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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > liminfresuz | Structured version Visualization version GIF version |
Description: If the real part of the domain of a function is a subset of the integers, the inferior limit doesn't change when the function is restricted to an upper set of integers. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
Ref | Expression |
---|---|
liminfresuz.m | β’ (π β π β β€) |
liminfresuz.z | β’ π = (β€β₯βπ) |
liminfresuz.f | β’ (π β πΉ β π) |
liminfresuz.d | β’ (π β dom (πΉ βΎ β) β β€) |
Ref | Expression |
---|---|
liminfresuz | β’ (π β (lim infβ(πΉ βΎ π)) = (lim infβπΉ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rescom 5964 | . . . . 5 β’ ((πΉ βΎ π) βΎ β) = ((πΉ βΎ β) βΎ π) | |
2 | 1 | fveq2i 6846 | . . . 4 β’ (lim infβ((πΉ βΎ π) βΎ β)) = (lim infβ((πΉ βΎ β) βΎ π)) |
3 | 2 | a1i 11 | . . 3 β’ (π β (lim infβ((πΉ βΎ π) βΎ β)) = (lim infβ((πΉ βΎ β) βΎ π))) |
4 | relres 5967 | . . . . . . . . . 10 β’ Rel (πΉ βΎ β) | |
5 | 4 | a1i 11 | . . . . . . . . 9 β’ (π β Rel (πΉ βΎ β)) |
6 | liminfresuz.d | . . . . . . . . 9 β’ (π β dom (πΉ βΎ β) β β€) | |
7 | relssres 5979 | . . . . . . . . 9 β’ ((Rel (πΉ βΎ β) β§ dom (πΉ βΎ β) β β€) β ((πΉ βΎ β) βΎ β€) = (πΉ βΎ β)) | |
8 | 5, 6, 7 | syl2anc 585 | . . . . . . . 8 β’ (π β ((πΉ βΎ β) βΎ β€) = (πΉ βΎ β)) |
9 | 8 | eqcomd 2739 | . . . . . . 7 β’ (π β (πΉ βΎ β) = ((πΉ βΎ β) βΎ β€)) |
10 | 9 | reseq1d 5937 | . . . . . 6 β’ (π β ((πΉ βΎ β) βΎ (π[,)+β)) = (((πΉ βΎ β) βΎ β€) βΎ (π[,)+β))) |
11 | resres 5951 | . . . . . . 7 β’ (((πΉ βΎ β) βΎ β€) βΎ (π[,)+β)) = ((πΉ βΎ β) βΎ (β€ β© (π[,)+β))) | |
12 | 11 | a1i 11 | . . . . . 6 β’ (π β (((πΉ βΎ β) βΎ β€) βΎ (π[,)+β)) = ((πΉ βΎ β) βΎ (β€ β© (π[,)+β)))) |
13 | liminfresuz.m | . . . . . . . . 9 β’ (π β π β β€) | |
14 | liminfresuz.z | . . . . . . . . 9 β’ π = (β€β₯βπ) | |
15 | 13, 14 | uzinico 43884 | . . . . . . . 8 β’ (π β π = (β€ β© (π[,)+β))) |
16 | 15 | eqcomd 2739 | . . . . . . 7 β’ (π β (β€ β© (π[,)+β)) = π) |
17 | 16 | reseq2d 5938 | . . . . . 6 β’ (π β ((πΉ βΎ β) βΎ (β€ β© (π[,)+β))) = ((πΉ βΎ β) βΎ π)) |
18 | 10, 12, 17 | 3eqtrrd 2778 | . . . . 5 β’ (π β ((πΉ βΎ β) βΎ π) = ((πΉ βΎ β) βΎ (π[,)+β))) |
19 | 18 | fveq2d 6847 | . . . 4 β’ (π β (lim infβ((πΉ βΎ β) βΎ π)) = (lim infβ((πΉ βΎ β) βΎ (π[,)+β)))) |
20 | 13 | zred 12612 | . . . . 5 β’ (π β π β β) |
21 | eqid 2733 | . . . . 5 β’ (π[,)+β) = (π[,)+β) | |
22 | liminfresuz.f | . . . . . 6 β’ (π β πΉ β π) | |
23 | 22 | resexd 5985 | . . . . 5 β’ (π β (πΉ βΎ β) β V) |
24 | 20, 21, 23 | liminfresico 44098 | . . . 4 β’ (π β (lim infβ((πΉ βΎ β) βΎ (π[,)+β))) = (lim infβ(πΉ βΎ β))) |
25 | 19, 24 | eqtrd 2773 | . . 3 β’ (π β (lim infβ((πΉ βΎ β) βΎ π)) = (lim infβ(πΉ βΎ β))) |
26 | 3, 25 | eqtrd 2773 | . 2 β’ (π β (lim infβ((πΉ βΎ π) βΎ β)) = (lim infβ(πΉ βΎ β))) |
27 | 22 | resexd 5985 | . . 3 β’ (π β (πΉ βΎ π) β V) |
28 | 27 | liminfresre 44106 | . 2 β’ (π β (lim infβ((πΉ βΎ π) βΎ β)) = (lim infβ(πΉ βΎ π))) |
29 | 22 | liminfresre 44106 | . 2 β’ (π β (lim infβ(πΉ βΎ β)) = (lim infβπΉ)) |
30 | 26, 28, 29 | 3eqtr3d 2781 | 1 β’ (π β (lim infβ(πΉ βΎ π)) = (lim infβπΉ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1542 β wcel 2107 Vcvv 3444 β© cin 3910 β wss 3911 dom cdm 5634 βΎ cres 5636 Rel wrel 5639 βcfv 6497 (class class class)co 7358 βcr 11055 +βcpnf 11191 β€cz 12504 β€β₯cuz 12768 [,)cico 13272 lim infclsi 44078 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 ax-pre-sup 11134 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-er 8651 df-en 8887 df-dom 8888 df-sdom 8889 df-sup 9383 df-inf 9384 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-div 11818 df-nn 12159 df-n0 12419 df-z 12505 df-uz 12769 df-q 12879 df-ico 13276 df-liminf 44079 |
This theorem is referenced by: liminfresuz2 44114 |
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