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Theorem iscnrm3l 47537
Description: Lemma for iscnrm3 47538. Given a topology 𝐽, if two separated sets can be separated by open neighborhoods, then all subspaces of the topology 𝐽 are normal, i.e., two disjoint closed sets can be separated by open neighborhoods. (Contributed by Zhi Wang, 5-Sep-2024.)
Assertion
Ref Expression
iscnrm3l (𝐽 ∈ Top → (∀𝑠 ∈ 𝒫 𝐽𝑡 ∈ 𝒫 𝐽(((𝑠 ∩ ((cls‘𝐽)‘𝑡)) = ∅ ∧ (((cls‘𝐽)‘𝑠) ∩ 𝑡) = ∅) → ∃𝑛𝐽𝑚𝐽 (𝑠𝑛𝑡𝑚 ∧ (𝑛𝑚) = ∅)) → ((𝑍 ∈ 𝒫 𝐽𝐶 ∈ (Clsd‘(𝐽t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽t 𝑍))) → ((𝐶𝐷) = ∅ → ∃𝑙 ∈ (𝐽t 𝑍)∃𝑘 ∈ (𝐽t 𝑍)(𝐶𝑙𝐷𝑘 ∧ (𝑙𝑘) = ∅)))))
Distinct variable groups:   𝐶,𝑘,𝑙,𝑚,𝑛   𝐷,𝑘,𝑙,𝑚,𝑛   𝑘,𝐽,𝑙,𝑚,𝑛   𝑘,𝑍,𝑙,𝑚,𝑛   𝐶,𝑠,𝑡,𝑚,𝑛   𝐷,𝑠,𝑡   𝐽,𝑠,𝑡
Allowed substitution hints:   𝑍(𝑡,𝑠)

Proof of Theorem iscnrm3l
StepHypRef Expression
1 simpl 483 . . . . 5 ((𝑠 = 𝐶𝑡 = 𝐷) → 𝑠 = 𝐶)
2 simpr 485 . . . . . 6 ((𝑠 = 𝐶𝑡 = 𝐷) → 𝑡 = 𝐷)
32fveq2d 6892 . . . . 5 ((𝑠 = 𝐶𝑡 = 𝐷) → ((cls‘𝐽)‘𝑡) = ((cls‘𝐽)‘𝐷))
41, 3ineq12d 4212 . . . 4 ((𝑠 = 𝐶𝑡 = 𝐷) → (𝑠 ∩ ((cls‘𝐽)‘𝑡)) = (𝐶 ∩ ((cls‘𝐽)‘𝐷)))
54eqeq1d 2734 . . 3 ((𝑠 = 𝐶𝑡 = 𝐷) → ((𝑠 ∩ ((cls‘𝐽)‘𝑡)) = ∅ ↔ (𝐶 ∩ ((cls‘𝐽)‘𝐷)) = ∅))
61fveq2d 6892 . . . . 5 ((𝑠 = 𝐶𝑡 = 𝐷) → ((cls‘𝐽)‘𝑠) = ((cls‘𝐽)‘𝐶))
76, 2ineq12d 4212 . . . 4 ((𝑠 = 𝐶𝑡 = 𝐷) → (((cls‘𝐽)‘𝑠) ∩ 𝑡) = (((cls‘𝐽)‘𝐶) ∩ 𝐷))
87eqeq1d 2734 . . 3 ((𝑠 = 𝐶𝑡 = 𝐷) → ((((cls‘𝐽)‘𝑠) ∩ 𝑡) = ∅ ↔ (((cls‘𝐽)‘𝐶) ∩ 𝐷) = ∅))
95, 8anbi12d 631 . 2 ((𝑠 = 𝐶𝑡 = 𝐷) → (((𝑠 ∩ ((cls‘𝐽)‘𝑡)) = ∅ ∧ (((cls‘𝐽)‘𝑠) ∩ 𝑡) = ∅) ↔ ((𝐶 ∩ ((cls‘𝐽)‘𝐷)) = ∅ ∧ (((cls‘𝐽)‘𝐶) ∩ 𝐷) = ∅)))
101sseq1d 4012 . . . 4 ((𝑠 = 𝐶𝑡 = 𝐷) → (𝑠𝑛𝐶𝑛))
112sseq1d 4012 . . . 4 ((𝑠 = 𝐶𝑡 = 𝐷) → (𝑡𝑚𝐷𝑚))
1210, 113anbi12d 1437 . . 3 ((𝑠 = 𝐶𝑡 = 𝐷) → ((𝑠𝑛𝑡𝑚 ∧ (𝑛𝑚) = ∅) ↔ (𝐶𝑛𝐷𝑚 ∧ (𝑛𝑚) = ∅)))
13122rexbidv 3219 . 2 ((𝑠 = 𝐶𝑡 = 𝐷) → (∃𝑛𝐽𝑚𝐽 (𝑠𝑛𝑡𝑚 ∧ (𝑛𝑚) = ∅) ↔ ∃𝑛𝐽𝑚𝐽 (𝐶𝑛𝐷𝑚 ∧ (𝑛𝑚) = ∅)))
14 iscnrm3llem1 47535 . 2 ((𝐽 ∈ Top ∧ (𝑍 ∈ 𝒫 𝐽𝐶 ∈ (Clsd‘(𝐽t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽t 𝑍))) ∧ (𝐶𝐷) = ∅) → (𝐶 ∈ 𝒫 𝐽𝐷 ∈ 𝒫 𝐽))
15 simp1 1136 . . . 4 ((𝐽 ∈ Top ∧ (𝑍 ∈ 𝒫 𝐽𝐶 ∈ (Clsd‘(𝐽t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽t 𝑍))) ∧ (𝐶𝐷) = ∅) → 𝐽 ∈ Top)
16 eqidd 2733 . . . 4 ((𝐽 ∈ Top ∧ (𝑍 ∈ 𝒫 𝐽𝐶 ∈ (Clsd‘(𝐽t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽t 𝑍))) ∧ (𝐶𝐷) = ∅) → 𝐽 = 𝐽)
17 simp21 1206 . . . . 5 ((𝐽 ∈ Top ∧ (𝑍 ∈ 𝒫 𝐽𝐶 ∈ (Clsd‘(𝐽t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽t 𝑍))) ∧ (𝐶𝐷) = ∅) → 𝑍 ∈ 𝒫 𝐽)
1817elpwid 4610 . . . 4 ((𝐽 ∈ Top ∧ (𝑍 ∈ 𝒫 𝐽𝐶 ∈ (Clsd‘(𝐽t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽t 𝑍))) ∧ (𝐶𝐷) = ∅) → 𝑍 𝐽)
19 eqidd 2733 . . . 4 ((𝐽 ∈ Top ∧ (𝑍 ∈ 𝒫 𝐽𝐶 ∈ (Clsd‘(𝐽t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽t 𝑍))) ∧ (𝐶𝐷) = ∅) → (𝐽t 𝑍) = (𝐽t 𝑍))
20 simp22 1207 . . . 4 ((𝐽 ∈ Top ∧ (𝑍 ∈ 𝒫 𝐽𝐶 ∈ (Clsd‘(𝐽t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽t 𝑍))) ∧ (𝐶𝐷) = ∅) → 𝐶 ∈ (Clsd‘(𝐽t 𝑍)))
21 simp3 1138 . . . 4 ((𝐽 ∈ Top ∧ (𝑍 ∈ 𝒫 𝐽𝐶 ∈ (Clsd‘(𝐽t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽t 𝑍))) ∧ (𝐶𝐷) = ∅) → (𝐶𝐷) = ∅)
22 simp23 1208 . . . 4 ((𝐽 ∈ Top ∧ (𝑍 ∈ 𝒫 𝐽𝐶 ∈ (Clsd‘(𝐽t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽t 𝑍))) ∧ (𝐶𝐷) = ∅) → 𝐷 ∈ (Clsd‘(𝐽t 𝑍)))
2315, 16, 18, 19, 20, 21, 22restclssep 47501 . . 3 ((𝐽 ∈ Top ∧ (𝑍 ∈ 𝒫 𝐽𝐶 ∈ (Clsd‘(𝐽t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽t 𝑍))) ∧ (𝐶𝐷) = ∅) → ((𝐶 ∩ ((cls‘𝐽)‘𝐷)) = ∅ ∧ (((cls‘𝐽)‘𝐶) ∩ 𝐷) = ∅))
24 iscnrm3llem2 47536 . . 3 ((𝐽 ∈ Top ∧ (𝑍 ∈ 𝒫 𝐽𝐶 ∈ (Clsd‘(𝐽t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽t 𝑍))) ∧ (𝐶𝐷) = ∅) → (∃𝑛𝐽𝑚𝐽 (𝐶𝑛𝐷𝑚 ∧ (𝑛𝑚) = ∅) → ∃𝑙 ∈ (𝐽t 𝑍)∃𝑘 ∈ (𝐽t 𝑍)(𝐶𝑙𝐷𝑘 ∧ (𝑙𝑘) = ∅)))
2523, 24embantd 59 . 2 ((𝐽 ∈ Top ∧ (𝑍 ∈ 𝒫 𝐽𝐶 ∈ (Clsd‘(𝐽t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽t 𝑍))) ∧ (𝐶𝐷) = ∅) → ((((𝐶 ∩ ((cls‘𝐽)‘𝐷)) = ∅ ∧ (((cls‘𝐽)‘𝐶) ∩ 𝐷) = ∅) → ∃𝑛𝐽𝑚𝐽 (𝐶𝑛𝐷𝑚 ∧ (𝑛𝑚) = ∅)) → ∃𝑙 ∈ (𝐽t 𝑍)∃𝑘 ∈ (𝐽t 𝑍)(𝐶𝑙𝐷𝑘 ∧ (𝑙𝑘) = ∅)))
269, 13, 14, 25iscnrm3lem5 47523 1 (𝐽 ∈ Top → (∀𝑠 ∈ 𝒫 𝐽𝑡 ∈ 𝒫 𝐽(((𝑠 ∩ ((cls‘𝐽)‘𝑡)) = ∅ ∧ (((cls‘𝐽)‘𝑠) ∩ 𝑡) = ∅) → ∃𝑛𝐽𝑚𝐽 (𝑠𝑛𝑡𝑚 ∧ (𝑛𝑚) = ∅)) → ((𝑍 ∈ 𝒫 𝐽𝐶 ∈ (Clsd‘(𝐽t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽t 𝑍))) → ((𝐶𝐷) = ∅ → ∃𝑙 ∈ (𝐽t 𝑍)∃𝑘 ∈ (𝐽t 𝑍)(𝐶𝑙𝐷𝑘 ∧ (𝑙𝑘) = ∅)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1087   = wceq 1541  wcel 2106  wral 3061  wrex 3070  cin 3946  wss 3947  c0 4321  𝒫 cpw 4601   cuni 4907  cfv 6540  (class class class)co 7405  t crest 17362  Topctop 22386  Clsdccld 22511  clsccl 22513
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-iin 4999  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7852  df-1st 7971  df-2nd 7972  df-en 8936  df-fin 8939  df-fi 9402  df-rest 17364  df-topgen 17385  df-top 22387  df-topon 22404  df-bases 22440  df-cld 22514  df-cls 22516
This theorem is referenced by:  iscnrm3  47538
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