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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > limsupresicompt | Structured version Visualization version GIF version |
Description: The superior 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 |
---|---|
limsupresicompt.a | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
limsupresicompt.m | ⊢ (𝜑 → 𝑀 ∈ ℝ) |
limsupresicompt.z | ⊢ 𝑍 = (𝑀[,)+∞) |
Ref | Expression |
---|---|
limsupresicompt | ⊢ (𝜑 → (lim sup‘(𝑥 ∈ 𝐴 ↦ 𝐵)) = (lim sup‘(𝑥 ∈ (𝐴 ∩ 𝑍) ↦ 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | limsupresicompt.m | . . 3 ⊢ (𝜑 → 𝑀 ∈ ℝ) | |
2 | limsupresicompt.z | . . 3 ⊢ 𝑍 = (𝑀[,)+∞) | |
3 | limsupresicompt.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
4 | 3 | mptexd 7230 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V) |
5 | 1, 2, 4 | limsupresico 45011 | . 2 ⊢ (𝜑 → (lim sup‘((𝑥 ∈ 𝐴 ↦ 𝐵) ↾ 𝑍)) = (lim sup‘(𝑥 ∈ 𝐴 ↦ 𝐵))) |
6 | resmpt3 6036 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐵) ↾ 𝑍) = (𝑥 ∈ (𝐴 ∩ 𝑍) ↦ 𝐵) | |
7 | 6 | a1i 11 | . . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵) ↾ 𝑍) = (𝑥 ∈ (𝐴 ∩ 𝑍) ↦ 𝐵)) |
8 | 7 | fveq2d 6895 | . 2 ⊢ (𝜑 → (lim sup‘((𝑥 ∈ 𝐴 ↦ 𝐵) ↾ 𝑍)) = (lim sup‘(𝑥 ∈ (𝐴 ∩ 𝑍) ↦ 𝐵))) |
9 | 5, 8 | eqtr3d 2769 | 1 ⊢ (𝜑 → (lim sup‘(𝑥 ∈ 𝐴 ↦ 𝐵)) = (lim sup‘(𝑥 ∈ (𝐴 ∩ 𝑍) ↦ 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 Vcvv 3469 ∩ cin 3943 ↦ cmpt 5225 ↾ cres 5674 ‘cfv 6542 (class class class)co 7414 ℝcr 11129 +∞cpnf 11267 [,)cico 13350 lim supclsp 15438 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11186 ax-resscn 11187 ax-1cn 11188 ax-icn 11189 ax-addcl 11190 ax-addrcl 11191 ax-mulcl 11192 ax-mulrcl 11193 ax-mulcom 11194 ax-addass 11195 ax-mulass 11196 ax-distr 11197 ax-i2m1 11198 ax-1ne0 11199 ax-1rid 11200 ax-rnegex 11201 ax-rrecex 11202 ax-cnre 11203 ax-pre-lttri 11204 ax-pre-lttrn 11205 ax-pre-ltadd 11206 ax-pre-mulgt0 11207 ax-pre-sup 11208 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-er 8718 df-en 8956 df-dom 8957 df-sdom 8958 df-sup 9457 df-inf 9458 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11468 df-neg 11469 df-div 11894 df-nn 12235 df-n0 12495 df-z 12581 df-uz 12845 df-q 12955 df-ico 13354 df-limsup 15439 |
This theorem is referenced by: liminfval4 45100 liminfval3 45101 limsupval4 45105 |
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