 3484	. . . . . 6 ⊢ 𝑦 ∈ V
11	7	elrnmpt 5969	. . . . . 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 216	. . . 4 ⊢ (𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐶) → ∃𝑥 ∈ 𝐴 𝑦 = 𝐶)
14	13	adantl 481	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐶)) → ∃𝑥 ∈ 𝐴 𝑦 = 𝐶)
15		nfmpt1 5250	. . . . . . 7 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐵)
16	15	nfrn 5963	. . . . . 6 ⊢ Ⅎ𝑥ran (𝑥 ∈ 𝐴 ↦ 𝐵)
17		nfv 1914	. . . . . 6 ⊢ Ⅎ𝑥 𝑧 ≤ 𝑦
18	16, 17	nfrexw 3313	. . . . 5 ⊢ Ⅎ𝑥∃𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝑦
19		simpr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
20	4	elrnmpt1 5971	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ ℝ*) → 𝐵 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵))
21	19, 5, 20	syl2anc 584	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵))
22	21	3adant3 1133	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶) → 𝐵 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵))
23		infrnmptle.l	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ≤ 𝐶)
24	23	3adant3 1133	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶) → 𝐵 ≤ 𝐶)
25		id 22	. . . . . . . . . 10 ⊢ (𝑦 = 𝐶 → 𝑦 = 𝐶)
26	25	eqcomd 2743	. . . . . . . . 9 ⊢ (𝑦 = 𝐶 → 𝐶 = 𝑦)
27	26	3ad2ant3 1136	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶) → 𝐶 = 𝑦)
28	24, 27	breqtrd 5169	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶) → 𝐵 ≤ 𝑦)
29		breq1 5146	. . . . . . . 8 ⊢ (𝑧 = 𝐵 → (𝑧 ≤ 𝑦 ↔ 𝐵 ≤ 𝑦))
30	29	rspcev 3622	. . . . . . 7 ⊢ ((𝐵 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ∧ 𝐵 ≤ 𝑦) → ∃𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝑦)
31	22, 28, 30	syl2anc 584	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶) → ∃𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝑦)
32	31	3exp 1120	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → (𝑦 = 𝐶 → ∃𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝑦)))
33	3, 18, 32	rexlimd 3266	. . . 4 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝑦 = 𝐶 → ∃𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝑦))
34	33	adantr 480	. . 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 45425	1 ⊢ (𝜑 → inf(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ*, < ) ≤ inf(ran (𝑥 ∈ 𝐴 ↦ 𝐶), ℝ*, < ))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1540  Ⅎwnf 1783   ∈ wcel 2108  ∃wrex 3070  Vcvv 3480   class class class wbr 5143   ↦ cmpt 5225  ran crn 5686  infcinf 9481  ℝ^*cxr 11294   < clt 11295   ≤ cle 11296

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-po 5592  df-so 5593  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-sup 9482  df-inf 9483  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495

This theorem is referenced by:  limsupres  45720