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Theorem lhprelat3N 38553
Description: The Hilbert lattice is relatively atomic with respect to co-atoms (lattice hyperplanes). Dual version of hlrelat3 37925. (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 486 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ 𝑋 < π‘Œ) ∧ 𝑝 ∈ (Atomsβ€˜πΎ)) β†’ 𝑝 ∈ (Atomsβ€˜πΎ))
2 simpll1 1213 . . . . . 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 2733 . . . . . . . 8 (Atomsβ€˜πΎ) = (Atomsβ€˜πΎ)
53, 4atbase 37801 . . . . . . 7 (𝑝 ∈ (Atomsβ€˜πΎ) β†’ 𝑝 ∈ 𝐡)
65adantl 483 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ 𝑋 < π‘Œ) ∧ 𝑝 ∈ (Atomsβ€˜πΎ)) β†’ 𝑝 ∈ 𝐡)
7 eqid 2733 . . . . . . 7 (ocβ€˜πΎ) = (ocβ€˜πΎ)
8 lhprelat3.h . . . . . . 7 𝐻 = (LHypβ€˜πΎ)
93, 7, 4, 8lhpoc2N 38528 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑝 ∈ 𝐡) β†’ (𝑝 ∈ (Atomsβ€˜πΎ) ↔ ((ocβ€˜πΎ)β€˜π‘) ∈ 𝐻))
102, 6, 9syl2anc 585 . . . . 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 482 . . 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 37874 . . . . . . . . 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 37876 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ 𝑋 < π‘Œ) ∧ 𝑝 ∈ (Atomsβ€˜πΎ)) β†’ 𝐾 ∈ Lat)
16 simpll3 1215 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ 𝑋 < π‘Œ) ∧ 𝑝 ∈ (Atomsβ€˜πΎ)) β†’ π‘Œ ∈ 𝐡)
173, 7opoccl 37706 . . . . . . . . . 10 ((𝐾 ∈ OP ∧ 𝑝 ∈ 𝐡) β†’ ((ocβ€˜πΎ)β€˜π‘) ∈ 𝐡)
1814, 6, 17syl2anc 585 . . . . . . . . 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 18337 . . . . . . . . 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 1372 . . . . . . . 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 37789 . . . . . . . 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 1372 . . . . . . 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 37873 . . . . . . . . . 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 2733 . . . . . . . . . 10 (joinβ€˜πΎ) = (joinβ€˜πΎ)
283, 27, 19, 7oldmm3N 37731 . . . . . . . . 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 1372 . . . . . . . 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 5121 . . . . . . 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 1214 . . . . . . . 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 37712 . . . . . . . 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 1372 . . . . . . 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 5119 . . . . . . 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 632 . . . . 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 478 . . . 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 463 . . 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 7369 . . . . . 6 (𝑀 = ((ocβ€˜πΎ)β€˜π‘) β†’ (π‘Œ ∧ 𝑀) = (π‘Œ ∧ ((ocβ€˜πΎ)β€˜π‘)))
4241breq2d 5121 . . . . 5 (𝑀 = ((ocβ€˜πΎ)β€˜π‘) β†’ (𝑋 ≀ (π‘Œ ∧ 𝑀) ↔ 𝑋 ≀ (π‘Œ ∧ ((ocβ€˜πΎ)β€˜π‘))))
4341breq1d 5119 . . . . 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 632 . . . 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 3583 . . 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 585 . 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 1192 . . 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 1194 . . . 4 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ 𝑋 < π‘Œ) β†’ π‘Œ ∈ 𝐡)
503, 7opoccl 37706 . . . 4 ((𝐾 ∈ OP ∧ π‘Œ ∈ 𝐡) β†’ ((ocβ€˜πΎ)β€˜π‘Œ) ∈ 𝐡)
5148, 49, 50syl2anc 585 . . 3 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ 𝑋 < π‘Œ) β†’ ((ocβ€˜πΎ)β€˜π‘Œ) ∈ 𝐡)
52 simpl2 1193 . . . 4 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ 𝑋 < π‘Œ) β†’ 𝑋 ∈ 𝐡)
533, 7opoccl 37706 . . . 4 ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐡) β†’ ((ocβ€˜πΎ)β€˜π‘‹) ∈ 𝐡)
5448, 52, 53syl2anc 585 . . 3 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ 𝑋 < π‘Œ) β†’ ((ocβ€˜πΎ)β€˜π‘‹) ∈ 𝐡)
55 simpr 486 . . . 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 37716 . . . . 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 1372 . . . 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 37925 . . 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 1374 . 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 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆƒwrex 3070   class class class wbr 5109  β€˜cfv 6500  (class class class)co 7361  Basecbs 17091  lecple 17148  occoc 17149  ltcplt 18205  joincjn 18208  meetcmee 18209  Latclat 18328  OPcops 37684  OLcol 37686   β‹– ccvr 37774  Atomscatm 37775  HLchlt 37862  LHypclh 38497
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 2704  ax-rep 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
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 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7317  df-ov 7364  df-oprab 7365  df-proset 18192  df-poset 18210  df-plt 18227  df-lub 18243  df-glb 18244  df-join 18245  df-meet 18246  df-p0 18322  df-p1 18323  df-lat 18329  df-clat 18396  df-oposet 37688  df-ol 37690  df-oml 37691  df-covers 37778  df-ats 37779  df-atl 37810  df-cvlat 37834  df-hlat 37863  df-lhyp 38501
This theorem is referenced by: (None)
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