oduprs - Mathbox for Thierry Arnoux

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Theorem oduprs 32389

Description: Being a proset is a self-dual property. (Contributed by Thierry Arnoux, 13-Sep-2018.)

**Hypothesis**
Ref	Expression
oduprs.d	⊢ 𝐷 = (ODual‘𝐾)

**Assertion**
Ref	Expression
oduprs	⊢ (𝐾 ∈ Proset → 𝐷 ∈ Proset )

Proof of Theorem oduprs Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2732	. . . . . . . . . . . . . 14 ⊢ (Base‘𝐾) = (Base‘𝐾)
2		eqid 2732	. . . . . . . . . . . . . 14 ⊢ (le‘𝐾) = (le‘𝐾)
3	1, 2	isprs 18254	. . . . . . . . . . . . 13 ⊢ (𝐾 ∈ Proset ↔ (𝐾 ∈ V ∧ ∀𝑥 ∈ (Base‘𝐾)∀𝑦 ∈ (Base‘𝐾)∀𝑧 ∈ (Base‘𝐾)(𝑥(le‘𝐾)𝑥 ∧ ((𝑥(le‘𝐾)𝑦 ∧ 𝑦(le‘𝐾)𝑧) → 𝑥(le‘𝐾)𝑧))))
4	3	simprbi 497	. . . . . . . . . . . 12 ⊢ (𝐾 ∈ Proset → ∀𝑥 ∈ (Base‘𝐾)∀𝑦 ∈ (Base‘𝐾)∀𝑧 ∈ (Base‘𝐾)(𝑥(le‘𝐾)𝑥 ∧ ((𝑥(le‘𝐾)𝑦 ∧ 𝑦(le‘𝐾)𝑧) → 𝑥(le‘𝐾)𝑧)))
5	4	r19.21bi 3248	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ Proset ∧ 𝑥 ∈ (Base‘𝐾)) → ∀𝑦 ∈ (Base‘𝐾)∀𝑧 ∈ (Base‘𝐾)(𝑥(le‘𝐾)𝑥 ∧ ((𝑥(le‘𝐾)𝑦 ∧ 𝑦(le‘𝐾)𝑧) → 𝑥(le‘𝐾)𝑧)))
6	5	r19.21bi 3248	. . . . . . . . . 10 ⊢ (((𝐾 ∈ Proset ∧ 𝑥 ∈ (Base‘𝐾)) ∧ 𝑦 ∈ (Base‘𝐾)) → ∀𝑧 ∈ (Base‘𝐾)(𝑥(le‘𝐾)𝑥 ∧ ((𝑥(le‘𝐾)𝑦 ∧ 𝑦(le‘𝐾)𝑧) → 𝑥(le‘𝐾)𝑧)))
7	6	r19.21bi 3248	. . . . . . . . 9 ⊢ ((((𝐾 ∈ Proset ∧ 𝑥 ∈ (Base‘𝐾)) ∧ 𝑦 ∈ (Base‘𝐾)) ∧ 𝑧 ∈ (Base‘𝐾)) → (𝑥(le‘𝐾)𝑥 ∧ ((𝑥(le‘𝐾)𝑦 ∧ 𝑦(le‘𝐾)𝑧) → 𝑥(le‘𝐾)𝑧)))
8	7	simpld 495	. . . . . . . 8 ⊢ ((((𝐾 ∈ Proset ∧ 𝑥 ∈ (Base‘𝐾)) ∧ 𝑦 ∈ (Base‘𝐾)) ∧ 𝑧 ∈ (Base‘𝐾)) → 𝑥(le‘𝐾)𝑥)
9		vex 3478	. . . . . . . . 9 ⊢ 𝑥 ∈ V
10	9, 9	brcnv 5882	. . . . . . . 8 ⊢ (𝑥^◡(le‘𝐾)𝑥 ↔ 𝑥(le‘𝐾)𝑥)
11	8, 10	sylibr 233	. . . . . . 7 ⊢ ((((𝐾 ∈ Proset ∧ 𝑥 ∈ (Base‘𝐾)) ∧ 𝑦 ∈ (Base‘𝐾)) ∧ 𝑧 ∈ (Base‘𝐾)) → 𝑥^◡(le‘𝐾)𝑥)
12	1, 2	isprs 18254	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐾 ∈ Proset ↔ (𝐾 ∈ V ∧ ∀𝑧 ∈ (Base‘𝐾)∀𝑦 ∈ (Base‘𝐾)∀𝑥 ∈ (Base‘𝐾)(𝑧(le‘𝐾)𝑧 ∧ ((𝑧(le‘𝐾)𝑦 ∧ 𝑦(le‘𝐾)𝑥) → 𝑧(le‘𝐾)𝑥))))
13	12	simprbi 497	. . . . . . . . . . . . . . . . 17 ⊢ (𝐾 ∈ Proset → ∀𝑧 ∈ (Base‘𝐾)∀𝑦 ∈ (Base‘𝐾)∀𝑥 ∈ (Base‘𝐾)(𝑧(le‘𝐾)𝑧 ∧ ((𝑧(le‘𝐾)𝑦 ∧ 𝑦(le‘𝐾)𝑥) → 𝑧(le‘𝐾)𝑥)))
14	13	r19.21bi 3248	. . . . . . . . . . . . . . . 16 ⊢ ((𝐾 ∈ Proset ∧ 𝑧 ∈ (Base‘𝐾)) → ∀𝑦 ∈ (Base‘𝐾)∀𝑥 ∈ (Base‘𝐾)(𝑧(le‘𝐾)𝑧 ∧ ((𝑧(le‘𝐾)𝑦 ∧ 𝑦(le‘𝐾)𝑥) → 𝑧(le‘𝐾)𝑥)))
15	14	r19.21bi 3248	. . . . . . . . . . . . . . 15 ⊢ (((𝐾 ∈ Proset ∧ 𝑧 ∈ (Base‘𝐾)) ∧ 𝑦 ∈ (Base‘𝐾)) → ∀𝑥 ∈ (Base‘𝐾)(𝑧(le‘𝐾)𝑧 ∧ ((𝑧(le‘𝐾)𝑦 ∧ 𝑦(le‘𝐾)𝑥) → 𝑧(le‘𝐾)𝑥)))
16	15	r19.21bi 3248	. . . . . . . . . . . . . 14 ⊢ ((((𝐾 ∈ Proset ∧ 𝑧 ∈ (Base‘𝐾)) ∧ 𝑦 ∈ (Base‘𝐾)) ∧ 𝑥 ∈ (Base‘𝐾)) → (𝑧(le‘𝐾)𝑧 ∧ ((𝑧(le‘𝐾)𝑦 ∧ 𝑦(le‘𝐾)𝑥) → 𝑧(le‘𝐾)𝑥)))
17	16	simprd 496	. . . . . . . . . . . . 13 ⊢ ((((𝐾 ∈ Proset ∧ 𝑧 ∈ (Base‘𝐾)) ∧ 𝑦 ∈ (Base‘𝐾)) ∧ 𝑥 ∈ (Base‘𝐾)) → ((𝑧(le‘𝐾)𝑦 ∧ 𝑦(le‘𝐾)𝑥) → 𝑧(le‘𝐾)𝑥))
18	17	an32s 650	. . . . . . . . . . . 12 ⊢ ((((𝐾 ∈ Proset ∧ 𝑧 ∈ (Base‘𝐾)) ∧ 𝑥 ∈ (Base‘𝐾)) ∧ 𝑦 ∈ (Base‘𝐾)) → ((𝑧(le‘𝐾)𝑦 ∧ 𝑦(le‘𝐾)𝑥) → 𝑧(le‘𝐾)𝑥))
19	18	ex 413	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ Proset ∧ 𝑧 ∈ (Base‘𝐾)) ∧ 𝑥 ∈ (Base‘𝐾)) → (𝑦 ∈ (Base‘𝐾) → ((𝑧(le‘𝐾)𝑦 ∧ 𝑦(le‘𝐾)𝑥) → 𝑧(le‘𝐾)𝑥)))
20	19	an32s 650	. . . . . . . . . 10 ⊢ (((𝐾 ∈ Proset ∧ 𝑥 ∈ (Base‘𝐾)) ∧ 𝑧 ∈ (Base‘𝐾)) → (𝑦 ∈ (Base‘𝐾) → ((𝑧(le‘𝐾)𝑦 ∧ 𝑦(le‘𝐾)𝑥) → 𝑧(le‘𝐾)𝑥)))
21	20	imp 407	. . . . . . . . 9 ⊢ ((((𝐾 ∈ Proset ∧ 𝑥 ∈ (Base‘𝐾)) ∧ 𝑧 ∈ (Base‘𝐾)) ∧ 𝑦 ∈ (Base‘𝐾)) → ((𝑧(le‘𝐾)𝑦 ∧ 𝑦(le‘𝐾)𝑥) → 𝑧(le‘𝐾)𝑥))
22	21	an32s 650	. . . . . . . 8 ⊢ ((((𝐾 ∈ Proset ∧ 𝑥 ∈ (Base‘𝐾)) ∧ 𝑦 ∈ (Base‘𝐾)) ∧ 𝑧 ∈ (Base‘𝐾)) → ((𝑧(le‘𝐾)𝑦 ∧ 𝑦(le‘𝐾)𝑥) → 𝑧(le‘𝐾)𝑥))
23		vex 3478	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
24	9, 23	brcnv 5882	. . . . . . . . 9 ⊢ (𝑥^◡(le‘𝐾)𝑦 ↔ 𝑦(le‘𝐾)𝑥)
25		vex 3478	. . . . . . . . . 10 ⊢ 𝑧 ∈ V
26	23, 25	brcnv 5882	. . . . . . . . 9 ⊢ (𝑦^◡(le‘𝐾)𝑧 ↔ 𝑧(le‘𝐾)𝑦)
27	24, 26	anbi12ci 628	. . . . . . . 8 ⊢ ((𝑥^◡(le‘𝐾)𝑦 ∧ 𝑦^◡(le‘𝐾)𝑧) ↔ (𝑧(le‘𝐾)𝑦 ∧ 𝑦(le‘𝐾)𝑥))
28	9, 25	brcnv 5882	. . . . . . . 8 ⊢ (𝑥^◡(le‘𝐾)𝑧 ↔ 𝑧(le‘𝐾)𝑥)
29	22, 27, 28	3imtr4g 295	. . . . . . 7 ⊢ ((((𝐾 ∈ Proset ∧ 𝑥 ∈ (Base‘𝐾)) ∧ 𝑦 ∈ (Base‘𝐾)) ∧ 𝑧 ∈ (Base‘𝐾)) → ((𝑥^◡(le‘𝐾)𝑦 ∧ 𝑦^◡(le‘𝐾)𝑧) → 𝑥^◡(le‘𝐾)𝑧))
30	11, 29	jca 512	. . . . . 6 ⊢ ((((𝐾 ∈ Proset ∧ 𝑥 ∈ (Base‘𝐾)) ∧ 𝑦 ∈ (Base‘𝐾)) ∧ 𝑧 ∈ (Base‘𝐾)) → (𝑥^◡(le‘𝐾)𝑥 ∧ ((𝑥^◡(le‘𝐾)𝑦 ∧ 𝑦^◡(le‘𝐾)𝑧) → 𝑥^◡(le‘𝐾)𝑧)))
31	30	ralrimiva 3146	. . . . 5 ⊢ (((𝐾 ∈ Proset ∧ 𝑥 ∈ (Base‘𝐾)) ∧ 𝑦 ∈ (Base‘𝐾)) → ∀𝑧 ∈ (Base‘𝐾)(𝑥^◡(le‘𝐾)𝑥 ∧ ((𝑥^◡(le‘𝐾)𝑦 ∧ 𝑦^◡(le‘𝐾)𝑧) → 𝑥^◡(le‘𝐾)𝑧)))
32	31	ralrimiva 3146	. . . 4 ⊢ ((𝐾 ∈ Proset ∧ 𝑥 ∈ (Base‘𝐾)) → ∀𝑦 ∈ (Base‘𝐾)∀𝑧 ∈ (Base‘𝐾)(𝑥^◡(le‘𝐾)𝑥 ∧ ((𝑥^◡(le‘𝐾)𝑦 ∧ 𝑦^◡(le‘𝐾)𝑧) → 𝑥^◡(le‘𝐾)𝑧)))
33	32	ralrimiva 3146	. . 3 ⊢ (𝐾 ∈ Proset → ∀𝑥 ∈ (Base‘𝐾)∀𝑦 ∈ (Base‘𝐾)∀𝑧 ∈ (Base‘𝐾)(𝑥^◡(le‘𝐾)𝑥 ∧ ((𝑥^◡(le‘𝐾)𝑦 ∧ 𝑦^◡(le‘𝐾)𝑧) → 𝑥^◡(le‘𝐾)𝑧)))
34		oduprs.d	. . . 4 ⊢ 𝐷 = (ODual‘𝐾)
35	34	fvexi 6905	. . 3 ⊢ 𝐷 ∈ V
36	33, 35	jctil 520	. 2 ⊢ (𝐾 ∈ Proset → (𝐷 ∈ V ∧ ∀𝑥 ∈ (Base‘𝐾)∀𝑦 ∈ (Base‘𝐾)∀𝑧 ∈ (Base‘𝐾)(𝑥^◡(le‘𝐾)𝑥 ∧ ((𝑥^◡(le‘𝐾)𝑦 ∧ 𝑦^◡(le‘𝐾)𝑧) → 𝑥^◡(le‘𝐾)𝑧))))
37	34, 1	odubas 18248	. . 3 ⊢ (Base‘𝐾) = (Base‘𝐷)
38	34, 2	oduleval 18246	. . 3 ⊢ ^◡(le‘𝐾) = (le‘𝐷)
39	37, 38	isprs 18254	. 2 ⊢ (𝐷 ∈ Proset ↔ (𝐷 ∈ V ∧ ∀𝑥 ∈ (Base‘𝐾)∀𝑦 ∈ (Base‘𝐾)∀𝑧 ∈ (Base‘𝐾)(𝑥^◡(le‘𝐾)𝑥 ∧ ((𝑥^◡(le‘𝐾)𝑦 ∧ 𝑦^◡(le‘𝐾)𝑧) → 𝑥^◡(le‘𝐾)𝑧))))
40	36, 39	sylibr 233	1 ⊢ (𝐾 ∈ Proset → 𝐷 ∈ Proset )

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∀wral 3061 Vcvv 3474 class class class wbr 5148 ^◡ccnv 5675 ‘cfv 6543 Basecbs 17148 lecple 17208 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 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 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 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 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-odu 18244 df-proset 18252

This theorem is referenced by: mgccnv 32424 ordtcnvNEW 33186

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