| 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 33942 | . . . 4 ⊢ (TopOpen‘(ℝ*𝑠 ↾s (0[,]+∞))) = ((ordTop‘ ≤ ) ↾t (0[,]+∞)) | |
| 3 | 1, 2 | eqtri 2765 | . . 3 ⊢ 𝐽 = ((ordTop‘ ≤ ) ↾t (0[,]+∞)) |
| 4 | letopon 23213 | . . . 4 ⊢ (ordTop‘ ≤ ) ∈ (TopOn‘ℝ*) | |
| 5 | iccssxr 13470 | . . . 4 ⊢ (0[,]+∞) ⊆ ℝ* | |
| 6 | resttopon 23169 | . . . 4 ⊢ (((ordTop‘ ≤ ) ∈ (TopOn‘ℝ*) ∧ (0[,]+∞) ⊆ ℝ*) → ((ordTop‘ ≤ ) ↾t (0[,]+∞)) ∈ (TopOn‘(0[,]+∞))) | |
| 7 | 4, 5, 6 | mp2an 692 | . . 3 ⊢ ((ordTop‘ ≤ ) ↾t (0[,]+∞)) ∈ (TopOn‘(0[,]+∞)) |
| 8 | 3, 7 | eqeltri 2837 | . 2 ⊢ 𝐽 ∈ (TopOn‘(0[,]+∞)) |
| 9 | lmlimxrge0.f | . 2 ⊢ (𝜑 → 𝐹:ℕ⟶𝑋) | |
| 10 | lmlimxrge0.p | . 2 ⊢ (𝜑 → 𝑃 ∈ 𝑋) | |
| 11 | fvex 6919 | . . . 4 ⊢ (ordTop‘ ≤ ) ∈ V | |
| 12 | lmlimxrge0.x | . . . . 5 ⊢ 𝑋 ⊆ (0[,)+∞) | |
| 13 | icossicc 13476 | . . . . 5 ⊢ (0[,)+∞) ⊆ (0[,]+∞) | |
| 14 | 12, 13 | sstri 3993 | . . . 4 ⊢ 𝑋 ⊆ (0[,]+∞) |
| 15 | ovex 7464 | . . . 4 ⊢ (0[,]+∞) ∈ V | |
| 16 | restabs 23173 | . . . 4 ⊢ (((ordTop‘ ≤ ) ∈ V ∧ 𝑋 ⊆ (0[,]+∞) ∧ (0[,]+∞) ∈ V) → (((ordTop‘ ≤ ) ↾t (0[,]+∞)) ↾t 𝑋) = ((ordTop‘ ≤ ) ↾t 𝑋)) | |
| 17 | 11, 14, 15, 16 | mp3an 1463 | . . 3 ⊢ (((ordTop‘ ≤ ) ↾t (0[,]+∞)) ↾t 𝑋) = ((ordTop‘ ≤ ) ↾t 𝑋) |
| 18 | 3 | oveq1i 7441 | . . 3 ⊢ (𝐽 ↾t 𝑋) = (((ordTop‘ ≤ ) ↾t (0[,]+∞)) ↾t 𝑋) |
| 19 | rge0ssre 13496 | . . . . 5 ⊢ (0[,)+∞) ⊆ ℝ | |
| 20 | 12, 19 | sstri 3993 | . . . 4 ⊢ 𝑋 ⊆ ℝ |
| 21 | eqid 2737 | . . . . 5 ⊢ (TopOpen‘ℂfld) = (TopOpen‘ℂfld) | |
| 22 | eqid 2737 | . . . . 5 ⊢ (ordTop‘ ≤ ) = (ordTop‘ ≤ ) | |
| 23 | 21, 22 | xrrest2 24830 | . . . 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 2775 | . 2 ⊢ (𝐽 ↾t 𝑋) = ((TopOpen‘ℂfld) ↾t 𝑋) |
| 26 | ax-resscn 11212 | . . 3 ⊢ ℝ ⊆ ℂ | |
| 27 | 20, 26 | sstri 3993 | . 2 ⊢ 𝑋 ⊆ ℂ |
| 28 | 8, 9, 10, 25, 27 | lmlim 33946 | 1 ⊢ (𝜑 → (𝐹(⇝𝑡‘𝐽)𝑃 ↔ 𝐹 ⇝ 𝑃)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ⊆ wss 3951 class class class wbr 5143 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 ℂcc 11153 ℝcr 11154 0cc0 11155 +∞cpnf 11292 ℝ*cxr 11294 ≤ cle 11296 ℕcn 12266 [,)cico 13389 [,]cicc 13390 ⇝ cli 15520 ↾s cress 17274 ↾t crest 17465 TopOpenctopn 17466 ordTopcordt 17544 ℝ*𝑠cxrs 17545 ℂfldccnfld 21364 TopOnctopon 22916 ⇝𝑡clm 23234 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-map 8868 df-pm 8869 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fi 9451 df-sup 9482 df-inf 9483 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-q 12991 df-rp 13035 df-xneg 13154 df-xadd 13155 df-xmul 13156 df-ioo 13391 df-ioc 13392 df-ico 13393 df-icc 13394 df-fz 13548 df-seq 14043 df-exp 14103 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-clim 15524 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-starv 17312 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-rest 17467 df-topn 17468 df-topgen 17488 df-ordt 17546 df-xrs 17547 df-ps 18611 df-tsr 18612 df-psmet 21356 df-xmet 21357 df-met 21358 df-bl 21359 df-mopn 21360 df-cnfld 21365 df-top 22900 df-topon 22917 df-topsp 22939 df-bases 22953 df-lm 23237 df-xms 24330 df-ms 24331 |
| This theorem is referenced by: esumcvg 34087 dstfrvclim1 34480 |
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