Theorem ordtcnvNEW 33910

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 3451	. . . . . . . . . . . . 13 ⊢ 𝑦 ∈ V
2		vex 3451	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ V
3	1, 2	brcnv 5846	. . . . . . . . . . . 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 3413	. . . . . . . . 9 ⊢ (𝐾 ∈ Proset → {𝑦 ∈ 𝐵 ∣ ¬ 𝑦◡ ≤ 𝑥} = {𝑦 ∈ 𝐵 ∣ ¬ 𝑥 ≤ 𝑦})
7	6	mpteq2dv 5201	. . . . . . . 8 ⊢ (𝐾 ∈ Proset → (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑦◡ ≤ 𝑥}) = (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑥 ≤ 𝑦}))
8	7	rneqd 5902	. . . . . . 7 ⊢ (𝐾 ∈ Proset → ran (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑦◡ ≤ 𝑥}) = ran (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑥 ≤ 𝑦}))
9	2, 1	brcnv 5846	. . . . . . . . . . . 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 3413	. . . . . . . . 9 ⊢ (𝐾 ∈ Proset → {𝑦 ∈ 𝐵 ∣ ¬ 𝑥◡ ≤ 𝑦} = {𝑦 ∈ 𝐵 ∣ ¬ 𝑦 ≤ 𝑥})
13	12	mpteq2dv 5201	. . . . . . . 8 ⊢ (𝐾 ∈ Proset → (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑥◡ ≤ 𝑦}) = (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑦 ≤ 𝑥}))
14	13	rneqd 5902	. . . . . . 7 ⊢ (𝐾 ∈ Proset → ran (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑥◡ ≤ 𝑦}) = ran (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑦 ≤ 𝑥}))
15	8, 14	uneq12d 4132	. . . . . 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 4121	. . . . . 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 2780	. . . . 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 4131	. . . 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 6862	. . 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 6862	. 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 2729	. . . 4 ⊢ (ODual‘𝐾) = (ODual‘𝐾)
22	21	oduprs 18261	. . 3 ⊢ (𝐾 ∈ Proset → (ODual‘𝐾) ∈ Proset )
23		ordtNEW.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐾)
24	21, 23	odubas 18252	. . . 4 ⊢ 𝐵 = (Base‘(ODual‘𝐾))
25		ordtNEW.l	. . . . . 6 ⊢ ≤ = ((le‘𝐾) ∩ (𝐵 × 𝐵))
26	25	cnveqi 5838	. . . . 5 ⊢ ◡ ≤ = ◡((le‘𝐾) ∩ (𝐵 × 𝐵))
27		cnvin 6117	. . . . 5 ⊢ ◡((le‘𝐾) ∩ (𝐵 × 𝐵)) = (◡(le‘𝐾) ∩ ◡(𝐵 × 𝐵))
28		eqid 2729	. . . . . . 7 ⊢ (le‘𝐾) = (le‘𝐾)
29	21, 28	oduleval 18250	. . . . . 6 ⊢ ◡(le‘𝐾) = (le‘(ODual‘𝐾))
30		cnvxp 6130	. . . . . 6 ⊢ ◡(𝐵 × 𝐵) = (𝐵 × 𝐵)
31	29, 30	ineq12i 4181	. . . . 5 ⊢ (◡(le‘𝐾) ∩ ◡(𝐵 × 𝐵)) = ((le‘(ODual‘𝐾)) ∩ (𝐵 × 𝐵))
32	26, 27, 31	3eqtri 2756	. . . 4 ⊢ ◡ ≤ = ((le‘(ODual‘𝐾)) ∩ (𝐵 × 𝐵))
33		eqid 2729	. . . 4 ⊢ ran (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑦◡ ≤ 𝑥}) = ran (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑦◡ ≤ 𝑥})
34		eqid 2729	. . . 4 ⊢ ran (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑥◡ ≤ 𝑦}) = ran (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑥◡ ≤ 𝑦})
35	24, 32, 33, 34	ordtprsval 33908	. . 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 2729	. . 3 ⊢ ran (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑦 ≤ 𝑥}) = ran (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑦 ≤ 𝑥})
38		eqid 2729	. . 3 ⊢ ran (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑥 ≤ 𝑦}) = ran (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑥 ≤ 𝑦})
39	23, 25, 37, 38	ordtprsval 33908	. 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 2774	1 ⊢ (𝐾 ∈ Proset → (ordTop‘◡ ≤ ) = (ordTop‘ ≤ ))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   = wceq 1540   ∈ wcel 2109  {crab 3405   ∪ cun 3912   ∩ cin 3913  {csn 4589   class class class wbr 5107   ↦ cmpt 5188   × cxp 5636  ^◡ccnv 5637  ran crn 5639  ‘cfv 6511  ficfi 9361  Basecbs 17179  lecple 17227  topGenctg 17400  ordTopcordt 17462  ODualcodu 18247   Proset cproset 18253

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-cnex 11124  ax-resscn 11125  ax-1cn 11126  ax-icn 11127  ax-addcl 11128  ax-addrcl 11129  ax-mulcl 11130  ax-mulrcl 11131  ax-mulcom 11132  ax-addass 11133  ax-mulass 11134  ax-distr 11135  ax-i2m1 11136  ax-1ne0 11137  ax-1rid 11138  ax-rnegex 11139  ax-rrecex 11140  ax-cnre 11141  ax-pre-lttri 11142  ax-pre-lttrn 11143  ax-pre-ltadd 11144  ax-pre-mulgt0 11145

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-om 7843  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-er 8671  df-en 8919  df-dom 8920  df-sdom 8921  df-pnf 11210  df-mnf 11211  df-xr 11212  df-ltxr 11213  df-le 11214  df-sub 11407  df-neg 11408  df-nn 12187  df-2 12249  df-3 12250  df-4 12251  df-5 12252  df-6 12253  df-7 12254  df-8 12255  df-9 12256  df-dec 12650  df-sets 17134  df-slot 17152  df-ndx 17164  df-base 17180  df-ple 17240  df-ordt 17464  df-odu 18248  df-proset 18255

This theorem is referenced by:  ordtrest2NEW  33913