Theorem iscnrm3l 49310

Description: Lemma for iscnrm3 49311. 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 482	. . . . 5 ⊢ ((𝑠 = 𝐶 ∧ 𝑡 = 𝐷) → 𝑠 = 𝐶)
2		simpr 484	. . . . . 6 ⊢ ((𝑠 = 𝐶 ∧ 𝑡 = 𝐷) → 𝑡 = 𝐷)
3	2	fveq2d 6846	. . . . 5 ⊢ ((𝑠 = 𝐶 ∧ 𝑡 = 𝐷) → ((cls‘𝐽)‘𝑡) = ((cls‘𝐽)‘𝐷))
4	1, 3	ineq12d 4175	. . . 4 ⊢ ((𝑠 = 𝐶 ∧ 𝑡 = 𝐷) → (𝑠 ∩ ((cls‘𝐽)‘𝑡)) = (𝐶 ∩ ((cls‘𝐽)‘𝐷)))
5	4	eqeq1d 2739	. . 3 ⊢ ((𝑠 = 𝐶 ∧ 𝑡 = 𝐷) → ((𝑠 ∩ ((cls‘𝐽)‘𝑡)) = ∅ ↔ (𝐶 ∩ ((cls‘𝐽)‘𝐷)) = ∅))
6	1	fveq2d 6846	. . . . 5 ⊢ ((𝑠 = 𝐶 ∧ 𝑡 = 𝐷) → ((cls‘𝐽)‘𝑠) = ((cls‘𝐽)‘𝐶))
7	6, 2	ineq12d 4175	. . . 4 ⊢ ((𝑠 = 𝐶 ∧ 𝑡 = 𝐷) → (((cls‘𝐽)‘𝑠) ∩ 𝑡) = (((cls‘𝐽)‘𝐶) ∩ 𝐷))
8	7	eqeq1d 2739	. . 3 ⊢ ((𝑠 = 𝐶 ∧ 𝑡 = 𝐷) → ((((cls‘𝐽)‘𝑠) ∩ 𝑡) = ∅ ↔ (((cls‘𝐽)‘𝐶) ∩ 𝐷) = ∅))
9	5, 8	anbi12d 633	. 2 ⊢ ((𝑠 = 𝐶 ∧ 𝑡 = 𝐷) → (((𝑠 ∩ ((cls‘𝐽)‘𝑡)) = ∅ ∧ (((cls‘𝐽)‘𝑠) ∩ 𝑡) = ∅) ↔ ((𝐶 ∩ ((cls‘𝐽)‘𝐷)) = ∅ ∧ (((cls‘𝐽)‘𝐶) ∩ 𝐷) = ∅)))
10	1	sseq1d 3967	. . . 4 ⊢ ((𝑠 = 𝐶 ∧ 𝑡 = 𝐷) → (𝑠 ⊆ 𝑛 ↔ 𝐶 ⊆ 𝑛))
11	2	sseq1d 3967	. . . 4 ⊢ ((𝑠 = 𝐶 ∧ 𝑡 = 𝐷) → (𝑡 ⊆ 𝑚 ↔ 𝐷 ⊆ 𝑚))
12	10, 11	3anbi12d 1440	. . 3 ⊢ ((𝑠 = 𝐶 ∧ 𝑡 = 𝐷) → ((𝑠 ⊆ 𝑛 ∧ 𝑡 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅) ↔ (𝐶 ⊆ 𝑛 ∧ 𝐷 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)))
13	12	2rexbidv 3203	. 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 49308	. 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 1137	. . . 4 ⊢ ((𝐽 ∈ Top ∧ (𝑍 ∈ 𝒫 ∪ 𝐽 ∧ 𝐶 ∈ (Clsd‘(𝐽 ↾t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽 ↾t 𝑍))) ∧ (𝐶 ∩ 𝐷) = ∅) → 𝐽 ∈ Top)
16		eqidd 2738	. . . 4 ⊢ ((𝐽 ∈ Top ∧ (𝑍 ∈ 𝒫 ∪ 𝐽 ∧ 𝐶 ∈ (Clsd‘(𝐽 ↾t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽 ↾t 𝑍))) ∧ (𝐶 ∩ 𝐷) = ∅) → ∪ 𝐽 = ∪ 𝐽)
17		simp21 1208	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ (𝑍 ∈ 𝒫 ∪ 𝐽 ∧ 𝐶 ∈ (Clsd‘(𝐽 ↾t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽 ↾t 𝑍))) ∧ (𝐶 ∩ 𝐷) = ∅) → 𝑍 ∈ 𝒫 ∪ 𝐽)
18	17	elpwid 4565	. . . 4 ⊢ ((𝐽 ∈ Top ∧ (𝑍 ∈ 𝒫 ∪ 𝐽 ∧ 𝐶 ∈ (Clsd‘(𝐽 ↾t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽 ↾t 𝑍))) ∧ (𝐶 ∩ 𝐷) = ∅) → 𝑍 ⊆ ∪ 𝐽)
19		eqidd 2738	. . . 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 1209	. . . 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 1139	. . . 4 ⊢ ((𝐽 ∈ Top ∧ (𝑍 ∈ 𝒫 ∪ 𝐽 ∧ 𝐶 ∈ (Clsd‘(𝐽 ↾t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽 ↾t 𝑍))) ∧ (𝐶 ∩ 𝐷) = ∅) → (𝐶 ∩ 𝐷) = ∅)
22		simp23 1210	. . . 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 49275	. . 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 49309	. . 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 49296	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 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3052  ∃wrex 3062   ∩ cin 3902   ⊆ wss 3903  ∅c0 4287  𝒫 cpw 4556  ∪ cuni 4865  ‘cfv 6500  (class class class)co 7368   ↾_t crest 17352  Topctop 22849  Clsdccld 22972  clsccl 22974

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4905  df-iun 4950  df-iin 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7371  df-oprab 7372  df-mpo 7373  df-om 7819  df-1st 7943  df-2nd 7944  df-en 8896  df-fin 8899  df-fi 9326  df-rest 17354  df-topgen 17375  df-top 22850  df-topon 22867  df-bases 22902  df-cld 22975  df-cls 22977

This theorem is referenced by:  iscnrm3  49311