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Theorem ordtcnvNEW 29760
Description: The order dual generates the same topology as the original order. (Contributed by Mario Carneiro, 3-Sep-2015.) (Revised by Thierry Arnoux, 13-Sep-2018.)
Hypotheses
Ref Expression
ordtNEW.b 𝐵 = (Base‘𝐾)
ordtNEW.l = ((le‘𝐾) ∩ (𝐵 × 𝐵))
Assertion
Ref Expression
ordtcnvNEW (𝐾 ∈ Preset → (ordTop‘ ) = (ordTop‘ ))

Proof of Theorem ordtcnvNEW
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 3189 . . . . . . . . . . . . 13 𝑦 ∈ V
2 vex 3189 . . . . . . . . . . . . 13 𝑥 ∈ V
31, 2brcnv 5267 . . . . . . . . . . . 12 (𝑦 𝑥𝑥 𝑦)
43a1i 11 . . . . . . . . . . 11 (𝐾 ∈ Preset → (𝑦 𝑥𝑥 𝑦))
54notbid 308 . . . . . . . . . 10 (𝐾 ∈ Preset → (¬ 𝑦 𝑥 ↔ ¬ 𝑥 𝑦))
65rabbidv 3177 . . . . . . . . 9 (𝐾 ∈ Preset → {𝑦𝐵 ∣ ¬ 𝑦 𝑥} = {𝑦𝐵 ∣ ¬ 𝑥 𝑦})
76mpteq2dv 4707 . . . . . . . 8 (𝐾 ∈ Preset → (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑦 𝑥}) = (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑥 𝑦}))
87rneqd 5315 . . . . . . 7 (𝐾 ∈ Preset → ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑦 𝑥}) = ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑥 𝑦}))
92, 1brcnv 5267 . . . . . . . . . . . 12 (𝑥 𝑦𝑦 𝑥)
109a1i 11 . . . . . . . . . . 11 (𝐾 ∈ Preset → (𝑥 𝑦𝑦 𝑥))
1110notbid 308 . . . . . . . . . 10 (𝐾 ∈ Preset → (¬ 𝑥 𝑦 ↔ ¬ 𝑦 𝑥))
1211rabbidv 3177 . . . . . . . . 9 (𝐾 ∈ Preset → {𝑦𝐵 ∣ ¬ 𝑥 𝑦} = {𝑦𝐵 ∣ ¬ 𝑦 𝑥})
1312mpteq2dv 4707 . . . . . . . 8 (𝐾 ∈ Preset → (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑥 𝑦}) = (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑦 𝑥}))
1413rneqd 5315 . . . . . . 7 (𝐾 ∈ Preset → ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑥 𝑦}) = ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑦 𝑥}))
158, 14uneq12d 3748 . . . . . 6 (𝐾 ∈ Preset → (ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑦 𝑥}) ∪ ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑥 𝑦})) = (ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑥 𝑦}) ∪ ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑦 𝑥})))
16 uncom 3737 . . . . . 6 (ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑥 𝑦}) ∪ ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑦 𝑥})) = (ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑦 𝑥}) ∪ ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑥 𝑦}))
1715, 16syl6eq 2671 . . . . 5 (𝐾 ∈ Preset → (ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑦 𝑥}) ∪ ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑥 𝑦})) = (ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑦 𝑥}) ∪ ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑥 𝑦})))
1817uneq2d 3747 . . . 4 (𝐾 ∈ Preset → ({𝐵} ∪ (ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑦 𝑥}) ∪ ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑥 𝑦}))) = ({𝐵} ∪ (ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑦 𝑥}) ∪ ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑥 𝑦}))))
1918fveq2d 6154 . . 3 (𝐾 ∈ Preset → (fi‘({𝐵} ∪ (ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑦 𝑥}) ∪ ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑥 𝑦})))) = (fi‘({𝐵} ∪ (ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑦 𝑥}) ∪ ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑥 𝑦})))))
2019fveq2d 6154 . 2 (𝐾 ∈ Preset → (topGen‘(fi‘({𝐵} ∪ (ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑦 𝑥}) ∪ ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑥 𝑦}))))) = (topGen‘(fi‘({𝐵} ∪ (ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑦 𝑥}) ∪ ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑥 𝑦}))))))
21 eqid 2621 . . . 4 (ODual‘𝐾) = (ODual‘𝐾)
2221oduprs 29453 . . 3 (𝐾 ∈ Preset → (ODual‘𝐾) ∈ Preset )
23 ordtNEW.b . . . . 5 𝐵 = (Base‘𝐾)
2421, 23odubas 17057 . . . 4 𝐵 = (Base‘(ODual‘𝐾))
25 ordtNEW.l . . . . . 6 = ((le‘𝐾) ∩ (𝐵 × 𝐵))
2625cnveqi 5259 . . . . 5 = ((le‘𝐾) ∩ (𝐵 × 𝐵))
27 cnvin 5501 . . . . 5 ((le‘𝐾) ∩ (𝐵 × 𝐵)) = ((le‘𝐾) ∩ (𝐵 × 𝐵))
28 eqid 2621 . . . . . . 7 (le‘𝐾) = (le‘𝐾)
2921, 28oduleval 17055 . . . . . 6 (le‘𝐾) = (le‘(ODual‘𝐾))
30 cnvxp 5512 . . . . . 6 (𝐵 × 𝐵) = (𝐵 × 𝐵)
3129, 30ineq12i 3792 . . . . 5 ((le‘𝐾) ∩ (𝐵 × 𝐵)) = ((le‘(ODual‘𝐾)) ∩ (𝐵 × 𝐵))
3226, 27, 313eqtri 2647 . . . 4 = ((le‘(ODual‘𝐾)) ∩ (𝐵 × 𝐵))
33 eqid 2621 . . . 4 ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑦 𝑥}) = ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑦 𝑥})
34 eqid 2621 . . . 4 ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑥 𝑦}) = ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑥 𝑦})
3524, 32, 33, 34ordtprsval 29758 . . 3 ((ODual‘𝐾) ∈ Preset → (ordTop‘ ) = (topGen‘(fi‘({𝐵} ∪ (ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑦 𝑥}) ∪ ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑥 𝑦}))))))
3622, 35syl 17 . 2 (𝐾 ∈ Preset → (ordTop‘ ) = (topGen‘(fi‘({𝐵} ∪ (ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑦 𝑥}) ∪ ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑥 𝑦}))))))
37 eqid 2621 . . 3 ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑦 𝑥}) = ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑦 𝑥})
38 eqid 2621 . . 3 ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑥 𝑦}) = ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑥 𝑦})
3923, 25, 37, 38ordtprsval 29758 . 2 (𝐾 ∈ Preset → (ordTop‘ ) = (topGen‘(fi‘({𝐵} ∪ (ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑦 𝑥}) ∪ ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑥 𝑦}))))))
4020, 36, 393eqtr4d 2665 1 (𝐾 ∈ Preset → (ordTop‘ ) = (ordTop‘ ))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196   = wceq 1480  wcel 1987  {crab 2911  cun 3554  cin 3555  {csn 4150   class class class wbr 4615  cmpt 4675   × cxp 5074  ccnv 5075  ran crn 5077  cfv 5849  ficfi 8263  Basecbs 15784  lecple 15872  topGenctg 16022  ordTopcordt 16083   Preset cpreset 16850  ODualcodu 17052
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4743  ax-nul 4751  ax-pow 4805  ax-pr 4869  ax-un 6905  ax-cnex 9939  ax-resscn 9940  ax-1cn 9941  ax-icn 9942  ax-addcl 9943  ax-addrcl 9944  ax-mulcl 9945  ax-mulrcl 9946  ax-mulcom 9947  ax-addass 9948  ax-mulass 9949  ax-distr 9950  ax-i2m1 9951  ax-1ne0 9952  ax-1rid 9953  ax-rnegex 9954  ax-rrecex 9955  ax-cnre 9956  ax-pre-lttri 9957  ax-pre-lttrn 9958  ax-pre-ltadd 9959  ax-pre-mulgt0 9960
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3419  df-csb 3516  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-pss 3572  df-nul 3894  df-if 4061  df-pw 4134  df-sn 4151  df-pr 4153  df-tp 4155  df-op 4157  df-uni 4405  df-iun 4489  df-br 4616  df-opab 4676  df-mpt 4677  df-tr 4715  df-eprel 4987  df-id 4991  df-po 4997  df-so 4998  df-fr 5035  df-we 5037  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-rn 5087  df-res 5088  df-ima 5089  df-pred 5641  df-ord 5687  df-on 5688  df-lim 5689  df-suc 5690  df-iota 5812  df-fun 5851  df-fn 5852  df-f 5853  df-f1 5854  df-fo 5855  df-f1o 5856  df-fv 5857  df-riota 6568  df-ov 6610  df-oprab 6611  df-mpt2 6612  df-om 7016  df-wrecs 7355  df-recs 7416  df-rdg 7454  df-er 7690  df-en 7903  df-dom 7904  df-sdom 7905  df-pnf 10023  df-mnf 10024  df-xr 10025  df-ltxr 10026  df-le 10027  df-sub 10215  df-neg 10216  df-nn 10968  df-2 11026  df-3 11027  df-4 11028  df-5 11029  df-6 11030  df-7 11031  df-8 11032  df-9 11033  df-dec 11441  df-ndx 15787  df-slot 15788  df-base 15789  df-sets 15790  df-ple 15885  df-ordt 16085  df-preset 16852  df-odu 17053
This theorem is referenced by:  ordtrest2NEW  29763
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