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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > xlimclim2lem | Structured version Visualization version GIF version |
Description: Lemma for xlimclim2 40861. Here it is additionally assumed that the sequence will eventually become (and stay) real. (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
Ref | Expression |
---|---|
xlimclim2lem.z | ⊢ 𝑍 = (ℤ≥‘𝑀) |
xlimclim2lem.f | ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*) |
xlimclim2lem.a | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
xlimclim2lem.r | ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶ℝ) |
Ref | Expression |
---|---|
xlimclim2lem | ⊢ (𝜑 → (𝐹~~>*𝐴 ↔ 𝐹 ⇝ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xlimclim2lem.z | . . . . . 6 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
2 | xlimclim2lem.f | . . . . . 6 ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*) | |
3 | 1, 2 | fuzxrpmcn 40849 | . . . . 5 ⊢ (𝜑 → 𝐹 ∈ (ℝ* ↑pm ℂ)) |
4 | 3 | ad2antrr 719 | . . . 4 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶ℝ) → 𝐹 ∈ (ℝ* ↑pm ℂ)) |
5 | 1 | eluzelz2 40422 | . . . . 5 ⊢ (𝑗 ∈ 𝑍 → 𝑗 ∈ ℤ) |
6 | 5 | ad2antlr 720 | . . . 4 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶ℝ) → 𝑗 ∈ ℤ) |
7 | 4, 6 | xlimres 40842 | . . 3 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶ℝ) → (𝐹~~>*𝐴 ↔ (𝐹 ↾ (ℤ≥‘𝑗))~~>*𝐴)) |
8 | eqid 2825 | . . . 4 ⊢ (ℤ≥‘𝑗) = (ℤ≥‘𝑗) | |
9 | simpr 479 | . . . 4 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶ℝ) → (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶ℝ) | |
10 | xlimclim2lem.a | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
11 | 10 | ad2antrr 719 | . . . 4 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶ℝ) → 𝐴 ∈ ℝ) |
12 | 6, 8, 9, 11 | xlimclim 40845 | . . 3 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶ℝ) → ((𝐹 ↾ (ℤ≥‘𝑗))~~>*𝐴 ↔ (𝐹 ↾ (ℤ≥‘𝑗)) ⇝ 𝐴)) |
13 | 1 | fvexi 6447 | . . . . . . 7 ⊢ 𝑍 ∈ V |
14 | 13 | a1i 11 | . . . . . 6 ⊢ (𝜑 → 𝑍 ∈ V) |
15 | 2, 14 | fexd 40111 | . . . . 5 ⊢ (𝜑 → 𝐹 ∈ V) |
16 | climres 14683 | . . . . 5 ⊢ ((𝑗 ∈ ℤ ∧ 𝐹 ∈ V) → ((𝐹 ↾ (ℤ≥‘𝑗)) ⇝ 𝐴 ↔ 𝐹 ⇝ 𝐴)) | |
17 | 5, 15, 16 | syl2anr 592 | . . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → ((𝐹 ↾ (ℤ≥‘𝑗)) ⇝ 𝐴 ↔ 𝐹 ⇝ 𝐴)) |
18 | 17 | adantr 474 | . . 3 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶ℝ) → ((𝐹 ↾ (ℤ≥‘𝑗)) ⇝ 𝐴 ↔ 𝐹 ⇝ 𝐴)) |
19 | 7, 12, 18 | 3bitrd 297 | . 2 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶ℝ) → (𝐹~~>*𝐴 ↔ 𝐹 ⇝ 𝐴)) |
20 | xlimclim2lem.r | . 2 ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶ℝ) | |
21 | 19, 20 | r19.29a 3288 | 1 ⊢ (𝜑 → (𝐹~~>*𝐴 ↔ 𝐹 ⇝ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 = wceq 1658 ∈ wcel 2166 ∃wrex 3118 Vcvv 3414 class class class wbr 4873 ↾ cres 5344 ⟶wf 6119 ‘cfv 6123 (class class class)co 6905 ↑pm cpm 8123 ℂcc 10250 ℝcr 10251 ℝ*cxr 10390 ℤcz 11704 ℤ≥cuz 11968 ⇝ cli 14592 ~~>*clsxlim 40839 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-rep 4994 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-cnex 10308 ax-resscn 10309 ax-1cn 10310 ax-icn 10311 ax-addcl 10312 ax-addrcl 10313 ax-mulcl 10314 ax-mulrcl 10315 ax-mulcom 10316 ax-addass 10317 ax-mulass 10318 ax-distr 10319 ax-i2m1 10320 ax-1ne0 10321 ax-1rid 10322 ax-rnegex 10323 ax-rrecex 10324 ax-cnre 10325 ax-pre-lttri 10326 ax-pre-lttrn 10327 ax-pre-ltadd 10328 ax-pre-mulgt0 10329 ax-pre-sup 10330 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4659 df-int 4698 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-pred 5920 df-ord 5966 df-on 5967 df-lim 5968 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-riota 6866 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-om 7327 df-1st 7428 df-2nd 7429 df-wrecs 7672 df-recs 7734 df-rdg 7772 df-1o 7826 df-oadd 7830 df-er 8009 df-map 8124 df-pm 8125 df-en 8223 df-dom 8224 df-sdom 8225 df-fin 8226 df-fi 8586 df-sup 8617 df-inf 8618 df-pnf 10393 df-mnf 10394 df-xr 10395 df-ltxr 10396 df-le 10397 df-sub 10587 df-neg 10588 df-div 11010 df-nn 11351 df-2 11414 df-3 11415 df-4 11416 df-5 11417 df-6 11418 df-7 11419 df-8 11420 df-9 11421 df-n0 11619 df-z 11705 df-dec 11822 df-uz 11969 df-q 12072 df-rp 12113 df-xneg 12232 df-xadd 12233 df-xmul 12234 df-ioo 12467 df-ioc 12468 df-ico 12469 df-icc 12470 df-fz 12620 df-fl 12888 df-seq 13096 df-exp 13155 df-cj 14216 df-re 14217 df-im 14218 df-sqrt 14352 df-abs 14353 df-clim 14596 df-rlim 14597 df-struct 16224 df-ndx 16225 df-slot 16226 df-base 16228 df-plusg 16318 df-mulr 16319 df-starv 16320 df-tset 16324 df-ple 16325 df-ds 16327 df-unif 16328 df-rest 16436 df-topn 16437 df-topgen 16457 df-ordt 16514 df-ps 17553 df-tsr 17554 df-psmet 20098 df-xmet 20099 df-met 20100 df-bl 20101 df-mopn 20102 df-cnfld 20107 df-top 21069 df-topon 21086 df-topsp 21108 df-bases 21121 df-lm 21404 df-xms 22495 df-ms 22496 df-xlim 40840 |
This theorem is referenced by: xlimclim2 40861 |
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