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Theorem lhprelat3N 38906
Description: The Hilbert lattice is relatively atomic with respect to co-atoms (lattice hyperplanes). Dual version of hlrelat3 38278. (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 485 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ 𝑋 < π‘Œ) ∧ 𝑝 ∈ (Atomsβ€˜πΎ)) β†’ 𝑝 ∈ (Atomsβ€˜πΎ))
2 simpll1 1212 . . . . . 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 2732 . . . . . . . 8 (Atomsβ€˜πΎ) = (Atomsβ€˜πΎ)
53, 4atbase 38154 . . . . . . 7 (𝑝 ∈ (Atomsβ€˜πΎ) β†’ 𝑝 ∈ 𝐡)
65adantl 482 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ 𝑋 < π‘Œ) ∧ 𝑝 ∈ (Atomsβ€˜πΎ)) β†’ 𝑝 ∈ 𝐡)
7 eqid 2732 . . . . . . 7 (ocβ€˜πΎ) = (ocβ€˜πΎ)
8 lhprelat3.h . . . . . . 7 𝐻 = (LHypβ€˜πΎ)
93, 7, 4, 8lhpoc2N 38881 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑝 ∈ 𝐡) β†’ (𝑝 ∈ (Atomsβ€˜πΎ) ↔ ((ocβ€˜πΎ)β€˜π‘) ∈ 𝐻))
102, 6, 9syl2anc 584 . . . . 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 481 . . 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 38227 . . . . . . . . 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 38229 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ 𝑋 < π‘Œ) ∧ 𝑝 ∈ (Atomsβ€˜πΎ)) β†’ 𝐾 ∈ Lat)
16 simpll3 1214 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ 𝑋 < π‘Œ) ∧ 𝑝 ∈ (Atomsβ€˜πΎ)) β†’ π‘Œ ∈ 𝐡)
173, 7opoccl 38059 . . . . . . . . . 10 ((𝐾 ∈ OP ∧ 𝑝 ∈ 𝐡) β†’ ((ocβ€˜πΎ)β€˜π‘) ∈ 𝐡)
1814, 6, 17syl2anc 584 . . . . . . . . 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 18392 . . . . . . . . 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 1371 . . . . . . . 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 38142 . . . . . . . 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 1371 . . . . . . 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 38226 . . . . . . . . . 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 2732 . . . . . . . . . 10 (joinβ€˜πΎ) = (joinβ€˜πΎ)
283, 27, 19, 7oldmm3N 38084 . . . . . . . . 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 1371 . . . . . . . 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 5160 . . . . . . 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 1213 . . . . . . . 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 38065 . . . . . . . 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 1371 . . . . . . 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 5158 . . . . . . 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 631 . . . . 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 477 . . . 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 462 . . 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 7416 . . . . . 6 (𝑀 = ((ocβ€˜πΎ)β€˜π‘) β†’ (π‘Œ ∧ 𝑀) = (π‘Œ ∧ ((ocβ€˜πΎ)β€˜π‘)))
4241breq2d 5160 . . . . 5 (𝑀 = ((ocβ€˜πΎ)β€˜π‘) β†’ (𝑋 ≀ (π‘Œ ∧ 𝑀) ↔ 𝑋 ≀ (π‘Œ ∧ ((ocβ€˜πΎ)β€˜π‘))))
4341breq1d 5158 . . . . 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 631 . . . 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 3612 . . 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 584 . 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 1191 . . 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 1193 . . . 4 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ 𝑋 < π‘Œ) β†’ π‘Œ ∈ 𝐡)
503, 7opoccl 38059 . . . 4 ((𝐾 ∈ OP ∧ π‘Œ ∈ 𝐡) β†’ ((ocβ€˜πΎ)β€˜π‘Œ) ∈ 𝐡)
5148, 49, 50syl2anc 584 . . 3 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ 𝑋 < π‘Œ) β†’ ((ocβ€˜πΎ)β€˜π‘Œ) ∈ 𝐡)
52 simpl2 1192 . . . 4 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ 𝑋 < π‘Œ) β†’ 𝑋 ∈ 𝐡)
533, 7opoccl 38059 . . . 4 ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐡) β†’ ((ocβ€˜πΎ)β€˜π‘‹) ∈ 𝐡)
5448, 52, 53syl2anc 584 . . 3 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ 𝑋 < π‘Œ) β†’ ((ocβ€˜πΎ)β€˜π‘‹) ∈ 𝐡)
55 simpr 485 . . . 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 38069 . . . . 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 1371 . . . 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 38278 . . 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 1373 . 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 3162 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 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  βˆƒwrex 3070   class class class wbr 5148  β€˜cfv 6543  (class class class)co 7408  Basecbs 17143  lecple 17203  occoc 17204  ltcplt 18260  joincjn 18263  meetcmee 18264  Latclat 18383  OPcops 38037  OLcol 38039   β‹– ccvr 38127  Atomscatm 38128  HLchlt 38215  LHypclh 38850
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  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 7364  df-ov 7411  df-oprab 7412  df-proset 18247  df-poset 18265  df-plt 18282  df-lub 18298  df-glb 18299  df-join 18300  df-meet 18301  df-p0 18377  df-p1 18378  df-lat 18384  df-clat 18451  df-oposet 38041  df-ol 38043  df-oml 38044  df-covers 38131  df-ats 38132  df-atl 38163  df-cvlat 38187  df-hlat 38216  df-lhyp 38854
This theorem is referenced by: (None)
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