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Mirrors > Home > MPE Home > Th. List > Mathboxes > kur14lem2 | Structured version Visualization version GIF version |
Description: Lemma for kur14 34202. Write interior in terms of closure and complement: ππ΄ = ππππ΄ where π is complement and π is closure. (Contributed by Mario Carneiro, 11-Feb-2015.) |
Ref | Expression |
---|---|
kur14lem.j | β’ π½ β Top |
kur14lem.x | β’ π = βͺ π½ |
kur14lem.k | β’ πΎ = (clsβπ½) |
kur14lem.i | β’ πΌ = (intβπ½) |
kur14lem.a | β’ π΄ β π |
Ref | Expression |
---|---|
kur14lem2 | β’ (πΌβπ΄) = (π β (πΎβ(π β π΄))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | kur14lem.j | . . 3 β’ π½ β Top | |
2 | kur14lem.a | . . 3 β’ π΄ β π | |
3 | kur14lem.x | . . . 4 β’ π = βͺ π½ | |
4 | 3 | ntrval2 22554 | . . 3 β’ ((π½ β Top β§ π΄ β π) β ((intβπ½)βπ΄) = (π β ((clsβπ½)β(π β π΄)))) |
5 | 1, 2, 4 | mp2an 690 | . 2 β’ ((intβπ½)βπ΄) = (π β ((clsβπ½)β(π β π΄))) |
6 | kur14lem.i | . . 3 β’ πΌ = (intβπ½) | |
7 | 6 | fveq1i 6892 | . 2 β’ (πΌβπ΄) = ((intβπ½)βπ΄) |
8 | kur14lem.k | . . . 4 β’ πΎ = (clsβπ½) | |
9 | 8 | fveq1i 6892 | . . 3 β’ (πΎβ(π β π΄)) = ((clsβπ½)β(π β π΄)) |
10 | 9 | difeq2i 4119 | . 2 β’ (π β (πΎβ(π β π΄))) = (π β ((clsβπ½)β(π β π΄))) |
11 | 5, 7, 10 | 3eqtr4i 2770 | 1 β’ (πΌβπ΄) = (π β (πΎβ(π β π΄))) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1541 β wcel 2106 β cdif 3945 β wss 3948 βͺ cuni 4908 βcfv 6543 Topctop 22394 intcnt 22520 clsccl 22521 |
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 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-top 22395 df-cld 22522 df-ntr 22523 df-cls 22524 |
This theorem is referenced by: kur14lem6 34197 kur14lem7 34198 |
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