ordtcnvNEW - Mathbox for Thierry Arnoux

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Theorem ordtcnvNEW 33198

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 3476	. . . . . . . . . . . . 13 ⊢ 𝑦 ∈ V
2		vex 3476	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ V
3	1, 2	brcnv 5881	. . . . . . . . . . . 12 ⊢ (𝑦^◡ ≤ 𝑥 ↔ 𝑥 ≤ 𝑦)
4	3	a1i 11	. . . . . . . . . . 11 ⊢ (𝐾 ∈ Proset → (𝑦^◡ ≤ 𝑥 ↔ 𝑥 ≤ 𝑦))
5	4	notbid 317	. . . . . . . . . 10 ⊢ (𝐾 ∈ Proset → (¬ 𝑦^◡ ≤ 𝑥 ↔ ¬ 𝑥 ≤ 𝑦))
6	5	rabbidv 3438	. . . . . . . . 9 ⊢ (𝐾 ∈ Proset → {𝑦 ∈ 𝐵 ∣ ¬ 𝑦^◡ ≤ 𝑥} = {𝑦 ∈ 𝐵 ∣ ¬ 𝑥 ≤ 𝑦})
7	6	mpteq2dv 5249	. . . . . . . 8 ⊢ (𝐾 ∈ Proset → (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑦^◡ ≤ 𝑥}) = (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑥 ≤ 𝑦}))
8	7	rneqd 5936	. . . . . . 7 ⊢ (𝐾 ∈ Proset → ran (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑦^◡ ≤ 𝑥}) = ran (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑥 ≤ 𝑦}))
9	2, 1	brcnv 5881	. . . . . . . . . . . 12 ⊢ (𝑥^◡ ≤ 𝑦 ↔ 𝑦 ≤ 𝑥)
10	9	a1i 11	. . . . . . . . . . 11 ⊢ (𝐾 ∈ Proset → (𝑥^◡ ≤ 𝑦 ↔ 𝑦 ≤ 𝑥))
11	10	notbid 317	. . . . . . . . . 10 ⊢ (𝐾 ∈ Proset → (¬ 𝑥^◡ ≤ 𝑦 ↔ ¬ 𝑦 ≤ 𝑥))
12	11	rabbidv 3438	. . . . . . . . 9 ⊢ (𝐾 ∈ Proset → {𝑦 ∈ 𝐵 ∣ ¬ 𝑥^◡ ≤ 𝑦} = {𝑦 ∈ 𝐵 ∣ ¬ 𝑦 ≤ 𝑥})
13	12	mpteq2dv 5249	. . . . . . . 8 ⊢ (𝐾 ∈ Proset → (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑥^◡ ≤ 𝑦}) = (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑦 ≤ 𝑥}))
14	13	rneqd 5936	. . . . . . 7 ⊢ (𝐾 ∈ Proset → ran (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑥^◡ ≤ 𝑦}) = ran (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑦 ≤ 𝑥}))
15	8, 14	uneq12d 4163	. . . . . 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 4152	. . . . . 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 2786	. . . . 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 4162	. . . 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 6894	. . 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 6894	. 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 2730	. . . 4 ⊢ (ODual‘𝐾) = (ODual‘𝐾)
22	21	oduprs 32401	. . 3 ⊢ (𝐾 ∈ Proset → (ODual‘𝐾) ∈ Proset )
23		ordtNEW.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐾)
24	21, 23	odubas 18248	. . . 4 ⊢ 𝐵 = (Base‘(ODual‘𝐾))
25		ordtNEW.l	. . . . . 6 ⊢ ≤ = ((le‘𝐾) ∩ (𝐵 × 𝐵))
26	25	cnveqi 5873	. . . . 5 ⊢ ^◡ ≤ = ^◡((le‘𝐾) ∩ (𝐵 × 𝐵))
27		cnvin 6143	. . . . 5 ⊢ ^◡((le‘𝐾) ∩ (𝐵 × 𝐵)) = (^◡(le‘𝐾) ∩ ^◡(𝐵 × 𝐵))
28		eqid 2730	. . . . . . 7 ⊢ (le‘𝐾) = (le‘𝐾)
29	21, 28	oduleval 18246	. . . . . 6 ⊢ ^◡(le‘𝐾) = (le‘(ODual‘𝐾))
30		cnvxp 6155	. . . . . 6 ⊢ ^◡(𝐵 × 𝐵) = (𝐵 × 𝐵)
31	29, 30	ineq12i 4209	. . . . 5 ⊢ (^◡(le‘𝐾) ∩ ^◡(𝐵 × 𝐵)) = ((le‘(ODual‘𝐾)) ∩ (𝐵 × 𝐵))
32	26, 27, 31	3eqtri 2762	. . . 4 ⊢ ^◡ ≤ = ((le‘(ODual‘𝐾)) ∩ (𝐵 × 𝐵))
33		eqid 2730	. . . 4 ⊢ ran (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑦^◡ ≤ 𝑥}) = ran (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑦^◡ ≤ 𝑥})
34		eqid 2730	. . . 4 ⊢ ran (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑥^◡ ≤ 𝑦}) = ran (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑥^◡ ≤ 𝑦})
35	24, 32, 33, 34	ordtprsval 33196	. . 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 2730	. . 3 ⊢ ran (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑦 ≤ 𝑥}) = ran (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑦 ≤ 𝑥})
38		eqid 2730	. . 3 ⊢ ran (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑥 ≤ 𝑦}) = ran (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑥 ≤ 𝑦})
39	23, 25, 37, 38	ordtprsval 33196	. 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 2780	1 ⊢ (𝐾 ∈ Proset → (ordTop‘^◡ ≤ ) = (ordTop‘ ≤ ))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 = wceq 1539 ∈ wcel 2104 {crab 3430 ∪ cun 3945 ∩ cin 3946 {csn 4627 class class class wbr 5147 ↦ cmpt 5230 × cxp 5673 ^◡ccnv 5674 ran crn 5676 ‘cfv 6542 ficfi 9407 Basecbs 17148 lecple 17208 topGenctg 17387 ordTopcordt 17449 ODualcodu 18243 Proset cproset 18250

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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189

This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-dec 12682 df-sets 17101 df-slot 17119 df-ndx 17131 df-base 17149 df-ple 17221 df-ordt 17451 df-odu 18244 df-proset 18252

This theorem is referenced by: ordtrest2NEW 33201

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