MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  limcfval Structured version   Visualization version   GIF version

Theorem limcfval 24157
Description: Value and set bounds on the limit operator. (Contributed by Mario Carneiro, 25-Dec-2016.)
Hypotheses
Ref Expression
limcval.j 𝐽 = (𝐾t (𝐴 ∪ {𝐵}))
limcval.k 𝐾 = (TopOpen‘ℂfld)
Assertion
Ref Expression
limcfval ((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐹 lim 𝐵) = {𝑦 ∣ (𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝑦, (𝐹𝑧))) ∈ ((𝐽 CnP 𝐾)‘𝐵)} ∧ (𝐹 lim 𝐵) ⊆ ℂ))
Distinct variable groups:   𝑦,𝑧,𝐴   𝑦,𝐵,𝑧   𝑦,𝐹,𝑧   𝑦,𝐾,𝑧   𝑦,𝐽
Allowed substitution hint:   𝐽(𝑧)

Proof of Theorem limcfval
Dummy variables 𝑓 𝑗 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-limc 24151 . . . 4 lim = (𝑓 ∈ (ℂ ↑pm ℂ), 𝑥 ∈ ℂ ↦ {𝑦[(TopOpen‘ℂfld) / 𝑗](𝑧 ∈ (dom 𝑓 ∪ {𝑥}) ↦ if(𝑧 = 𝑥, 𝑦, (𝑓𝑧))) ∈ (((𝑗t (dom 𝑓 ∪ {𝑥})) CnP 𝑗)‘𝑥)})
21a1i 11 . . 3 ((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ) → lim = (𝑓 ∈ (ℂ ↑pm ℂ), 𝑥 ∈ ℂ ↦ {𝑦[(TopOpen‘ℂfld) / 𝑗](𝑧 ∈ (dom 𝑓 ∪ {𝑥}) ↦ if(𝑧 = 𝑥, 𝑦, (𝑓𝑧))) ∈ (((𝑗t (dom 𝑓 ∪ {𝑥})) CnP 𝑗)‘𝑥)}))
3 fvexd 6560 . . . . 5 (((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑓 = 𝐹𝑥 = 𝐵)) → (TopOpen‘ℂfld) ∈ V)
4 simplrl 773 . . . . . . . . . 10 ((((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑓 = 𝐹𝑥 = 𝐵)) ∧ 𝑗 = (TopOpen‘ℂfld)) → 𝑓 = 𝐹)
54dmeqd 5667 . . . . . . . . 9 ((((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑓 = 𝐹𝑥 = 𝐵)) ∧ 𝑗 = (TopOpen‘ℂfld)) → dom 𝑓 = dom 𝐹)
6 simpll1 1205 . . . . . . . . . 10 ((((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑓 = 𝐹𝑥 = 𝐵)) ∧ 𝑗 = (TopOpen‘ℂfld)) → 𝐹:𝐴⟶ℂ)
76fdmd 6398 . . . . . . . . 9 ((((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑓 = 𝐹𝑥 = 𝐵)) ∧ 𝑗 = (TopOpen‘ℂfld)) → dom 𝐹 = 𝐴)
85, 7eqtrd 2833 . . . . . . . 8 ((((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑓 = 𝐹𝑥 = 𝐵)) ∧ 𝑗 = (TopOpen‘ℂfld)) → dom 𝑓 = 𝐴)
9 simplrr 774 . . . . . . . . 9 ((((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑓 = 𝐹𝑥 = 𝐵)) ∧ 𝑗 = (TopOpen‘ℂfld)) → 𝑥 = 𝐵)
109sneqd 4490 . . . . . . . 8 ((((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑓 = 𝐹𝑥 = 𝐵)) ∧ 𝑗 = (TopOpen‘ℂfld)) → {𝑥} = {𝐵})
118, 10uneq12d 4067 . . . . . . 7 ((((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑓 = 𝐹𝑥 = 𝐵)) ∧ 𝑗 = (TopOpen‘ℂfld)) → (dom 𝑓 ∪ {𝑥}) = (𝐴 ∪ {𝐵}))
129eqeq2d 2807 . . . . . . . 8 ((((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑓 = 𝐹𝑥 = 𝐵)) ∧ 𝑗 = (TopOpen‘ℂfld)) → (𝑧 = 𝑥𝑧 = 𝐵))
134fveq1d 6547 . . . . . . . 8 ((((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑓 = 𝐹𝑥 = 𝐵)) ∧ 𝑗 = (TopOpen‘ℂfld)) → (𝑓𝑧) = (𝐹𝑧))
1412, 13ifbieq2d 4412 . . . . . . 7 ((((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑓 = 𝐹𝑥 = 𝐵)) ∧ 𝑗 = (TopOpen‘ℂfld)) → if(𝑧 = 𝑥, 𝑦, (𝑓𝑧)) = if(𝑧 = 𝐵, 𝑦, (𝐹𝑧)))
1511, 14mpteq12dv 5052 . . . . . 6 ((((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑓 = 𝐹𝑥 = 𝐵)) ∧ 𝑗 = (TopOpen‘ℂfld)) → (𝑧 ∈ (dom 𝑓 ∪ {𝑥}) ↦ if(𝑧 = 𝑥, 𝑦, (𝑓𝑧))) = (𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝑦, (𝐹𝑧))))
16 simpr 485 . . . . . . . . . . 11 ((((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑓 = 𝐹𝑥 = 𝐵)) ∧ 𝑗 = (TopOpen‘ℂfld)) → 𝑗 = (TopOpen‘ℂfld))
17 limcval.k . . . . . . . . . . 11 𝐾 = (TopOpen‘ℂfld)
1816, 17syl6eqr 2851 . . . . . . . . . 10 ((((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑓 = 𝐹𝑥 = 𝐵)) ∧ 𝑗 = (TopOpen‘ℂfld)) → 𝑗 = 𝐾)
1918, 11oveq12d 7041 . . . . . . . . 9 ((((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑓 = 𝐹𝑥 = 𝐵)) ∧ 𝑗 = (TopOpen‘ℂfld)) → (𝑗t (dom 𝑓 ∪ {𝑥})) = (𝐾t (𝐴 ∪ {𝐵})))
20 limcval.j . . . . . . . . 9 𝐽 = (𝐾t (𝐴 ∪ {𝐵}))
2119, 20syl6eqr 2851 . . . . . . . 8 ((((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑓 = 𝐹𝑥 = 𝐵)) ∧ 𝑗 = (TopOpen‘ℂfld)) → (𝑗t (dom 𝑓 ∪ {𝑥})) = 𝐽)
2221, 18oveq12d 7041 . . . . . . 7 ((((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑓 = 𝐹𝑥 = 𝐵)) ∧ 𝑗 = (TopOpen‘ℂfld)) → ((𝑗t (dom 𝑓 ∪ {𝑥})) CnP 𝑗) = (𝐽 CnP 𝐾))
2322, 9fveq12d 6552 . . . . . 6 ((((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑓 = 𝐹𝑥 = 𝐵)) ∧ 𝑗 = (TopOpen‘ℂfld)) → (((𝑗t (dom 𝑓 ∪ {𝑥})) CnP 𝑗)‘𝑥) = ((𝐽 CnP 𝐾)‘𝐵))
2415, 23eleq12d 2879 . . . . 5 ((((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑓 = 𝐹𝑥 = 𝐵)) ∧ 𝑗 = (TopOpen‘ℂfld)) → ((𝑧 ∈ (dom 𝑓 ∪ {𝑥}) ↦ if(𝑧 = 𝑥, 𝑦, (𝑓𝑧))) ∈ (((𝑗t (dom 𝑓 ∪ {𝑥})) CnP 𝑗)‘𝑥) ↔ (𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝑦, (𝐹𝑧))) ∈ ((𝐽 CnP 𝐾)‘𝐵)))
253, 24sbcied 3748 . . . 4 (((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑓 = 𝐹𝑥 = 𝐵)) → ([(TopOpen‘ℂfld) / 𝑗](𝑧 ∈ (dom 𝑓 ∪ {𝑥}) ↦ if(𝑧 = 𝑥, 𝑦, (𝑓𝑧))) ∈ (((𝑗t (dom 𝑓 ∪ {𝑥})) CnP 𝑗)‘𝑥) ↔ (𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝑦, (𝐹𝑧))) ∈ ((𝐽 CnP 𝐾)‘𝐵)))
2625abbidv 2862 . . 3 (((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝑓 = 𝐹𝑥 = 𝐵)) → {𝑦[(TopOpen‘ℂfld) / 𝑗](𝑧 ∈ (dom 𝑓 ∪ {𝑥}) ↦ if(𝑧 = 𝑥, 𝑦, (𝑓𝑧))) ∈ (((𝑗t (dom 𝑓 ∪ {𝑥})) CnP 𝑗)‘𝑥)} = {𝑦 ∣ (𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝑦, (𝐹𝑧))) ∈ ((𝐽 CnP 𝐾)‘𝐵)})
27 cnex 10471 . . . . 5 ℂ ∈ V
28 elpm2r 8281 . . . . 5 (((ℂ ∈ V ∧ ℂ ∈ V) ∧ (𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ)) → 𝐹 ∈ (ℂ ↑pm ℂ))
2927, 27, 28mpanl12 698 . . . 4 ((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ) → 𝐹 ∈ (ℂ ↑pm ℂ))
30293adant3 1125 . . 3 ((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ) → 𝐹 ∈ (ℂ ↑pm ℂ))
31 simp3 1131 . . 3 ((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ) → 𝐵 ∈ ℂ)
32 eqid 2797 . . . . . 6 (𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝑦, (𝐹𝑧))) = (𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝑦, (𝐹𝑧)))
3320, 17, 32limcvallem 24156 . . . . 5 ((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ) → ((𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝑦, (𝐹𝑧))) ∈ ((𝐽 CnP 𝐾)‘𝐵) → 𝑦 ∈ ℂ))
3433abssdv 3972 . . . 4 ((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ) → {𝑦 ∣ (𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝑦, (𝐹𝑧))) ∈ ((𝐽 CnP 𝐾)‘𝐵)} ⊆ ℂ)
3527ssex 5123 . . . 4 ({𝑦 ∣ (𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝑦, (𝐹𝑧))) ∈ ((𝐽 CnP 𝐾)‘𝐵)} ⊆ ℂ → {𝑦 ∣ (𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝑦, (𝐹𝑧))) ∈ ((𝐽 CnP 𝐾)‘𝐵)} ∈ V)
3634, 35syl 17 . . 3 ((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ) → {𝑦 ∣ (𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝑦, (𝐹𝑧))) ∈ ((𝐽 CnP 𝐾)‘𝐵)} ∈ V)
372, 26, 30, 31, 36ovmpod 7165 . 2 ((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ) → (𝐹 lim 𝐵) = {𝑦 ∣ (𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝑦, (𝐹𝑧))) ∈ ((𝐽 CnP 𝐾)‘𝐵)})
3837, 34eqsstrd 3932 . 2 ((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ) → (𝐹 lim 𝐵) ⊆ ℂ)
3937, 38jca 512 1 ((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐹 lim 𝐵) = {𝑦 ∣ (𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝑦, (𝐹𝑧))) ∈ ((𝐽 CnP 𝐾)‘𝐵)} ∧ (𝐹 lim 𝐵) ⊆ ℂ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1080   = wceq 1525  wcel 2083  {cab 2777  Vcvv 3440  [wsbc 3711  cun 3863  wss 3865  ifcif 4387  {csn 4478  cmpt 5047  dom cdm 5450  wf 6228  cfv 6232  (class class class)co 7023  cmpo 7025  pm cpm 8264  cc 10388  t crest 16527  TopOpenctopn 16528  fldccnfld 20231   CnP ccnp 21521   lim climc 24147
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771  ax-rep 5088  ax-sep 5101  ax-nul 5108  ax-pow 5164  ax-pr 5228  ax-un 7326  ax-cnex 10446  ax-resscn 10447  ax-1cn 10448  ax-icn 10449  ax-addcl 10450  ax-addrcl 10451  ax-mulcl 10452  ax-mulrcl 10453  ax-mulcom 10454  ax-addass 10455  ax-mulass 10456  ax-distr 10457  ax-i2m1 10458  ax-1ne0 10459  ax-1rid 10460  ax-rnegex 10461  ax-rrecex 10462  ax-cnre 10463  ax-pre-lttri 10464  ax-pre-lttrn 10465  ax-pre-ltadd 10466  ax-pre-mulgt0 10467  ax-pre-sup 10468
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-mo 2578  df-eu 2614  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ne 2987  df-nel 3093  df-ral 3112  df-rex 3113  df-reu 3114  df-rmo 3115  df-rab 3116  df-v 3442  df-sbc 3712  df-csb 3818  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-pss 3882  df-nul 4218  df-if 4388  df-pw 4461  df-sn 4479  df-pr 4481  df-tp 4483  df-op 4485  df-uni 4752  df-int 4789  df-iun 4833  df-br 4969  df-opab 5031  df-mpt 5048  df-tr 5071  df-id 5355  df-eprel 5360  df-po 5369  df-so 5370  df-fr 5409  df-we 5411  df-xp 5456  df-rel 5457  df-cnv 5458  df-co 5459  df-dm 5460  df-rn 5461  df-res 5462  df-ima 5463  df-pred 6030  df-ord 6076  df-on 6077  df-lim 6078  df-suc 6079  df-iota 6196  df-fun 6234  df-fn 6235  df-f 6236  df-f1 6237  df-fo 6238  df-f1o 6239  df-fv 6240  df-riota 6984  df-ov 7026  df-oprab 7027  df-mpo 7028  df-om 7444  df-1st 7552  df-2nd 7553  df-wrecs 7805  df-recs 7867  df-rdg 7905  df-1o 7960  df-oadd 7964  df-er 8146  df-map 8265  df-pm 8266  df-en 8365  df-dom 8366  df-sdom 8367  df-fin 8368  df-fi 8728  df-sup 8759  df-inf 8760  df-pnf 10530  df-mnf 10531  df-xr 10532  df-ltxr 10533  df-le 10534  df-sub 10725  df-neg 10726  df-div 11152  df-nn 11493  df-2 11554  df-3 11555  df-4 11556  df-5 11557  df-6 11558  df-7 11559  df-8 11560  df-9 11561  df-n0 11752  df-z 11836  df-dec 11953  df-uz 12098  df-q 12202  df-rp 12244  df-xneg 12361  df-xadd 12362  df-xmul 12363  df-fz 12747  df-seq 13224  df-exp 13284  df-cj 14296  df-re 14297  df-im 14298  df-sqrt 14432  df-abs 14433  df-struct 16318  df-ndx 16319  df-slot 16320  df-base 16322  df-plusg 16411  df-mulr 16412  df-starv 16413  df-tset 16417  df-ple 16418  df-ds 16420  df-unif 16421  df-rest 16529  df-topn 16530  df-topgen 16550  df-psmet 20223  df-xmet 20224  df-met 20225  df-bl 20226  df-mopn 20227  df-cnfld 20232  df-top 21190  df-topon 21207  df-topsp 21229  df-bases 21242  df-cnp 21524  df-xms 22617  df-ms 22618  df-limc 24151
This theorem is referenced by:  ellimc  24158  limccl  24160
  Copyright terms: Public domain W3C validator