Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > infrnmptle | Structured version Visualization version GIF version |
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.) |
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) |
Ref | Expression |
---|---|
infrnmptle | ⊢ (𝜑 → inf(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ*, < ) ≤ inf(ran (𝑥 ∈ 𝐴 ↦ 𝐶), ℝ*, < )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1914 | . 2 ⊢ Ⅎ𝑦𝜑 | |
2 | nfv 1914 | . 2 ⊢ Ⅎ𝑧𝜑 | |
3 | infrnmptle.x | . . 3 ⊢ Ⅎ𝑥𝜑 | |
4 | eqid 2824 | . . 3 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
5 | infrnmptle.b | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ*) | |
6 | 3, 4, 5 | rnmptssd 41464 | . 2 ⊢ (𝜑 → ran (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ ℝ*) |
7 | eqid 2824 | . . 3 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐴 ↦ 𝐶) | |
8 | infrnmptle.c | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ ℝ*) | |
9 | 3, 7, 8 | rnmptssd 41464 | . 2 ⊢ (𝜑 → ran (𝑥 ∈ 𝐴 ↦ 𝐶) ⊆ ℝ*) |
10 | vex 3500 | . . . . . 6 ⊢ 𝑦 ∈ V | |
11 | 7 | elrnmpt 5831 | . . . . . 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 5167 | . . . . . . 7 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐵) | |
16 | 15 | nfrn 5827 | . . . . . 6 ⊢ Ⅎ𝑥ran (𝑥 ∈ 𝐴 ↦ 𝐵) |
17 | nfv 1914 | . . . . . 6 ⊢ Ⅎ𝑥 𝑧 ≤ 𝑦 | |
18 | 16, 17 | nfrex 3312 | . . . . 5 ⊢ Ⅎ𝑥∃𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝑦 |
19 | simpr 487 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴) | |
20 | 4 | elrnmpt1 5833 | . . . . . . . . 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 2830 | . . . . . . . . 9 ⊢ (𝑦 = 𝐶 → 𝐶 = 𝑦) |
27 | 26 | 3ad2ant3 1131 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶) → 𝐶 = 𝑦) |
28 | 24, 27 | breqtrd 5095 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶) → 𝐵 ≤ 𝑦) |
29 | breq1 5072 | . . . . . . . 8 ⊢ (𝑧 = 𝐵 → (𝑧 ≤ 𝑦 ↔ 𝐵 ≤ 𝑦)) | |
30 | 29 | rspcev 3626 | . . . . . . 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 3320 | . . . 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 41694 | 1 ⊢ (𝜑 → inf(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ*, < ) ≤ inf(ran (𝑥 ∈ 𝐴 ↦ 𝐶), ℝ*, < )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1536 Ⅎwnf 1783 ∈ wcel 2113 ∃wrex 3142 Vcvv 3497 class class class wbr 5069 ↦ cmpt 5149 ran crn 5559 infcinf 8908 ℝ*cxr 10677 < clt 10678 ≤ cle 10679 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 ax-pre-sup 10618 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-po 5477 df-so 5478 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-er 8292 df-en 8513 df-dom 8514 df-sdom 8515 df-sup 8909 df-inf 8910 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 |
This theorem is referenced by: limsupres 41992 |
Copyright terms: Public domain | W3C validator |