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Theorem lhprelat3N 39569
Description: The Hilbert lattice is relatively atomic with respect to co-atoms (lattice hyperplanes). Dual version of hlrelat3 38941. (Contributed by NM, 20-Jun-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
lhprelat3.b 𝐡 = (Baseβ€˜πΎ)
lhprelat3.l ≀ = (leβ€˜πΎ)
lhprelat3.s < = (ltβ€˜πΎ)
lhprelat3.m ∧ = (meetβ€˜πΎ)
lhprelat3.c 𝐢 = ( β‹– β€˜πΎ)
lhprelat3.h 𝐻 = (LHypβ€˜πΎ)
Assertion
Ref Expression
lhprelat3N (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ 𝑋 < π‘Œ) β†’ βˆƒπ‘€ ∈ 𝐻 (𝑋 ≀ (π‘Œ ∧ 𝑀) ∧ (π‘Œ ∧ 𝑀)πΆπ‘Œ))
Distinct variable groups:   𝑀,𝐢   𝑀,𝐻   𝑀,𝐾   𝑀, ≀   𝑀, ∧   𝑀,𝑋   𝑀,π‘Œ
Allowed substitution hints:   𝐡(𝑀)   < (𝑀)
Proof of Theorem lhprelat3N
Dummy variable 𝑝 is distinct from all other variables.
StepHypRef Expression
1 simpr 483 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ 𝑋 < π‘Œ) ∧ 𝑝 ∈ (Atomsβ€˜πΎ)) β†’ 𝑝 ∈ (Atomsβ€˜πΎ))
2 simpll1 1209 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ 𝑋 < π‘Œ) ∧ 𝑝 ∈ (Atomsβ€˜πΎ)) β†’ 𝐾 ∈ HL)
3 lhprelat3.b . . . . . . . 8 𝐡 = (Baseβ€˜πΎ)
4 eqid 2725 . . . . . . . 8 (Atomsβ€˜πΎ) = (Atomsβ€˜πΎ)
53, 4atbase 38817 . . . . . . 7 (𝑝 ∈ (Atomsβ€˜πΎ) β†’ 𝑝 ∈ 𝐡)
65adantl 480 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ 𝑋 < π‘Œ) ∧ 𝑝 ∈ (Atomsβ€˜πΎ)) β†’ 𝑝 ∈ 𝐡)
7 eqid 2725 . . . . . . 7 (ocβ€˜πΎ) = (ocβ€˜πΎ)
8 lhprelat3.h . . . . . . 7 𝐻 = (LHypβ€˜πΎ)
93, 7, 4, 8lhpoc2N 39544 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑝 ∈ 𝐡) β†’ (𝑝 ∈ (Atomsβ€˜πΎ) ↔ ((ocβ€˜πΎ)β€˜π‘) ∈ 𝐻))
102, 6, 9syl2anc 582 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ 𝑋 < π‘Œ) ∧ 𝑝 ∈ (Atomsβ€˜πΎ)) β†’ (𝑝 ∈ (Atomsβ€˜πΎ) ↔ ((ocβ€˜πΎ)β€˜π‘) ∈ 𝐻))
111, 10mpbid 231 . . . 4 ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ 𝑋 < π‘Œ) ∧ 𝑝 ∈ (Atomsβ€˜πΎ)) β†’ ((ocβ€˜πΎ)β€˜π‘) ∈ 𝐻)
1211adantr 479 . . 3 (((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ 𝑋 < π‘Œ) ∧ 𝑝 ∈ (Atomsβ€˜πΎ)) ∧ (((ocβ€˜πΎ)β€˜π‘Œ)𝐢(((ocβ€˜πΎ)β€˜π‘Œ)(joinβ€˜πΎ)𝑝) ∧ (((ocβ€˜πΎ)β€˜π‘Œ)(joinβ€˜πΎ)𝑝) ≀ ((ocβ€˜πΎ)β€˜π‘‹))) β†’ ((ocβ€˜πΎ)β€˜π‘) ∈ 𝐻)
13 hlop 38890 . . . . . . . . 9 (𝐾 ∈ HL β†’ 𝐾 ∈ OP)
142, 13syl 17 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ 𝑋 < π‘Œ) ∧ 𝑝 ∈ (Atomsβ€˜πΎ)) β†’ 𝐾 ∈ OP)
152hllatd 38892 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ 𝑋 < π‘Œ) ∧ 𝑝 ∈ (Atomsβ€˜πΎ)) β†’ 𝐾 ∈ Lat)
16 simpll3 1211 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ 𝑋 < π‘Œ) ∧ 𝑝 ∈ (Atomsβ€˜πΎ)) β†’ π‘Œ ∈ 𝐡)
173, 7opoccl 38722 . . . . . . . . . 10 ((𝐾 ∈ OP ∧ 𝑝 ∈ 𝐡) β†’ ((ocβ€˜πΎ)β€˜π‘) ∈ 𝐡)
1814, 6, 17syl2anc 582 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ 𝑋 < π‘Œ) ∧ 𝑝 ∈ (Atomsβ€˜πΎ)) β†’ ((ocβ€˜πΎ)β€˜π‘) ∈ 𝐡)
19 lhprelat3.m . . . . . . . . . 10 ∧ = (meetβ€˜πΎ)
203, 19latmcl 18431 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ π‘Œ ∈ 𝐡 ∧ ((ocβ€˜πΎ)β€˜π‘) ∈ 𝐡) β†’ (π‘Œ ∧ ((ocβ€˜πΎ)β€˜π‘)) ∈ 𝐡)
2115, 16, 18, 20syl3anc 1368 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ 𝑋 < π‘Œ) ∧ 𝑝 ∈ (Atomsβ€˜πΎ)) β†’ (π‘Œ ∧ ((ocβ€˜πΎ)β€˜π‘)) ∈ 𝐡)
22 lhprelat3.c . . . . . . . . 9 𝐢 = ( β‹– β€˜πΎ)
233, 7, 22cvrcon3b 38805 . . . . . . . 8 ((𝐾 ∈ OP ∧ (π‘Œ ∧ ((ocβ€˜πΎ)β€˜π‘)) ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ ((π‘Œ ∧ ((ocβ€˜πΎ)β€˜π‘))πΆπ‘Œ ↔ ((ocβ€˜πΎ)β€˜π‘Œ)𝐢((ocβ€˜πΎ)β€˜(π‘Œ ∧ ((ocβ€˜πΎ)β€˜π‘)))))
2414, 21, 16, 23syl3anc 1368 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ 𝑋 < π‘Œ) ∧ 𝑝 ∈ (Atomsβ€˜πΎ)) β†’ ((π‘Œ ∧ ((ocβ€˜πΎ)β€˜π‘))πΆπ‘Œ ↔ ((ocβ€˜πΎ)β€˜π‘Œ)𝐢((ocβ€˜πΎ)β€˜(π‘Œ ∧ ((ocβ€˜πΎ)β€˜π‘)))))
25 hlol 38889 . . . . . . . . . 10 (𝐾 ∈ HL β†’ 𝐾 ∈ OL)
262, 25syl 17 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ 𝑋 < π‘Œ) ∧ 𝑝 ∈ (Atomsβ€˜πΎ)) β†’ 𝐾 ∈ OL)
27 eqid 2725 . . . . . . . . . 10 (joinβ€˜πΎ) = (joinβ€˜πΎ)
283, 27, 19, 7oldmm3N 38747 . . . . . . . . 9 ((𝐾 ∈ OL ∧ π‘Œ ∈ 𝐡 ∧ 𝑝 ∈ 𝐡) β†’ ((ocβ€˜πΎ)β€˜(π‘Œ ∧ ((ocβ€˜πΎ)β€˜π‘))) = (((ocβ€˜πΎ)β€˜π‘Œ)(joinβ€˜πΎ)𝑝))
2926, 16, 6, 28syl3anc 1368 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ 𝑋 < π‘Œ) ∧ 𝑝 ∈ (Atomsβ€˜πΎ)) β†’ ((ocβ€˜πΎ)β€˜(π‘Œ ∧ ((ocβ€˜πΎ)β€˜π‘))) = (((ocβ€˜πΎ)β€˜π‘Œ)(joinβ€˜πΎ)𝑝))
3029breq2d 5155 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ 𝑋 < π‘Œ) ∧ 𝑝 ∈ (Atomsβ€˜πΎ)) β†’ (((ocβ€˜πΎ)β€˜π‘Œ)𝐢((ocβ€˜πΎ)β€˜(π‘Œ ∧ ((ocβ€˜πΎ)β€˜π‘))) ↔ ((ocβ€˜πΎ)β€˜π‘Œ)𝐢(((ocβ€˜πΎ)β€˜π‘Œ)(joinβ€˜πΎ)𝑝)))
3124, 30bitr2d 279 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ 𝑋 < π‘Œ) ∧ 𝑝 ∈ (Atomsβ€˜πΎ)) β†’ (((ocβ€˜πΎ)β€˜π‘Œ)𝐢(((ocβ€˜πΎ)β€˜π‘Œ)(joinβ€˜πΎ)𝑝) ↔ (π‘Œ ∧ ((ocβ€˜πΎ)β€˜π‘))πΆπ‘Œ))
32 simpll2 1210 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ 𝑋 < π‘Œ) ∧ 𝑝 ∈ (Atomsβ€˜πΎ)) β†’ 𝑋 ∈ 𝐡)
33 lhprelat3.l . . . . . . . . 9 ≀ = (leβ€˜πΎ)
343, 33, 7oplecon3b 38728 . . . . . . . 8 ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐡 ∧ (π‘Œ ∧ ((ocβ€˜πΎ)β€˜π‘)) ∈ 𝐡) β†’ (𝑋 ≀ (π‘Œ ∧ ((ocβ€˜πΎ)β€˜π‘)) ↔ ((ocβ€˜πΎ)β€˜(π‘Œ ∧ ((ocβ€˜πΎ)β€˜π‘))) ≀ ((ocβ€˜πΎ)β€˜π‘‹)))
3514, 32, 21, 34syl3anc 1368 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ 𝑋 < π‘Œ) ∧ 𝑝 ∈ (Atomsβ€˜πΎ)) β†’ (𝑋 ≀ (π‘Œ ∧ ((ocβ€˜πΎ)β€˜π‘)) ↔ ((ocβ€˜πΎ)β€˜(π‘Œ ∧ ((ocβ€˜πΎ)β€˜π‘))) ≀ ((ocβ€˜πΎ)β€˜π‘‹)))
3629breq1d 5153 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ 𝑋 < π‘Œ) ∧ 𝑝 ∈ (Atomsβ€˜πΎ)) β†’ (((ocβ€˜πΎ)β€˜(π‘Œ ∧ ((ocβ€˜πΎ)β€˜π‘))) ≀ ((ocβ€˜πΎ)β€˜π‘‹) ↔ (((ocβ€˜πΎ)β€˜π‘Œ)(joinβ€˜πΎ)𝑝) ≀ ((ocβ€˜πΎ)β€˜π‘‹)))
3735, 36bitr2d 279 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ 𝑋 < π‘Œ) ∧ 𝑝 ∈ (Atomsβ€˜πΎ)) β†’ ((((ocβ€˜πΎ)β€˜π‘Œ)(joinβ€˜πΎ)𝑝) ≀ ((ocβ€˜πΎ)β€˜π‘‹) ↔ 𝑋 ≀ (π‘Œ ∧ ((ocβ€˜πΎ)β€˜π‘))))
3831, 37anbi12d 630 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ 𝑋 < π‘Œ) ∧ 𝑝 ∈ (Atomsβ€˜πΎ)) β†’ ((((ocβ€˜πΎ)β€˜π‘Œ)𝐢(((ocβ€˜πΎ)β€˜π‘Œ)(joinβ€˜πΎ)𝑝) ∧ (((ocβ€˜πΎ)β€˜π‘Œ)(joinβ€˜πΎ)𝑝) ≀ ((ocβ€˜πΎ)β€˜π‘‹)) ↔ ((π‘Œ ∧ ((ocβ€˜πΎ)β€˜π‘))πΆπ‘Œ ∧ 𝑋 ≀ (π‘Œ ∧ ((ocβ€˜πΎ)β€˜π‘)))))
3938biimpa 475 . . . 4 (((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ 𝑋 < π‘Œ) ∧ 𝑝 ∈ (Atomsβ€˜πΎ)) ∧ (((ocβ€˜πΎ)β€˜π‘Œ)𝐢(((ocβ€˜πΎ)β€˜π‘Œ)(joinβ€˜πΎ)𝑝) ∧ (((ocβ€˜πΎ)β€˜π‘Œ)(joinβ€˜πΎ)𝑝) ≀ ((ocβ€˜πΎ)β€˜π‘‹))) β†’ ((π‘Œ ∧ ((ocβ€˜πΎ)β€˜π‘))πΆπ‘Œ ∧ 𝑋 ≀ (π‘Œ ∧ ((ocβ€˜πΎ)β€˜π‘))))
4039ancomd 460 . . 3 (((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ 𝑋 < π‘Œ) ∧ 𝑝 ∈ (Atomsβ€˜πΎ)) ∧ (((ocβ€˜πΎ)β€˜π‘Œ)𝐢(((ocβ€˜πΎ)β€˜π‘Œ)(joinβ€˜πΎ)𝑝) ∧ (((ocβ€˜πΎ)β€˜π‘Œ)(joinβ€˜πΎ)𝑝) ≀ ((ocβ€˜πΎ)β€˜π‘‹))) β†’ (𝑋 ≀ (π‘Œ ∧ ((ocβ€˜πΎ)β€˜π‘)) ∧ (π‘Œ ∧ ((ocβ€˜πΎ)β€˜π‘))πΆπ‘Œ))
41 oveq2 7424 . . . . . 6 (𝑀 = ((ocβ€˜πΎ)β€˜π‘) β†’ (π‘Œ ∧ 𝑀) = (π‘Œ ∧ ((ocβ€˜πΎ)β€˜π‘)))
4241breq2d 5155 . . . . 5 (𝑀 = ((ocβ€˜πΎ)β€˜π‘) β†’ (𝑋 ≀ (π‘Œ ∧ 𝑀) ↔ 𝑋 ≀ (π‘Œ ∧ ((ocβ€˜πΎ)β€˜π‘))))
4341breq1d 5153 . . . . 5 (𝑀 = ((ocβ€˜πΎ)β€˜π‘) β†’ ((π‘Œ ∧ 𝑀)πΆπ‘Œ ↔ (π‘Œ ∧ ((ocβ€˜πΎ)β€˜π‘))πΆπ‘Œ))
4442, 43anbi12d 630 . . . 4 (𝑀 = ((ocβ€˜πΎ)β€˜π‘) β†’ ((𝑋 ≀ (π‘Œ ∧ 𝑀) ∧ (π‘Œ ∧ 𝑀)πΆπ‘Œ) ↔ (𝑋 ≀ (π‘Œ ∧ ((ocβ€˜πΎ)β€˜π‘)) ∧ (π‘Œ ∧ ((ocβ€˜πΎ)β€˜π‘))πΆπ‘Œ)))
4544rspcev 3601 . . 3 ((((ocβ€˜πΎ)β€˜π‘) ∈ 𝐻 ∧ (𝑋 ≀ (π‘Œ ∧ ((ocβ€˜πΎ)β€˜π‘)) ∧ (π‘Œ ∧ ((ocβ€˜πΎ)β€˜π‘))πΆπ‘Œ)) β†’ βˆƒπ‘€ ∈ 𝐻 (𝑋 ≀ (π‘Œ ∧ 𝑀) ∧ (π‘Œ ∧ 𝑀)πΆπ‘Œ))
4612, 40, 45syl2anc 582 . 2 (((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ 𝑋 < π‘Œ) ∧ 𝑝 ∈ (Atomsβ€˜πΎ)) ∧ (((ocβ€˜πΎ)β€˜π‘Œ)𝐢(((ocβ€˜πΎ)β€˜π‘Œ)(joinβ€˜πΎ)𝑝) ∧ (((ocβ€˜πΎ)β€˜π‘Œ)(joinβ€˜πΎ)𝑝) ≀ ((ocβ€˜πΎ)β€˜π‘‹))) β†’ βˆƒπ‘€ ∈ 𝐻 (𝑋 ≀ (π‘Œ ∧ 𝑀) ∧ (π‘Œ ∧ 𝑀)πΆπ‘Œ))
47 simpl1 1188 . . 3 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ 𝑋 < π‘Œ) β†’ 𝐾 ∈ HL)
4847, 13syl 17 . . . 4 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ 𝑋 < π‘Œ) β†’ 𝐾 ∈ OP)
49 simpl3 1190 . . . 4 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ 𝑋 < π‘Œ) β†’ π‘Œ ∈ 𝐡)
503, 7opoccl 38722 . . . 4 ((𝐾 ∈ OP ∧ π‘Œ ∈ 𝐡) β†’ ((ocβ€˜πΎ)β€˜π‘Œ) ∈ 𝐡)
5148, 49, 50syl2anc 582 . . 3 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ 𝑋 < π‘Œ) β†’ ((ocβ€˜πΎ)β€˜π‘Œ) ∈ 𝐡)
52 simpl2 1189 . . . 4 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ 𝑋 < π‘Œ) β†’ 𝑋 ∈ 𝐡)
533, 7opoccl 38722 . . . 4 ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐡) β†’ ((ocβ€˜πΎ)β€˜π‘‹) ∈ 𝐡)
5448, 52, 53syl2anc 582 . . 3 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ 𝑋 < π‘Œ) β†’ ((ocβ€˜πΎ)β€˜π‘‹) ∈ 𝐡)
55 simpr 483 . . . 4 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ 𝑋 < π‘Œ) β†’ 𝑋 < π‘Œ)
56 lhprelat3.s . . . . . 6 < = (ltβ€˜πΎ)
573, 56, 7opltcon3b 38732 . . . . 5 ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝑋 < π‘Œ ↔ ((ocβ€˜πΎ)β€˜π‘Œ) < ((ocβ€˜πΎ)β€˜π‘‹)))
5848, 52, 49, 57syl3anc 1368 . . . 4 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ 𝑋 < π‘Œ) β†’ (𝑋 < π‘Œ ↔ ((ocβ€˜πΎ)β€˜π‘Œ) < ((ocβ€˜πΎ)β€˜π‘‹)))
5955, 58mpbid 231 . . 3 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ 𝑋 < π‘Œ) β†’ ((ocβ€˜πΎ)β€˜π‘Œ) < ((ocβ€˜πΎ)β€˜π‘‹))
603, 33, 56, 27, 22, 4hlrelat3 38941 . . 3 (((𝐾 ∈ HL ∧ ((ocβ€˜πΎ)β€˜π‘Œ) ∈ 𝐡 ∧ ((ocβ€˜πΎ)β€˜π‘‹) ∈ 𝐡) ∧ ((ocβ€˜πΎ)β€˜π‘Œ) < ((ocβ€˜πΎ)β€˜π‘‹)) β†’ βˆƒπ‘ ∈ (Atomsβ€˜πΎ)(((ocβ€˜πΎ)β€˜π‘Œ)𝐢(((ocβ€˜πΎ)β€˜π‘Œ)(joinβ€˜πΎ)𝑝) ∧ (((ocβ€˜πΎ)β€˜π‘Œ)(joinβ€˜πΎ)𝑝) ≀ ((ocβ€˜πΎ)β€˜π‘‹)))
6147, 51, 54, 59, 60syl31anc 1370 . 2 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ 𝑋 < π‘Œ) β†’ βˆƒπ‘ ∈ (Atomsβ€˜πΎ)(((ocβ€˜πΎ)β€˜π‘Œ)𝐢(((ocβ€˜πΎ)β€˜π‘Œ)(joinβ€˜πΎ)𝑝) ∧ (((ocβ€˜πΎ)β€˜π‘Œ)(joinβ€˜πΎ)𝑝) ≀ ((ocβ€˜πΎ)β€˜π‘‹)))
6246, 61r19.29a 3152 1 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ 𝑋 < π‘Œ) β†’ βˆƒπ‘€ ∈ 𝐻 (𝑋 ≀ (π‘Œ ∧ 𝑀) ∧ (π‘Œ ∧ 𝑀)πΆπ‘Œ))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  βˆƒwrex 3060   class class class wbr 5143  β€˜cfv 6543  (class class class)co 7416  Basecbs 17179  lecple 17239  occoc 17240  ltcplt 18299  joincjn 18302  meetcmee 18303  Latclat 18422  OPcops 38700  OLcol 38702   β‹– ccvr 38790  Atomscatm 38791  HLchlt 38878  LHypclh 39513
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7738
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3769  df-csb 3885  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7372  df-ov 7419  df-oprab 7420  df-proset 18286  df-poset 18304  df-plt 18321  df-lub 18337  df-glb 18338  df-join 18339  df-meet 18340  df-p0 18416  df-p1 18417  df-lat 18423  df-clat 18490  df-oposet 38704  df-ol 38706  df-oml 38707  df-covers 38794  df-ats 38795  df-atl 38826  df-cvlat 38850  df-hlat 38879  df-lhyp 39517
This theorem is referenced by: (None)
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