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Mirrors > Home > MPE Home > Th. List > Mathboxes > neicvgfv | Structured version Visualization version GIF version |
Description: The value of the neighborhoods (convergents) in terms of the convergents (neighborhoods) function. (Contributed by RP, 27-Jun-2021.) |
Ref | Expression |
---|---|
neicvg.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)}))) |
neicvg.p | β’ π = (π β V β¦ (π β (π« π βm π« π) β¦ (π β π« π β¦ (π β (πβ(π β π)))))) |
neicvg.d | β’ π· = (πβπ΅) |
neicvg.f | β’ πΉ = (π« π΅ππ΅) |
neicvg.g | β’ πΊ = (π΅ππ« π΅) |
neicvg.h | β’ π» = (πΉ β (π· β πΊ)) |
neicvg.r | β’ (π β ππ»π) |
neicvgfv.x | β’ (π β π β π΅) |
Ref | Expression |
---|---|
neicvgfv | β’ (π β (πβπ) = {π β π« π΅ β£ Β¬ (π΅ β π ) β (πβπ)}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfin5 3949 | . 2 β’ (π« π΅ β© (πβπ)) = {π β π« π΅ β£ π β (πβπ)} | |
2 | neicvg.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 | neicvg.p | . . . . . . 7 β’ π = (π β V β¦ (π β (π« π βm π« π) β¦ (π β π« π β¦ (π β (πβ(π β π)))))) | |
4 | neicvg.d | . . . . . . 7 β’ π· = (πβπ΅) | |
5 | neicvg.f | . . . . . . 7 β’ πΉ = (π« π΅ππ΅) | |
6 | neicvg.g | . . . . . . 7 β’ πΊ = (π΅ππ« π΅) | |
7 | neicvg.h | . . . . . . 7 β’ π» = (πΉ β (π· β πΊ)) | |
8 | neicvg.r | . . . . . . 7 β’ (π β ππ»π) | |
9 | 2, 3, 4, 5, 6, 7, 8 | neicvgnex 43418 | . . . . . 6 β’ (π β π β (π« π« π΅ βm π΅)) |
10 | elmapi 8840 | . . . . . 6 β’ (π β (π« π« π΅ βm π΅) β π:π΅βΆπ« π« π΅) | |
11 | 9, 10 | syl 17 | . . . . 5 β’ (π β π:π΅βΆπ« π« π΅) |
12 | neicvgfv.x | . . . . 5 β’ (π β π β π΅) | |
13 | 11, 12 | ffvelcdmd 7078 | . . . 4 β’ (π β (πβπ) β π« π« π΅) |
14 | 13 | elpwid 4604 | . . 3 β’ (π β (πβπ) β π« π΅) |
15 | sseqin2 4208 | . . 3 β’ ((πβπ) β π« π΅ β (π« π΅ β© (πβπ)) = (πβπ)) | |
16 | 14, 15 | sylib 217 | . 2 β’ (π β (π« π΅ β© (πβπ)) = (πβπ)) |
17 | 8 | adantr 480 | . . . 4 β’ ((π β§ π β π« π΅) β ππ»π) |
18 | 12 | adantr 480 | . . . 4 β’ ((π β§ π β π« π΅) β π β π΅) |
19 | simpr 484 | . . . 4 β’ ((π β§ π β π« π΅) β π β π« π΅) | |
20 | 2, 3, 4, 5, 6, 7, 17, 18, 19 | neicvgel1 43419 | . . 3 β’ ((π β§ π β π« π΅) β (π β (πβπ) β Β¬ (π΅ β π ) β (πβπ))) |
21 | 20 | rabbidva 3431 | . 2 β’ (π β {π β π« π΅ β£ π β (πβπ)} = {π β π« π΅ β£ Β¬ (π΅ β π ) β (πβπ)}) |
22 | 1, 16, 21 | 3eqtr3a 2788 | 1 β’ (π β (πβπ) = {π β π« π΅ β£ Β¬ (π΅ β π ) β (πβπ)}) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β§ wa 395 = wceq 1533 β wcel 2098 {crab 3424 Vcvv 3466 β cdif 3938 β© cin 3940 β wss 3941 π« cpw 4595 class class class wbr 5139 β¦ cmpt 5222 β ccom 5671 βΆwf 6530 βcfv 6534 (class class class)co 7402 β cmpo 7404 βm cmap 8817 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-id 5565 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-ov 7405 df-oprab 7406 df-mpo 7407 df-1st 7969 df-2nd 7970 df-map 8819 |
This theorem is referenced by: (None) |
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