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Theorem cdlemd3 38666
Description: Part of proof of Lemma D in [Crawley] p. 113. The 𝑅 β‰  𝑃 requirement is not mentioned in their proof. (Contributed by NM, 29-May-2012.)
Hypotheses
Ref Expression
cdlemd3.l ≀ = (leβ€˜πΎ)
cdlemd3.j ∨ = (joinβ€˜πΎ)
cdlemd3.a 𝐴 = (Atomsβ€˜πΎ)
cdlemd3.h 𝐻 = (LHypβ€˜πΎ)
Assertion
Ref Expression
cdlemd3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ (𝑃 β‰  𝑄 ∧ 𝑅 ≀ (𝑃 ∨ 𝑄) ∧ 𝑅 β‰  𝑃)) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ (𝑃 ∨ 𝑄))) β†’ Β¬ 𝑅 ≀ (𝑃 ∨ 𝑆))

Proof of Theorem cdlemd3
StepHypRef Expression
1 simp33 1212 . 2 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ (𝑃 β‰  𝑄 ∧ 𝑅 ≀ (𝑃 ∨ 𝑄) ∧ 𝑅 β‰  𝑃)) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ (𝑃 ∨ 𝑄))) β†’ Β¬ 𝑆 ≀ (𝑃 ∨ 𝑄))
2 simp1l 1198 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ (𝑃 β‰  𝑄 ∧ 𝑅 ≀ (𝑃 ∨ 𝑄) ∧ 𝑅 β‰  𝑃)) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ (𝑃 ∨ 𝑄))) β†’ 𝐾 ∈ HL)
3 simp31 1210 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ (𝑃 β‰  𝑄 ∧ 𝑅 ≀ (𝑃 ∨ 𝑄) ∧ 𝑅 β‰  𝑃)) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ (𝑃 ∨ 𝑄))) β†’ 𝑅 ∈ 𝐴)
4 simp32 1211 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ (𝑃 β‰  𝑄 ∧ 𝑅 ≀ (𝑃 ∨ 𝑄) ∧ 𝑅 β‰  𝑃)) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ (𝑃 ∨ 𝑄))) β†’ 𝑆 ∈ 𝐴)
5 simp21l 1291 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ (𝑃 β‰  𝑄 ∧ 𝑅 ≀ (𝑃 ∨ 𝑄) ∧ 𝑅 β‰  𝑃)) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ (𝑃 ∨ 𝑄))) β†’ 𝑃 ∈ 𝐴)
6 simp233 1320 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ (𝑃 β‰  𝑄 ∧ 𝑅 ≀ (𝑃 ∨ 𝑄) ∧ 𝑅 β‰  𝑃)) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ (𝑃 ∨ 𝑄))) β†’ 𝑅 β‰  𝑃)
7 cdlemd3.l . . . . 5 ≀ = (leβ€˜πΎ)
8 cdlemd3.j . . . . 5 ∨ = (joinβ€˜πΎ)
9 cdlemd3.a . . . . 5 𝐴 = (Atomsβ€˜πΎ)
107, 8, 9hlatexch1 37861 . . . 4 ((𝐾 ∈ HL ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴) ∧ 𝑅 β‰  𝑃) β†’ (𝑅 ≀ (𝑃 ∨ 𝑆) β†’ 𝑆 ≀ (𝑃 ∨ 𝑅)))
112, 3, 4, 5, 6, 10syl131anc 1384 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ (𝑃 β‰  𝑄 ∧ 𝑅 ≀ (𝑃 ∨ 𝑄) ∧ 𝑅 β‰  𝑃)) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ (𝑃 ∨ 𝑄))) β†’ (𝑅 ≀ (𝑃 ∨ 𝑆) β†’ 𝑆 ≀ (𝑃 ∨ 𝑅)))
12 simp22l 1293 . . . . . 6 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ (𝑃 β‰  𝑄 ∧ 𝑅 ≀ (𝑃 ∨ 𝑄) ∧ 𝑅 β‰  𝑃)) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ (𝑃 ∨ 𝑄))) β†’ 𝑄 ∈ 𝐴)
137, 8, 9hlatlej1 37840 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) β†’ 𝑃 ≀ (𝑃 ∨ 𝑄))
142, 5, 12, 13syl3anc 1372 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ (𝑃 β‰  𝑄 ∧ 𝑅 ≀ (𝑃 ∨ 𝑄) ∧ 𝑅 β‰  𝑃)) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ (𝑃 ∨ 𝑄))) β†’ 𝑃 ≀ (𝑃 ∨ 𝑄))
15 simp232 1319 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ (𝑃 β‰  𝑄 ∧ 𝑅 ≀ (𝑃 ∨ 𝑄) ∧ 𝑅 β‰  𝑃)) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ (𝑃 ∨ 𝑄))) β†’ 𝑅 ≀ (𝑃 ∨ 𝑄))
162hllatd 37829 . . . . . 6 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ (𝑃 β‰  𝑄 ∧ 𝑅 ≀ (𝑃 ∨ 𝑄) ∧ 𝑅 β‰  𝑃)) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ (𝑃 ∨ 𝑄))) β†’ 𝐾 ∈ Lat)
17 eqid 2737 . . . . . . . 8 (Baseβ€˜πΎ) = (Baseβ€˜πΎ)
1817, 9atbase 37754 . . . . . . 7 (𝑃 ∈ 𝐴 β†’ 𝑃 ∈ (Baseβ€˜πΎ))
195, 18syl 17 . . . . . 6 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ (𝑃 β‰  𝑄 ∧ 𝑅 ≀ (𝑃 ∨ 𝑄) ∧ 𝑅 β‰  𝑃)) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ (𝑃 ∨ 𝑄))) β†’ 𝑃 ∈ (Baseβ€˜πΎ))
2017, 9atbase 37754 . . . . . . 7 (𝑅 ∈ 𝐴 β†’ 𝑅 ∈ (Baseβ€˜πΎ))
213, 20syl 17 . . . . . 6 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ (𝑃 β‰  𝑄 ∧ 𝑅 ≀ (𝑃 ∨ 𝑄) ∧ 𝑅 β‰  𝑃)) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ (𝑃 ∨ 𝑄))) β†’ 𝑅 ∈ (Baseβ€˜πΎ))
2217, 9atbase 37754 . . . . . . . 8 (𝑄 ∈ 𝐴 β†’ 𝑄 ∈ (Baseβ€˜πΎ))
2312, 22syl 17 . . . . . . 7 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ (𝑃 β‰  𝑄 ∧ 𝑅 ≀ (𝑃 ∨ 𝑄) ∧ 𝑅 β‰  𝑃)) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ (𝑃 ∨ 𝑄))) β†’ 𝑄 ∈ (Baseβ€˜πΎ))
2417, 8latjcl 18329 . . . . . . 7 ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Baseβ€˜πΎ) ∧ 𝑄 ∈ (Baseβ€˜πΎ)) β†’ (𝑃 ∨ 𝑄) ∈ (Baseβ€˜πΎ))
2516, 19, 23, 24syl3anc 1372 . . . . . 6 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ (𝑃 β‰  𝑄 ∧ 𝑅 ≀ (𝑃 ∨ 𝑄) ∧ 𝑅 β‰  𝑃)) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ (𝑃 ∨ 𝑄))) β†’ (𝑃 ∨ 𝑄) ∈ (Baseβ€˜πΎ))
2617, 7, 8latjle12 18340 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑃 ∈ (Baseβ€˜πΎ) ∧ 𝑅 ∈ (Baseβ€˜πΎ) ∧ (𝑃 ∨ 𝑄) ∈ (Baseβ€˜πΎ))) β†’ ((𝑃 ≀ (𝑃 ∨ 𝑄) ∧ 𝑅 ≀ (𝑃 ∨ 𝑄)) ↔ (𝑃 ∨ 𝑅) ≀ (𝑃 ∨ 𝑄)))
2716, 19, 21, 25, 26syl13anc 1373 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ (𝑃 β‰  𝑄 ∧ 𝑅 ≀ (𝑃 ∨ 𝑄) ∧ 𝑅 β‰  𝑃)) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ (𝑃 ∨ 𝑄))) β†’ ((𝑃 ≀ (𝑃 ∨ 𝑄) ∧ 𝑅 ≀ (𝑃 ∨ 𝑄)) ↔ (𝑃 ∨ 𝑅) ≀ (𝑃 ∨ 𝑄)))
2814, 15, 27mpbi2and 711 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ (𝑃 β‰  𝑄 ∧ 𝑅 ≀ (𝑃 ∨ 𝑄) ∧ 𝑅 β‰  𝑃)) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ (𝑃 ∨ 𝑄))) β†’ (𝑃 ∨ 𝑅) ≀ (𝑃 ∨ 𝑄))
2917, 9atbase 37754 . . . . . 6 (𝑆 ∈ 𝐴 β†’ 𝑆 ∈ (Baseβ€˜πΎ))
304, 29syl 17 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ (𝑃 β‰  𝑄 ∧ 𝑅 ≀ (𝑃 ∨ 𝑄) ∧ 𝑅 β‰  𝑃)) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ (𝑃 ∨ 𝑄))) β†’ 𝑆 ∈ (Baseβ€˜πΎ))
3117, 8latjcl 18329 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Baseβ€˜πΎ) ∧ 𝑅 ∈ (Baseβ€˜πΎ)) β†’ (𝑃 ∨ 𝑅) ∈ (Baseβ€˜πΎ))
3216, 19, 21, 31syl3anc 1372 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ (𝑃 β‰  𝑄 ∧ 𝑅 ≀ (𝑃 ∨ 𝑄) ∧ 𝑅 β‰  𝑃)) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ (𝑃 ∨ 𝑄))) β†’ (𝑃 ∨ 𝑅) ∈ (Baseβ€˜πΎ))
3317, 7lattr 18334 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑆 ∈ (Baseβ€˜πΎ) ∧ (𝑃 ∨ 𝑅) ∈ (Baseβ€˜πΎ) ∧ (𝑃 ∨ 𝑄) ∈ (Baseβ€˜πΎ))) β†’ ((𝑆 ≀ (𝑃 ∨ 𝑅) ∧ (𝑃 ∨ 𝑅) ≀ (𝑃 ∨ 𝑄)) β†’ 𝑆 ≀ (𝑃 ∨ 𝑄)))
3416, 30, 32, 25, 33syl13anc 1373 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ (𝑃 β‰  𝑄 ∧ 𝑅 ≀ (𝑃 ∨ 𝑄) ∧ 𝑅 β‰  𝑃)) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ (𝑃 ∨ 𝑄))) β†’ ((𝑆 ≀ (𝑃 ∨ 𝑅) ∧ (𝑃 ∨ 𝑅) ≀ (𝑃 ∨ 𝑄)) β†’ 𝑆 ≀ (𝑃 ∨ 𝑄)))
3528, 34mpan2d 693 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ (𝑃 β‰  𝑄 ∧ 𝑅 ≀ (𝑃 ∨ 𝑄) ∧ 𝑅 β‰  𝑃)) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ (𝑃 ∨ 𝑄))) β†’ (𝑆 ≀ (𝑃 ∨ 𝑅) β†’ 𝑆 ≀ (𝑃 ∨ 𝑄)))
3611, 35syld 47 . 2 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ (𝑃 β‰  𝑄 ∧ 𝑅 ≀ (𝑃 ∨ 𝑄) ∧ 𝑅 β‰  𝑃)) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ (𝑃 ∨ 𝑄))) β†’ (𝑅 ≀ (𝑃 ∨ 𝑆) β†’ 𝑆 ≀ (𝑃 ∨ 𝑄)))
371, 36mtod 197 1 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ (𝑃 β‰  𝑄 ∧ 𝑅 ≀ (𝑃 ∨ 𝑄) ∧ 𝑅 β‰  𝑃)) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ (𝑃 ∨ 𝑄))) β†’ Β¬ 𝑅 ≀ (𝑃 ∨ 𝑆))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   β‰  wne 2944   class class class wbr 5106  β€˜cfv 6497  (class class class)co 7358  Basecbs 17084  lecple 17141  joincjn 18201  Latclat 18321  Atomscatm 37728  HLchlt 37815  LHypclh 38450
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-reu 3355  df-rab 3409  df-v 3448  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-ov 7361  df-oprab 7362  df-proset 18185  df-poset 18203  df-plt 18220  df-lub 18236  df-glb 18237  df-join 18238  df-meet 18239  df-p0 18315  df-lat 18322  df-covers 37731  df-ats 37732  df-atl 37763  df-cvlat 37787  df-hlat 37816
This theorem is referenced by:  cdlemd4  38667
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