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Theorem ltrncnv 35933
Description: The converse of a lattice translation is a lattice translation. (Contributed by NM, 10-May-2013.)
Hypotheses
Ref Expression
ltrncnv.h 𝐻 = (LHyp‘𝐾)
ltrncnv.t 𝑇 = ((LTrn‘𝐾)‘𝑊)
Assertion
Ref Expression
ltrncnv (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) → 𝐹𝑇)

Proof of Theorem ltrncnv
Dummy variables 𝑞 𝑝 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ltrncnv.h . . . 4 𝐻 = (LHyp‘𝐾)
2 eqid 2758 . . . 4 ((LDil‘𝐾)‘𝑊) = ((LDil‘𝐾)‘𝑊)
3 ltrncnv.t . . . 4 𝑇 = ((LTrn‘𝐾)‘𝑊)
41, 2, 3ltrnldil 35909 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) → 𝐹 ∈ ((LDil‘𝐾)‘𝑊))
51, 2ldilcnv 35902 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹 ∈ ((LDil‘𝐾)‘𝑊)) → 𝐹 ∈ ((LDil‘𝐾)‘𝑊))
64, 5syldan 488 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) → 𝐹 ∈ ((LDil‘𝐾)‘𝑊))
7 simp1 1131 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → ((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇))
8 simp1l 1240 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → (𝐾 ∈ HL ∧ 𝑊𝐻))
9 simp1r 1241 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → 𝐹𝑇)
10 simp2l 1242 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → 𝑝 ∈ (Atoms‘𝐾))
11 simp3l 1244 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → ¬ 𝑝(le‘𝐾)𝑊)
12 eqid 2758 . . . . . . . 8 (le‘𝐾) = (le‘𝐾)
13 eqid 2758 . . . . . . . 8 (Atoms‘𝐾) = (Atoms‘𝐾)
1412, 13, 1, 3ltrncnvel 35929 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ ¬ 𝑝(le‘𝐾)𝑊)) → ((𝐹𝑝) ∈ (Atoms‘𝐾) ∧ ¬ (𝐹𝑝)(le‘𝐾)𝑊))
158, 9, 10, 11, 14syl112anc 1481 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → ((𝐹𝑝) ∈ (Atoms‘𝐾) ∧ ¬ (𝐹𝑝)(le‘𝐾)𝑊))
16 simp2r 1243 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → 𝑞 ∈ (Atoms‘𝐾))
17 simp3r 1245 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → ¬ 𝑞(le‘𝐾)𝑊)
1812, 13, 1, 3ltrncnvel 35929 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ ¬ 𝑞(le‘𝐾)𝑊)) → ((𝐹𝑞) ∈ (Atoms‘𝐾) ∧ ¬ (𝐹𝑞)(le‘𝐾)𝑊))
198, 9, 16, 17, 18syl112anc 1481 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → ((𝐹𝑞) ∈ (Atoms‘𝐾) ∧ ¬ (𝐹𝑞)(le‘𝐾)𝑊))
20 eqid 2758 . . . . . . 7 (join‘𝐾) = (join‘𝐾)
21 eqid 2758 . . . . . . 7 (meet‘𝐾) = (meet‘𝐾)
2212, 20, 21, 13, 1, 3ltrnu 35908 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) ∧ ((𝐹𝑝) ∈ (Atoms‘𝐾) ∧ ¬ (𝐹𝑝)(le‘𝐾)𝑊) ∧ ((𝐹𝑞) ∈ (Atoms‘𝐾) ∧ ¬ (𝐹𝑞)(le‘𝐾)𝑊)) → (((𝐹𝑝)(join‘𝐾)(𝐹‘(𝐹𝑝)))(meet‘𝐾)𝑊) = (((𝐹𝑞)(join‘𝐾)(𝐹‘(𝐹𝑞)))(meet‘𝐾)𝑊))
237, 15, 19, 22syl3anc 1477 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → (((𝐹𝑝)(join‘𝐾)(𝐹‘(𝐹𝑝)))(meet‘𝐾)𝑊) = (((𝐹𝑞)(join‘𝐾)(𝐹‘(𝐹𝑞)))(meet‘𝐾)𝑊))
24 eqid 2758 . . . . . . . . . . 11 (Base‘𝐾) = (Base‘𝐾)
2524, 1, 3ltrn1o 35911 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) → 𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
26253ad2ant1 1128 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → 𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
2724, 13atbase 35077 . . . . . . . . . 10 (𝑝 ∈ (Atoms‘𝐾) → 𝑝 ∈ (Base‘𝐾))
2810, 27syl 17 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → 𝑝 ∈ (Base‘𝐾))
29 f1ocnvfv2 6694 . . . . . . . . 9 ((𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾) ∧ 𝑝 ∈ (Base‘𝐾)) → (𝐹‘(𝐹𝑝)) = 𝑝)
3026, 28, 29syl2anc 696 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → (𝐹‘(𝐹𝑝)) = 𝑝)
3130oveq2d 6827 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → ((𝐹𝑝)(join‘𝐾)(𝐹‘(𝐹𝑝))) = ((𝐹𝑝)(join‘𝐾)𝑝))
32 simp1ll 1303 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → 𝐾 ∈ HL)
3312, 13, 1, 3ltrncnvat 35928 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝑝 ∈ (Atoms‘𝐾)) → (𝐹𝑝) ∈ (Atoms‘𝐾))
348, 9, 10, 33syl3anc 1477 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → (𝐹𝑝) ∈ (Atoms‘𝐾))
3520, 13hlatjcom 35155 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝐹𝑝) ∈ (Atoms‘𝐾) ∧ 𝑝 ∈ (Atoms‘𝐾)) → ((𝐹𝑝)(join‘𝐾)𝑝) = (𝑝(join‘𝐾)(𝐹𝑝)))
3632, 34, 10, 35syl3anc 1477 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → ((𝐹𝑝)(join‘𝐾)𝑝) = (𝑝(join‘𝐾)(𝐹𝑝)))
3731, 36eqtrd 2792 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → ((𝐹𝑝)(join‘𝐾)(𝐹‘(𝐹𝑝))) = (𝑝(join‘𝐾)(𝐹𝑝)))
3837oveq1d 6826 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → (((𝐹𝑝)(join‘𝐾)(𝐹‘(𝐹𝑝)))(meet‘𝐾)𝑊) = ((𝑝(join‘𝐾)(𝐹𝑝))(meet‘𝐾)𝑊))
3924, 13atbase 35077 . . . . . . . . . 10 (𝑞 ∈ (Atoms‘𝐾) → 𝑞 ∈ (Base‘𝐾))
4016, 39syl 17 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → 𝑞 ∈ (Base‘𝐾))
41 f1ocnvfv2 6694 . . . . . . . . 9 ((𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾) ∧ 𝑞 ∈ (Base‘𝐾)) → (𝐹‘(𝐹𝑞)) = 𝑞)
4226, 40, 41syl2anc 696 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → (𝐹‘(𝐹𝑞)) = 𝑞)
4342oveq2d 6827 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → ((𝐹𝑞)(join‘𝐾)(𝐹‘(𝐹𝑞))) = ((𝐹𝑞)(join‘𝐾)𝑞))
4412, 13, 1, 3ltrncnvat 35928 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝑞 ∈ (Atoms‘𝐾)) → (𝐹𝑞) ∈ (Atoms‘𝐾))
458, 9, 16, 44syl3anc 1477 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → (𝐹𝑞) ∈ (Atoms‘𝐾))
4620, 13hlatjcom 35155 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝐹𝑞) ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) → ((𝐹𝑞)(join‘𝐾)𝑞) = (𝑞(join‘𝐾)(𝐹𝑞)))
4732, 45, 16, 46syl3anc 1477 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → ((𝐹𝑞)(join‘𝐾)𝑞) = (𝑞(join‘𝐾)(𝐹𝑞)))
4843, 47eqtrd 2792 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → ((𝐹𝑞)(join‘𝐾)(𝐹‘(𝐹𝑞))) = (𝑞(join‘𝐾)(𝐹𝑞)))
4948oveq1d 6826 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → (((𝐹𝑞)(join‘𝐾)(𝐹‘(𝐹𝑞)))(meet‘𝐾)𝑊) = ((𝑞(join‘𝐾)(𝐹𝑞))(meet‘𝐾)𝑊))
5023, 38, 493eqtr3d 2800 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → ((𝑝(join‘𝐾)(𝐹𝑝))(meet‘𝐾)𝑊) = ((𝑞(join‘𝐾)(𝐹𝑞))(meet‘𝐾)𝑊))
51503exp 1113 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) → ((𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) → ((¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊) → ((𝑝(join‘𝐾)(𝐹𝑝))(meet‘𝐾)𝑊) = ((𝑞(join‘𝐾)(𝐹𝑞))(meet‘𝐾)𝑊))))
5251ralrimivv 3106 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) → ∀𝑝 ∈ (Atoms‘𝐾)∀𝑞 ∈ (Atoms‘𝐾)((¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊) → ((𝑝(join‘𝐾)(𝐹𝑝))(meet‘𝐾)𝑊) = ((𝑞(join‘𝐾)(𝐹𝑞))(meet‘𝐾)𝑊)))
5312, 20, 21, 13, 1, 2, 3isltrn 35906 . . 3 ((𝐾 ∈ HL ∧ 𝑊𝐻) → (𝐹𝑇 ↔ (𝐹 ∈ ((LDil‘𝐾)‘𝑊) ∧ ∀𝑝 ∈ (Atoms‘𝐾)∀𝑞 ∈ (Atoms‘𝐾)((¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊) → ((𝑝(join‘𝐾)(𝐹𝑝))(meet‘𝐾)𝑊) = ((𝑞(join‘𝐾)(𝐹𝑞))(meet‘𝐾)𝑊)))))
5453adantr 472 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) → (𝐹𝑇 ↔ (𝐹 ∈ ((LDil‘𝐾)‘𝑊) ∧ ∀𝑝 ∈ (Atoms‘𝐾)∀𝑞 ∈ (Atoms‘𝐾)((¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊) → ((𝑝(join‘𝐾)(𝐹𝑝))(meet‘𝐾)𝑊) = ((𝑞(join‘𝐾)(𝐹𝑞))(meet‘𝐾)𝑊)))))
556, 52, 54mpbir2and 995 1 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) → 𝐹𝑇)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383  w3a 1072   = wceq 1630  wcel 2137  wral 3048   class class class wbr 4802  ccnv 5263  1-1-ontowf1o 6046  cfv 6047  (class class class)co 6811  Basecbs 16057  lecple 16148  joincjn 17143  meetcmee 17144  Atomscatm 35051  HLchlt 35138  LHypclh 35771  LDilcldil 35887  LTrncltrn 35888
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 1986  ax-6 2052  ax-7 2088  ax-8 2139  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738  ax-rep 4921  ax-sep 4931  ax-nul 4939  ax-pow 4990  ax-pr 5053  ax-un 7112
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2045  df-eu 2609  df-mo 2610  df-clab 2745  df-cleq 2751  df-clel 2754  df-nfc 2889  df-ne 2931  df-ral 3053  df-rex 3054  df-reu 3055  df-rab 3057  df-v 3340  df-sbc 3575  df-csb 3673  df-dif 3716  df-un 3718  df-in 3720  df-ss 3727  df-nul 4057  df-if 4229  df-pw 4302  df-sn 4320  df-pr 4322  df-op 4326  df-uni 4587  df-iun 4672  df-br 4803  df-opab 4863  df-mpt 4880  df-id 5172  df-xp 5270  df-rel 5271  df-cnv 5272  df-co 5273  df-dm 5274  df-rn 5275  df-res 5276  df-ima 5277  df-iota 6010  df-fun 6049  df-fn 6050  df-f 6051  df-f1 6052  df-fo 6053  df-f1o 6054  df-fv 6055  df-riota 6772  df-ov 6814  df-oprab 6815  df-mpt2 6816  df-map 8023  df-preset 17127  df-poset 17145  df-plt 17157  df-lub 17173  df-glb 17174  df-join 17175  df-p0 17238  df-lat 17245  df-oposet 34964  df-ol 34966  df-oml 34967  df-covers 35054  df-ats 35055  df-atl 35086  df-cvlat 35110  df-hlat 35139  df-lhyp 35775  df-laut 35776  df-ldil 35891  df-ltrn 35892
This theorem is referenced by:  trlcnv  35953  trlcocnv  36508  trlcoabs2N  36510  trlcoat  36511  trlcocnvat  36512  trlcone  36516  cdlemg46  36523  tgrpgrplem  36537  tendoicl  36584  cdlemh1  36603  cdlemh2  36604  cdlemh  36605  cdlemi2  36607  cdlemi  36608  cdlemk2  36620  cdlemk3  36621  cdlemk4  36622  cdlemk8  36626  cdlemk9  36627  cdlemk9bN  36628  cdlemkvcl  36630  cdlemk10  36631  cdlemk11  36637  cdlemk12  36638  cdlemk14  36642  cdlemk11u  36659  cdlemk12u  36660  cdlemk37  36702  cdlemkfid1N  36709  cdlemkid1  36710  cdlemkid2  36712  tendocnv  36810  tendospcanN  36812  dvhgrp  36896  cdlemn8  36993  dihopelvalcpre  37037  dih1  37075  dihglbcpreN  37089  dihjatcclem3  37209  dihjatcclem4  37210
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