Theorem iscnrm3l 47104

Description: Lemma for iscnrm3 47105. 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 ∪ 𝐽∀𝑡 ∈ 𝒫 ∪ 𝐽(((𝑠 ∩ ((cls‘𝐽)‘𝑡)) = ∅ ∧ (((cls‘𝐽)‘𝑠) ∩ 𝑡) = ∅) → ∃𝑛 ∈ 𝐽 ∃𝑚 ∈ 𝐽 (𝑠 ⊆ 𝑛 ∧ 𝑡 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) → ((𝑍 ∈ 𝒫 ∪ 𝐽 ∧ 𝐶 ∈ (Clsd‘(𝐽 ↾t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽 ↾t 𝑍))) → ((𝐶 ∩ 𝐷) = ∅ → ∃𝑙 ∈ (𝐽 ↾t 𝑍)∃𝑘 ∈ (𝐽 ↾t 𝑍)(𝐶 ⊆ 𝑙 ∧ 𝐷 ⊆ 𝑘 ∧ (𝑙 ∩ 𝑘) = ∅)))))

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

Step	Hyp	Ref	Expression
1		simpl 483	. . . . 5 ⊢ ((𝑠 = 𝐶 ∧ 𝑡 = 𝐷) → 𝑠 = 𝐶)
2		simpr 485	. . . . . 6 ⊢ ((𝑠 = 𝐶 ∧ 𝑡 = 𝐷) → 𝑡 = 𝐷)
3	2	fveq2d 6851	. . . . 5 ⊢ ((𝑠 = 𝐶 ∧ 𝑡 = 𝐷) → ((cls‘𝐽)‘𝑡) = ((cls‘𝐽)‘𝐷))
4	1, 3	ineq12d 4178	. . . 4 ⊢ ((𝑠 = 𝐶 ∧ 𝑡 = 𝐷) → (𝑠 ∩ ((cls‘𝐽)‘𝑡)) = (𝐶 ∩ ((cls‘𝐽)‘𝐷)))
5	4	eqeq1d 2733	. . 3 ⊢ ((𝑠 = 𝐶 ∧ 𝑡 = 𝐷) → ((𝑠 ∩ ((cls‘𝐽)‘𝑡)) = ∅ ↔ (𝐶 ∩ ((cls‘𝐽)‘𝐷)) = ∅))
6	1	fveq2d 6851	. . . . 5 ⊢ ((𝑠 = 𝐶 ∧ 𝑡 = 𝐷) → ((cls‘𝐽)‘𝑠) = ((cls‘𝐽)‘𝐶))
7	6, 2	ineq12d 4178	. . . 4 ⊢ ((𝑠 = 𝐶 ∧ 𝑡 = 𝐷) → (((cls‘𝐽)‘𝑠) ∩ 𝑡) = (((cls‘𝐽)‘𝐶) ∩ 𝐷))
8	7	eqeq1d 2733	. . 3 ⊢ ((𝑠 = 𝐶 ∧ 𝑡 = 𝐷) → ((((cls‘𝐽)‘𝑠) ∩ 𝑡) = ∅ ↔ (((cls‘𝐽)‘𝐶) ∩ 𝐷) = ∅))
9	5, 8	anbi12d 631	. 2 ⊢ ((𝑠 = 𝐶 ∧ 𝑡 = 𝐷) → (((𝑠 ∩ ((cls‘𝐽)‘𝑡)) = ∅ ∧ (((cls‘𝐽)‘𝑠) ∩ 𝑡) = ∅) ↔ ((𝐶 ∩ ((cls‘𝐽)‘𝐷)) = ∅ ∧ (((cls‘𝐽)‘𝐶) ∩ 𝐷) = ∅)))
10	1	sseq1d 3978	. . . 4 ⊢ ((𝑠 = 𝐶 ∧ 𝑡 = 𝐷) → (𝑠 ⊆ 𝑛 ↔ 𝐶 ⊆ 𝑛))
11	2	sseq1d 3978	. . . 4 ⊢ ((𝑠 = 𝐶 ∧ 𝑡 = 𝐷) → (𝑡 ⊆ 𝑚 ↔ 𝐷 ⊆ 𝑚))
12	10, 11	3anbi12d 1437	. . 3 ⊢ ((𝑠 = 𝐶 ∧ 𝑡 = 𝐷) → ((𝑠 ⊆ 𝑛 ∧ 𝑡 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅) ↔ (𝐶 ⊆ 𝑛 ∧ 𝐷 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)))
13	12	2rexbidv 3209	. 2 ⊢ ((𝑠 = 𝐶 ∧ 𝑡 = 𝐷) → (∃𝑛 ∈ 𝐽 ∃𝑚 ∈ 𝐽 (𝑠 ⊆ 𝑛 ∧ 𝑡 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅) ↔ ∃𝑛 ∈ 𝐽 ∃𝑚 ∈ 𝐽 (𝐶 ⊆ 𝑛 ∧ 𝐷 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)))
14		iscnrm3llem1 47102	. 2 ⊢ ((𝐽 ∈ Top ∧ (𝑍 ∈ 𝒫 ∪ 𝐽 ∧ 𝐶 ∈ (Clsd‘(𝐽 ↾t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽 ↾t 𝑍))) ∧ (𝐶 ∩ 𝐷) = ∅) → (𝐶 ∈ 𝒫 ∪ 𝐽 ∧ 𝐷 ∈ 𝒫 ∪ 𝐽))
15		simp1 1136	. . . 4 ⊢ ((𝐽 ∈ Top ∧ (𝑍 ∈ 𝒫 ∪ 𝐽 ∧ 𝐶 ∈ (Clsd‘(𝐽 ↾t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽 ↾t 𝑍))) ∧ (𝐶 ∩ 𝐷) = ∅) → 𝐽 ∈ Top)
16		eqidd 2732	. . . 4 ⊢ ((𝐽 ∈ Top ∧ (𝑍 ∈ 𝒫 ∪ 𝐽 ∧ 𝐶 ∈ (Clsd‘(𝐽 ↾t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽 ↾t 𝑍))) ∧ (𝐶 ∩ 𝐷) = ∅) → ∪ 𝐽 = ∪ 𝐽)
17		simp21 1206	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ (𝑍 ∈ 𝒫 ∪ 𝐽 ∧ 𝐶 ∈ (Clsd‘(𝐽 ↾t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽 ↾t 𝑍))) ∧ (𝐶 ∩ 𝐷) = ∅) → 𝑍 ∈ 𝒫 ∪ 𝐽)
18	17	elpwid 4574	. . . 4 ⊢ ((𝐽 ∈ Top ∧ (𝑍 ∈ 𝒫 ∪ 𝐽 ∧ 𝐶 ∈ (Clsd‘(𝐽 ↾t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽 ↾t 𝑍))) ∧ (𝐶 ∩ 𝐷) = ∅) → 𝑍 ⊆ ∪ 𝐽)
19		eqidd 2732	. . . 4 ⊢ ((𝐽 ∈ Top ∧ (𝑍 ∈ 𝒫 ∪ 𝐽 ∧ 𝐶 ∈ (Clsd‘(𝐽 ↾t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽 ↾t 𝑍))) ∧ (𝐶 ∩ 𝐷) = ∅) → (𝐽 ↾t 𝑍) = (𝐽 ↾t 𝑍))
20		simp22 1207	. . . 4 ⊢ ((𝐽 ∈ Top ∧ (𝑍 ∈ 𝒫 ∪ 𝐽 ∧ 𝐶 ∈ (Clsd‘(𝐽 ↾t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽 ↾t 𝑍))) ∧ (𝐶 ∩ 𝐷) = ∅) → 𝐶 ∈ (Clsd‘(𝐽 ↾t 𝑍)))
21		simp3 1138	. . . 4 ⊢ ((𝐽 ∈ Top ∧ (𝑍 ∈ 𝒫 ∪ 𝐽 ∧ 𝐶 ∈ (Clsd‘(𝐽 ↾t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽 ↾t 𝑍))) ∧ (𝐶 ∩ 𝐷) = ∅) → (𝐶 ∩ 𝐷) = ∅)
22		simp23 1208	. . . 4 ⊢ ((𝐽 ∈ Top ∧ (𝑍 ∈ 𝒫 ∪ 𝐽 ∧ 𝐶 ∈ (Clsd‘(𝐽 ↾t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽 ↾t 𝑍))) ∧ (𝐶 ∩ 𝐷) = ∅) → 𝐷 ∈ (Clsd‘(𝐽 ↾t 𝑍)))
23	15, 16, 18, 19, 20, 21, 22	restclssep 47068	. . 3 ⊢ ((𝐽 ∈ Top ∧ (𝑍 ∈ 𝒫 ∪ 𝐽 ∧ 𝐶 ∈ (Clsd‘(𝐽 ↾t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽 ↾t 𝑍))) ∧ (𝐶 ∩ 𝐷) = ∅) → ((𝐶 ∩ ((cls‘𝐽)‘𝐷)) = ∅ ∧ (((cls‘𝐽)‘𝐶) ∩ 𝐷) = ∅))
24		iscnrm3llem2 47103	. . 3 ⊢ ((𝐽 ∈ Top ∧ (𝑍 ∈ 𝒫 ∪ 𝐽 ∧ 𝐶 ∈ (Clsd‘(𝐽 ↾t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽 ↾t 𝑍))) ∧ (𝐶 ∩ 𝐷) = ∅) → (∃𝑛 ∈ 𝐽 ∃𝑚 ∈ 𝐽 (𝐶 ⊆ 𝑛 ∧ 𝐷 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅) → ∃𝑙 ∈ (𝐽 ↾t 𝑍)∃𝑘 ∈ (𝐽 ↾t 𝑍)(𝐶 ⊆ 𝑙 ∧ 𝐷 ⊆ 𝑘 ∧ (𝑙 ∩ 𝑘) = ∅)))
25	23, 24	embantd 59	. 2 ⊢ ((𝐽 ∈ Top ∧ (𝑍 ∈ 𝒫 ∪ 𝐽 ∧ 𝐶 ∈ (Clsd‘(𝐽 ↾t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽 ↾t 𝑍))) ∧ (𝐶 ∩ 𝐷) = ∅) → ((((𝐶 ∩ ((cls‘𝐽)‘𝐷)) = ∅ ∧ (((cls‘𝐽)‘𝐶) ∩ 𝐷) = ∅) → ∃𝑛 ∈ 𝐽 ∃𝑚 ∈ 𝐽 (𝐶 ⊆ 𝑛 ∧ 𝐷 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) → ∃𝑙 ∈ (𝐽 ↾t 𝑍)∃𝑘 ∈ (𝐽 ↾t 𝑍)(𝐶 ⊆ 𝑙 ∧ 𝐷 ⊆ 𝑘 ∧ (𝑙 ∩ 𝑘) = ∅)))
26	9, 13, 14, 25	iscnrm3lem5 47090	1 ⊢ (𝐽 ∈ Top → (∀𝑠 ∈ 𝒫 ∪ 𝐽∀𝑡 ∈ 𝒫 ∪ 𝐽(((𝑠 ∩ ((cls‘𝐽)‘𝑡)) = ∅ ∧ (((cls‘𝐽)‘𝑠) ∩ 𝑡) = ∅) → ∃𝑛 ∈ 𝐽 ∃𝑚 ∈ 𝐽 (𝑠 ⊆ 𝑛 ∧ 𝑡 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) → ((𝑍 ∈ 𝒫 ∪ 𝐽 ∧ 𝐶 ∈ (Clsd‘(𝐽 ↾t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽 ↾t 𝑍))) → ((𝐶 ∩ 𝐷) = ∅ → ∃𝑙 ∈ (𝐽 ↾t 𝑍)∃𝑘 ∈ (𝐽 ↾t 𝑍)(𝐶 ⊆ 𝑙 ∧ 𝐷 ⊆ 𝑘 ∧ (𝑙 ∩ 𝑘) = ∅)))))

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  ∀wral 3060  ∃wrex 3069   ∩ cin 3912   ⊆ wss 3913  ∅c0 4287  𝒫 cpw 4565  ∪ cuni 4870  ‘cfv 6501  (class class class)co 7362   ↾_t crest 17316  Topctop 22279  Clsdccld 22404  clsccl 22406

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 2702  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677

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 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3352  df-rab 3406  df-v 3448  df-sbc 3743  df-csb 3859  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-int 4913  df-iun 4961  df-iin 4962  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7808  df-1st 7926  df-2nd 7927  df-en 8891  df-fin 8894  df-fi 9356  df-rest 17318  df-topgen 17339  df-top 22280  df-topon 22297  df-bases 22333  df-cld 22407  df-cls 22409

This theorem is referenced by:  iscnrm3  47105