Theorem ordtcnvNEW 33919

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 3484	. . . . . . . . . . . . 13 ⊢ 𝑦 ∈ V
2		vex 3484	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ V
3	1, 2	brcnv 5893	. . . . . . . . . . . 12 ⊢ (𝑦◡ ≤ 𝑥 ↔ 𝑥 ≤ 𝑦)
4	3	a1i 11	. . . . . . . . . . 11 ⊢ (𝐾 ∈ Proset → (𝑦◡ ≤ 𝑥 ↔ 𝑥 ≤ 𝑦))
5	4	notbid 318	. . . . . . . . . 10 ⊢ (𝐾 ∈ Proset → (¬ 𝑦◡ ≤ 𝑥 ↔ ¬ 𝑥 ≤ 𝑦))
6	5	rabbidv 3444	. . . . . . . . 9 ⊢ (𝐾 ∈ Proset → {𝑦 ∈ 𝐵 ∣ ¬ 𝑦◡ ≤ 𝑥} = {𝑦 ∈ 𝐵 ∣ ¬ 𝑥 ≤ 𝑦})
7	6	mpteq2dv 5244	. . . . . . . 8 ⊢ (𝐾 ∈ Proset → (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑦◡ ≤ 𝑥}) = (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑥 ≤ 𝑦}))
8	7	rneqd 5949	. . . . . . 7 ⊢ (𝐾 ∈ Proset → ran (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑦◡ ≤ 𝑥}) = ran (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑥 ≤ 𝑦}))
9	2, 1	brcnv 5893	. . . . . . . . . . . 12 ⊢ (𝑥◡ ≤ 𝑦 ↔ 𝑦 ≤ 𝑥)
10	9	a1i 11	. . . . . . . . . . 11 ⊢ (𝐾 ∈ Proset → (𝑥◡ ≤ 𝑦 ↔ 𝑦 ≤ 𝑥))
11	10	notbid 318	. . . . . . . . . 10 ⊢ (𝐾 ∈ Proset → (¬ 𝑥◡ ≤ 𝑦 ↔ ¬ 𝑦 ≤ 𝑥))
12	11	rabbidv 3444	. . . . . . . . 9 ⊢ (𝐾 ∈ Proset → {𝑦 ∈ 𝐵 ∣ ¬ 𝑥◡ ≤ 𝑦} = {𝑦 ∈ 𝐵 ∣ ¬ 𝑦 ≤ 𝑥})
13	12	mpteq2dv 5244	. . . . . . . 8 ⊢ (𝐾 ∈ Proset → (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑥◡ ≤ 𝑦}) = (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑦 ≤ 𝑥}))
14	13	rneqd 5949	. . . . . . 7 ⊢ (𝐾 ∈ Proset → ran (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑥◡ ≤ 𝑦}) = ran (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑦 ≤ 𝑥}))
15	8, 14	uneq12d 4169	. . . . . 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 4158	. . . . . 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 2793	. . . . 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 4168	. . . 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 6910	. . 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 6910	. 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 2737	. . . 4 ⊢ (ODual‘𝐾) = (ODual‘𝐾)
22	21	oduprs 18346	. . 3 ⊢ (𝐾 ∈ Proset → (ODual‘𝐾) ∈ Proset )
23		ordtNEW.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐾)
24	21, 23	odubas 18336	. . . 4 ⊢ 𝐵 = (Base‘(ODual‘𝐾))
25		ordtNEW.l	. . . . . 6 ⊢ ≤ = ((le‘𝐾) ∩ (𝐵 × 𝐵))
26	25	cnveqi 5885	. . . . 5 ⊢ ◡ ≤ = ◡((le‘𝐾) ∩ (𝐵 × 𝐵))
27		cnvin 6164	. . . . 5 ⊢ ◡((le‘𝐾) ∩ (𝐵 × 𝐵)) = (◡(le‘𝐾) ∩ ◡(𝐵 × 𝐵))
28		eqid 2737	. . . . . . 7 ⊢ (le‘𝐾) = (le‘𝐾)
29	21, 28	oduleval 18334	. . . . . 6 ⊢ ◡(le‘𝐾) = (le‘(ODual‘𝐾))
30		cnvxp 6177	. . . . . 6 ⊢ ◡(𝐵 × 𝐵) = (𝐵 × 𝐵)
31	29, 30	ineq12i 4218	. . . . 5 ⊢ (◡(le‘𝐾) ∩ ◡(𝐵 × 𝐵)) = ((le‘(ODual‘𝐾)) ∩ (𝐵 × 𝐵))
32	26, 27, 31	3eqtri 2769	. . . 4 ⊢ ◡ ≤ = ((le‘(ODual‘𝐾)) ∩ (𝐵 × 𝐵))
33		eqid 2737	. . . 4 ⊢ ran (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑦◡ ≤ 𝑥}) = ran (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑦◡ ≤ 𝑥})
34		eqid 2737	. . . 4 ⊢ ran (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑥◡ ≤ 𝑦}) = ran (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑥◡ ≤ 𝑦})
35	24, 32, 33, 34	ordtprsval 33917	. . 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 2737	. . 3 ⊢ ran (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑦 ≤ 𝑥}) = ran (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑦 ≤ 𝑥})
38		eqid 2737	. . 3 ⊢ ran (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑥 ≤ 𝑦}) = ran (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑥 ≤ 𝑦})
39	23, 25, 37, 38	ordtprsval 33917	. 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 2787	1 ⊢ (𝐾 ∈ Proset → (ordTop‘◡ ≤ ) = (ordTop‘ ≤ ))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   = wceq 1540   ∈ wcel 2108  {crab 3436   ∪ cun 3949   ∩ cin 3950  {csn 4626   class class class wbr 5143   ↦ cmpt 5225   × cxp 5683  ^◡ccnv 5684  ran crn 5686  ‘cfv 6561  ficfi 9450  Basecbs 17247  lecple 17304  topGenctg 17482  ordTopcordt 17544  ODualcodu 18331   Proset cproset 18338

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-5 12332  df-6 12333  df-7 12334  df-8 12335  df-9 12336  df-dec 12734  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ple 17317  df-ordt 17546  df-odu 18332  df-proset 18340

This theorem is referenced by:  ordtrest2NEW  33922