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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > liminfresicompt | Structured version Visualization version GIF version |
Description: The inferior limit doesn't change when a function is restricted to the upper part of the reals. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
Ref | Expression |
---|---|
liminfresicompt.1 | ⊢ (𝜑 → 𝑀 ∈ ℝ) |
liminfresicompt.2 | ⊢ 𝑍 = (𝑀[,)+∞) |
liminfresicompt.3 | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
Ref | Expression |
---|---|
liminfresicompt | ⊢ (𝜑 → (lim inf‘(𝑥 ∈ (𝐴 ∩ 𝑍) ↦ 𝐵)) = (lim inf‘(𝑥 ∈ 𝐴 ↦ 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | resmpt3 5787 | . . . . 5 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐵) ↾ 𝑍) = (𝑥 ∈ (𝐴 ∩ 𝑍) ↦ 𝐵) | |
2 | 1 | eqcomi 2804 | . . . 4 ⊢ (𝑥 ∈ (𝐴 ∩ 𝑍) ↦ 𝐵) = ((𝑥 ∈ 𝐴 ↦ 𝐵) ↾ 𝑍) |
3 | 2 | a1i 11 | . . 3 ⊢ (𝜑 → (𝑥 ∈ (𝐴 ∩ 𝑍) ↦ 𝐵) = ((𝑥 ∈ 𝐴 ↦ 𝐵) ↾ 𝑍)) |
4 | 3 | fveq2d 6542 | . 2 ⊢ (𝜑 → (lim inf‘(𝑥 ∈ (𝐴 ∩ 𝑍) ↦ 𝐵)) = (lim inf‘((𝑥 ∈ 𝐴 ↦ 𝐵) ↾ 𝑍))) |
5 | liminfresicompt.1 | . . 3 ⊢ (𝜑 → 𝑀 ∈ ℝ) | |
6 | liminfresicompt.2 | . . 3 ⊢ 𝑍 = (𝑀[,)+∞) | |
7 | liminfresicompt.3 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
8 | 7 | mptexd 6853 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V) |
9 | 5, 6, 8 | liminfresico 41594 | . 2 ⊢ (𝜑 → (lim inf‘((𝑥 ∈ 𝐴 ↦ 𝐵) ↾ 𝑍)) = (lim inf‘(𝑥 ∈ 𝐴 ↦ 𝐵))) |
10 | 4, 9 | eqtrd 2831 | 1 ⊢ (𝜑 → (lim inf‘(𝑥 ∈ (𝐴 ∩ 𝑍) ↦ 𝐵)) = (lim inf‘(𝑥 ∈ 𝐴 ↦ 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1522 ∈ wcel 2081 Vcvv 3437 ∩ cin 3858 ↦ cmpt 5041 ↾ cres 5445 ‘cfv 6225 (class class class)co 7016 ℝcr 10382 +∞cpnf 10518 [,)cico 12590 lim infclsi 41574 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-rep 5081 ax-sep 5094 ax-nul 5101 ax-pow 5157 ax-pr 5221 ax-un 7319 ax-cnex 10439 ax-resscn 10440 ax-1cn 10441 ax-icn 10442 ax-addcl 10443 ax-addrcl 10444 ax-mulcl 10445 ax-mulrcl 10446 ax-mulcom 10447 ax-addass 10448 ax-mulass 10449 ax-distr 10450 ax-i2m1 10451 ax-1ne0 10452 ax-1rid 10453 ax-rnegex 10454 ax-rrecex 10455 ax-cnre 10456 ax-pre-lttri 10457 ax-pre-lttrn 10458 ax-pre-ltadd 10459 ax-pre-mulgt0 10460 ax-pre-sup 10461 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-nel 3091 df-ral 3110 df-rex 3111 df-reu 3112 df-rmo 3113 df-rab 3114 df-v 3439 df-sbc 3707 df-csb 3812 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-pss 3876 df-nul 4212 df-if 4382 df-pw 4455 df-sn 4473 df-pr 4475 df-tp 4477 df-op 4479 df-uni 4746 df-iun 4827 df-br 4963 df-opab 5025 df-mpt 5042 df-tr 5064 df-id 5348 df-eprel 5353 df-po 5362 df-so 5363 df-fr 5402 df-we 5404 df-xp 5449 df-rel 5450 df-cnv 5451 df-co 5452 df-dm 5453 df-rn 5454 df-res 5455 df-ima 5456 df-pred 6023 df-ord 6069 df-on 6070 df-lim 6071 df-suc 6072 df-iota 6189 df-fun 6227 df-fn 6228 df-f 6229 df-f1 6230 df-fo 6231 df-f1o 6232 df-fv 6233 df-riota 6977 df-ov 7019 df-oprab 7020 df-mpo 7021 df-om 7437 df-1st 7545 df-2nd 7546 df-wrecs 7798 df-recs 7860 df-rdg 7898 df-er 8139 df-en 8358 df-dom 8359 df-sdom 8360 df-sup 8752 df-inf 8753 df-pnf 10523 df-mnf 10524 df-xr 10525 df-ltxr 10526 df-le 10527 df-sub 10719 df-neg 10720 df-div 11146 df-nn 11487 df-n0 11746 df-z 11830 df-uz 12094 df-q 12198 df-ico 12594 df-liminf 41575 |
This theorem is referenced by: liminfval4 41612 liminfval3 41613 |
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