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Theorem ltrncnv 40128
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 2734 . . . 4 ((LDil‘𝐾)‘𝑊) = ((LDil‘𝐾)‘𝑊)
3 ltrncnv.t . . . 4 𝑇 = ((LTrn‘𝐾)‘𝑊)
41, 2, 3ltrnldil 40104 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) → 𝐹 ∈ ((LDil‘𝐾)‘𝑊))
51, 2ldilcnv 40097 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹 ∈ ((LDil‘𝐾)‘𝑊)) → 𝐹 ∈ ((LDil‘𝐾)‘𝑊))
64, 5syldan 591 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) → 𝐹 ∈ ((LDil‘𝐾)‘𝑊))
7 simp1 1135 . . . . . 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 1196 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → (𝐾 ∈ HL ∧ 𝑊𝐻))
9 simp1r 1197 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → 𝐹𝑇)
10 simp2l 1198 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → 𝑝 ∈ (Atoms‘𝐾))
11 simp3l 1200 . . . . . . 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 2734 . . . . . . . 8 (le‘𝐾) = (le‘𝐾)
13 eqid 2734 . . . . . . . 8 (Atoms‘𝐾) = (Atoms‘𝐾)
1412, 13, 1, 3ltrncnvel 40124 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ ¬ 𝑝(le‘𝐾)𝑊)) → ((𝐹𝑝) ∈ (Atoms‘𝐾) ∧ ¬ (𝐹𝑝)(le‘𝐾)𝑊))
158, 9, 10, 11, 14syl112anc 1373 . . . . . 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 1199 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → 𝑞 ∈ (Atoms‘𝐾))
17 simp3r 1201 . . . . . . 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 40124 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ ¬ 𝑞(le‘𝐾)𝑊)) → ((𝐹𝑞) ∈ (Atoms‘𝐾) ∧ ¬ (𝐹𝑞)(le‘𝐾)𝑊))
198, 9, 16, 17, 18syl112anc 1373 . . . . . 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 2734 . . . . . . 7 (join‘𝐾) = (join‘𝐾)
21 eqid 2734 . . . . . . 7 (meet‘𝐾) = (meet‘𝐾)
2212, 20, 21, 13, 1, 3ltrnu 40103 . . . . . 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 1370 . . . . 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 2734 . . . . . . . . . . 11 (Base‘𝐾) = (Base‘𝐾)
2524, 1, 3ltrn1o 40106 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) → 𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
26253ad2ant1 1132 . . . . . . . . 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 39270 . . . . . . . . . 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 7296 . . . . . . . . 9 ((𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾) ∧ 𝑝 ∈ (Base‘𝐾)) → (𝐹‘(𝐹𝑝)) = 𝑝)
3026, 28, 29syl2anc 584 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → (𝐹‘(𝐹𝑝)) = 𝑝)
3130oveq2d 7446 . . . . . . 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 1235 . . . . . . . 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 40123 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝑝 ∈ (Atoms‘𝐾)) → (𝐹𝑝) ∈ (Atoms‘𝐾))
348, 9, 10, 33syl3anc 1370 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → (𝐹𝑝) ∈ (Atoms‘𝐾))
3520, 13hlatjcom 39349 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝐹𝑝) ∈ (Atoms‘𝐾) ∧ 𝑝 ∈ (Atoms‘𝐾)) → ((𝐹𝑝)(join‘𝐾)𝑝) = (𝑝(join‘𝐾)(𝐹𝑝)))
3632, 34, 10, 35syl3anc 1370 . . . . . . 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 2774 . . . . . 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 7445 . . . . 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 39270 . . . . . . . . . 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 7296 . . . . . . . . 9 ((𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾) ∧ 𝑞 ∈ (Base‘𝐾)) → (𝐹‘(𝐹𝑞)) = 𝑞)
4226, 40, 41syl2anc 584 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → (𝐹‘(𝐹𝑞)) = 𝑞)
4342oveq2d 7446 . . . . . . 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 40123 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝑞 ∈ (Atoms‘𝐾)) → (𝐹𝑞) ∈ (Atoms‘𝐾))
458, 9, 16, 44syl3anc 1370 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → (𝐹𝑞) ∈ (Atoms‘𝐾))
4620, 13hlatjcom 39349 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝐹𝑞) ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) → ((𝐹𝑞)(join‘𝐾)𝑞) = (𝑞(join‘𝐾)(𝐹𝑞)))
4732, 45, 16, 46syl3anc 1370 . . . . . . 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 2774 . . . . . 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 7445 . . . . 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 2782 . . . 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 1118 . . 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 3197 . 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 40101 . . 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 480 . 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 713 1 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) → 𝐹𝑇)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  w3a 1086   = wceq 1536  wcel 2105  wral 3058   class class class wbr 5147  ccnv 5687  1-1-ontowf1o 6561  cfv 6562  (class class class)co 7430  Basecbs 17244  lecple 17304  joincjn 18368  meetcmee 18369  Atomscatm 39244  HLchlt 39331  LHypclh 39966  LDilcldil 40082  LTrncltrn 40083
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-map 8866  df-proset 18351  df-poset 18370  df-plt 18387  df-lub 18403  df-glb 18404  df-join 18405  df-p0 18482  df-lat 18489  df-oposet 39157  df-ol 39159  df-oml 39160  df-covers 39247  df-ats 39248  df-atl 39279  df-cvlat 39303  df-hlat 39332  df-lhyp 39970  df-laut 39971  df-ldil 40086  df-ltrn 40087
This theorem is referenced by:  trlcnv  40147  trlcocnv  40702  trlcoabs2N  40704  trlcoat  40705  trlcocnvat  40706  trlcone  40710  cdlemg46  40717  tgrpgrplem  40731  tendoicl  40778  cdlemh1  40797  cdlemh2  40798  cdlemh  40799  cdlemi2  40801  cdlemi  40802  cdlemk2  40814  cdlemk3  40815  cdlemk4  40816  cdlemk8  40820  cdlemk9  40821  cdlemk9bN  40822  cdlemkvcl  40824  cdlemk10  40825  cdlemk11  40831  cdlemk12  40832  cdlemk14  40836  cdlemk11u  40853  cdlemk12u  40854  cdlemk37  40896  cdlemkfid1N  40903  cdlemkid1  40904  cdlemkid2  40906  tendocnv  41003  tendospcanN  41005  dvhgrp  41089  cdlemn8  41186  dihopelvalcpre  41230  dih1  41268  dihglbcpreN  41282  dihjatcclem3  41402  dihjatcclem4  41403
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