Theorem ordtcnvNEW 34113

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.

Step	Hyp	Ref	Expression
1		vex 3435	. . . . . . . . . . . . 13 ⊢ 𝑦 ∈ V
2		vex 3435	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ V
3	1, 2	brcnv 5825	. . . . . . . . . . . 12 ⊢ (𝑦◡ ≤ 𝑥 ↔ 𝑥 ≤ 𝑦)
4	3	a1i 11	. . . . . . . . . . 11 ⊢ (𝐾 ∈ Proset → (𝑦◡ ≤ 𝑥 ↔ 𝑥 ≤ 𝑦))
5	4	notbid 319	. . . . . . . . . 10 ⊢ (𝐾 ∈ Proset → (¬ 𝑦◡ ≤ 𝑥 ↔ ¬ 𝑥 ≤ 𝑦))
6	5	rabbidv 3398	. . . . . . . . 9 ⊢ (𝐾 ∈ Proset → {𝑦 ∈ 𝐵 ∣ ¬ 𝑦◡ ≤ 𝑥} = {𝑦 ∈ 𝐵 ∣ ¬ 𝑥 ≤ 𝑦})
7	6	mpteq2dv 5167	. . . . . . . 8 ⊢ (𝐾 ∈ Proset → (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑦◡ ≤ 𝑥}) = (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑥 ≤ 𝑦}))
8	7	rneqd 5881	. . . . . . 7 ⊢ (𝐾 ∈ Proset → ran (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑦◡ ≤ 𝑥}) = ran (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑥 ≤ 𝑦}))
9	2, 1	brcnv 5825	. . . . . . . . . . . 12 ⊢ (𝑥◡ ≤ 𝑦 ↔ 𝑦 ≤ 𝑥)
10	9	a1i 11	. . . . . . . . . . 11 ⊢ (𝐾 ∈ Proset → (𝑥◡ ≤ 𝑦 ↔ 𝑦 ≤ 𝑥))
11	10	notbid 319	. . . . . . . . . 10 ⊢ (𝐾 ∈ Proset → (¬ 𝑥◡ ≤ 𝑦 ↔ ¬ 𝑦 ≤ 𝑥))
12	11	rabbidv 3398	. . . . . . . . 9 ⊢ (𝐾 ∈ Proset → {𝑦 ∈ 𝐵 ∣ ¬ 𝑥◡ ≤ 𝑦} = {𝑦 ∈ 𝐵 ∣ ¬ 𝑦 ≤ 𝑥})
13	12	mpteq2dv 5167	. . . . . . . 8 ⊢ (𝐾 ∈ Proset → (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑥◡ ≤ 𝑦}) = (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑦 ≤ 𝑥}))
14	13	rneqd 5881	. . . . . . 7 ⊢ (𝐾 ∈ Proset → ran (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑥◡ ≤ 𝑦}) = ran (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑦 ≤ 𝑥}))
15	8, 14	uneq12d 4100	. . . . . 6 ⊢ (𝐾 ∈ Proset → (ran (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑦◡ ≤ 𝑥}) ∪ ran (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑥◡ ≤ 𝑦})) = (ran (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑥 ≤ 𝑦}) ∪ ran (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑦 ≤ 𝑥})))
16		uncom 4089	. . . . . 6 ⊢ (ran (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑥 ≤ 𝑦}) ∪ ran (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑦 ≤ 𝑥})) = (ran (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑦 ≤ 𝑥}) ∪ ran (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑥 ≤ 𝑦}))
17	15, 16	eqtrdi 2790	. . . . 5 ⊢ (𝐾 ∈ Proset → (ran (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑦◡ ≤ 𝑥}) ∪ ran (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑥◡ ≤ 𝑦})) = (ran (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑦 ≤ 𝑥}) ∪ ran (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑥 ≤ 𝑦})))
18	17	uneq2d 4099	. . . 4 ⊢ (𝐾 ∈ Proset → ({𝐵} ∪ (ran (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑦◡ ≤ 𝑥}) ∪ ran (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑥◡ ≤ 𝑦}))) = ({𝐵} ∪ (ran (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑦 ≤ 𝑥}) ∪ ran (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑥 ≤ 𝑦}))))
19	18	fveq2d 6832	. . 3 ⊢ (𝐾 ∈ Proset → (fi‘({𝐵} ∪ (ran (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑦◡ ≤ 𝑥}) ∪ ran (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑥◡ ≤ 𝑦})))) = (fi‘({𝐵} ∪ (ran (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑦 ≤ 𝑥}) ∪ ran (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑥 ≤ 𝑦})))))
20	19	fveq2d 6832	. 2 ⊢ (𝐾 ∈ Proset → (topGen‘(fi‘({𝐵} ∪ (ran (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑦◡ ≤ 𝑥}) ∪ ran (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑥◡ ≤ 𝑦}))))) = (topGen‘(fi‘({𝐵} ∪ (ran (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑦 ≤ 𝑥}) ∪ ran (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑥 ≤ 𝑦}))))))
21		eqid 2739	. . . 4 ⊢ (ODual‘𝐾) = (ODual‘𝐾)
22	21	oduprs 18258	. . 3 ⊢ (𝐾 ∈ Proset → (ODual‘𝐾) ∈ Proset )
23		ordtNEW.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐾)
24	21, 23	odubas 18249	. . . 4 ⊢ 𝐵 = (Base‘(ODual‘𝐾))
25		ordtNEW.l	. . . . . 6 ⊢ ≤ = ((le‘𝐾) ∩ (𝐵 × 𝐵))
26	25	cnveqi 5817	. . . . 5 ⊢ ◡ ≤ = ◡((le‘𝐾) ∩ (𝐵 × 𝐵))
27		cnvin 6096	. . . . 5 ⊢ ◡((le‘𝐾) ∩ (𝐵 × 𝐵)) = (◡(le‘𝐾) ∩ ◡(𝐵 × 𝐵))
28		eqid 2739	. . . . . . 7 ⊢ (le‘𝐾) = (le‘𝐾)
29	21, 28	oduleval 18247	. . . . . 6 ⊢ ◡(le‘𝐾) = (le‘(ODual‘𝐾))
30		cnvxp 6109	. . . . . 6 ⊢ ◡(𝐵 × 𝐵) = (𝐵 × 𝐵)
31	29, 30	ineq12i 4148	. . . . 5 ⊢ (◡(le‘𝐾) ∩ ◡(𝐵 × 𝐵)) = ((le‘(ODual‘𝐾)) ∩ (𝐵 × 𝐵))
32	26, 27, 31	3eqtri 2766	. . . 4 ⊢ ◡ ≤ = ((le‘(ODual‘𝐾)) ∩ (𝐵 × 𝐵))
33		eqid 2739	. . . 4 ⊢ ran (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑦◡ ≤ 𝑥}) = ran (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑦◡ ≤ 𝑥})
34		eqid 2739	. . . 4 ⊢ ran (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑥◡ ≤ 𝑦}) = ran (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑥◡ ≤ 𝑦})
35	24, 32, 33, 34	ordtprsval 34111	. . 3 ⊢ ((ODual‘𝐾) ∈ Proset → (ordTop‘◡ ≤ ) = (topGen‘(fi‘({𝐵} ∪ (ran (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑦◡ ≤ 𝑥}) ∪ ran (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑥◡ ≤ 𝑦}))))))
36	22, 35	syl 17	. 2 ⊢ (𝐾 ∈ Proset → (ordTop‘◡ ≤ ) = (topGen‘(fi‘({𝐵} ∪ (ran (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑦◡ ≤ 𝑥}) ∪ ran (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑥◡ ≤ 𝑦}))))))
37		eqid 2739	. . 3 ⊢ ran (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑦 ≤ 𝑥}) = ran (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑦 ≤ 𝑥})
38		eqid 2739	. . 3 ⊢ ran (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑥 ≤ 𝑦}) = ran (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑥 ≤ 𝑦})
39	23, 25, 37, 38	ordtprsval 34111	. 2 ⊢ (𝐾 ∈ Proset → (ordTop‘ ≤ ) = (topGen‘(fi‘({𝐵} ∪ (ran (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑦 ≤ 𝑥}) ∪ ran (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑥 ≤ 𝑦}))))))
40	20, 36, 39	3eqtr4d 2784	1 ⊢ (𝐾 ∈ Proset → (ordTop‘◡ ≤ ) = (ordTop‘ ≤ ))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 207   = wceq 1547   ∈ wcel 2119  {crab 3391   ∪ cun 3881   ∩ cin 3882  {csn 4556   class class class wbr 5073   ↦ cmpt 5154   × cxp 5617  ^◡ccnv 5618  ran crn 5620  ‘cfv 6486  ficfi 9314  Basecbs 17171  lecple 17219  topGenctg 17392  ordTopcordt 17455  ODualcodu 18244   Proset cproset 18250

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5219  ax-nul 5229  ax-pow 5295  ax-pr 5363  ax-un 7679  ax-cnex 11086  ax-resscn 11087  ax-1cn 11088  ax-icn 11089  ax-addcl 11090  ax-addrcl 11091  ax-mulcl 11092  ax-mulrcl 11093  ax-mulcom 11094  ax-addass 11095  ax-mulass 11096  ax-distr 11097  ax-i2m1 11098  ax-1ne0 11099  ax-1rid 11100  ax-rnegex 11101  ax-rrecex 11102  ax-cnre 11103  ax-pre-lttri 11104  ax-pre-lttrn 11105  ax-pre-ltadd 11106  ax-pre-mulgt0 11107

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-nel 3039  df-ral 3054  df-rex 3064  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4263  df-if 4456  df-pw 4532  df-sn 4557  df-pr 4559  df-op 4563  df-uni 4840  df-iun 4924  df-br 5074  df-opab 5136  df-mpt 5155  df-tr 5181  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-riota 7314  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7808  df-2nd 7933  df-frecs 8222  df-wrecs 8253  df-recs 8302  df-rdg 8340  df-er 8634  df-en 8885  df-dom 8886  df-sdom 8887  df-pnf 11173  df-mnf 11174  df-xr 11175  df-ltxr 11176  df-le 11177  df-sub 11371  df-neg 11372  df-nn 12167  df-2 12236  df-3 12237  df-4 12238  df-5 12239  df-6 12240  df-7 12241  df-8 12242  df-9 12243  df-dec 12637  df-sets 17126  df-slot 17144  df-ndx 17156  df-base 17172  df-ple 17232  df-ordt 17457  df-odu 18245  df-proset 18252

This theorem is referenced by:  ordtrest2NEW  34116