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Theorem ordtcnvNEW 30306
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 3354 . . . . . . . . . . . . 13 𝑦 ∈ V
2 vex 3354 . . . . . . . . . . . . 13 𝑥 ∈ V
31, 2brcnv 5443 . . . . . . . . . . . 12 (𝑦 𝑥𝑥 𝑦)
43a1i 11 . . . . . . . . . . 11 (𝐾 ∈ Preset → (𝑦 𝑥𝑥 𝑦))
54notbid 307 . . . . . . . . . 10 (𝐾 ∈ Preset → (¬ 𝑦 𝑥 ↔ ¬ 𝑥 𝑦))
65rabbidv 3339 . . . . . . . . 9 (𝐾 ∈ Preset → {𝑦𝐵 ∣ ¬ 𝑦 𝑥} = {𝑦𝐵 ∣ ¬ 𝑥 𝑦})
76mpteq2dv 4879 . . . . . . . 8 (𝐾 ∈ Preset → (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑦 𝑥}) = (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑥 𝑦}))
87rneqd 5491 . . . . . . 7 (𝐾 ∈ Preset → ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑦 𝑥}) = ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑥 𝑦}))
92, 1brcnv 5443 . . . . . . . . . . . 12 (𝑥 𝑦𝑦 𝑥)
109a1i 11 . . . . . . . . . . 11 (𝐾 ∈ Preset → (𝑥 𝑦𝑦 𝑥))
1110notbid 307 . . . . . . . . . 10 (𝐾 ∈ Preset → (¬ 𝑥 𝑦 ↔ ¬ 𝑦 𝑥))
1211rabbidv 3339 . . . . . . . . 9 (𝐾 ∈ Preset → {𝑦𝐵 ∣ ¬ 𝑥 𝑦} = {𝑦𝐵 ∣ ¬ 𝑦 𝑥})
1312mpteq2dv 4879 . . . . . . . 8 (𝐾 ∈ Preset → (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑥 𝑦}) = (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑦 𝑥}))
1413rneqd 5491 . . . . . . 7 (𝐾 ∈ Preset → ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑥 𝑦}) = ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑦 𝑥}))
158, 14uneq12d 3919 . . . . . 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 3908 . . . . . 6 (ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑥 𝑦}) ∪ ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑦 𝑥})) = (ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑦 𝑥}) ∪ ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑥 𝑦}))
1715, 16syl6eq 2821 . . . . 5 (𝐾 ∈ Preset → (ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑦 𝑥}) ∪ ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑥 𝑦})) = (ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑦 𝑥}) ∪ ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑥 𝑦})))
1817uneq2d 3918 . . . 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 6336 . . 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 6336 . 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 2771 . . . 4 (ODual‘𝐾) = (ODual‘𝐾)
2221oduprs 29996 . . 3 (𝐾 ∈ Preset → (ODual‘𝐾) ∈ Preset )
23 ordtNEW.b . . . . 5 𝐵 = (Base‘𝐾)
2421, 23odubas 17341 . . . 4 𝐵 = (Base‘(ODual‘𝐾))
25 ordtNEW.l . . . . . 6 = ((le‘𝐾) ∩ (𝐵 × 𝐵))
2625cnveqi 5435 . . . . 5 = ((le‘𝐾) ∩ (𝐵 × 𝐵))
27 cnvin 5681 . . . . 5 ((le‘𝐾) ∩ (𝐵 × 𝐵)) = ((le‘𝐾) ∩ (𝐵 × 𝐵))
28 eqid 2771 . . . . . . 7 (le‘𝐾) = (le‘𝐾)
2921, 28oduleval 17339 . . . . . 6 (le‘𝐾) = (le‘(ODual‘𝐾))
30 cnvxp 5692 . . . . . 6 (𝐵 × 𝐵) = (𝐵 × 𝐵)
3129, 30ineq12i 3963 . . . . 5 ((le‘𝐾) ∩ (𝐵 × 𝐵)) = ((le‘(ODual‘𝐾)) ∩ (𝐵 × 𝐵))
3226, 27, 313eqtri 2797 . . . 4 = ((le‘(ODual‘𝐾)) ∩ (𝐵 × 𝐵))
33 eqid 2771 . . . 4 ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑦 𝑥}) = ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑦 𝑥})
34 eqid 2771 . . . 4 ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑥 𝑦}) = ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑥 𝑦})
3524, 32, 33, 34ordtprsval 30304 . . 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 2771 . . 3 ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑦 𝑥}) = ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑦 𝑥})
38 eqid 2771 . . 3 ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑥 𝑦}) = ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑥 𝑦})
3923, 25, 37, 38ordtprsval 30304 . 2 (𝐾 ∈ Preset → (ordTop‘ ) = (topGen‘(fi‘({𝐵} ∪ (ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑦 𝑥}) ∪ ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑥 𝑦}))))))
4020, 36, 393eqtr4d 2815 1 (𝐾 ∈ Preset → (ordTop‘ ) = (ordTop‘ ))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196   = wceq 1631  wcel 2145  {crab 3065  cun 3721  cin 3722  {csn 4316   class class class wbr 4786  cmpt 4863   × cxp 5247  ccnv 5248  ran crn 5250  cfv 6031  ficfi 8472  Basecbs 16064  lecple 16156  topGenctg 16306  ordTopcordt 16367   Preset cpreset 17134  ODualcodu 17336
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-cnex 10194  ax-resscn 10195  ax-1cn 10196  ax-icn 10197  ax-addcl 10198  ax-addrcl 10199  ax-mulcl 10200  ax-mulrcl 10201  ax-mulcom 10202  ax-addass 10203  ax-mulass 10204  ax-distr 10205  ax-i2m1 10206  ax-1ne0 10207  ax-1rid 10208  ax-rnegex 10209  ax-rrecex 10210  ax-cnre 10211  ax-pre-lttri 10212  ax-pre-lttrn 10213  ax-pre-ltadd 10214  ax-pre-mulgt0 10215
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6754  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-om 7213  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-er 7896  df-en 8110  df-dom 8111  df-sdom 8112  df-pnf 10278  df-mnf 10279  df-xr 10280  df-ltxr 10281  df-le 10282  df-sub 10470  df-neg 10471  df-nn 11223  df-2 11281  df-3 11282  df-4 11283  df-5 11284  df-6 11285  df-7 11286  df-8 11287  df-9 11288  df-dec 11696  df-ndx 16067  df-slot 16068  df-base 16070  df-sets 16071  df-ple 16169  df-ordt 16369  df-preset 17136  df-odu 17337
This theorem is referenced by:  ordtrest2NEW  30309
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