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Mirrors > Home > MPE Home > Th. List > Mathboxes > kur14lem2 | Structured version Visualization version GIF version |
Description: Lemma for kur14 33874. 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 22425 | . . 3 β’ ((π½ β Top β§ π΄ β π) β ((intβπ½)βπ΄) = (π β ((clsβπ½)β(π β π΄)))) |
5 | 1, 2, 4 | mp2an 691 | . 2 β’ ((intβπ½)βπ΄) = (π β ((clsβπ½)β(π β π΄))) |
6 | kur14lem.i | . . 3 β’ πΌ = (intβπ½) | |
7 | 6 | fveq1i 6847 | . 2 β’ (πΌβπ΄) = ((intβπ½)βπ΄) |
8 | kur14lem.k | . . . 4 β’ πΎ = (clsβπ½) | |
9 | 8 | fveq1i 6847 | . . 3 β’ (πΎβ(π β π΄)) = ((clsβπ½)β(π β π΄)) |
10 | 9 | difeq2i 4083 | . 2 β’ (π β (πΎβ(π β π΄))) = (π β ((clsβπ½)β(π β π΄))) |
11 | 5, 7, 10 | 3eqtr4i 2771 | 1 β’ (πΌβπ΄) = (π β (πΎβ(π β π΄))) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 β wcel 2107 β cdif 3911 β wss 3914 βͺ cuni 4869 βcfv 6500 Topctop 22265 intcnt 22391 clsccl 22392 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5246 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-int 4912 df-iun 4960 df-iin 4961 df-br 5110 df-opab 5172 df-mpt 5193 df-id 5535 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-top 22266 df-cld 22393 df-ntr 22394 df-cls 22395 |
This theorem is referenced by: kur14lem6 33869 kur14lem7 33870 |
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