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Theorem ltrncnv 34244
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 2610 . . . 4 ((LDil‘𝐾)‘𝑊) = ((LDil‘𝐾)‘𝑊)
3 ltrncnv.t . . . 4 𝑇 = ((LTrn‘𝐾)‘𝑊)
41, 2, 3ltrnldil 34220 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) → 𝐹 ∈ ((LDil‘𝐾)‘𝑊))
51, 2ldilcnv 34213 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹 ∈ ((LDil‘𝐾)‘𝑊)) → 𝐹 ∈ ((LDil‘𝐾)‘𝑊))
64, 5syldan 486 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) → 𝐹 ∈ ((LDil‘𝐾)‘𝑊))
7 simp1 1054 . . . . . 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 1078 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → (𝐾 ∈ HL ∧ 𝑊𝐻))
9 simp1r 1079 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → 𝐹𝑇)
10 simp2l 1080 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → 𝑝 ∈ (Atoms‘𝐾))
11 simp3l 1082 . . . . . . 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 2610 . . . . . . . 8 (le‘𝐾) = (le‘𝐾)
13 eqid 2610 . . . . . . . 8 (Atoms‘𝐾) = (Atoms‘𝐾)
1412, 13, 1, 3ltrncnvel 34240 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ ¬ 𝑝(le‘𝐾)𝑊)) → ((𝐹𝑝) ∈ (Atoms‘𝐾) ∧ ¬ (𝐹𝑝)(le‘𝐾)𝑊))
158, 9, 10, 11, 14syl112anc 1322 . . . . . 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 1081 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → 𝑞 ∈ (Atoms‘𝐾))
17 simp3r 1083 . . . . . . 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 34240 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ ¬ 𝑞(le‘𝐾)𝑊)) → ((𝐹𝑞) ∈ (Atoms‘𝐾) ∧ ¬ (𝐹𝑞)(le‘𝐾)𝑊))
198, 9, 16, 17, 18syl112anc 1322 . . . . . 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 2610 . . . . . . 7 (join‘𝐾) = (join‘𝐾)
21 eqid 2610 . . . . . . 7 (meet‘𝐾) = (meet‘𝐾)
2212, 20, 21, 13, 1, 3ltrnu 34219 . . . . . 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 1318 . . . . 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 2610 . . . . . . . . . . 11 (Base‘𝐾) = (Base‘𝐾)
2524, 1, 3ltrn1o 34222 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) → 𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
26253ad2ant1 1075 . . . . . . . . 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 33388 . . . . . . . . . 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 6411 . . . . . . . . 9 ((𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾) ∧ 𝑝 ∈ (Base‘𝐾)) → (𝐹‘(𝐹𝑝)) = 𝑝)
3026, 28, 29syl2anc 691 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → (𝐹‘(𝐹𝑝)) = 𝑝)
3130oveq2d 6543 . . . . . . 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 1117 . . . . . . . 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 34239 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝑝 ∈ (Atoms‘𝐾)) → (𝐹𝑝) ∈ (Atoms‘𝐾))
348, 9, 10, 33syl3anc 1318 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → (𝐹𝑝) ∈ (Atoms‘𝐾))
3520, 13hlatjcom 33466 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝐹𝑝) ∈ (Atoms‘𝐾) ∧ 𝑝 ∈ (Atoms‘𝐾)) → ((𝐹𝑝)(join‘𝐾)𝑝) = (𝑝(join‘𝐾)(𝐹𝑝)))
3632, 34, 10, 35syl3anc 1318 . . . . . . 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 2644 . . . . . 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 6542 . . . . 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 33388 . . . . . . . . . 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 6411 . . . . . . . . 9 ((𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾) ∧ 𝑞 ∈ (Base‘𝐾)) → (𝐹‘(𝐹𝑞)) = 𝑞)
4226, 40, 41syl2anc 691 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → (𝐹‘(𝐹𝑞)) = 𝑞)
4342oveq2d 6543 . . . . . . 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 34239 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝑞 ∈ (Atoms‘𝐾)) → (𝐹𝑞) ∈ (Atoms‘𝐾))
458, 9, 16, 44syl3anc 1318 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → (𝐹𝑞) ∈ (Atoms‘𝐾))
4620, 13hlatjcom 33466 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝐹𝑞) ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) → ((𝐹𝑞)(join‘𝐾)𝑞) = (𝑞(join‘𝐾)(𝐹𝑞)))
4732, 45, 16, 46syl3anc 1318 . . . . . . 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 2644 . . . . . 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 6542 . . . . 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 2652 . . . 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 1256 . . 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 2953 . 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 34217 . . 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 959 1 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) → 𝐹𝑇)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 195  wa 383  w3a 1031   = wceq 1475  wcel 1977  wral 2896   class class class wbr 4578  ccnv 5027  1-1-ontowf1o 5789  cfv 5790  (class class class)co 6527  Basecbs 15644  lecple 15724  joincjn 16716  meetcmee 16717  Atomscatm 33362  HLchlt 33449  LHypclh 34082  LDilcldil 34198  LTrncltrn 34199
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4694  ax-sep 4704  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6825
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4368  df-iun 4452  df-br 4579  df-opab 4639  df-mpt 4640  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-riota 6489  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-map 7724  df-preset 16700  df-poset 16718  df-plt 16730  df-lub 16746  df-glb 16747  df-join 16748  df-p0 16811  df-lat 16818  df-oposet 33275  df-ol 33277  df-oml 33278  df-covers 33365  df-ats 33366  df-atl 33397  df-cvlat 33421  df-hlat 33450  df-lhyp 34086  df-laut 34087  df-ldil 34202  df-ltrn 34203
This theorem is referenced by:  trlcnv  34264  trlcocnv  34820  trlcoabs2N  34822  trlcoat  34823  trlcocnvat  34824  trlcone  34828  cdlemg46  34835  tgrpgrplem  34849  tendoicl  34896  cdlemh1  34915  cdlemh2  34916  cdlemh  34917  cdlemi2  34919  cdlemi  34920  cdlemk2  34932  cdlemk3  34933  cdlemk4  34934  cdlemk8  34938  cdlemk9  34939  cdlemk9bN  34940  cdlemkvcl  34942  cdlemk10  34943  cdlemk11  34949  cdlemk12  34950  cdlemk14  34954  cdlemk11u  34971  cdlemk12u  34972  cdlemk37  35014  cdlemkfid1N  35021  cdlemkid1  35022  cdlemkid2  35024  tendocnv  35122  tendospcanN  35124  dvhgrp  35208  cdlemn8  35305  dihopelvalcpre  35349  dih1  35387  dihglbcpreN  35401  dihjatcclem3  35521  dihjatcclem4  35522
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