![]() |
Mathbox for Richard Penner |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > ntrneifv4 | Structured version Visualization version GIF version |
Description: The value of the interior (closure) expressed in terms of the neighbors (convergents) function. (Contributed by RP, 26-Jun-2021.) |
Ref | Expression |
---|---|
ntrnei.o | β’ π = (π β V, π β V β¦ (π β (π« π βm π) β¦ (π β π β¦ {π β π β£ π β (πβπ)}))) |
ntrnei.f | β’ πΉ = (π« π΅ππ΅) |
ntrnei.r | β’ (π β πΌπΉπ) |
ntrneifv.s | β’ (π β π β π« π΅) |
Ref | Expression |
---|---|
ntrneifv4 | β’ (π β (πΌβπ) = {π₯ β π΅ β£ π β (πβπ₯)}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfin5 3943 | . 2 β’ (π΅ β© (πΌβπ)) = {π₯ β π΅ β£ π₯ β (πΌβπ)} | |
2 | ntrnei.o | . . . . . . 7 β’ π = (π β V, π β V β¦ (π β (π« π βm π) β¦ (π β π β¦ {π β π β£ π β (πβπ)}))) | |
3 | ntrnei.f | . . . . . . 7 β’ πΉ = (π« π΅ππ΅) | |
4 | ntrnei.r | . . . . . . 7 β’ (π β πΌπΉπ) | |
5 | 2, 3, 4 | ntrneiiex 42510 | . . . . . 6 β’ (π β πΌ β (π« π΅ βm π« π΅)) |
6 | elmapi 8816 | . . . . . 6 β’ (πΌ β (π« π΅ βm π« π΅) β πΌ:π« π΅βΆπ« π΅) | |
7 | 5, 6 | syl 17 | . . . . 5 β’ (π β πΌ:π« π΅βΆπ« π΅) |
8 | ntrneifv.s | . . . . 5 β’ (π β π β π« π΅) | |
9 | 7, 8 | ffvelcdmd 7063 | . . . 4 β’ (π β (πΌβπ) β π« π΅) |
10 | 9 | elpwid 4596 | . . 3 β’ (π β (πΌβπ) β π΅) |
11 | sseqin2 4202 | . . 3 β’ ((πΌβπ) β π΅ β (π΅ β© (πΌβπ)) = (πΌβπ)) | |
12 | 10, 11 | sylib 217 | . 2 β’ (π β (π΅ β© (πΌβπ)) = (πΌβπ)) |
13 | 4 | adantr 481 | . . . 4 β’ ((π β§ π₯ β π΅) β πΌπΉπ) |
14 | simpr 485 | . . . 4 β’ ((π β§ π₯ β π΅) β π₯ β π΅) | |
15 | 8 | adantr 481 | . . . 4 β’ ((π β§ π₯ β π΅) β π β π« π΅) |
16 | 2, 3, 13, 14, 15 | ntrneiel 42515 | . . 3 β’ ((π β§ π₯ β π΅) β (π₯ β (πΌβπ) β π β (πβπ₯))) |
17 | 16 | rabbidva 3432 | . 2 β’ (π β {π₯ β π΅ β£ π₯ β (πΌβπ)} = {π₯ β π΅ β£ π β (πβπ₯)}) |
18 | 1, 12, 17 | 3eqtr3a 2795 | 1 β’ (π β (πΌβπ) = {π₯ β π΅ β£ π β (πβπ₯)}) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 396 = wceq 1541 β wcel 2106 {crab 3425 Vcvv 3466 β© cin 3934 β wss 3935 π« cpw 4587 class class class wbr 5132 β¦ cmpt 5215 βΆwf 6519 βcfv 6523 (class class class)co 7384 β cmpo 7386 βm cmap 8794 |
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 2702 ax-rep 5269 ax-sep 5283 ax-nul 5290 ax-pow 5347 ax-pr 5411 ax-un 7699 |
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 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3372 df-rab 3426 df-v 3468 df-sbc 3765 df-csb 3881 df-dif 3938 df-un 3940 df-in 3942 df-ss 3952 df-nul 4310 df-if 4514 df-pw 4589 df-sn 4614 df-pr 4616 df-op 4620 df-uni 4893 df-iun 4983 df-br 5133 df-opab 5195 df-mpt 5216 df-id 5558 df-xp 5666 df-rel 5667 df-cnv 5668 df-co 5669 df-dm 5670 df-rn 5671 df-res 5672 df-ima 5673 df-iota 6475 df-fun 6525 df-fn 6526 df-f 6527 df-f1 6528 df-fo 6529 df-f1o 6530 df-fv 6531 df-ov 7387 df-oprab 7388 df-mpo 7389 df-1st 7948 df-2nd 7949 df-map 8796 |
This theorem is referenced by: ntrneiel2 42520 |
Copyright terms: Public domain | W3C validator |