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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > odutos | Structured version Visualization version GIF version |
Description: Being a toset is a self-dual property. (Contributed by Thierry Arnoux, 13-Sep-2018.) |
Ref | Expression |
---|---|
odutos.d | β’ π· = (ODualβπΎ) |
Ref | Expression |
---|---|
odutos | β’ (πΎ β Toset β π· β Toset) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tospos 18412 | . . 3 β’ (πΎ β Toset β πΎ β Poset) | |
2 | odutos.d | . . . 4 β’ π· = (ODualβπΎ) | |
3 | 2 | odupos 18320 | . . 3 β’ (πΎ β Poset β π· β Poset) |
4 | 1, 3 | syl 17 | . 2 β’ (πΎ β Toset β π· β Poset) |
5 | eqid 2728 | . . . . . . 7 β’ (BaseβπΎ) = (BaseβπΎ) | |
6 | eqid 2728 | . . . . . . 7 β’ (leβπΎ) = (leβπΎ) | |
7 | 5, 6 | tleile 18413 | . . . . . 6 β’ ((πΎ β Toset β§ π¦ β (BaseβπΎ) β§ π₯ β (BaseβπΎ)) β (π¦(leβπΎ)π₯ β¨ π₯(leβπΎ)π¦)) |
8 | vex 3475 | . . . . . . . 8 β’ π₯ β V | |
9 | vex 3475 | . . . . . . . 8 β’ π¦ β V | |
10 | 8, 9 | brcnv 5885 | . . . . . . 7 β’ (π₯β‘(leβπΎ)π¦ β π¦(leβπΎ)π₯) |
11 | 9, 8 | brcnv 5885 | . . . . . . 7 β’ (π¦β‘(leβπΎ)π₯ β π₯(leβπΎ)π¦) |
12 | 10, 11 | orbi12i 913 | . . . . . 6 β’ ((π₯β‘(leβπΎ)π¦ β¨ π¦β‘(leβπΎ)π₯) β (π¦(leβπΎ)π₯ β¨ π₯(leβπΎ)π¦)) |
13 | 7, 12 | sylibr 233 | . . . . 5 β’ ((πΎ β Toset β§ π¦ β (BaseβπΎ) β§ π₯ β (BaseβπΎ)) β (π₯β‘(leβπΎ)π¦ β¨ π¦β‘(leβπΎ)π₯)) |
14 | 13 | 3com23 1124 | . . . 4 β’ ((πΎ β Toset β§ π₯ β (BaseβπΎ) β§ π¦ β (BaseβπΎ)) β (π₯β‘(leβπΎ)π¦ β¨ π¦β‘(leβπΎ)π₯)) |
15 | 14 | 3expb 1118 | . . 3 β’ ((πΎ β Toset β§ (π₯ β (BaseβπΎ) β§ π¦ β (BaseβπΎ))) β (π₯β‘(leβπΎ)π¦ β¨ π¦β‘(leβπΎ)π₯)) |
16 | 15 | ralrimivva 3197 | . 2 β’ (πΎ β Toset β βπ₯ β (BaseβπΎ)βπ¦ β (BaseβπΎ)(π₯β‘(leβπΎ)π¦ β¨ π¦β‘(leβπΎ)π₯)) |
17 | 2, 5 | odubas 18283 | . . 3 β’ (BaseβπΎ) = (Baseβπ·) |
18 | 2, 6 | oduleval 18281 | . . 3 β’ β‘(leβπΎ) = (leβπ·) |
19 | 17, 18 | istos 18410 | . 2 β’ (π· β Toset β (π· β Poset β§ βπ₯ β (BaseβπΎ)βπ¦ β (BaseβπΎ)(π₯β‘(leβπΎ)π¦ β¨ π¦β‘(leβπΎ)π₯))) |
20 | 4, 16, 19 | sylanbrc 582 | 1 β’ (πΎ β Toset β π· β Toset) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β¨ wo 846 β§ w3a 1085 = wceq 1534 β wcel 2099 βwral 3058 class class class wbr 5148 β‘ccnv 5677 βcfv 6548 Basecbs 17180 lecple 17240 ODualcodu 18278 Posetcpo 18299 Tosetctos 18408 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-2nd 7994 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-nn 12244 df-2 12306 df-3 12307 df-4 12308 df-5 12309 df-6 12310 df-7 12311 df-8 12312 df-9 12313 df-dec 12709 df-sets 17133 df-slot 17151 df-ndx 17163 df-base 17181 df-ple 17253 df-odu 18279 df-proset 18287 df-poset 18305 df-toset 18409 |
This theorem is referenced by: ordtrest2NEW 33524 |
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