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Theorem ordtcnvNEW 32501
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 ∈ Proset → (ordTop‘ ) = (ordTop‘ ))

Proof of Theorem ordtcnvNEW
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 3449 . . . . . . . . . . . . 13 𝑦 ∈ V
2 vex 3449 . . . . . . . . . . . . 13 𝑥 ∈ V
31, 2brcnv 5838 . . . . . . . . . . . 12 (𝑦 𝑥𝑥 𝑦)
43a1i 11 . . . . . . . . . . 11 (𝐾 ∈ Proset → (𝑦 𝑥𝑥 𝑦))
54notbid 317 . . . . . . . . . 10 (𝐾 ∈ Proset → (¬ 𝑦 𝑥 ↔ ¬ 𝑥 𝑦))
65rabbidv 3415 . . . . . . . . 9 (𝐾 ∈ Proset → {𝑦𝐵 ∣ ¬ 𝑦 𝑥} = {𝑦𝐵 ∣ ¬ 𝑥 𝑦})
76mpteq2dv 5207 . . . . . . . 8 (𝐾 ∈ Proset → (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑦 𝑥}) = (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑥 𝑦}))
87rneqd 5893 . . . . . . 7 (𝐾 ∈ Proset → ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑦 𝑥}) = ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑥 𝑦}))
92, 1brcnv 5838 . . . . . . . . . . . 12 (𝑥 𝑦𝑦 𝑥)
109a1i 11 . . . . . . . . . . 11 (𝐾 ∈ Proset → (𝑥 𝑦𝑦 𝑥))
1110notbid 317 . . . . . . . . . 10 (𝐾 ∈ Proset → (¬ 𝑥 𝑦 ↔ ¬ 𝑦 𝑥))
1211rabbidv 3415 . . . . . . . . 9 (𝐾 ∈ Proset → {𝑦𝐵 ∣ ¬ 𝑥 𝑦} = {𝑦𝐵 ∣ ¬ 𝑦 𝑥})
1312mpteq2dv 5207 . . . . . . . 8 (𝐾 ∈ Proset → (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑥 𝑦}) = (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑦 𝑥}))
1413rneqd 5893 . . . . . . 7 (𝐾 ∈ Proset → ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑥 𝑦}) = ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑦 𝑥}))
158, 14uneq12d 4124 . . . . . 6 (𝐾 ∈ Proset → (ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑦 𝑥}) ∪ ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑥 𝑦})) = (ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑥 𝑦}) ∪ ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑦 𝑥})))
16 uncom 4113 . . . . . 6 (ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑥 𝑦}) ∪ ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑦 𝑥})) = (ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑦 𝑥}) ∪ ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑥 𝑦}))
1715, 16eqtrdi 2792 . . . . 5 (𝐾 ∈ Proset → (ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑦 𝑥}) ∪ ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑥 𝑦})) = (ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑦 𝑥}) ∪ ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑥 𝑦})))
1817uneq2d 4123 . . . 4 (𝐾 ∈ Proset → ({𝐵} ∪ (ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑦 𝑥}) ∪ ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑥 𝑦}))) = ({𝐵} ∪ (ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑦 𝑥}) ∪ ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑥 𝑦}))))
1918fveq2d 6846 . . 3 (𝐾 ∈ Proset → (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 6846 . 2 (𝐾 ∈ Proset → (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 2736 . . . 4 (ODual‘𝐾) = (ODual‘𝐾)
2221oduprs 31824 . . 3 (𝐾 ∈ Proset → (ODual‘𝐾) ∈ Proset )
23 ordtNEW.b . . . . 5 𝐵 = (Base‘𝐾)
2421, 23odubas 18180 . . . 4 𝐵 = (Base‘(ODual‘𝐾))
25 ordtNEW.l . . . . . 6 = ((le‘𝐾) ∩ (𝐵 × 𝐵))
2625cnveqi 5830 . . . . 5 = ((le‘𝐾) ∩ (𝐵 × 𝐵))
27 cnvin 6097 . . . . 5 ((le‘𝐾) ∩ (𝐵 × 𝐵)) = ((le‘𝐾) ∩ (𝐵 × 𝐵))
28 eqid 2736 . . . . . . 7 (le‘𝐾) = (le‘𝐾)
2921, 28oduleval 18178 . . . . . 6 (le‘𝐾) = (le‘(ODual‘𝐾))
30 cnvxp 6109 . . . . . 6 (𝐵 × 𝐵) = (𝐵 × 𝐵)
3129, 30ineq12i 4170 . . . . 5 ((le‘𝐾) ∩ (𝐵 × 𝐵)) = ((le‘(ODual‘𝐾)) ∩ (𝐵 × 𝐵))
3226, 27, 313eqtri 2768 . . . 4 = ((le‘(ODual‘𝐾)) ∩ (𝐵 × 𝐵))
33 eqid 2736 . . . 4 ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑦 𝑥}) = ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑦 𝑥})
34 eqid 2736 . . . 4 ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑥 𝑦}) = ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑥 𝑦})
3524, 32, 33, 34ordtprsval 32499 . . 3 ((ODual‘𝐾) ∈ Proset → (ordTop‘ ) = (topGen‘(fi‘({𝐵} ∪ (ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑦 𝑥}) ∪ ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑥 𝑦}))))))
3622, 35syl 17 . 2 (𝐾 ∈ Proset → (ordTop‘ ) = (topGen‘(fi‘({𝐵} ∪ (ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑦 𝑥}) ∪ ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑥 𝑦}))))))
37 eqid 2736 . . 3 ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑦 𝑥}) = ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑦 𝑥})
38 eqid 2736 . . 3 ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑥 𝑦}) = ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑥 𝑦})
3923, 25, 37, 38ordtprsval 32499 . 2 (𝐾 ∈ Proset → (ordTop‘ ) = (topGen‘(fi‘({𝐵} ∪ (ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑦 𝑥}) ∪ ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑥 𝑦}))))))
4020, 36, 393eqtr4d 2786 1 (𝐾 ∈ Proset → (ordTop‘ ) = (ordTop‘ ))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205   = wceq 1541  wcel 2106  {crab 3407  cun 3908  cin 3909  {csn 4586   class class class wbr 5105  cmpt 5188   × cxp 5631  ccnv 5632  ran crn 5634  cfv 6496  ficfi 9346  Basecbs 17083  lecple 17140  topGenctg 17319  ordTopcordt 17381  ODualcodu 18175   Proset cproset 18182
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 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
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 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-3 12217  df-4 12218  df-5 12219  df-6 12220  df-7 12221  df-8 12222  df-9 12223  df-dec 12619  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ple 17153  df-ordt 17383  df-odu 18176  df-proset 18184
This theorem is referenced by:  ordtrest2NEW  32504
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