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Mirrors > Home > MPE Home > Th. List > Mathboxes > iscnrm3rlem3 | Structured version Visualization version GIF version |
Description: Lemma for iscnrm3r 46300. The designed subspace is a subset of the original set; the two sets are closed sets in the subspace. (Contributed by Zhi Wang, 5-Sep-2024.) |
Ref | Expression |
---|---|
iscnrm3rlem3 | ⊢ ((𝐽 ∈ Top ∧ (𝑆 ∈ 𝒫 ∪ 𝐽 ∧ 𝑇 ∈ 𝒫 ∪ 𝐽)) → ((∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))) ∈ 𝒫 ∪ 𝐽 ∧ (((cls‘𝐽)‘𝑆) ∖ ((cls‘𝐽)‘𝑇)) ∈ (Clsd‘(𝐽 ↾t (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))))) ∧ (((cls‘𝐽)‘𝑇) ∖ ((cls‘𝐽)‘𝑆)) ∈ (Clsd‘(𝐽 ↾t (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uniexg 7625 | . . . 4 ⊢ (𝐽 ∈ Top → ∪ 𝐽 ∈ V) | |
2 | difssd 4073 | . . . 4 ⊢ (𝐽 ∈ Top → (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))) ⊆ ∪ 𝐽) | |
3 | 1, 2 | sselpwd 5259 | . . 3 ⊢ (𝐽 ∈ Top → (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))) ∈ 𝒫 ∪ 𝐽) |
4 | 3 | adantr 482 | . 2 ⊢ ((𝐽 ∈ Top ∧ (𝑆 ∈ 𝒫 ∪ 𝐽 ∧ 𝑇 ∈ 𝒫 ∪ 𝐽)) → (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))) ∈ 𝒫 ∪ 𝐽) |
5 | simpl 484 | . . 3 ⊢ ((𝐽 ∈ Top ∧ (𝑆 ∈ 𝒫 ∪ 𝐽 ∧ 𝑇 ∈ 𝒫 ∪ 𝐽)) → 𝐽 ∈ Top) | |
6 | simprl 769 | . . . 4 ⊢ ((𝐽 ∈ Top ∧ (𝑆 ∈ 𝒫 ∪ 𝐽 ∧ 𝑇 ∈ 𝒫 ∪ 𝐽)) → 𝑆 ∈ 𝒫 ∪ 𝐽) | |
7 | 6 | elpwid 4548 | . . 3 ⊢ ((𝐽 ∈ Top ∧ (𝑆 ∈ 𝒫 ∪ 𝐽 ∧ 𝑇 ∈ 𝒫 ∪ 𝐽)) → 𝑆 ⊆ ∪ 𝐽) |
8 | 5, 7 | iscnrm3rlem2 46293 | . 2 ⊢ ((𝐽 ∈ Top ∧ (𝑆 ∈ 𝒫 ∪ 𝐽 ∧ 𝑇 ∈ 𝒫 ∪ 𝐽)) → (((cls‘𝐽)‘𝑆) ∖ ((cls‘𝐽)‘𝑇)) ∈ (Clsd‘(𝐽 ↾t (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇)))))) |
9 | simprr 771 | . . . . 5 ⊢ ((𝐽 ∈ Top ∧ (𝑆 ∈ 𝒫 ∪ 𝐽 ∧ 𝑇 ∈ 𝒫 ∪ 𝐽)) → 𝑇 ∈ 𝒫 ∪ 𝐽) | |
10 | 9 | elpwid 4548 | . . . 4 ⊢ ((𝐽 ∈ Top ∧ (𝑆 ∈ 𝒫 ∪ 𝐽 ∧ 𝑇 ∈ 𝒫 ∪ 𝐽)) → 𝑇 ⊆ ∪ 𝐽) |
11 | 5, 10 | iscnrm3rlem2 46293 | . . 3 ⊢ ((𝐽 ∈ Top ∧ (𝑆 ∈ 𝒫 ∪ 𝐽 ∧ 𝑇 ∈ 𝒫 ∪ 𝐽)) → (((cls‘𝐽)‘𝑇) ∖ ((cls‘𝐽)‘𝑆)) ∈ (Clsd‘(𝐽 ↾t (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑇) ∩ ((cls‘𝐽)‘𝑆)))))) |
12 | incom 4141 | . . . . . 6 ⊢ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇)) = (((cls‘𝐽)‘𝑇) ∩ ((cls‘𝐽)‘𝑆)) | |
13 | 12 | difeq2i 4060 | . . . . 5 ⊢ (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))) = (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑇) ∩ ((cls‘𝐽)‘𝑆))) |
14 | 13 | oveq2i 7318 | . . . 4 ⊢ (𝐽 ↾t (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇)))) = (𝐽 ↾t (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑇) ∩ ((cls‘𝐽)‘𝑆)))) |
15 | 14 | fveq2i 6807 | . . 3 ⊢ (Clsd‘(𝐽 ↾t (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))))) = (Clsd‘(𝐽 ↾t (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑇) ∩ ((cls‘𝐽)‘𝑆))))) |
16 | 11, 15 | eleqtrrdi 2848 | . 2 ⊢ ((𝐽 ∈ Top ∧ (𝑆 ∈ 𝒫 ∪ 𝐽 ∧ 𝑇 ∈ 𝒫 ∪ 𝐽)) → (((cls‘𝐽)‘𝑇) ∖ ((cls‘𝐽)‘𝑆)) ∈ (Clsd‘(𝐽 ↾t (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇)))))) |
17 | 4, 8, 16 | 3jca 1128 | 1 ⊢ ((𝐽 ∈ Top ∧ (𝑆 ∈ 𝒫 ∪ 𝐽 ∧ 𝑇 ∈ 𝒫 ∪ 𝐽)) → ((∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))) ∈ 𝒫 ∪ 𝐽 ∧ (((cls‘𝐽)‘𝑆) ∖ ((cls‘𝐽)‘𝑇)) ∈ (Clsd‘(𝐽 ↾t (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))))) ∧ (((cls‘𝐽)‘𝑇) ∖ ((cls‘𝐽)‘𝑆)) ∈ (Clsd‘(𝐽 ↾t (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))))))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∧ w3a 1087 ∈ wcel 2104 Vcvv 3437 ∖ cdif 3889 ∩ cin 3891 𝒫 cpw 4539 ∪ cuni 4844 ‘cfv 6458 (class class class)co 7307 ↾t crest 17176 Topctop 22087 Clsdccld 22212 clsccl 22214 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2707 ax-rep 5218 ax-sep 5232 ax-nul 5239 ax-pow 5297 ax-pr 5361 ax-un 7620 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 846 df-3or 1088 df-3an 1089 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3286 df-rab 3287 df-v 3439 df-sbc 3722 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4566 df-pr 4568 df-op 4572 df-uni 4845 df-int 4887 df-iun 4933 df-iin 4934 df-br 5082 df-opab 5144 df-mpt 5165 df-tr 5199 df-id 5500 df-eprel 5506 df-po 5514 df-so 5515 df-fr 5555 df-we 5557 df-xp 5606 df-rel 5607 df-cnv 5608 df-co 5609 df-dm 5610 df-rn 5611 df-res 5612 df-ima 5613 df-ord 6284 df-on 6285 df-lim 6286 df-suc 6287 df-iota 6410 df-fun 6460 df-fn 6461 df-f 6462 df-f1 6463 df-fo 6464 df-f1o 6465 df-fv 6466 df-ov 7310 df-oprab 7311 df-mpo 7312 df-om 7745 df-1st 7863 df-2nd 7864 df-en 8765 df-fin 8768 df-fi 9214 df-rest 17178 df-topgen 17199 df-top 22088 df-topon 22105 df-bases 22141 df-cld 22215 df-cls 22217 |
This theorem is referenced by: iscnrm3r 46300 |
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