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Theorem iscnrm3l 48821
Description: Lemma for iscnrm3 48822. 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 482 . . . . 5 ((𝑠 = 𝐶𝑡 = 𝐷) → 𝑠 = 𝐶)
2 simpr 484 . . . . . 6 ((𝑠 = 𝐶𝑡 = 𝐷) → 𝑡 = 𝐷)
32fveq2d 6908 . . . . 5 ((𝑠 = 𝐶𝑡 = 𝐷) → ((cls‘𝐽)‘𝑡) = ((cls‘𝐽)‘𝐷))
41, 3ineq12d 4220 . . . 4 ((𝑠 = 𝐶𝑡 = 𝐷) → (𝑠 ∩ ((cls‘𝐽)‘𝑡)) = (𝐶 ∩ ((cls‘𝐽)‘𝐷)))
54eqeq1d 2738 . . 3 ((𝑠 = 𝐶𝑡 = 𝐷) → ((𝑠 ∩ ((cls‘𝐽)‘𝑡)) = ∅ ↔ (𝐶 ∩ ((cls‘𝐽)‘𝐷)) = ∅))
61fveq2d 6908 . . . . 5 ((𝑠 = 𝐶𝑡 = 𝐷) → ((cls‘𝐽)‘𝑠) = ((cls‘𝐽)‘𝐶))
76, 2ineq12d 4220 . . . 4 ((𝑠 = 𝐶𝑡 = 𝐷) → (((cls‘𝐽)‘𝑠) ∩ 𝑡) = (((cls‘𝐽)‘𝐶) ∩ 𝐷))
87eqeq1d 2738 . . 3 ((𝑠 = 𝐶𝑡 = 𝐷) → ((((cls‘𝐽)‘𝑠) ∩ 𝑡) = ∅ ↔ (((cls‘𝐽)‘𝐶) ∩ 𝐷) = ∅))
95, 8anbi12d 632 . 2 ((𝑠 = 𝐶𝑡 = 𝐷) → (((𝑠 ∩ ((cls‘𝐽)‘𝑡)) = ∅ ∧ (((cls‘𝐽)‘𝑠) ∩ 𝑡) = ∅) ↔ ((𝐶 ∩ ((cls‘𝐽)‘𝐷)) = ∅ ∧ (((cls‘𝐽)‘𝐶) ∩ 𝐷) = ∅)))
101sseq1d 4014 . . . 4 ((𝑠 = 𝐶𝑡 = 𝐷) → (𝑠𝑛𝐶𝑛))
112sseq1d 4014 . . . 4 ((𝑠 = 𝐶𝑡 = 𝐷) → (𝑡𝑚𝐷𝑚))
1210, 113anbi12d 1439 . . 3 ((𝑠 = 𝐶𝑡 = 𝐷) → ((𝑠𝑛𝑡𝑚 ∧ (𝑛𝑚) = ∅) ↔ (𝐶𝑛𝐷𝑚 ∧ (𝑛𝑚) = ∅)))
13122rexbidv 3221 . 2 ((𝑠 = 𝐶𝑡 = 𝐷) → (∃𝑛𝐽𝑚𝐽 (𝑠𝑛𝑡𝑚 ∧ (𝑛𝑚) = ∅) ↔ ∃𝑛𝐽𝑚𝐽 (𝐶𝑛𝐷𝑚 ∧ (𝑛𝑚) = ∅)))
14 iscnrm3llem1 48819 . 2 ((𝐽 ∈ Top ∧ (𝑍 ∈ 𝒫 𝐽𝐶 ∈ (Clsd‘(𝐽t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽t 𝑍))) ∧ (𝐶𝐷) = ∅) → (𝐶 ∈ 𝒫 𝐽𝐷 ∈ 𝒫 𝐽))
15 simp1 1137 . . . 4 ((𝐽 ∈ Top ∧ (𝑍 ∈ 𝒫 𝐽𝐶 ∈ (Clsd‘(𝐽t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽t 𝑍))) ∧ (𝐶𝐷) = ∅) → 𝐽 ∈ Top)
16 eqidd 2737 . . . 4 ((𝐽 ∈ Top ∧ (𝑍 ∈ 𝒫 𝐽𝐶 ∈ (Clsd‘(𝐽t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽t 𝑍))) ∧ (𝐶𝐷) = ∅) → 𝐽 = 𝐽)
17 simp21 1207 . . . . 5 ((𝐽 ∈ Top ∧ (𝑍 ∈ 𝒫 𝐽𝐶 ∈ (Clsd‘(𝐽t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽t 𝑍))) ∧ (𝐶𝐷) = ∅) → 𝑍 ∈ 𝒫 𝐽)
1817elpwid 4607 . . . 4 ((𝐽 ∈ Top ∧ (𝑍 ∈ 𝒫 𝐽𝐶 ∈ (Clsd‘(𝐽t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽t 𝑍))) ∧ (𝐶𝐷) = ∅) → 𝑍 𝐽)
19 eqidd 2737 . . . 4 ((𝐽 ∈ Top ∧ (𝑍 ∈ 𝒫 𝐽𝐶 ∈ (Clsd‘(𝐽t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽t 𝑍))) ∧ (𝐶𝐷) = ∅) → (𝐽t 𝑍) = (𝐽t 𝑍))
20 simp22 1208 . . . 4 ((𝐽 ∈ Top ∧ (𝑍 ∈ 𝒫 𝐽𝐶 ∈ (Clsd‘(𝐽t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽t 𝑍))) ∧ (𝐶𝐷) = ∅) → 𝐶 ∈ (Clsd‘(𝐽t 𝑍)))
21 simp3 1139 . . . 4 ((𝐽 ∈ Top ∧ (𝑍 ∈ 𝒫 𝐽𝐶 ∈ (Clsd‘(𝐽t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽t 𝑍))) ∧ (𝐶𝐷) = ∅) → (𝐶𝐷) = ∅)
22 simp23 1209 . . . 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 48786 . . 3 ((𝐽 ∈ Top ∧ (𝑍 ∈ 𝒫 𝐽𝐶 ∈ (Clsd‘(𝐽t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽t 𝑍))) ∧ (𝐶𝐷) = ∅) → ((𝐶 ∩ ((cls‘𝐽)‘𝐷)) = ∅ ∧ (((cls‘𝐽)‘𝐶) ∩ 𝐷) = ∅))
24 iscnrm3llem2 48820 . . 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 48807 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 395  w3a 1087   = wceq 1540  wcel 2108  wral 3060  wrex 3069  cin 3949  wss 3950  c0 4332  𝒫 cpw 4598   cuni 4905  cfv 6559  (class class class)co 7429  t crest 17461  Topctop 22889  Clsdccld 23014  clsccl 23016
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 2707  ax-rep 5277  ax-sep 5294  ax-nul 5304  ax-pow 5363  ax-pr 5430  ax-un 7751
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 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4906  df-int 4945  df-iun 4991  df-iin 4992  df-br 5142  df-opab 5204  df-mpt 5224  df-tr 5258  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5635  df-we 5637  df-xp 5689  df-rel 5690  df-cnv 5691  df-co 5692  df-dm 5693  df-rn 5694  df-res 5695  df-ima 5696  df-ord 6385  df-on 6386  df-lim 6387  df-suc 6388  df-iota 6512  df-fun 6561  df-fn 6562  df-f 6563  df-f1 6564  df-fo 6565  df-f1o 6566  df-fv 6567  df-ov 7432  df-oprab 7433  df-mpo 7434  df-om 7884  df-1st 8010  df-2nd 8011  df-en 8982  df-fin 8985  df-fi 9447  df-rest 17463  df-topgen 17484  df-top 22890  df-topon 22907  df-bases 22943  df-cld 23017  df-cls 23019
This theorem is referenced by:  iscnrm3  48822
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