Theorem ordtcnvNEW 34084

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 3434	. . . . . . . . . . . . 13 ⊢ 𝑦 ∈ V
2		vex 3434	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ V
3	1, 2	brcnv 5833	. . . . . . . . . . . 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 3397	. . . . . . . . 9 ⊢ (𝐾 ∈ Proset → {𝑦 ∈ 𝐵 ∣ ¬ 𝑦◡ ≤ 𝑥} = {𝑦 ∈ 𝐵 ∣ ¬ 𝑥 ≤ 𝑦})
7	6	mpteq2dv 5180	. . . . . . . 8 ⊢ (𝐾 ∈ Proset → (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑦◡ ≤ 𝑥}) = (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑥 ≤ 𝑦}))
8	7	rneqd 5889	. . . . . . 7 ⊢ (𝐾 ∈ Proset → ran (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑦◡ ≤ 𝑥}) = ran (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑥 ≤ 𝑦}))
9	2, 1	brcnv 5833	. . . . . . . . . . . 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 3397	. . . . . . . . 9 ⊢ (𝐾 ∈ Proset → {𝑦 ∈ 𝐵 ∣ ¬ 𝑥◡ ≤ 𝑦} = {𝑦 ∈ 𝐵 ∣ ¬ 𝑦 ≤ 𝑥})
13	12	mpteq2dv 5180	. . . . . . . 8 ⊢ (𝐾 ∈ Proset → (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑥◡ ≤ 𝑦}) = (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑦 ≤ 𝑥}))
14	13	rneqd 5889	. . . . . . 7 ⊢ (𝐾 ∈ Proset → ran (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑥◡ ≤ 𝑦}) = ran (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑦 ≤ 𝑥}))
15	8, 14	uneq12d 4110	. . . . . 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 4099	. . . . . 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 2788	. . . . 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 4109	. . . 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 6840	. . 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 6840	. 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 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 5825	. . . . 5 ⊢ ◡ ≤ = ◡((le‘𝐾) ∩ (𝐵 × 𝐵))
27		cnvin 6104	. . . . 5 ⊢ ◡((le‘𝐾) ∩ (𝐵 × 𝐵)) = (◡(le‘𝐾) ∩ ◡(𝐵 × 𝐵))
28		eqid 2737	. . . . . . 7 ⊢ (le‘𝐾) = (le‘𝐾)
29	21, 28	oduleval 18250	. . . . . 6 ⊢ ◡(le‘𝐾) = (le‘(ODual‘𝐾))
30		cnvxp 6117	. . . . . 6 ⊢ ◡(𝐵 × 𝐵) = (𝐵 × 𝐵)
31	29, 30	ineq12i 4159	. . . . 5 ⊢ (◡(le‘𝐾) ∩ ◡(𝐵 × 𝐵)) = ((le‘(ODual‘𝐾)) ∩ (𝐵 × 𝐵))
32	26, 27, 31	3eqtri 2764	. . . 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 34082	. . 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 34082	. 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 2782	1 ⊢ (𝐾 ∈ Proset → (ordTop‘◡ ≤ ) = (ordTop‘ ≤ ))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   = wceq 1542   ∈ wcel 2114  {crab 3390   ∪ cun 3888   ∩ cin 3889  {csn 4568   class class class wbr 5086   ↦ cmpt 5167   × cxp 5624  ^◡ccnv 5625  ran crn 5627  ‘cfv 6494  ficfi 9318  Basecbs 17174  lecple 17222  topGenctg 17395  ordTopcordt 17458  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 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5232  ax-nul 5242  ax-pow 5304  ax-pr 5372  ax-un 7684  ax-cnex 11089  ax-resscn 11090  ax-1cn 11091  ax-icn 11092  ax-addcl 11093  ax-addrcl 11094  ax-mulcl 11095  ax-mulrcl 11096  ax-mulcom 11097  ax-addass 11098  ax-mulass 11099  ax-distr 11100  ax-i2m1 11101  ax-1ne0 11102  ax-1rid 11103  ax-rnegex 11104  ax-rrecex 11105  ax-cnre 11106  ax-pre-lttri 11107  ax-pre-lttrn 11108  ax-pre-ltadd 11109  ax-pre-mulgt0 11110

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5521  df-eprel 5526  df-po 5534  df-so 5535  df-fr 5579  df-we 5581  df-xp 5632  df-rel 5633  df-cnv 5634  df-co 5635  df-dm 5636  df-rn 5637  df-res 5638  df-ima 5639  df-pred 6261  df-ord 6322  df-on 6323  df-lim 6324  df-suc 6325  df-iota 6450  df-fun 6496  df-fn 6497  df-f 6498  df-f1 6499  df-fo 6500  df-f1o 6501  df-fv 6502  df-riota 7319  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7813  df-2nd 7938  df-frecs 8226  df-wrecs 8257  df-recs 8306  df-rdg 8344  df-er 8638  df-en 8889  df-dom 8890  df-sdom 8891  df-pnf 11176  df-mnf 11177  df-xr 11178  df-ltxr 11179  df-le 11180  df-sub 11374  df-neg 11375  df-nn 12170  df-2 12239  df-3 12240  df-4 12241  df-5 12242  df-6 12243  df-7 12244  df-8 12245  df-9 12246  df-dec 12640  df-sets 17129  df-slot 17147  df-ndx 17159  df-base 17175  df-ple 17235  df-ordt 17460  df-odu 18248  df-proset 18255

This theorem is referenced by:  ordtrest2NEW  34087