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Theorem doca2N 36929
Description: Double orthocomplement of partial isomorphism A. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
doca2.h 𝐻 = (LHyp‘𝐾)
doca2.i 𝐼 = ((DIsoA‘𝐾)‘𝑊)
doca2.n = ((ocA‘𝐾)‘𝑊)
Assertion
Ref Expression
doca2N (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋 ∈ dom 𝐼) → ( ‘( ‘(𝐼𝑋))) = (𝐼𝑋))

Proof of Theorem doca2N
StepHypRef Expression
1 hlol 35163 . . . . . . . . . . . . 13 (𝐾 ∈ HL → 𝐾 ∈ OL)
21ad2antrr 697 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋 ∈ dom 𝐼) → 𝐾 ∈ OL)
3 eqid 2770 . . . . . . . . . . . . 13 (Base‘𝐾) = (Base‘𝐾)
4 doca2.h . . . . . . . . . . . . 13 𝐻 = (LHyp‘𝐾)
5 doca2.i . . . . . . . . . . . . 13 𝐼 = ((DIsoA‘𝐾)‘𝑊)
63, 4, 5diadmclN 36840 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋 ∈ dom 𝐼) → 𝑋 ∈ (Base‘𝐾))
73, 4lhpbase 35799 . . . . . . . . . . . . 13 (𝑊𝐻𝑊 ∈ (Base‘𝐾))
87ad2antlr 698 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋 ∈ dom 𝐼) → 𝑊 ∈ (Base‘𝐾))
9 eqid 2770 . . . . . . . . . . . . 13 (join‘𝐾) = (join‘𝐾)
10 eqid 2770 . . . . . . . . . . . . 13 (meet‘𝐾) = (meet‘𝐾)
11 eqid 2770 . . . . . . . . . . . . 13 (oc‘𝐾) = (oc‘𝐾)
123, 9, 10, 11oldmm1 35019 . . . . . . . . . . . 12 ((𝐾 ∈ OL ∧ 𝑋 ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((oc‘𝐾)‘(𝑋(meet‘𝐾)𝑊)) = (((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑊)))
132, 6, 8, 12syl3anc 1475 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋 ∈ dom 𝐼) → ((oc‘𝐾)‘(𝑋(meet‘𝐾)𝑊)) = (((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑊)))
1413oveq1d 6807 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋 ∈ dom 𝐼) → (((oc‘𝐾)‘(𝑋(meet‘𝐾)𝑊))(meet‘𝐾)𝑊) = ((((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊))
1514eqcomd 2776 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋 ∈ dom 𝐼) → ((((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊) = (((oc‘𝐾)‘(𝑋(meet‘𝐾)𝑊))(meet‘𝐾)𝑊))
1615fveq2d 6336 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋 ∈ dom 𝐼) → ((oc‘𝐾)‘((((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)) = ((oc‘𝐾)‘(((oc‘𝐾)‘(𝑋(meet‘𝐾)𝑊))(meet‘𝐾)𝑊)))
17 hllat 35165 . . . . . . . . . . 11 (𝐾 ∈ HL → 𝐾 ∈ Lat)
1817ad2antrr 697 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋 ∈ dom 𝐼) → 𝐾 ∈ Lat)
193, 10latmcl 17259 . . . . . . . . . 10 ((𝐾 ∈ Lat ∧ 𝑋 ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → (𝑋(meet‘𝐾)𝑊) ∈ (Base‘𝐾))
2018, 6, 8, 19syl3anc 1475 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋 ∈ dom 𝐼) → (𝑋(meet‘𝐾)𝑊) ∈ (Base‘𝐾))
213, 9, 10, 11oldmm2 35020 . . . . . . . . 9 ((𝐾 ∈ OL ∧ (𝑋(meet‘𝐾)𝑊) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((oc‘𝐾)‘(((oc‘𝐾)‘(𝑋(meet‘𝐾)𝑊))(meet‘𝐾)𝑊)) = ((𝑋(meet‘𝐾)𝑊)(join‘𝐾)((oc‘𝐾)‘𝑊)))
222, 20, 8, 21syl3anc 1475 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋 ∈ dom 𝐼) → ((oc‘𝐾)‘(((oc‘𝐾)‘(𝑋(meet‘𝐾)𝑊))(meet‘𝐾)𝑊)) = ((𝑋(meet‘𝐾)𝑊)(join‘𝐾)((oc‘𝐾)‘𝑊)))
2316, 22eqtrd 2804 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋 ∈ dom 𝐼) → ((oc‘𝐾)‘((((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)) = ((𝑋(meet‘𝐾)𝑊)(join‘𝐾)((oc‘𝐾)‘𝑊)))
2423oveq1d 6807 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋 ∈ dom 𝐼) → (((oc‘𝐾)‘((((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊))(join‘𝐾)((oc‘𝐾)‘𝑊)) = (((𝑋(meet‘𝐾)𝑊)(join‘𝐾)((oc‘𝐾)‘𝑊))(join‘𝐾)((oc‘𝐾)‘𝑊)))
25 hlop 35164 . . . . . . . . . 10 (𝐾 ∈ HL → 𝐾 ∈ OP)
2625ad2antrr 697 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋 ∈ dom 𝐼) → 𝐾 ∈ OP)
273, 11opoccl 34996 . . . . . . . . 9 ((𝐾 ∈ OP ∧ 𝑊 ∈ (Base‘𝐾)) → ((oc‘𝐾)‘𝑊) ∈ (Base‘𝐾))
2826, 8, 27syl2anc 565 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋 ∈ dom 𝐼) → ((oc‘𝐾)‘𝑊) ∈ (Base‘𝐾))
293, 9latjass 17302 . . . . . . . 8 ((𝐾 ∈ Lat ∧ ((𝑋(meet‘𝐾)𝑊) ∈ (Base‘𝐾) ∧ ((oc‘𝐾)‘𝑊) ∈ (Base‘𝐾) ∧ ((oc‘𝐾)‘𝑊) ∈ (Base‘𝐾))) → (((𝑋(meet‘𝐾)𝑊)(join‘𝐾)((oc‘𝐾)‘𝑊))(join‘𝐾)((oc‘𝐾)‘𝑊)) = ((𝑋(meet‘𝐾)𝑊)(join‘𝐾)(((oc‘𝐾)‘𝑊)(join‘𝐾)((oc‘𝐾)‘𝑊))))
3018, 20, 28, 28, 29syl13anc 1477 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋 ∈ dom 𝐼) → (((𝑋(meet‘𝐾)𝑊)(join‘𝐾)((oc‘𝐾)‘𝑊))(join‘𝐾)((oc‘𝐾)‘𝑊)) = ((𝑋(meet‘𝐾)𝑊)(join‘𝐾)(((oc‘𝐾)‘𝑊)(join‘𝐾)((oc‘𝐾)‘𝑊))))
313, 9latjidm 17281 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ ((oc‘𝐾)‘𝑊) ∈ (Base‘𝐾)) → (((oc‘𝐾)‘𝑊)(join‘𝐾)((oc‘𝐾)‘𝑊)) = ((oc‘𝐾)‘𝑊))
3218, 28, 31syl2anc 565 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋 ∈ dom 𝐼) → (((oc‘𝐾)‘𝑊)(join‘𝐾)((oc‘𝐾)‘𝑊)) = ((oc‘𝐾)‘𝑊))
3332oveq2d 6808 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋 ∈ dom 𝐼) → ((𝑋(meet‘𝐾)𝑊)(join‘𝐾)(((oc‘𝐾)‘𝑊)(join‘𝐾)((oc‘𝐾)‘𝑊))) = ((𝑋(meet‘𝐾)𝑊)(join‘𝐾)((oc‘𝐾)‘𝑊)))
3430, 33eqtrd 2804 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋 ∈ dom 𝐼) → (((𝑋(meet‘𝐾)𝑊)(join‘𝐾)((oc‘𝐾)‘𝑊))(join‘𝐾)((oc‘𝐾)‘𝑊)) = ((𝑋(meet‘𝐾)𝑊)(join‘𝐾)((oc‘𝐾)‘𝑊)))
3524, 34eqtrd 2804 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋 ∈ dom 𝐼) → (((oc‘𝐾)‘((((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊))(join‘𝐾)((oc‘𝐾)‘𝑊)) = ((𝑋(meet‘𝐾)𝑊)(join‘𝐾)((oc‘𝐾)‘𝑊)))
3635oveq1d 6807 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋 ∈ dom 𝐼) → ((((oc‘𝐾)‘((((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊))(join‘𝐾)((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊) = (((𝑋(meet‘𝐾)𝑊)(join‘𝐾)((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊))
37 hloml 35159 . . . . . 6 (𝐾 ∈ HL → 𝐾 ∈ OML)
3837ad2antrr 697 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋 ∈ dom 𝐼) → 𝐾 ∈ OML)
39 eqid 2770 . . . . . . 7 (le‘𝐾) = (le‘𝐾)
403, 39, 10latmle2 17284 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑋 ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → (𝑋(meet‘𝐾)𝑊)(le‘𝐾)𝑊)
4118, 6, 8, 40syl3anc 1475 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋 ∈ dom 𝐼) → (𝑋(meet‘𝐾)𝑊)(le‘𝐾)𝑊)
423, 39, 9, 10, 11omlspjN 35063 . . . . 5 ((𝐾 ∈ OML ∧ ((𝑋(meet‘𝐾)𝑊) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) ∧ (𝑋(meet‘𝐾)𝑊)(le‘𝐾)𝑊) → (((𝑋(meet‘𝐾)𝑊)(join‘𝐾)((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊) = (𝑋(meet‘𝐾)𝑊))
4338, 20, 8, 41, 42syl121anc 1480 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋 ∈ dom 𝐼) → (((𝑋(meet‘𝐾)𝑊)(join‘𝐾)((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊) = (𝑋(meet‘𝐾)𝑊))
4439, 4, 5diadmleN 36841 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋 ∈ dom 𝐼) → 𝑋(le‘𝐾)𝑊)
453, 39, 10latleeqm1 17286 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑋 ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → (𝑋(le‘𝐾)𝑊 ↔ (𝑋(meet‘𝐾)𝑊) = 𝑋))
4618, 6, 8, 45syl3anc 1475 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋 ∈ dom 𝐼) → (𝑋(le‘𝐾)𝑊 ↔ (𝑋(meet‘𝐾)𝑊) = 𝑋))
4744, 46mpbid 222 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋 ∈ dom 𝐼) → (𝑋(meet‘𝐾)𝑊) = 𝑋)
4836, 43, 473eqtrrd 2809 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋 ∈ dom 𝐼) → 𝑋 = ((((oc‘𝐾)‘((((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊))(join‘𝐾)((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊))
4948fveq2d 6336 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋 ∈ dom 𝐼) → (𝐼𝑋) = (𝐼‘((((oc‘𝐾)‘((((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊))(join‘𝐾)((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)))
503, 11opoccl 34996 . . . . . . 7 ((𝐾 ∈ OP ∧ 𝑋 ∈ (Base‘𝐾)) → ((oc‘𝐾)‘𝑋) ∈ (Base‘𝐾))
5126, 6, 50syl2anc 565 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋 ∈ dom 𝐼) → ((oc‘𝐾)‘𝑋) ∈ (Base‘𝐾))
523, 9latjcl 17258 . . . . . 6 ((𝐾 ∈ Lat ∧ ((oc‘𝐾)‘𝑋) ∈ (Base‘𝐾) ∧ ((oc‘𝐾)‘𝑊) ∈ (Base‘𝐾)) → (((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑊)) ∈ (Base‘𝐾))
5318, 51, 28, 52syl3anc 1475 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋 ∈ dom 𝐼) → (((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑊)) ∈ (Base‘𝐾))
543, 10latmcl 17259 . . . . 5 ((𝐾 ∈ Lat ∧ (((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑊)) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊) ∈ (Base‘𝐾))
5518, 53, 8, 54syl3anc 1475 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋 ∈ dom 𝐼) → ((((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊) ∈ (Base‘𝐾))
563, 39, 10latmle2 17284 . . . . 5 ((𝐾 ∈ Lat ∧ (((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑊)) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)(le‘𝐾)𝑊)
5718, 53, 8, 56syl3anc 1475 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋 ∈ dom 𝐼) → ((((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)(le‘𝐾)𝑊)
583, 39, 4, 5diaeldm 36839 . . . . 5 ((𝐾 ∈ HL ∧ 𝑊𝐻) → (((((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊) ∈ dom 𝐼 ↔ (((((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊) ∈ (Base‘𝐾) ∧ ((((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)(le‘𝐾)𝑊)))
5958adantr 466 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋 ∈ dom 𝐼) → (((((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊) ∈ dom 𝐼 ↔ (((((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊) ∈ (Base‘𝐾) ∧ ((((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)(le‘𝐾)𝑊)))
6055, 57, 59mpbir2and 684 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋 ∈ dom 𝐼) → ((((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊) ∈ dom 𝐼)
61 eqid 2770 . . . 4 ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊)
62 doca2.n . . . 4 = ((ocA‘𝐾)‘𝑊)
639, 10, 11, 4, 61, 5, 62diaocN 36928 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊) ∈ dom 𝐼) → (𝐼‘((((oc‘𝐾)‘((((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊))(join‘𝐾)((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)) = ( ‘(𝐼‘((((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊))))
6460, 63syldan 571 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋 ∈ dom 𝐼) → (𝐼‘((((oc‘𝐾)‘((((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊))(join‘𝐾)((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)) = ( ‘(𝐼‘((((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊))))
659, 10, 11, 4, 61, 5, 62diaocN 36928 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋 ∈ dom 𝐼) → (𝐼‘((((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)) = ( ‘(𝐼𝑋)))
6665fveq2d 6336 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋 ∈ dom 𝐼) → ( ‘(𝐼‘((((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊))) = ( ‘( ‘(𝐼𝑋))))
6749, 64, 663eqtrrd 2809 1 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋 ∈ dom 𝐼) → ( ‘( ‘(𝐼𝑋))) = (𝐼𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382   = wceq 1630  wcel 2144   class class class wbr 4784  dom cdm 5249  cfv 6031  (class class class)co 6792  Basecbs 16063  lecple 16155  occoc 16156  joincjn 17151  meetcmee 17152  Latclat 17252  OPcops 34974  OLcol 34976  OMLcoml 34977  HLchlt 35152  LHypclh 35785  LTrncltrn 35902  DIsoAcdia 36831  ocAcocaN 36922
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095  ax-riotaBAD 34754
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1071  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-nel 3046  df-ral 3065  df-rex 3066  df-reu 3067  df-rmo 3068  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-int 4610  df-iun 4654  df-iin 4655  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6753  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-1st 7314  df-2nd 7315  df-undef 7550  df-map 8010  df-preset 17135  df-poset 17153  df-plt 17165  df-lub 17181  df-glb 17182  df-join 17183  df-meet 17184  df-p0 17246  df-p1 17247  df-lat 17253  df-clat 17315  df-oposet 34978  df-cmtN 34979  df-ol 34980  df-oml 34981  df-covers 35068  df-ats 35069  df-atl 35100  df-cvlat 35124  df-hlat 35153  df-llines 35299  df-lplanes 35300  df-lvols 35301  df-lines 35302  df-psubsp 35304  df-pmap 35305  df-padd 35597  df-lhyp 35789  df-laut 35790  df-ldil 35905  df-ltrn 35906  df-trl 35961  df-disoa 36832  df-docaN 36923
This theorem is referenced by:  doca3N  36930
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