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Mirrors > Home > ILE Home > Th. List > ioo2bl | GIF version |
Description: An open interval of reals in terms of a ball. (Contributed by NM, 18-May-2007.) (Revised by Mario Carneiro, 28-Aug-2015.) |
Ref | Expression |
---|---|
remet.1 | β’ π· = ((abs β β ) βΎ (β Γ β)) |
Ref | Expression |
---|---|
ioo2bl | β’ ((π΄ β β β§ π΅ β β) β (π΄(,)π΅) = (((π΄ + π΅) / 2)(ballβπ·)((π΅ β π΄) / 2))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | readdcl 7912 | . . . . 5 β’ ((π΅ β β β§ π΄ β β) β (π΅ + π΄) β β) | |
2 | 1 | ancoms 268 | . . . 4 β’ ((π΄ β β β§ π΅ β β) β (π΅ + π΄) β β) |
3 | 2 | rehalfcld 9136 | . . 3 β’ ((π΄ β β β§ π΅ β β) β ((π΅ + π΄) / 2) β β) |
4 | resubcl 8195 | . . . . 5 β’ ((π΅ β β β§ π΄ β β) β (π΅ β π΄) β β) | |
5 | 4 | ancoms 268 | . . . 4 β’ ((π΄ β β β§ π΅ β β) β (π΅ β π΄) β β) |
6 | 5 | rehalfcld 9136 | . . 3 β’ ((π΄ β β β§ π΅ β β) β ((π΅ β π΄) / 2) β β) |
7 | remet.1 | . . . 4 β’ π· = ((abs β β ) βΎ (β Γ β)) | |
8 | 7 | bl2ioo 13535 | . . 3 β’ ((((π΅ + π΄) / 2) β β β§ ((π΅ β π΄) / 2) β β) β (((π΅ + π΄) / 2)(ballβπ·)((π΅ β π΄) / 2)) = ((((π΅ + π΄) / 2) β ((π΅ β π΄) / 2))(,)(((π΅ + π΄) / 2) + ((π΅ β π΄) / 2)))) |
9 | 3, 6, 8 | syl2anc 411 | . 2 β’ ((π΄ β β β§ π΅ β β) β (((π΅ + π΄) / 2)(ballβπ·)((π΅ β π΄) / 2)) = ((((π΅ + π΄) / 2) β ((π΅ β π΄) / 2))(,)(((π΅ + π΄) / 2) + ((π΅ β π΄) / 2)))) |
10 | recn 7919 | . . . . 5 β’ (π΅ β β β π΅ β β) | |
11 | recn 7919 | . . . . 5 β’ (π΄ β β β π΄ β β) | |
12 | addcom 8068 | . . . . 5 β’ ((π΅ β β β§ π΄ β β) β (π΅ + π΄) = (π΄ + π΅)) | |
13 | 10, 11, 12 | syl2anr 290 | . . . 4 β’ ((π΄ β β β§ π΅ β β) β (π΅ + π΄) = (π΄ + π΅)) |
14 | 13 | oveq1d 5880 | . . 3 β’ ((π΄ β β β§ π΅ β β) β ((π΅ + π΄) / 2) = ((π΄ + π΅) / 2)) |
15 | 14 | oveq1d 5880 | . 2 β’ ((π΄ β β β§ π΅ β β) β (((π΅ + π΄) / 2)(ballβπ·)((π΅ β π΄) / 2)) = (((π΄ + π΅) / 2)(ballβπ·)((π΅ β π΄) / 2))) |
16 | halfaddsub 9124 | . . . . 5 β’ ((π΅ β β β§ π΄ β β) β ((((π΅ + π΄) / 2) + ((π΅ β π΄) / 2)) = π΅ β§ (((π΅ + π΄) / 2) β ((π΅ β π΄) / 2)) = π΄)) | |
17 | 10, 11, 16 | syl2anr 290 | . . . 4 β’ ((π΄ β β β§ π΅ β β) β ((((π΅ + π΄) / 2) + ((π΅ β π΄) / 2)) = π΅ β§ (((π΅ + π΄) / 2) β ((π΅ β π΄) / 2)) = π΄)) |
18 | 17 | simprd 114 | . . 3 β’ ((π΄ β β β§ π΅ β β) β (((π΅ + π΄) / 2) β ((π΅ β π΄) / 2)) = π΄) |
19 | 17 | simpld 112 | . . 3 β’ ((π΄ β β β§ π΅ β β) β (((π΅ + π΄) / 2) + ((π΅ β π΄) / 2)) = π΅) |
20 | 18, 19 | oveq12d 5883 | . 2 β’ ((π΄ β β β§ π΅ β β) β ((((π΅ + π΄) / 2) β ((π΅ β π΄) / 2))(,)(((π΅ + π΄) / 2) + ((π΅ β π΄) / 2))) = (π΄(,)π΅)) |
21 | 9, 15, 20 | 3eqtr3rd 2217 | 1 β’ ((π΄ β β β§ π΅ β β) β (π΄(,)π΅) = (((π΄ + π΅) / 2)(ballβπ·)((π΅ β π΄) / 2))) |
Colors of variables: wff set class |
Syntax hints: β wi 4 β§ wa 104 = wceq 1353 β wcel 2146 Γ cxp 4618 βΎ cres 4622 β ccom 4624 βcfv 5208 (class class class)co 5865 βcc 7784 βcr 7785 + caddc 7789 β cmin 8102 / cdiv 8601 2c2 8941 (,)cioo 9857 abscabs 10973 ballcbl 12975 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1445 ax-7 1446 ax-gen 1447 ax-ie1 1491 ax-ie2 1492 ax-8 1502 ax-10 1503 ax-11 1504 ax-i12 1505 ax-bndl 1507 ax-4 1508 ax-17 1524 ax-i9 1528 ax-ial 1532 ax-i5r 1533 ax-13 2148 ax-14 2149 ax-ext 2157 ax-coll 4113 ax-sep 4116 ax-nul 4124 ax-pow 4169 ax-pr 4203 ax-un 4427 ax-setind 4530 ax-iinf 4581 ax-cnex 7877 ax-resscn 7878 ax-1cn 7879 ax-1re 7880 ax-icn 7881 ax-addcl 7882 ax-addrcl 7883 ax-mulcl 7884 ax-mulrcl 7885 ax-addcom 7886 ax-mulcom 7887 ax-addass 7888 ax-mulass 7889 ax-distr 7890 ax-i2m1 7891 ax-0lt1 7892 ax-1rid 7893 ax-0id 7894 ax-rnegex 7895 ax-precex 7896 ax-cnre 7897 ax-pre-ltirr 7898 ax-pre-ltwlin 7899 ax-pre-lttrn 7900 ax-pre-apti 7901 ax-pre-ltadd 7902 ax-pre-mulgt0 7903 ax-pre-mulext 7904 ax-arch 7905 ax-caucvg 7906 |
This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1459 df-sb 1761 df-eu 2027 df-mo 2028 df-clab 2162 df-cleq 2168 df-clel 2171 df-nfc 2306 df-ne 2346 df-nel 2441 df-ral 2458 df-rex 2459 df-reu 2460 df-rmo 2461 df-rab 2462 df-v 2737 df-sbc 2961 df-csb 3056 df-dif 3129 df-un 3131 df-in 3133 df-ss 3140 df-nul 3421 df-if 3533 df-pw 3574 df-sn 3595 df-pr 3596 df-op 3598 df-uni 3806 df-int 3841 df-iun 3884 df-br 3999 df-opab 4060 df-mpt 4061 df-tr 4097 df-id 4287 df-po 4290 df-iso 4291 df-iord 4360 df-on 4362 df-ilim 4363 df-suc 4365 df-iom 4584 df-xp 4626 df-rel 4627 df-cnv 4628 df-co 4629 df-dm 4630 df-rn 4631 df-res 4632 df-ima 4633 df-iota 5170 df-fun 5210 df-fn 5211 df-f 5212 df-f1 5213 df-fo 5214 df-f1o 5215 df-fv 5216 df-riota 5821 df-ov 5868 df-oprab 5869 df-mpo 5870 df-1st 6131 df-2nd 6132 df-recs 6296 df-frec 6382 df-map 6640 df-pnf 7968 df-mnf 7969 df-xr 7970 df-ltxr 7971 df-le 7972 df-sub 8104 df-neg 8105 df-reap 8506 df-ap 8513 df-div 8602 df-inn 8891 df-2 8949 df-3 8950 df-4 8951 df-n0 9148 df-z 9225 df-uz 9500 df-rp 9623 df-xadd 9742 df-ioo 9861 df-seqfrec 10414 df-exp 10488 df-cj 10818 df-re 10819 df-im 10820 df-rsqrt 10974 df-abs 10975 df-psmet 12980 df-xmet 12981 df-met 12982 df-bl 12983 |
This theorem is referenced by: ioo2blex 13537 |
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