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Theorem lhprelat3N 39424
Description: The Hilbert lattice is relatively atomic with respect to co-atoms (lattice hyperplanes). Dual version of hlrelat3 38796. (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 484 . . . . 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 2726 . . . . . . . 8 (Atomsβ€˜πΎ) = (Atomsβ€˜πΎ)
53, 4atbase 38672 . . . . . . 7 (𝑝 ∈ (Atomsβ€˜πΎ) β†’ 𝑝 ∈ 𝐡)
65adantl 481 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ 𝑋 < π‘Œ) ∧ 𝑝 ∈ (Atomsβ€˜πΎ)) β†’ 𝑝 ∈ 𝐡)
7 eqid 2726 . . . . . . 7 (ocβ€˜πΎ) = (ocβ€˜πΎ)
8 lhprelat3.h . . . . . . 7 𝐻 = (LHypβ€˜πΎ)
93, 7, 4, 8lhpoc2N 39399 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑝 ∈ 𝐡) β†’ (𝑝 ∈ (Atomsβ€˜πΎ) ↔ ((ocβ€˜πΎ)β€˜π‘) ∈ 𝐻))
102, 6, 9syl2anc 583 . . . . 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 480 . . 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 38745 . . . . . . . . 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 38747 . . . . . . . . 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 38577 . . . . . . . . . 10 ((𝐾 ∈ OP ∧ 𝑝 ∈ 𝐡) β†’ ((ocβ€˜πΎ)β€˜π‘) ∈ 𝐡)
1814, 6, 17syl2anc 583 . . . . . . . . 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 18405 . . . . . . . . 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 38660 . . . . . . . 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 38744 . . . . . . . . . 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 2726 . . . . . . . . . 10 (joinβ€˜πΎ) = (joinβ€˜πΎ)
283, 27, 19, 7oldmm3N 38602 . . . . . . . . 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 5153 . . . . . . 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 280 . . . . . 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 38583 . . . . . . . 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 5151 . . . . . . 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 280 . . . . . 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 476 . . . 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 461 . . 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 7413 . . . . . 6 (𝑀 = ((ocβ€˜πΎ)β€˜π‘) β†’ (π‘Œ ∧ 𝑀) = (π‘Œ ∧ ((ocβ€˜πΎ)β€˜π‘)))
4241breq2d 5153 . . . . 5 (𝑀 = ((ocβ€˜πΎ)β€˜π‘) β†’ (𝑋 ≀ (π‘Œ ∧ 𝑀) ↔ 𝑋 ≀ (π‘Œ ∧ ((ocβ€˜πΎ)β€˜π‘))))
4341breq1d 5151 . . . . 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 3606 . . 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 583 . 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 38577 . . . 4 ((𝐾 ∈ OP ∧ π‘Œ ∈ 𝐡) β†’ ((ocβ€˜πΎ)β€˜π‘Œ) ∈ 𝐡)
5148, 49, 50syl2anc 583 . . 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 38577 . . . 4 ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐡) β†’ ((ocβ€˜πΎ)β€˜π‘‹) ∈ 𝐡)
5448, 52, 53syl2anc 583 . . 3 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ 𝑋 < π‘Œ) β†’ ((ocβ€˜πΎ)β€˜π‘‹) ∈ 𝐡)
55 simpr 484 . . . 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 38587 . . . . 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 38796 . . 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 3156 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 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  βˆƒwrex 3064   class class class wbr 5141  β€˜cfv 6537  (class class class)co 7405  Basecbs 17153  lecple 17213  occoc 17214  ltcplt 18273  joincjn 18276  meetcmee 18277  Latclat 18396  OPcops 38555  OLcol 38557   β‹– ccvr 38645  Atomscatm 38646  HLchlt 38733  LHypclh 39368
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 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  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 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7361  df-ov 7408  df-oprab 7409  df-proset 18260  df-poset 18278  df-plt 18295  df-lub 18311  df-glb 18312  df-join 18313  df-meet 18314  df-p0 18390  df-p1 18391  df-lat 18397  df-clat 18464  df-oposet 38559  df-ol 38561  df-oml 38562  df-covers 38649  df-ats 38650  df-atl 38681  df-cvlat 38705  df-hlat 38734  df-lhyp 39372
This theorem is referenced by: (None)
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