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Mirrors > Home > MPE Home > Th. List > leordtval | Structured version Visualization version GIF version |
Description: The topology of the extended reals. (Contributed by Mario Carneiro, 3-Sep-2015.) |
Ref | Expression |
---|---|
leordtval.1 | β’ π΄ = ran (π₯ β β* β¦ (π₯(,]+β)) |
leordtval.2 | β’ π΅ = ran (π₯ β β* β¦ (-β[,)π₯)) |
leordtval.3 | β’ πΆ = ran (,) |
Ref | Expression |
---|---|
leordtval | β’ (ordTopβ β€ ) = (topGenβ((π΄ βͺ π΅) βͺ πΆ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | leordtval.1 | . . 3 β’ π΄ = ran (π₯ β β* β¦ (π₯(,]+β)) | |
2 | leordtval.2 | . . 3 β’ π΅ = ran (π₯ β β* β¦ (-β[,)π₯)) | |
3 | 1, 2 | leordtval2 22645 | . 2 β’ (ordTopβ β€ ) = (topGenβ(fiβ(π΄ βͺ π΅))) |
4 | letsr 18528 | . . . 4 β’ β€ β TosetRel | |
5 | ledm 18525 | . . . . 5 β’ β* = dom β€ | |
6 | 1 | leordtvallem1 22643 | . . . . 5 β’ π΄ = ran (π₯ β β* β¦ {π¦ β β* β£ Β¬ π¦ β€ π₯}) |
7 | 1, 2 | leordtvallem2 22644 | . . . . 5 β’ π΅ = ran (π₯ β β* β¦ {π¦ β β* β£ Β¬ π₯ β€ π¦}) |
8 | leordtval.3 | . . . . . 6 β’ πΆ = ran (,) | |
9 | df-ioo 13310 | . . . . . . . 8 β’ (,) = (π β β*, π β β* β¦ {π¦ β β* β£ (π < π¦ β§ π¦ < π)}) | |
10 | xrltnle 11263 | . . . . . . . . . . . 12 β’ ((π β β* β§ π¦ β β*) β (π < π¦ β Β¬ π¦ β€ π)) | |
11 | 10 | adantlr 713 | . . . . . . . . . . 11 β’ (((π β β* β§ π β β*) β§ π¦ β β*) β (π < π¦ β Β¬ π¦ β€ π)) |
12 | xrltnle 11263 | . . . . . . . . . . . . 13 β’ ((π¦ β β* β§ π β β*) β (π¦ < π β Β¬ π β€ π¦)) | |
13 | 12 | ancoms 459 | . . . . . . . . . . . 12 β’ ((π β β* β§ π¦ β β*) β (π¦ < π β Β¬ π β€ π¦)) |
14 | 13 | adantll 712 | . . . . . . . . . . 11 β’ (((π β β* β§ π β β*) β§ π¦ β β*) β (π¦ < π β Β¬ π β€ π¦)) |
15 | 11, 14 | anbi12d 631 | . . . . . . . . . 10 β’ (((π β β* β§ π β β*) β§ π¦ β β*) β ((π < π¦ β§ π¦ < π) β (Β¬ π¦ β€ π β§ Β¬ π β€ π¦))) |
16 | 15 | rabbidva 3438 | . . . . . . . . 9 β’ ((π β β* β§ π β β*) β {π¦ β β* β£ (π < π¦ β§ π¦ < π)} = {π¦ β β* β£ (Β¬ π¦ β€ π β§ Β¬ π β€ π¦)}) |
17 | 16 | mpoeq3ia 7471 | . . . . . . . 8 β’ (π β β*, π β β* β¦ {π¦ β β* β£ (π < π¦ β§ π¦ < π)}) = (π β β*, π β β* β¦ {π¦ β β* β£ (Β¬ π¦ β€ π β§ Β¬ π β€ π¦)}) |
18 | 9, 17 | eqtri 2759 | . . . . . . 7 β’ (,) = (π β β*, π β β* β¦ {π¦ β β* β£ (Β¬ π¦ β€ π β§ Β¬ π β€ π¦)}) |
19 | 18 | rneqi 5928 | . . . . . 6 β’ ran (,) = ran (π β β*, π β β* β¦ {π¦ β β* β£ (Β¬ π¦ β€ π β§ Β¬ π β€ π¦)}) |
20 | 8, 19 | eqtri 2759 | . . . . 5 β’ πΆ = ran (π β β*, π β β* β¦ {π¦ β β* β£ (Β¬ π¦ β€ π β§ Β¬ π β€ π¦)}) |
21 | 5, 6, 7, 20 | ordtbas2 22624 | . . . 4 β’ ( β€ β TosetRel β (fiβ(π΄ βͺ π΅)) = ((π΄ βͺ π΅) βͺ πΆ)) |
22 | 4, 21 | ax-mp 5 | . . 3 β’ (fiβ(π΄ βͺ π΅)) = ((π΄ βͺ π΅) βͺ πΆ) |
23 | 22 | fveq2i 6881 | . 2 β’ (topGenβ(fiβ(π΄ βͺ π΅))) = (topGenβ((π΄ βͺ π΅) βͺ πΆ)) |
24 | 3, 23 | eqtri 2759 | 1 β’ (ordTopβ β€ ) = (topGenβ((π΄ βͺ π΅) βͺ πΆ)) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wb 205 β§ wa 396 = wceq 1541 β wcel 2106 {crab 3431 βͺ cun 3942 class class class wbr 5141 β¦ cmpt 5224 ran crn 5670 βcfv 6532 (class class class)co 7393 β cmpo 7395 ficfi 9387 +βcpnf 11227 -βcmnf 11228 β*cxr 11229 < clt 11230 β€ cle 11231 (,)cioo 13306 (,]cioc 13307 [,)cico 13308 topGenctg 17365 ordTopcordt 17427 TosetRel ctsr 18500 |
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-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7708 ax-cnex 11148 ax-resscn 11149 ax-1cn 11150 ax-icn 11151 ax-addcl 11152 ax-addrcl 11153 ax-mulcl 11154 ax-mulrcl 11155 ax-mulcom 11156 ax-addass 11157 ax-mulass 11158 ax-distr 11159 ax-i2m1 11160 ax-1ne0 11161 ax-1rid 11162 ax-rnegex 11163 ax-rrecex 11164 ax-cnre 11165 ax-pre-lttri 11166 ax-pre-lttrn 11167 ax-pre-ltadd 11168 ax-pre-mulgt0 11169 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 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-nel 3046 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-ord 6356 df-on 6357 df-lim 6358 df-suc 6359 df-iota 6484 df-fun 6534 df-fn 6535 df-f 6536 df-f1 6537 df-fo 6538 df-f1o 6539 df-fv 6540 df-riota 7349 df-ov 7396 df-oprab 7397 df-mpo 7398 df-om 7839 df-1st 7957 df-2nd 7958 df-1o 8448 df-er 8686 df-en 8923 df-dom 8924 df-sdom 8925 df-fin 8926 df-fi 9388 df-pnf 11232 df-mnf 11233 df-xr 11234 df-ltxr 11235 df-le 11236 df-sub 11428 df-neg 11429 df-ioo 13310 df-ioc 13311 df-ico 13312 df-icc 13313 df-topgen 17371 df-ordt 17429 df-ps 18501 df-tsr 18502 df-top 22325 df-bases 22378 |
This theorem is referenced by: iocpnfordt 22648 icomnfordt 22649 iooordt 22650 pnfnei 22653 mnfnei 22654 xrtgioo 24251 |
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