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Theorem iscnrm3llem2 48747
Description: Lemma for iscnrm3l 48748. 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 44995.) (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 4022 . . 3 (𝑙 = (𝑛𝑍) → (𝐶𝑙𝐶 ⊆ (𝑛𝑍)))
2 ineq1 4221 . . . 4 (𝑙 = (𝑛𝑍) → (𝑙𝑘) = ((𝑛𝑍) ∩ 𝑘))
32eqeq1d 2737 . . 3 (𝑙 = (𝑛𝑍) → ((𝑙𝑘) = ∅ ↔ ((𝑛𝑍) ∩ 𝑘) = ∅))
41, 33anbi13d 1437 . 2 (𝑙 = (𝑛𝑍) → ((𝐶𝑙𝐷𝑘 ∧ (𝑙𝑘) = ∅) ↔ (𝐶 ⊆ (𝑛𝑍) ∧ 𝐷𝑘 ∧ ((𝑛𝑍) ∩ 𝑘) = ∅)))
5 sseq2 4022 . . 3 (𝑘 = (𝑚𝑍) → (𝐷𝑘𝐷 ⊆ (𝑚𝑍)))
6 ineq2 4222 . . . 4 (𝑘 = (𝑚𝑍) → ((𝑛𝑍) ∩ 𝑘) = ((𝑛𝑍) ∩ (𝑚𝑍)))
76eqeq1d 2737 . . 3 (𝑘 = (𝑚𝑍) → (((𝑛𝑍) ∩ 𝑘) = ∅ ↔ ((𝑛𝑍) ∩ (𝑚𝑍)) = ∅))
85, 73anbi23d 1438 . 2 (𝑘 = (𝑚𝑍) → ((𝐶 ⊆ (𝑛𝑍) ∧ 𝐷𝑘 ∧ ((𝑛𝑍) ∩ 𝑘) = ∅) ↔ (𝐶 ⊆ (𝑛𝑍) ∧ 𝐷 ⊆ (𝑚𝑍) ∧ ((𝑛𝑍) ∩ (𝑚𝑍)) = ∅)))
9 simp11 1202 . . . 4 (((𝐽 ∈ Top ∧ (𝑍 ∈ 𝒫 𝐽𝐶 ∈ (Clsd‘(𝐽t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽t 𝑍))) ∧ (𝐶𝐷) = ∅) ∧ (𝑛𝐽𝑚𝐽) ∧ (𝐶𝑛𝐷𝑚 ∧ (𝑛𝑚) = ∅)) → 𝐽 ∈ Top)
10 simp121 1304 . . . 4 (((𝐽 ∈ Top ∧ (𝑍 ∈ 𝒫 𝐽𝐶 ∈ (Clsd‘(𝐽t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽t 𝑍))) ∧ (𝐶𝐷) = ∅) ∧ (𝑛𝐽𝑚𝐽) ∧ (𝐶𝑛𝐷𝑚 ∧ (𝑛𝑚) = ∅)) → 𝑍 ∈ 𝒫 𝐽)
11 simp2l 1198 . . . 4 (((𝐽 ∈ Top ∧ (𝑍 ∈ 𝒫 𝐽𝐶 ∈ (Clsd‘(𝐽t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽t 𝑍))) ∧ (𝐶𝐷) = ∅) ∧ (𝑛𝐽𝑚𝐽) ∧ (𝐶𝑛𝐷𝑚 ∧ (𝑛𝑚) = ∅)) → 𝑛𝐽)
12 elrestr 17475 . . . 4 ((𝐽 ∈ Top ∧ 𝑍 ∈ 𝒫 𝐽𝑛𝐽) → (𝑛𝑍) ∈ (𝐽t 𝑍))
139, 10, 11, 12syl3anc 1370 . . 3 (((𝐽 ∈ Top ∧ (𝑍 ∈ 𝒫 𝐽𝐶 ∈ (Clsd‘(𝐽t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽t 𝑍))) ∧ (𝐶𝐷) = ∅) ∧ (𝑛𝐽𝑚𝐽) ∧ (𝐶𝑛𝐷𝑚 ∧ (𝑛𝑚) = ∅)) → (𝑛𝑍) ∈ (𝐽t 𝑍))
14 simp2r 1199 . . . 4 (((𝐽 ∈ Top ∧ (𝑍 ∈ 𝒫 𝐽𝐶 ∈ (Clsd‘(𝐽t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽t 𝑍))) ∧ (𝐶𝐷) = ∅) ∧ (𝑛𝐽𝑚𝐽) ∧ (𝐶𝑛𝐷𝑚 ∧ (𝑛𝑚) = ∅)) → 𝑚𝐽)
15 elrestr 17475 . . . 4 ((𝐽 ∈ Top ∧ 𝑍 ∈ 𝒫 𝐽𝑚𝐽) → (𝑚𝑍) ∈ (𝐽t 𝑍))
169, 10, 14, 15syl3anc 1370 . . 3 (((𝐽 ∈ Top ∧ (𝑍 ∈ 𝒫 𝐽𝐶 ∈ (Clsd‘(𝐽t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽t 𝑍))) ∧ (𝐶𝐷) = ∅) ∧ (𝑛𝐽𝑚𝐽) ∧ (𝐶𝑛𝐷𝑚 ∧ (𝑛𝑚) = ∅)) → (𝑚𝑍) ∈ (𝐽t 𝑍))
17 simp31 1208 . . . . 5 (((𝐽 ∈ Top ∧ (𝑍 ∈ 𝒫 𝐽𝐶 ∈ (Clsd‘(𝐽t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽t 𝑍))) ∧ (𝐶𝐷) = ∅) ∧ (𝑛𝐽𝑚𝐽) ∧ (𝐶𝑛𝐷𝑚 ∧ (𝑛𝑚) = ∅)) → 𝐶𝑛)
18 eqidd 2736 . . . . . 6 (((𝐽 ∈ Top ∧ (𝑍 ∈ 𝒫 𝐽𝐶 ∈ (Clsd‘(𝐽t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽t 𝑍))) ∧ (𝐶𝐷) = ∅) ∧ (𝑛𝐽𝑚𝐽) ∧ (𝐶𝑛𝐷𝑚 ∧ (𝑛𝑚) = ∅)) → 𝐽 = 𝐽)
1910elpwid 4614 . . . . . 6 (((𝐽 ∈ Top ∧ (𝑍 ∈ 𝒫 𝐽𝐶 ∈ (Clsd‘(𝐽t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽t 𝑍))) ∧ (𝐶𝐷) = ∅) ∧ (𝑛𝐽𝑚𝐽) ∧ (𝐶𝑛𝐷𝑚 ∧ (𝑛𝑚) = ∅)) → 𝑍 𝐽)
20 eqidd 2736 . . . . . 6 (((𝐽 ∈ Top ∧ (𝑍 ∈ 𝒫 𝐽𝐶 ∈ (Clsd‘(𝐽t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽t 𝑍))) ∧ (𝐶𝐷) = ∅) ∧ (𝑛𝐽𝑚𝐽) ∧ (𝐶𝑛𝐷𝑚 ∧ (𝑛𝑚) = ∅)) → (𝐽t 𝑍) = (𝐽t 𝑍))
21 simp122 1305 . . . . . 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 48709 . . . . 5 (((𝐽 ∈ Top ∧ (𝑍 ∈ 𝒫 𝐽𝐶 ∈ (Clsd‘(𝐽t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽t 𝑍))) ∧ (𝐶𝐷) = ∅) ∧ (𝑛𝐽𝑚𝐽) ∧ (𝐶𝑛𝐷𝑚 ∧ (𝑛𝑚) = ∅)) → 𝐶𝑍)
2317, 22ssind 4249 . . . 4 (((𝐽 ∈ Top ∧ (𝑍 ∈ 𝒫 𝐽𝐶 ∈ (Clsd‘(𝐽t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽t 𝑍))) ∧ (𝐶𝐷) = ∅) ∧ (𝑛𝐽𝑚𝐽) ∧ (𝐶𝑛𝐷𝑚 ∧ (𝑛𝑚) = ∅)) → 𝐶 ⊆ (𝑛𝑍))
24 simp32 1209 . . . . 5 (((𝐽 ∈ Top ∧ (𝑍 ∈ 𝒫 𝐽𝐶 ∈ (Clsd‘(𝐽t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽t 𝑍))) ∧ (𝐶𝐷) = ∅) ∧ (𝑛𝐽𝑚𝐽) ∧ (𝐶𝑛𝐷𝑚 ∧ (𝑛𝑚) = ∅)) → 𝐷𝑚)
25 simp123 1306 . . . . . 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 48709 . . . . 5 (((𝐽 ∈ Top ∧ (𝑍 ∈ 𝒫 𝐽𝐶 ∈ (Clsd‘(𝐽t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽t 𝑍))) ∧ (𝐶𝐷) = ∅) ∧ (𝑛𝐽𝑚𝐽) ∧ (𝐶𝑛𝐷𝑚 ∧ (𝑛𝑚) = ∅)) → 𝐷𝑍)
2724, 26ssind 4249 . . . 4 (((𝐽 ∈ Top ∧ (𝑍 ∈ 𝒫 𝐽𝐶 ∈ (Clsd‘(𝐽t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽t 𝑍))) ∧ (𝐶𝐷) = ∅) ∧ (𝑛𝐽𝑚𝐽) ∧ (𝐶𝑛𝐷𝑚 ∧ (𝑛𝑚) = ∅)) → 𝐷 ⊆ (𝑚𝑍))
28 inss1 4245 . . . . . . 7 (𝑛𝑍) ⊆ 𝑛
29 inss1 4245 . . . . . . 7 (𝑚𝑍) ⊆ 𝑚
30 ss2in 4253 . . . . . . 7 (((𝑛𝑍) ⊆ 𝑛 ∧ (𝑚𝑍) ⊆ 𝑚) → ((𝑛𝑍) ∩ (𝑚𝑍)) ⊆ (𝑛𝑚))
3128, 29, 30mp2an 692 . . . . . 6 ((𝑛𝑍) ∩ (𝑚𝑍)) ⊆ (𝑛𝑚)
32 simp33 1210 . . . . . 6 (((𝐽 ∈ Top ∧ (𝑍 ∈ 𝒫 𝐽𝐶 ∈ (Clsd‘(𝐽t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽t 𝑍))) ∧ (𝐶𝐷) = ∅) ∧ (𝑛𝐽𝑚𝐽) ∧ (𝐶𝑛𝐷𝑚 ∧ (𝑛𝑚) = ∅)) → (𝑛𝑚) = ∅)
3331, 32sseqtrid 4048 . . . . 5 (((𝐽 ∈ Top ∧ (𝑍 ∈ 𝒫 𝐽𝐶 ∈ (Clsd‘(𝐽t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽t 𝑍))) ∧ (𝐶𝐷) = ∅) ∧ (𝑛𝐽𝑚𝐽) ∧ (𝐶𝑛𝐷𝑚 ∧ (𝑛𝑚) = ∅)) → ((𝑛𝑍) ∩ (𝑚𝑍)) ⊆ ∅)
34 ss0 4408 . . . . 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 1127 . . 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 1127 . 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 48736 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 395  w3a 1086   = wceq 1537  wcel 2106  wrex 3068  cin 3962  wss 3963  c0 4339  𝒫 cpw 4605   cuni 4912  cfv 6563  (class class class)co 7431  t crest 17467  Topctop 22915  Clsdccld 23040
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-en 8985  df-fin 8988  df-fi 9449  df-rest 17469  df-topgen 17490  df-top 22916  df-topon 22933  df-bases 22969  df-cld 23043
This theorem is referenced by:  iscnrm3l  48748
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