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Theorem iscnrm3llem2 45925
Description: Lemma for iscnrm3l 45926. If there exist disjoint open neighborhoods in the orignal topology for two disjoint closed sets in a subspace, then they can be separated by open neighborhoods in the subspace topology. (Could shorten proof with ssin0 42284.) (Contributed by Zhi Wang, 5-Sep-2024.)
Assertion
Ref Expression
iscnrm3llem2 ((𝐽 ∈ Top ∧ (𝑍 ∈ 𝒫 𝐽𝐶 ∈ (Clsd‘(𝐽t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽t 𝑍))) ∧ (𝐶𝐷) = ∅) → (∃𝑛𝐽𝑚𝐽 (𝐶𝑛𝐷𝑚 ∧ (𝑛𝑚) = ∅) → ∃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

Proof of Theorem iscnrm3llem2
StepHypRef Expression
1 sseq2 3932 . . 3 (𝑙 = (𝑛𝑍) → (𝐶𝑙𝐶 ⊆ (𝑛𝑍)))
2 ineq1 4125 . . . 4 (𝑙 = (𝑛𝑍) → (𝑙𝑘) = ((𝑛𝑍) ∩ 𝑘))
32eqeq1d 2739 . . 3 (𝑙 = (𝑛𝑍) → ((𝑙𝑘) = ∅ ↔ ((𝑛𝑍) ∩ 𝑘) = ∅))
41, 33anbi13d 1440 . 2 (𝑙 = (𝑛𝑍) → ((𝐶𝑙𝐷𝑘 ∧ (𝑙𝑘) = ∅) ↔ (𝐶 ⊆ (𝑛𝑍) ∧ 𝐷𝑘 ∧ ((𝑛𝑍) ∩ 𝑘) = ∅)))
5 sseq2 3932 . . 3 (𝑘 = (𝑚𝑍) → (𝐷𝑘𝐷 ⊆ (𝑚𝑍)))
6 ineq2 4126 . . . 4 (𝑘 = (𝑚𝑍) → ((𝑛𝑍) ∩ 𝑘) = ((𝑛𝑍) ∩ (𝑚𝑍)))
76eqeq1d 2739 . . 3 (𝑘 = (𝑚𝑍) → (((𝑛𝑍) ∩ 𝑘) = ∅ ↔ ((𝑛𝑍) ∩ (𝑚𝑍)) = ∅))
85, 73anbi23d 1441 . 2 (𝑘 = (𝑚𝑍) → ((𝐶 ⊆ (𝑛𝑍) ∧ 𝐷𝑘 ∧ ((𝑛𝑍) ∩ 𝑘) = ∅) ↔ (𝐶 ⊆ (𝑛𝑍) ∧ 𝐷 ⊆ (𝑚𝑍) ∧ ((𝑛𝑍) ∩ (𝑚𝑍)) = ∅)))
9 simp11 1205 . . . 4 (((𝐽 ∈ Top ∧ (𝑍 ∈ 𝒫 𝐽𝐶 ∈ (Clsd‘(𝐽t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽t 𝑍))) ∧ (𝐶𝐷) = ∅) ∧ (𝑛𝐽𝑚𝐽) ∧ (𝐶𝑛𝐷𝑚 ∧ (𝑛𝑚) = ∅)) → 𝐽 ∈ Top)
10 simp121 1307 . . . 4 (((𝐽 ∈ Top ∧ (𝑍 ∈ 𝒫 𝐽𝐶 ∈ (Clsd‘(𝐽t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽t 𝑍))) ∧ (𝐶𝐷) = ∅) ∧ (𝑛𝐽𝑚𝐽) ∧ (𝐶𝑛𝐷𝑚 ∧ (𝑛𝑚) = ∅)) → 𝑍 ∈ 𝒫 𝐽)
11 simp2l 1201 . . . 4 (((𝐽 ∈ Top ∧ (𝑍 ∈ 𝒫 𝐽𝐶 ∈ (Clsd‘(𝐽t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽t 𝑍))) ∧ (𝐶𝐷) = ∅) ∧ (𝑛𝐽𝑚𝐽) ∧ (𝐶𝑛𝐷𝑚 ∧ (𝑛𝑚) = ∅)) → 𝑛𝐽)
12 elrestr 16938 . . . 4 ((𝐽 ∈ Top ∧ 𝑍 ∈ 𝒫 𝐽𝑛𝐽) → (𝑛𝑍) ∈ (𝐽t 𝑍))
139, 10, 11, 12syl3anc 1373 . . 3 (((𝐽 ∈ Top ∧ (𝑍 ∈ 𝒫 𝐽𝐶 ∈ (Clsd‘(𝐽t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽t 𝑍))) ∧ (𝐶𝐷) = ∅) ∧ (𝑛𝐽𝑚𝐽) ∧ (𝐶𝑛𝐷𝑚 ∧ (𝑛𝑚) = ∅)) → (𝑛𝑍) ∈ (𝐽t 𝑍))
14 simp2r 1202 . . . 4 (((𝐽 ∈ Top ∧ (𝑍 ∈ 𝒫 𝐽𝐶 ∈ (Clsd‘(𝐽t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽t 𝑍))) ∧ (𝐶𝐷) = ∅) ∧ (𝑛𝐽𝑚𝐽) ∧ (𝐶𝑛𝐷𝑚 ∧ (𝑛𝑚) = ∅)) → 𝑚𝐽)
15 elrestr 16938 . . . 4 ((𝐽 ∈ Top ∧ 𝑍 ∈ 𝒫 𝐽𝑚𝐽) → (𝑚𝑍) ∈ (𝐽t 𝑍))
169, 10, 14, 15syl3anc 1373 . . 3 (((𝐽 ∈ Top ∧ (𝑍 ∈ 𝒫 𝐽𝐶 ∈ (Clsd‘(𝐽t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽t 𝑍))) ∧ (𝐶𝐷) = ∅) ∧ (𝑛𝐽𝑚𝐽) ∧ (𝐶𝑛𝐷𝑚 ∧ (𝑛𝑚) = ∅)) → (𝑚𝑍) ∈ (𝐽t 𝑍))
17 simp31 1211 . . . . 5 (((𝐽 ∈ Top ∧ (𝑍 ∈ 𝒫 𝐽𝐶 ∈ (Clsd‘(𝐽t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽t 𝑍))) ∧ (𝐶𝐷) = ∅) ∧ (𝑛𝐽𝑚𝐽) ∧ (𝐶𝑛𝐷𝑚 ∧ (𝑛𝑚) = ∅)) → 𝐶𝑛)
18 eqidd 2738 . . . . . 6 (((𝐽 ∈ Top ∧ (𝑍 ∈ 𝒫 𝐽𝐶 ∈ (Clsd‘(𝐽t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽t 𝑍))) ∧ (𝐶𝐷) = ∅) ∧ (𝑛𝐽𝑚𝐽) ∧ (𝐶𝑛𝐷𝑚 ∧ (𝑛𝑚) = ∅)) → 𝐽 = 𝐽)
1910elpwid 4529 . . . . . 6 (((𝐽 ∈ Top ∧ (𝑍 ∈ 𝒫 𝐽𝐶 ∈ (Clsd‘(𝐽t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽t 𝑍))) ∧ (𝐶𝐷) = ∅) ∧ (𝑛𝐽𝑚𝐽) ∧ (𝐶𝑛𝐷𝑚 ∧ (𝑛𝑚) = ∅)) → 𝑍 𝐽)
20 eqidd 2738 . . . . . 6 (((𝐽 ∈ Top ∧ (𝑍 ∈ 𝒫 𝐽𝐶 ∈ (Clsd‘(𝐽t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽t 𝑍))) ∧ (𝐶𝐷) = ∅) ∧ (𝑛𝐽𝑚𝐽) ∧ (𝐶𝑛𝐷𝑚 ∧ (𝑛𝑚) = ∅)) → (𝐽t 𝑍) = (𝐽t 𝑍))
21 simp122 1308 . . . . . 6 (((𝐽 ∈ Top ∧ (𝑍 ∈ 𝒫 𝐽𝐶 ∈ (Clsd‘(𝐽t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽t 𝑍))) ∧ (𝐶𝐷) = ∅) ∧ (𝑛𝐽𝑚𝐽) ∧ (𝐶𝑛𝐷𝑚 ∧ (𝑛𝑚) = ∅)) → 𝐶 ∈ (Clsd‘(𝐽t 𝑍)))
229, 18, 19, 20, 21restcls2lem 45887 . . . . 5 (((𝐽 ∈ Top ∧ (𝑍 ∈ 𝒫 𝐽𝐶 ∈ (Clsd‘(𝐽t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽t 𝑍))) ∧ (𝐶𝐷) = ∅) ∧ (𝑛𝐽𝑚𝐽) ∧ (𝐶𝑛𝐷𝑚 ∧ (𝑛𝑚) = ∅)) → 𝐶𝑍)
2317, 22ssind 4152 . . . 4 (((𝐽 ∈ Top ∧ (𝑍 ∈ 𝒫 𝐽𝐶 ∈ (Clsd‘(𝐽t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽t 𝑍))) ∧ (𝐶𝐷) = ∅) ∧ (𝑛𝐽𝑚𝐽) ∧ (𝐶𝑛𝐷𝑚 ∧ (𝑛𝑚) = ∅)) → 𝐶 ⊆ (𝑛𝑍))
24 simp32 1212 . . . . 5 (((𝐽 ∈ Top ∧ (𝑍 ∈ 𝒫 𝐽𝐶 ∈ (Clsd‘(𝐽t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽t 𝑍))) ∧ (𝐶𝐷) = ∅) ∧ (𝑛𝐽𝑚𝐽) ∧ (𝐶𝑛𝐷𝑚 ∧ (𝑛𝑚) = ∅)) → 𝐷𝑚)
25 simp123 1309 . . . . . 6 (((𝐽 ∈ Top ∧ (𝑍 ∈ 𝒫 𝐽𝐶 ∈ (Clsd‘(𝐽t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽t 𝑍))) ∧ (𝐶𝐷) = ∅) ∧ (𝑛𝐽𝑚𝐽) ∧ (𝐶𝑛𝐷𝑚 ∧ (𝑛𝑚) = ∅)) → 𝐷 ∈ (Clsd‘(𝐽t 𝑍)))
269, 18, 19, 20, 25restcls2lem 45887 . . . . 5 (((𝐽 ∈ Top ∧ (𝑍 ∈ 𝒫 𝐽𝐶 ∈ (Clsd‘(𝐽t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽t 𝑍))) ∧ (𝐶𝐷) = ∅) ∧ (𝑛𝐽𝑚𝐽) ∧ (𝐶𝑛𝐷𝑚 ∧ (𝑛𝑚) = ∅)) → 𝐷𝑍)
2724, 26ssind 4152 . . . 4 (((𝐽 ∈ Top ∧ (𝑍 ∈ 𝒫 𝐽𝐶 ∈ (Clsd‘(𝐽t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽t 𝑍))) ∧ (𝐶𝐷) = ∅) ∧ (𝑛𝐽𝑚𝐽) ∧ (𝐶𝑛𝐷𝑚 ∧ (𝑛𝑚) = ∅)) → 𝐷 ⊆ (𝑚𝑍))
28 inss1 4148 . . . . . . 7 (𝑛𝑍) ⊆ 𝑛
29 inss1 4148 . . . . . . 7 (𝑚𝑍) ⊆ 𝑚
30 ss2in 4156 . . . . . . 7 (((𝑛𝑍) ⊆ 𝑛 ∧ (𝑚𝑍) ⊆ 𝑚) → ((𝑛𝑍) ∩ (𝑚𝑍)) ⊆ (𝑛𝑚))
3128, 29, 30mp2an 692 . . . . . 6 ((𝑛𝑍) ∩ (𝑚𝑍)) ⊆ (𝑛𝑚)
32 simp33 1213 . . . . . 6 (((𝐽 ∈ Top ∧ (𝑍 ∈ 𝒫 𝐽𝐶 ∈ (Clsd‘(𝐽t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽t 𝑍))) ∧ (𝐶𝐷) = ∅) ∧ (𝑛𝐽𝑚𝐽) ∧ (𝐶𝑛𝐷𝑚 ∧ (𝑛𝑚) = ∅)) → (𝑛𝑚) = ∅)
3331, 32sseqtrid 3958 . . . . 5 (((𝐽 ∈ Top ∧ (𝑍 ∈ 𝒫 𝐽𝐶 ∈ (Clsd‘(𝐽t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽t 𝑍))) ∧ (𝐶𝐷) = ∅) ∧ (𝑛𝐽𝑚𝐽) ∧ (𝐶𝑛𝐷𝑚 ∧ (𝑛𝑚) = ∅)) → ((𝑛𝑍) ∩ (𝑚𝑍)) ⊆ ∅)
34 ss0 4318 . . . . 5 (((𝑛𝑍) ∩ (𝑚𝑍)) ⊆ ∅ → ((𝑛𝑍) ∩ (𝑚𝑍)) = ∅)
3533, 34syl 17 . . . 4 (((𝐽 ∈ Top ∧ (𝑍 ∈ 𝒫 𝐽𝐶 ∈ (Clsd‘(𝐽t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽t 𝑍))) ∧ (𝐶𝐷) = ∅) ∧ (𝑛𝐽𝑚𝐽) ∧ (𝐶𝑛𝐷𝑚 ∧ (𝑛𝑚) = ∅)) → ((𝑛𝑍) ∩ (𝑚𝑍)) = ∅)
3623, 27, 353jca 1130 . . 3 (((𝐽 ∈ Top ∧ (𝑍 ∈ 𝒫 𝐽𝐶 ∈ (Clsd‘(𝐽t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽t 𝑍))) ∧ (𝐶𝐷) = ∅) ∧ (𝑛𝐽𝑚𝐽) ∧ (𝐶𝑛𝐷𝑚 ∧ (𝑛𝑚) = ∅)) → (𝐶 ⊆ (𝑛𝑍) ∧ 𝐷 ⊆ (𝑚𝑍) ∧ ((𝑛𝑍) ∩ (𝑚𝑍)) = ∅))
3713, 16, 363jca 1130 . 2 (((𝐽 ∈ Top ∧ (𝑍 ∈ 𝒫 𝐽𝐶 ∈ (Clsd‘(𝐽t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽t 𝑍))) ∧ (𝐶𝐷) = ∅) ∧ (𝑛𝐽𝑚𝐽) ∧ (𝐶𝑛𝐷𝑚 ∧ (𝑛𝑚) = ∅)) → ((𝑛𝑍) ∈ (𝐽t 𝑍) ∧ (𝑚𝑍) ∈ (𝐽t 𝑍) ∧ (𝐶 ⊆ (𝑛𝑍) ∧ 𝐷 ⊆ (𝑚𝑍) ∧ ((𝑛𝑍) ∩ (𝑚𝑍)) = ∅)))
384, 8, 37iscnrm3lem7 45914 1 ((𝐽 ∈ Top ∧ (𝑍 ∈ 𝒫 𝐽𝐶 ∈ (Clsd‘(𝐽t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽t 𝑍))) ∧ (𝐶𝐷) = ∅) → (∃𝑛𝐽𝑚𝐽 (𝐶𝑛𝐷𝑚 ∧ (𝑛𝑚) = ∅) → ∃𝑙 ∈ (𝐽t 𝑍)∃𝑘 ∈ (𝐽t 𝑍)(𝐶𝑙𝐷𝑘 ∧ (𝑙𝑘) = ∅)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1089   = wceq 1543  wcel 2110  wrex 3062  cin 3870  wss 3871  c0 4242  𝒫 cpw 4518   cuni 4824  cfv 6385  (class class class)co 7218  t crest 16930  Topctop 21795  Clsdccld 21918
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5184  ax-sep 5197  ax-nul 5204  ax-pow 5263  ax-pr 5327  ax-un 7528
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3415  df-sbc 3700  df-csb 3817  df-dif 3874  df-un 3876  df-in 3878  df-ss 3888  df-pss 3890  df-nul 4243  df-if 4445  df-pw 4520  df-sn 4547  df-pr 4549  df-tp 4551  df-op 4553  df-uni 4825  df-int 4865  df-iun 4911  df-br 5059  df-opab 5121  df-mpt 5141  df-tr 5167  df-id 5460  df-eprel 5465  df-po 5473  df-so 5474  df-fr 5514  df-we 5516  df-xp 5562  df-rel 5563  df-cnv 5564  df-co 5565  df-dm 5566  df-rn 5567  df-res 5568  df-ima 5569  df-ord 6221  df-on 6222  df-lim 6223  df-suc 6224  df-iota 6343  df-fun 6387  df-fn 6388  df-f 6389  df-f1 6390  df-fo 6391  df-f1o 6392  df-fv 6393  df-ov 7221  df-oprab 7222  df-mpo 7223  df-om 7650  df-1st 7766  df-2nd 7767  df-en 8632  df-fin 8635  df-fi 9032  df-rest 16932  df-topgen 16953  df-top 21796  df-topon 21813  df-bases 21848  df-cld 21921
This theorem is referenced by:  iscnrm3l  45926
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