Theorem infrnmptle 43191

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 2736	. . 3 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
5		infrnmptle.b	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ*)
6	3, 4, 5	rnmptssd 42955	. 2 ⊢ (𝜑 → ran (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ ℝ*)
7		eqid 2736	. . 3 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐴 ↦ 𝐶)
8		infrnmptle.c	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ ℝ*)
9	3, 7, 8	rnmptssd 42955	. 2 ⊢ (𝜑 → ran (𝑥 ∈ 𝐴 ↦ 𝐶) ⊆ ℝ*)
10		vex 3441	. . . . . 6 ⊢ 𝑦 ∈ V
11	7	elrnmpt 5877	. . . . . 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 215	. . . 4 ⊢ (𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐶) → ∃𝑥 ∈ 𝐴 𝑦 = 𝐶)
14	13	adantl 483	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐶)) → ∃𝑥 ∈ 𝐴 𝑦 = 𝐶)
15		nfmpt1 5189	. . . . . . 7 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐵)
16	15	nfrn 5873	. . . . . 6 ⊢ Ⅎ𝑥ran (𝑥 ∈ 𝐴 ↦ 𝐵)
17		nfv 1915	. . . . . 6 ⊢ Ⅎ𝑥 𝑧 ≤ 𝑦
18	16, 17	nfrexw 3293	. . . . 5 ⊢ Ⅎ𝑥∃𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝑦
19		simpr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
20	4	elrnmpt1 5879	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ ℝ*) → 𝐵 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵))
21	19, 5, 20	syl2anc 585	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵))
22	21	3adant3 1132	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶) → 𝐵 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵))
23		infrnmptle.l	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ≤ 𝐶)
24	23	3adant3 1132	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶) → 𝐵 ≤ 𝐶)
25		id 22	. . . . . . . . . 10 ⊢ (𝑦 = 𝐶 → 𝑦 = 𝐶)
26	25	eqcomd 2742	. . . . . . . . 9 ⊢ (𝑦 = 𝐶 → 𝐶 = 𝑦)
27	26	3ad2ant3 1135	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶) → 𝐶 = 𝑦)
28	24, 27	breqtrd 5107	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶) → 𝐵 ≤ 𝑦)
29		breq1 5084	. . . . . . . 8 ⊢ (𝑧 = 𝐵 → (𝑧 ≤ 𝑦 ↔ 𝐵 ≤ 𝑦))
30	29	rspcev 3566	. . . . . . 7 ⊢ ((𝐵 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ∧ 𝐵 ≤ 𝑦) → ∃𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝑦)
31	22, 28, 30	syl2anc 585	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶) → ∃𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝑦)
32	31	3exp 1119	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → (𝑦 = 𝐶 → ∃𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝑦)))
33	3, 18, 32	rexlimd 3246	. . . 4 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝑦 = 𝐶 → ∃𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝑦))
34	33	adantr 482	. . 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 43182	1 ⊢ (𝜑 → inf(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ*, < ) ≤ inf(ran (𝑥 ∈ 𝐴 ↦ 𝐶), ℝ*, < ))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1087   = wceq 1539  Ⅎwnf 1783   ∈ wcel 2104  ∃wrex 3071  Vcvv 3437   class class class wbr 5081   ↦ cmpt 5164  ran crn 5601  infcinf 9248  ℝ^*cxr 11058   < clt 11059   ≤ cle 11060

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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2707  ax-sep 5232  ax-nul 5239  ax-pow 5297  ax-pr 5361  ax-un 7620  ax-cnex 10977  ax-resscn 10978  ax-1cn 10979  ax-icn 10980  ax-addcl 10981  ax-addrcl 10982  ax-mulcl 10983  ax-mulrcl 10984  ax-mulcom 10985  ax-addass 10986  ax-mulass 10987  ax-distr 10988  ax-i2m1 10989  ax-1ne0 10990  ax-1rid 10991  ax-rnegex 10992  ax-rrecex 10993  ax-cnre 10994  ax-pre-lttri 10995  ax-pre-lttrn 10996  ax-pre-ltadd 10997  ax-pre-mulgt0 10998  ax-pre-sup 10999

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3or 1088  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3304  df-reu 3305  df-rab 3306  df-v 3439  df-sbc 3722  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4566  df-pr 4568  df-op 4572  df-uni 4845  df-br 5082  df-opab 5144  df-mpt 5165  df-id 5500  df-po 5514  df-so 5515  df-xp 5606  df-rel 5607  df-cnv 5608  df-co 5609  df-dm 5610  df-rn 5611  df-res 5612  df-ima 5613  df-iota 6410  df-fun 6460  df-fn 6461  df-f 6462  df-f1 6463  df-fo 6464  df-f1o 6465  df-fv 6466  df-riota 7264  df-ov 7310  df-oprab 7311  df-mpo 7312  df-er 8529  df-en 8765  df-dom 8766  df-sdom 8767  df-sup 9249  df-inf 9250  df-pnf 11061  df-mnf 11062  df-xr 11063  df-ltxr 11064  df-le 11065  df-sub 11257  df-neg 11258

This theorem is referenced by:  limsupres  43475