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Theorem iscnrm3llem2 46738
Description: Lemma for iscnrm3l 46739. If there exist disjoint open neighborhoods in the original 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 43028.) (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 3969 . . 3 (𝑙 = (𝑛𝑍) → (𝐶𝑙𝐶 ⊆ (𝑛𝑍)))
2 ineq1 4164 . . . 4 (𝑙 = (𝑛𝑍) → (𝑙𝑘) = ((𝑛𝑍) ∩ 𝑘))
32eqeq1d 2740 . . 3 (𝑙 = (𝑛𝑍) → ((𝑙𝑘) = ∅ ↔ ((𝑛𝑍) ∩ 𝑘) = ∅))
41, 33anbi13d 1439 . 2 (𝑙 = (𝑛𝑍) → ((𝐶𝑙𝐷𝑘 ∧ (𝑙𝑘) = ∅) ↔ (𝐶 ⊆ (𝑛𝑍) ∧ 𝐷𝑘 ∧ ((𝑛𝑍) ∩ 𝑘) = ∅)))
5 sseq2 3969 . . 3 (𝑘 = (𝑚𝑍) → (𝐷𝑘𝐷 ⊆ (𝑚𝑍)))
6 ineq2 4165 . . . 4 (𝑘 = (𝑚𝑍) → ((𝑛𝑍) ∩ 𝑘) = ((𝑛𝑍) ∩ (𝑚𝑍)))
76eqeq1d 2740 . . 3 (𝑘 = (𝑚𝑍) → (((𝑛𝑍) ∩ 𝑘) = ∅ ↔ ((𝑛𝑍) ∩ (𝑚𝑍)) = ∅))
85, 73anbi23d 1440 . 2 (𝑘 = (𝑚𝑍) → ((𝐶 ⊆ (𝑛𝑍) ∧ 𝐷𝑘 ∧ ((𝑛𝑍) ∩ 𝑘) = ∅) ↔ (𝐶 ⊆ (𝑛𝑍) ∧ 𝐷 ⊆ (𝑚𝑍) ∧ ((𝑛𝑍) ∩ (𝑚𝑍)) = ∅)))
9 simp11 1204 . . . 4 (((𝐽 ∈ Top ∧ (𝑍 ∈ 𝒫 𝐽𝐶 ∈ (Clsd‘(𝐽t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽t 𝑍))) ∧ (𝐶𝐷) = ∅) ∧ (𝑛𝐽𝑚𝐽) ∧ (𝐶𝑛𝐷𝑚 ∧ (𝑛𝑚) = ∅)) → 𝐽 ∈ Top)
10 simp121 1306 . . . 4 (((𝐽 ∈ Top ∧ (𝑍 ∈ 𝒫 𝐽𝐶 ∈ (Clsd‘(𝐽t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽t 𝑍))) ∧ (𝐶𝐷) = ∅) ∧ (𝑛𝐽𝑚𝐽) ∧ (𝐶𝑛𝐷𝑚 ∧ (𝑛𝑚) = ∅)) → 𝑍 ∈ 𝒫 𝐽)
11 simp2l 1200 . . . 4 (((𝐽 ∈ Top ∧ (𝑍 ∈ 𝒫 𝐽𝐶 ∈ (Clsd‘(𝐽t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽t 𝑍))) ∧ (𝐶𝐷) = ∅) ∧ (𝑛𝐽𝑚𝐽) ∧ (𝐶𝑛𝐷𝑚 ∧ (𝑛𝑚) = ∅)) → 𝑛𝐽)
12 elrestr 17245 . . . 4 ((𝐽 ∈ Top ∧ 𝑍 ∈ 𝒫 𝐽𝑛𝐽) → (𝑛𝑍) ∈ (𝐽t 𝑍))
139, 10, 11, 12syl3anc 1372 . . 3 (((𝐽 ∈ Top ∧ (𝑍 ∈ 𝒫 𝐽𝐶 ∈ (Clsd‘(𝐽t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽t 𝑍))) ∧ (𝐶𝐷) = ∅) ∧ (𝑛𝐽𝑚𝐽) ∧ (𝐶𝑛𝐷𝑚 ∧ (𝑛𝑚) = ∅)) → (𝑛𝑍) ∈ (𝐽t 𝑍))
14 simp2r 1201 . . . 4 (((𝐽 ∈ Top ∧ (𝑍 ∈ 𝒫 𝐽𝐶 ∈ (Clsd‘(𝐽t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽t 𝑍))) ∧ (𝐶𝐷) = ∅) ∧ (𝑛𝐽𝑚𝐽) ∧ (𝐶𝑛𝐷𝑚 ∧ (𝑛𝑚) = ∅)) → 𝑚𝐽)
15 elrestr 17245 . . . 4 ((𝐽 ∈ Top ∧ 𝑍 ∈ 𝒫 𝐽𝑚𝐽) → (𝑚𝑍) ∈ (𝐽t 𝑍))
169, 10, 14, 15syl3anc 1372 . . 3 (((𝐽 ∈ Top ∧ (𝑍 ∈ 𝒫 𝐽𝐶 ∈ (Clsd‘(𝐽t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽t 𝑍))) ∧ (𝐶𝐷) = ∅) ∧ (𝑛𝐽𝑚𝐽) ∧ (𝐶𝑛𝐷𝑚 ∧ (𝑛𝑚) = ∅)) → (𝑚𝑍) ∈ (𝐽t 𝑍))
17 simp31 1210 . . . . 5 (((𝐽 ∈ Top ∧ (𝑍 ∈ 𝒫 𝐽𝐶 ∈ (Clsd‘(𝐽t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽t 𝑍))) ∧ (𝐶𝐷) = ∅) ∧ (𝑛𝐽𝑚𝐽) ∧ (𝐶𝑛𝐷𝑚 ∧ (𝑛𝑚) = ∅)) → 𝐶𝑛)
18 eqidd 2739 . . . . . 6 (((𝐽 ∈ Top ∧ (𝑍 ∈ 𝒫 𝐽𝐶 ∈ (Clsd‘(𝐽t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽t 𝑍))) ∧ (𝐶𝐷) = ∅) ∧ (𝑛𝐽𝑚𝐽) ∧ (𝐶𝑛𝐷𝑚 ∧ (𝑛𝑚) = ∅)) → 𝐽 = 𝐽)
1910elpwid 4568 . . . . . 6 (((𝐽 ∈ Top ∧ (𝑍 ∈ 𝒫 𝐽𝐶 ∈ (Clsd‘(𝐽t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽t 𝑍))) ∧ (𝐶𝐷) = ∅) ∧ (𝑛𝐽𝑚𝐽) ∧ (𝐶𝑛𝐷𝑚 ∧ (𝑛𝑚) = ∅)) → 𝑍 𝐽)
20 eqidd 2739 . . . . . 6 (((𝐽 ∈ Top ∧ (𝑍 ∈ 𝒫 𝐽𝐶 ∈ (Clsd‘(𝐽t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽t 𝑍))) ∧ (𝐶𝐷) = ∅) ∧ (𝑛𝐽𝑚𝐽) ∧ (𝐶𝑛𝐷𝑚 ∧ (𝑛𝑚) = ∅)) → (𝐽t 𝑍) = (𝐽t 𝑍))
21 simp122 1307 . . . . . 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 46700 . . . . 5 (((𝐽 ∈ Top ∧ (𝑍 ∈ 𝒫 𝐽𝐶 ∈ (Clsd‘(𝐽t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽t 𝑍))) ∧ (𝐶𝐷) = ∅) ∧ (𝑛𝐽𝑚𝐽) ∧ (𝐶𝑛𝐷𝑚 ∧ (𝑛𝑚) = ∅)) → 𝐶𝑍)
2317, 22ssind 4191 . . . 4 (((𝐽 ∈ Top ∧ (𝑍 ∈ 𝒫 𝐽𝐶 ∈ (Clsd‘(𝐽t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽t 𝑍))) ∧ (𝐶𝐷) = ∅) ∧ (𝑛𝐽𝑚𝐽) ∧ (𝐶𝑛𝐷𝑚 ∧ (𝑛𝑚) = ∅)) → 𝐶 ⊆ (𝑛𝑍))
24 simp32 1211 . . . . 5 (((𝐽 ∈ Top ∧ (𝑍 ∈ 𝒫 𝐽𝐶 ∈ (Clsd‘(𝐽t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽t 𝑍))) ∧ (𝐶𝐷) = ∅) ∧ (𝑛𝐽𝑚𝐽) ∧ (𝐶𝑛𝐷𝑚 ∧ (𝑛𝑚) = ∅)) → 𝐷𝑚)
25 simp123 1308 . . . . . 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 46700 . . . . 5 (((𝐽 ∈ Top ∧ (𝑍 ∈ 𝒫 𝐽𝐶 ∈ (Clsd‘(𝐽t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽t 𝑍))) ∧ (𝐶𝐷) = ∅) ∧ (𝑛𝐽𝑚𝐽) ∧ (𝐶𝑛𝐷𝑚 ∧ (𝑛𝑚) = ∅)) → 𝐷𝑍)
2724, 26ssind 4191 . . . 4 (((𝐽 ∈ Top ∧ (𝑍 ∈ 𝒫 𝐽𝐶 ∈ (Clsd‘(𝐽t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽t 𝑍))) ∧ (𝐶𝐷) = ∅) ∧ (𝑛𝐽𝑚𝐽) ∧ (𝐶𝑛𝐷𝑚 ∧ (𝑛𝑚) = ∅)) → 𝐷 ⊆ (𝑚𝑍))
28 inss1 4187 . . . . . . 7 (𝑛𝑍) ⊆ 𝑛
29 inss1 4187 . . . . . . 7 (𝑚𝑍) ⊆ 𝑚
30 ss2in 4195 . . . . . . 7 (((𝑛𝑍) ⊆ 𝑛 ∧ (𝑚𝑍) ⊆ 𝑚) → ((𝑛𝑍) ∩ (𝑚𝑍)) ⊆ (𝑛𝑚))
3128, 29, 30mp2an 691 . . . . . 6 ((𝑛𝑍) ∩ (𝑚𝑍)) ⊆ (𝑛𝑚)
32 simp33 1212 . . . . . 6 (((𝐽 ∈ Top ∧ (𝑍 ∈ 𝒫 𝐽𝐶 ∈ (Clsd‘(𝐽t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽t 𝑍))) ∧ (𝐶𝐷) = ∅) ∧ (𝑛𝐽𝑚𝐽) ∧ (𝐶𝑛𝐷𝑚 ∧ (𝑛𝑚) = ∅)) → (𝑛𝑚) = ∅)
3331, 32sseqtrid 3995 . . . . 5 (((𝐽 ∈ Top ∧ (𝑍 ∈ 𝒫 𝐽𝐶 ∈ (Clsd‘(𝐽t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽t 𝑍))) ∧ (𝐶𝐷) = ∅) ∧ (𝑛𝐽𝑚𝐽) ∧ (𝐶𝑛𝐷𝑚 ∧ (𝑛𝑚) = ∅)) → ((𝑛𝑍) ∩ (𝑚𝑍)) ⊆ ∅)
34 ss0 4357 . . . . 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 1129 . . 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 1129 . 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 46727 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 397  w3a 1088   = wceq 1542  wcel 2107  wrex 3072  cin 3908  wss 3909  c0 4281  𝒫 cpw 4559   cuni 4864  cfv 6492  (class class class)co 7350  t crest 17237  Topctop 22165  Clsdccld 22290
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 2709  ax-rep 5241  ax-sep 5255  ax-nul 5262  ax-pow 5319  ax-pr 5383  ax-un 7663
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2888  df-ne 2943  df-ral 3064  df-rex 3073  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3739  df-csb 3855  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-pss 3928  df-nul 4282  df-if 4486  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4865  df-int 4907  df-iun 4955  df-br 5105  df-opab 5167  df-mpt 5188  df-tr 5222  df-id 5529  df-eprel 5535  df-po 5543  df-so 5544  df-fr 5586  df-we 5588  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-ord 6317  df-on 6318  df-lim 6319  df-suc 6320  df-iota 6444  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7353  df-oprab 7354  df-mpo 7355  df-om 7794  df-1st 7912  df-2nd 7913  df-en 8818  df-fin 8821  df-fi 9281  df-rest 17239  df-topgen 17260  df-top 22166  df-topon 22183  df-bases 22219  df-cld 22293
This theorem is referenced by:  iscnrm3l  46739
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