Theorem iscnrm3l 49441

Description: Lemma for iscnrm3 49442. 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 6831	. . . . 5 ⊢ ((𝑠 = 𝐶 ∧ 𝑡 = 𝐷) → ((cls‘𝐽)‘𝑡) = ((cls‘𝐽)‘𝐷))
4	1, 3	ineq12d 4150	. . . 4 ⊢ ((𝑠 = 𝐶 ∧ 𝑡 = 𝐷) → (𝑠 ∩ ((cls‘𝐽)‘𝑡)) = (𝐶 ∩ ((cls‘𝐽)‘𝐷)))
5	4	eqeq1d 2741	. . 3 ⊢ ((𝑠 = 𝐶 ∧ 𝑡 = 𝐷) → ((𝑠 ∩ ((cls‘𝐽)‘𝑡)) = ∅ ↔ (𝐶 ∩ ((cls‘𝐽)‘𝐷)) = ∅))
6	1	fveq2d 6831	. . . . 5 ⊢ ((𝑠 = 𝐶 ∧ 𝑡 = 𝐷) → ((cls‘𝐽)‘𝑠) = ((cls‘𝐽)‘𝐶))
7	6, 2	ineq12d 4150	. . . 4 ⊢ ((𝑠 = 𝐶 ∧ 𝑡 = 𝐷) → (((cls‘𝐽)‘𝑠) ∩ 𝑡) = (((cls‘𝐽)‘𝐶) ∩ 𝐷))
8	7	eqeq1d 2741	. . 3 ⊢ ((𝑠 = 𝐶 ∧ 𝑡 = 𝐷) → ((((cls‘𝐽)‘𝑠) ∩ 𝑡) = ∅ ↔ (((cls‘𝐽)‘𝐶) ∩ 𝐷) = ∅))
9	5, 8	anbi12d 638	. 2 ⊢ ((𝑠 = 𝐶 ∧ 𝑡 = 𝐷) → (((𝑠 ∩ ((cls‘𝐽)‘𝑡)) = ∅ ∧ (((cls‘𝐽)‘𝑠) ∩ 𝑡) = ∅) ↔ ((𝐶 ∩ ((cls‘𝐽)‘𝐷)) = ∅ ∧ (((cls‘𝐽)‘𝐶) ∩ 𝐷) = ∅)))
10	1	sseq1d 3946	. . . 4 ⊢ ((𝑠 = 𝐶 ∧ 𝑡 = 𝐷) → (𝑠 ⊆ 𝑛 ↔ 𝐶 ⊆ 𝑛))
11	2	sseq1d 3946	. . . 4 ⊢ ((𝑠 = 𝐶 ∧ 𝑡 = 𝐷) → (𝑡 ⊆ 𝑚 ↔ 𝐷 ⊆ 𝑚))
12	10, 11	3anbi12d 1445	. . 3 ⊢ ((𝑠 = 𝐶 ∧ 𝑡 = 𝐷) → ((𝑠 ⊆ 𝑛 ∧ 𝑡 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅) ↔ (𝐶 ⊆ 𝑛 ∧ 𝐷 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)))
13	12	2rexbidv 3204	. 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 49439	. 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 1142	. . . 4 ⊢ ((𝐽 ∈ Top ∧ (𝑍 ∈ 𝒫 ∪ 𝐽 ∧ 𝐶 ∈ (Clsd‘(𝐽 ↾t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽 ↾t 𝑍))) ∧ (𝐶 ∩ 𝐷) = ∅) → 𝐽 ∈ Top)
16		eqidd 2740	. . . 4 ⊢ ((𝐽 ∈ Top ∧ (𝑍 ∈ 𝒫 ∪ 𝐽 ∧ 𝐶 ∈ (Clsd‘(𝐽 ↾t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽 ↾t 𝑍))) ∧ (𝐶 ∩ 𝐷) = ∅) → ∪ 𝐽 = ∪ 𝐽)
17		simp21 1213	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ (𝑍 ∈ 𝒫 ∪ 𝐽 ∧ 𝐶 ∈ (Clsd‘(𝐽 ↾t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽 ↾t 𝑍))) ∧ (𝐶 ∩ 𝐷) = ∅) → 𝑍 ∈ 𝒫 ∪ 𝐽)
18	17	elpwid 4538	. . . 4 ⊢ ((𝐽 ∈ Top ∧ (𝑍 ∈ 𝒫 ∪ 𝐽 ∧ 𝐶 ∈ (Clsd‘(𝐽 ↾t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽 ↾t 𝑍))) ∧ (𝐶 ∩ 𝐷) = ∅) → 𝑍 ⊆ ∪ 𝐽)
19		eqidd 2740	. . . 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 1214	. . . 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 1144	. . . 4 ⊢ ((𝐽 ∈ Top ∧ (𝑍 ∈ 𝒫 ∪ 𝐽 ∧ 𝐶 ∈ (Clsd‘(𝐽 ↾t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽 ↾t 𝑍))) ∧ (𝐶 ∩ 𝐷) = ∅) → (𝐶 ∩ 𝐷) = ∅)
22		simp23 1215	. . . 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 49406	. . 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 49440	. . 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 49427	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 1092   = wceq 1547   ∈ wcel 2119  ∀wral 3053  ∃wrex 3063   ∩ cin 3882   ⊆ wss 3883  ∅c0 4261  𝒫 cpw 4529  ∪ cuni 4838  ‘cfv 6485  (class class class)co 7356   ↾_t crest 17374  Topctop 22876  Clsdccld 22999  clsccl 23001

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-int 4878  df-iun 4923  df-iin 4924  df-br 5073  df-opab 5135  df-mpt 5154  df-tr 5180  df-id 5513  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5571  df-we 5573  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-ord 6313  df-on 6314  df-lim 6315  df-suc 6316  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-1st 7931  df-2nd 7932  df-en 8884  df-fin 8887  df-fi 9314  df-rest 17376  df-topgen 17397  df-top 22877  df-topon 22894  df-bases 22929  df-cld 23002  df-cls 23004

This theorem is referenced by:  iscnrm3  49442