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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > lmbr3v | Structured version Visualization version GIF version |
Description: Express the binary relation "sequence πΉ converges to point π " in a metric space using an arbitrary upper set of integers. (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
Ref | Expression |
---|---|
lmbr3v.1 | β’ (π β π½ β (TopOnβπ)) |
Ref | Expression |
---|---|
lmbr3v | β’ (π β (πΉ(βπ‘βπ½)π β (πΉ β (π βpm β) β§ π β π β§ βπ’ β π½ (π β π’ β βπ β β€ βπ β (β€β₯βπ)(π β dom πΉ β§ (πΉβπ) β π’))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lmbr3v.1 | . . 3 β’ (π β π½ β (TopOnβπ)) | |
2 | eqid 2732 | . . 3 β’ (β€β₯β0) = (β€β₯β0) | |
3 | 0zd 12519 | . . 3 β’ (π β 0 β β€) | |
4 | 1, 2, 3 | lmbr2 22633 | . 2 β’ (π β (πΉ(βπ‘βπ½)π β (πΉ β (π βpm β) β§ π β π β§ βπ’ β π½ (π β π’ β βπ β (β€β₯β0)βπ β (β€β₯βπ)(π β dom πΉ β§ (πΉβπ) β π’))))) |
5 | 0z 12518 | . . . . . 6 β’ 0 β β€ | |
6 | 2 | rexuz3 15242 | . . . . . 6 β’ (0 β β€ β (βπ β (β€β₯β0)βπ β (β€β₯βπ)(π β dom πΉ β§ (πΉβπ) β π’) β βπ β β€ βπ β (β€β₯βπ)(π β dom πΉ β§ (πΉβπ) β π’))) |
7 | 5, 6 | ax-mp 5 | . . . . 5 β’ (βπ β (β€β₯β0)βπ β (β€β₯βπ)(π β dom πΉ β§ (πΉβπ) β π’) β βπ β β€ βπ β (β€β₯βπ)(π β dom πΉ β§ (πΉβπ) β π’)) |
8 | 7 | imbi2i 336 | . . . 4 β’ ((π β π’ β βπ β (β€β₯β0)βπ β (β€β₯βπ)(π β dom πΉ β§ (πΉβπ) β π’)) β (π β π’ β βπ β β€ βπ β (β€β₯βπ)(π β dom πΉ β§ (πΉβπ) β π’))) |
9 | 8 | ralbii 3093 | . . 3 β’ (βπ’ β π½ (π β π’ β βπ β (β€β₯β0)βπ β (β€β₯βπ)(π β dom πΉ β§ (πΉβπ) β π’)) β βπ’ β π½ (π β π’ β βπ β β€ βπ β (β€β₯βπ)(π β dom πΉ β§ (πΉβπ) β π’))) |
10 | 9 | 3anbi3i 1160 | . 2 β’ ((πΉ β (π βpm β) β§ π β π β§ βπ’ β π½ (π β π’ β βπ β (β€β₯β0)βπ β (β€β₯βπ)(π β dom πΉ β§ (πΉβπ) β π’))) β (πΉ β (π βpm β) β§ π β π β§ βπ’ β π½ (π β π’ β βπ β β€ βπ β (β€β₯βπ)(π β dom πΉ β§ (πΉβπ) β π’)))) |
11 | 4, 10 | bitrdi 287 | 1 β’ (π β (πΉ(βπ‘βπ½)π β (πΉ β (π βpm β) β§ π β π β§ βπ’ β π½ (π β π’ β βπ β β€ βπ β (β€β₯βπ)(π β dom πΉ β§ (πΉβπ) β π’))))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 397 β§ w3a 1088 β wcel 2107 βwral 3061 βwrex 3070 class class class wbr 5109 dom cdm 5637 βcfv 6500 (class class class)co 7361 βpm cpm 8772 βcc 11057 0cc0 11059 β€cz 12507 β€β₯cuz 12771 TopOnctopon 22282 βπ‘clm 22600 |
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 2703 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-addrcl 11120 ax-rnegex 11130 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 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-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-id 5535 df-po 5549 df-so 5550 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-ov 7364 df-oprab 7365 df-mpo 7366 df-1st 7925 df-2nd 7926 df-er 8654 df-pm 8774 df-en 8890 df-dom 8891 df-sdom 8892 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 df-neg 11396 df-z 12508 df-uz 12772 df-top 22266 df-topon 22283 df-lm 22603 |
This theorem is referenced by: lmbr3 44078 |
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