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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 3956 | . 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 42859 | . . . . . 6 β’ (π β π β (π« π« π΅ βm π΅)) |
10 | elmapi 8842 | . . . . . 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 7087 | . . . 4 β’ (π β (πβπ) β π« π« π΅) |
14 | 13 | elpwid 4611 | . . 3 β’ (π β (πβπ) β π« π΅) |
15 | sseqin2 4215 | . . 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 481 | . . . 4 β’ ((π β§ π β π« π΅) β ππ»π) |
18 | 12 | adantr 481 | . . . 4 β’ ((π β§ π β π« π΅) β π β π΅) |
19 | simpr 485 | . . . 4 β’ ((π β§ π β π« π΅) β π β π« π΅) | |
20 | 2, 3, 4, 5, 6, 7, 17, 18, 19 | neicvgel1 42860 | . . 3 β’ ((π β§ π β π« π΅) β (π β (πβπ) β Β¬ (π΅ β π ) β (πβπ))) |
21 | 20 | rabbidva 3439 | . 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 2796 | 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 396 = wceq 1541 β wcel 2106 {crab 3432 Vcvv 3474 β cdif 3945 β© cin 3947 β wss 3948 π« cpw 4602 class class class wbr 5148 β¦ cmpt 5231 β ccom 5680 βΆwf 6539 βcfv 6543 (class class class)co 7408 β cmpo 7410 βm cmap 8819 |
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-iun 4999 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-ov 7411 df-oprab 7412 df-mpo 7413 df-1st 7974 df-2nd 7975 df-map 8821 |
This theorem is referenced by: (None) |
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