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

Theorem hausflimlem 23483
Description: If 𝐴 and 𝐡 are both limits of the same filter, then all neighborhoods of 𝐴 and 𝐡 intersect. (Contributed by Mario Carneiro, 21-Sep-2015.)
Assertion
Ref Expression
hausflimlem (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝐡 ∈ (𝐽 fLim 𝐹)) ∧ (π‘ˆ ∈ 𝐽 ∧ 𝑉 ∈ 𝐽) ∧ (𝐴 ∈ π‘ˆ ∧ 𝐡 ∈ 𝑉)) β†’ (π‘ˆ ∩ 𝑉) β‰  βˆ…)

Proof of Theorem hausflimlem
StepHypRef Expression
1 simp1l 1198 . . 3 (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝐡 ∈ (𝐽 fLim 𝐹)) ∧ (π‘ˆ ∈ 𝐽 ∧ 𝑉 ∈ 𝐽) ∧ (𝐴 ∈ π‘ˆ ∧ 𝐡 ∈ 𝑉)) β†’ 𝐴 ∈ (𝐽 fLim 𝐹))
2 eqid 2733 . . . 4 βˆͺ 𝐽 = βˆͺ 𝐽
32flimfil 23473 . . 3 (𝐴 ∈ (𝐽 fLim 𝐹) β†’ 𝐹 ∈ (Filβ€˜βˆͺ 𝐽))
41, 3syl 17 . 2 (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝐡 ∈ (𝐽 fLim 𝐹)) ∧ (π‘ˆ ∈ 𝐽 ∧ 𝑉 ∈ 𝐽) ∧ (𝐴 ∈ π‘ˆ ∧ 𝐡 ∈ 𝑉)) β†’ 𝐹 ∈ (Filβ€˜βˆͺ 𝐽))
5 flimtop 23469 . . . . 5 (𝐴 ∈ (𝐽 fLim 𝐹) β†’ 𝐽 ∈ Top)
61, 5syl 17 . . . 4 (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝐡 ∈ (𝐽 fLim 𝐹)) ∧ (π‘ˆ ∈ 𝐽 ∧ 𝑉 ∈ 𝐽) ∧ (𝐴 ∈ π‘ˆ ∧ 𝐡 ∈ 𝑉)) β†’ 𝐽 ∈ Top)
7 simp2l 1200 . . . 4 (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝐡 ∈ (𝐽 fLim 𝐹)) ∧ (π‘ˆ ∈ 𝐽 ∧ 𝑉 ∈ 𝐽) ∧ (𝐴 ∈ π‘ˆ ∧ 𝐡 ∈ 𝑉)) β†’ π‘ˆ ∈ 𝐽)
8 simp3l 1202 . . . 4 (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝐡 ∈ (𝐽 fLim 𝐹)) ∧ (π‘ˆ ∈ 𝐽 ∧ 𝑉 ∈ 𝐽) ∧ (𝐴 ∈ π‘ˆ ∧ 𝐡 ∈ 𝑉)) β†’ 𝐴 ∈ π‘ˆ)
9 opnneip 22623 . . . 4 ((𝐽 ∈ Top ∧ π‘ˆ ∈ 𝐽 ∧ 𝐴 ∈ π‘ˆ) β†’ π‘ˆ ∈ ((neiβ€˜π½)β€˜{𝐴}))
106, 7, 8, 9syl3anc 1372 . . 3 (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝐡 ∈ (𝐽 fLim 𝐹)) ∧ (π‘ˆ ∈ 𝐽 ∧ 𝑉 ∈ 𝐽) ∧ (𝐴 ∈ π‘ˆ ∧ 𝐡 ∈ 𝑉)) β†’ π‘ˆ ∈ ((neiβ€˜π½)β€˜{𝐴}))
11 flimnei 23471 . . 3 ((𝐴 ∈ (𝐽 fLim 𝐹) ∧ π‘ˆ ∈ ((neiβ€˜π½)β€˜{𝐴})) β†’ π‘ˆ ∈ 𝐹)
121, 10, 11syl2anc 585 . 2 (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝐡 ∈ (𝐽 fLim 𝐹)) ∧ (π‘ˆ ∈ 𝐽 ∧ 𝑉 ∈ 𝐽) ∧ (𝐴 ∈ π‘ˆ ∧ 𝐡 ∈ 𝑉)) β†’ π‘ˆ ∈ 𝐹)
13 simp1r 1199 . . 3 (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝐡 ∈ (𝐽 fLim 𝐹)) ∧ (π‘ˆ ∈ 𝐽 ∧ 𝑉 ∈ 𝐽) ∧ (𝐴 ∈ π‘ˆ ∧ 𝐡 ∈ 𝑉)) β†’ 𝐡 ∈ (𝐽 fLim 𝐹))
14 simp2r 1201 . . . 4 (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝐡 ∈ (𝐽 fLim 𝐹)) ∧ (π‘ˆ ∈ 𝐽 ∧ 𝑉 ∈ 𝐽) ∧ (𝐴 ∈ π‘ˆ ∧ 𝐡 ∈ 𝑉)) β†’ 𝑉 ∈ 𝐽)
15 simp3r 1203 . . . 4 (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝐡 ∈ (𝐽 fLim 𝐹)) ∧ (π‘ˆ ∈ 𝐽 ∧ 𝑉 ∈ 𝐽) ∧ (𝐴 ∈ π‘ˆ ∧ 𝐡 ∈ 𝑉)) β†’ 𝐡 ∈ 𝑉)
16 opnneip 22623 . . . 4 ((𝐽 ∈ Top ∧ 𝑉 ∈ 𝐽 ∧ 𝐡 ∈ 𝑉) β†’ 𝑉 ∈ ((neiβ€˜π½)β€˜{𝐡}))
176, 14, 15, 16syl3anc 1372 . . 3 (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝐡 ∈ (𝐽 fLim 𝐹)) ∧ (π‘ˆ ∈ 𝐽 ∧ 𝑉 ∈ 𝐽) ∧ (𝐴 ∈ π‘ˆ ∧ 𝐡 ∈ 𝑉)) β†’ 𝑉 ∈ ((neiβ€˜π½)β€˜{𝐡}))
18 flimnei 23471 . . 3 ((𝐡 ∈ (𝐽 fLim 𝐹) ∧ 𝑉 ∈ ((neiβ€˜π½)β€˜{𝐡})) β†’ 𝑉 ∈ 𝐹)
1913, 17, 18syl2anc 585 . 2 (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝐡 ∈ (𝐽 fLim 𝐹)) ∧ (π‘ˆ ∈ 𝐽 ∧ 𝑉 ∈ 𝐽) ∧ (𝐴 ∈ π‘ˆ ∧ 𝐡 ∈ 𝑉)) β†’ 𝑉 ∈ 𝐹)
20 filinn0 23364 . 2 ((𝐹 ∈ (Filβ€˜βˆͺ 𝐽) ∧ π‘ˆ ∈ 𝐹 ∧ 𝑉 ∈ 𝐹) β†’ (π‘ˆ ∩ 𝑉) β‰  βˆ…)
214, 12, 19, 20syl3anc 1372 1 (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝐡 ∈ (𝐽 fLim 𝐹)) ∧ (π‘ˆ ∈ 𝐽 ∧ 𝑉 ∈ 𝐽) ∧ (𝐴 ∈ π‘ˆ ∧ 𝐡 ∈ 𝑉)) β†’ (π‘ˆ ∩ 𝑉) β‰  βˆ…)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   ∧ w3a 1088   ∈ wcel 2107   β‰  wne 2941   ∩ cin 3948  βˆ…c0 4323  {csn 4629  βˆͺ cuni 4909  β€˜cfv 6544  (class class class)co 7409  Topctop 22395  neicnei 22601  Filcfil 23349   fLim cflim 23438
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-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  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 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-fbas 20941  df-top 22396  df-nei 22602  df-fil 23350  df-flim 23443
This theorem is referenced by:  hausflimi  23484
  Copyright terms: Public domain W3C validator