Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > lmlimxrge0 | Structured version Visualization version GIF version |
Description: Relate a limit in the nonnegative extended reals to a complex limit, provided the considered function is a real function. (Contributed by Thierry Arnoux, 11-Jul-2017.) |
Ref | Expression |
---|---|
lmlimxrge0.j | ⊢ 𝐽 = (TopOpen‘(ℝ*𝑠 ↾s (0[,]+∞))) |
lmlimxrge0.f | ⊢ (𝜑 → 𝐹:ℕ⟶𝑋) |
lmlimxrge0.p | ⊢ (𝜑 → 𝑃 ∈ 𝑋) |
lmlimxrge0.x | ⊢ 𝑋 ⊆ (0[,)+∞) |
Ref | Expression |
---|---|
lmlimxrge0 | ⊢ (𝜑 → (𝐹(⇝𝑡‘𝐽)𝑃 ↔ 𝐹 ⇝ 𝑃)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lmlimxrge0.j | . . . 4 ⊢ 𝐽 = (TopOpen‘(ℝ*𝑠 ↾s (0[,]+∞))) | |
2 | xrge0topn 29317 | . . . 4 ⊢ (TopOpen‘(ℝ*𝑠 ↾s (0[,]+∞))) = ((ordTop‘ ≤ ) ↾t (0[,]+∞)) | |
3 | 1, 2 | eqtri 2632 | . . 3 ⊢ 𝐽 = ((ordTop‘ ≤ ) ↾t (0[,]+∞)) |
4 | letopon 20819 | . . . 4 ⊢ (ordTop‘ ≤ ) ∈ (TopOn‘ℝ*) | |
5 | iccssxr 12127 | . . . 4 ⊢ (0[,]+∞) ⊆ ℝ* | |
6 | resttopon 20775 | . . . 4 ⊢ (((ordTop‘ ≤ ) ∈ (TopOn‘ℝ*) ∧ (0[,]+∞) ⊆ ℝ*) → ((ordTop‘ ≤ ) ↾t (0[,]+∞)) ∈ (TopOn‘(0[,]+∞))) | |
7 | 4, 5, 6 | mp2an 704 | . . 3 ⊢ ((ordTop‘ ≤ ) ↾t (0[,]+∞)) ∈ (TopOn‘(0[,]+∞)) |
8 | 3, 7 | eqeltri 2684 | . 2 ⊢ 𝐽 ∈ (TopOn‘(0[,]+∞)) |
9 | lmlimxrge0.f | . 2 ⊢ (𝜑 → 𝐹:ℕ⟶𝑋) | |
10 | lmlimxrge0.p | . 2 ⊢ (𝜑 → 𝑃 ∈ 𝑋) | |
11 | fvex 6113 | . . . 4 ⊢ (ordTop‘ ≤ ) ∈ V | |
12 | lmlimxrge0.x | . . . . 5 ⊢ 𝑋 ⊆ (0[,)+∞) | |
13 | icossicc 12131 | . . . . 5 ⊢ (0[,)+∞) ⊆ (0[,]+∞) | |
14 | 12, 13 | sstri 3577 | . . . 4 ⊢ 𝑋 ⊆ (0[,]+∞) |
15 | ovex 6577 | . . . 4 ⊢ (0[,]+∞) ∈ V | |
16 | restabs 20779 | . . . 4 ⊢ (((ordTop‘ ≤ ) ∈ V ∧ 𝑋 ⊆ (0[,]+∞) ∧ (0[,]+∞) ∈ V) → (((ordTop‘ ≤ ) ↾t (0[,]+∞)) ↾t 𝑋) = ((ordTop‘ ≤ ) ↾t 𝑋)) | |
17 | 11, 14, 15, 16 | mp3an 1416 | . . 3 ⊢ (((ordTop‘ ≤ ) ↾t (0[,]+∞)) ↾t 𝑋) = ((ordTop‘ ≤ ) ↾t 𝑋) |
18 | 3 | oveq1i 6559 | . . 3 ⊢ (𝐽 ↾t 𝑋) = (((ordTop‘ ≤ ) ↾t (0[,]+∞)) ↾t 𝑋) |
19 | rge0ssre 12151 | . . . . 5 ⊢ (0[,)+∞) ⊆ ℝ | |
20 | 12, 19 | sstri 3577 | . . . 4 ⊢ 𝑋 ⊆ ℝ |
21 | eqid 2610 | . . . . 5 ⊢ (TopOpen‘ℂfld) = (TopOpen‘ℂfld) | |
22 | eqid 2610 | . . . . 5 ⊢ (ordTop‘ ≤ ) = (ordTop‘ ≤ ) | |
23 | 21, 22 | xrrest2 22419 | . . . 4 ⊢ (𝑋 ⊆ ℝ → ((TopOpen‘ℂfld) ↾t 𝑋) = ((ordTop‘ ≤ ) ↾t 𝑋)) |
24 | 20, 23 | ax-mp 5 | . . 3 ⊢ ((TopOpen‘ℂfld) ↾t 𝑋) = ((ordTop‘ ≤ ) ↾t 𝑋) |
25 | 17, 18, 24 | 3eqtr4i 2642 | . 2 ⊢ (𝐽 ↾t 𝑋) = ((TopOpen‘ℂfld) ↾t 𝑋) |
26 | ax-resscn 9872 | . . 3 ⊢ ℝ ⊆ ℂ | |
27 | 20, 26 | sstri 3577 | . 2 ⊢ 𝑋 ⊆ ℂ |
28 | 8, 9, 10, 25, 27 | lmlim 29321 | 1 ⊢ (𝜑 → (𝐹(⇝𝑡‘𝐽)𝑃 ↔ 𝐹 ⇝ 𝑃)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ⊆ wss 3540 class class class wbr 4583 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ℂcc 9813 ℝcr 9814 0cc0 9815 +∞cpnf 9950 ℝ*cxr 9952 ≤ cle 9954 ℕcn 10897 [,)cico 12048 [,]cicc 12049 ⇝ cli 14063 ↾s cress 15696 ↾t crest 15904 TopOpenctopn 15905 ordTopcordt 15982 ℝ*𝑠cxrs 15983 ℂfldccnfld 19567 TopOnctopon 20518 ⇝𝑡clm 20840 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-map 7746 df-pm 7747 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fi 8200 df-sup 8231 df-inf 8232 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-q 11665 df-rp 11709 df-xneg 11822 df-xadd 11823 df-xmul 11824 df-ioo 12050 df-ioc 12051 df-ico 12052 df-icc 12053 df-fz 12198 df-seq 12664 df-exp 12723 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-clim 14067 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-starv 15783 df-tset 15787 df-ple 15788 df-ds 15791 df-unif 15792 df-rest 15906 df-topn 15907 df-topgen 15927 df-ordt 15984 df-xrs 15985 df-ps 17023 df-tsr 17024 df-psmet 19559 df-xmet 19560 df-met 19561 df-bl 19562 df-mopn 19563 df-cnfld 19568 df-top 20521 df-bases 20522 df-topon 20523 df-topsp 20524 df-lm 20843 df-xms 21935 df-ms 21936 |
This theorem is referenced by: esumcvg 29475 dstfrvclim1 29866 |
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