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Theorem cdleme29b 39759
Description: Transform cdleme28 39757. (Compare cdleme25b 39738.) TODO: FIX COMMENT. (Contributed by NM, 7-Feb-2013.)
Hypotheses
Ref Expression
cdleme26.b 𝐡 = (Baseβ€˜πΎ)
cdleme26.l ≀ = (leβ€˜πΎ)
cdleme26.j ∨ = (joinβ€˜πΎ)
cdleme26.m ∧ = (meetβ€˜πΎ)
cdleme26.a 𝐴 = (Atomsβ€˜πΎ)
cdleme26.h 𝐻 = (LHypβ€˜πΎ)
cdleme27.u π‘ˆ = ((𝑃 ∨ 𝑄) ∧ π‘Š)
cdleme27.f 𝐹 = ((𝑠 ∨ π‘ˆ) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑠) ∧ π‘Š)))
cdleme27.z 𝑍 = ((𝑧 ∨ π‘ˆ) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑧) ∧ π‘Š)))
cdleme27.n 𝑁 = ((𝑃 ∨ 𝑄) ∧ (𝑍 ∨ ((𝑠 ∨ 𝑧) ∧ π‘Š)))
cdleme27.d 𝐷 = (℩𝑒 ∈ 𝐡 βˆ€π‘§ ∈ 𝐴 ((Β¬ 𝑧 ≀ π‘Š ∧ Β¬ 𝑧 ≀ (𝑃 ∨ 𝑄)) β†’ 𝑒 = 𝑁))
cdleme27.c 𝐢 = if(𝑠 ≀ (𝑃 ∨ 𝑄), 𝐷, 𝐹)
Assertion
Ref Expression
cdleme29b ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ 𝑃 β‰  𝑄 ∧ (𝑋 ∈ 𝐡 ∧ Β¬ 𝑋 ≀ π‘Š)) β†’ βˆƒπ‘£ ∈ 𝐡 βˆ€π‘  ∈ 𝐴 ((Β¬ 𝑠 ≀ π‘Š ∧ (𝑠 ∨ (𝑋 ∧ π‘Š)) = 𝑋) β†’ 𝑣 = (𝐢 ∨ (𝑋 ∧ π‘Š))))
Distinct variable groups:   𝑒,𝑠,𝑧,𝐴   𝐡,𝑠,𝑒,𝑧   𝑒,𝐹   𝐻,𝑠,𝑧   ∨ ,𝑠,𝑒,𝑧   𝐾,𝑠,𝑧   ≀ ,𝑠,𝑒,𝑧   ∧ ,𝑠,𝑒,𝑧   𝑒,𝑁   𝑃,𝑠,𝑒,𝑧   𝑄,𝑠,𝑒,𝑧   π‘ˆ,𝑠,𝑒,𝑧   π‘Š,𝑠,𝑒,𝑧   𝑋,𝑠   𝑣,𝐴   𝑣,𝐡   𝑣, ∨   𝑣, ≀   𝑣, ∧   𝑣,𝑃   𝑣,𝑄   𝑣,π‘ˆ   𝑣,π‘Š   𝑣,𝐢   𝑣,𝑠,𝑍,𝑒   𝑧,𝑣,𝑋
Allowed substitution hints:   𝐢(𝑧,𝑒,𝑠)   𝐷(𝑧,𝑣,𝑒,𝑠)   𝐹(𝑧,𝑣,𝑠)   𝐻(𝑣,𝑒)   𝐾(𝑣,𝑒)   𝑁(𝑧,𝑣,𝑠)   𝑋(𝑒)   𝑍(𝑧)
Proof of Theorem cdleme29b
Dummy variable 𝑑 is distinct from all other variables.
StepHypRef Expression
1 cdleme26.b . . 3 𝐡 = (Baseβ€˜πΎ)
2 cdleme26.l . . 3 ≀ = (leβ€˜πΎ)
3 cdleme26.j . . 3 ∨ = (joinβ€˜πΎ)
4 cdleme26.m . . 3 ∧ = (meetβ€˜πΎ)
5 cdleme26.a . . 3 𝐴 = (Atomsβ€˜πΎ)
6 cdleme26.h . . 3 𝐻 = (LHypβ€˜πΎ)
7 cdleme27.u . . 3 π‘ˆ = ((𝑃 ∨ 𝑄) ∧ π‘Š)
8 cdleme27.f . . 3 𝐹 = ((𝑠 ∨ π‘ˆ) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑠) ∧ π‘Š)))
9 cdleme27.z . . 3 𝑍 = ((𝑧 ∨ π‘ˆ) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑧) ∧ π‘Š)))
10 cdleme27.n . . 3 𝑁 = ((𝑃 ∨ 𝑄) ∧ (𝑍 ∨ ((𝑠 ∨ 𝑧) ∧ π‘Š)))
11 cdleme27.d . . 3 𝐷 = (℩𝑒 ∈ 𝐡 βˆ€π‘§ ∈ 𝐴 ((Β¬ 𝑧 ≀ π‘Š ∧ Β¬ 𝑧 ≀ (𝑃 ∨ 𝑄)) β†’ 𝑒 = 𝑁))
12 cdleme27.c . . 3 𝐢 = if(𝑠 ≀ (𝑃 ∨ 𝑄), 𝐷, 𝐹)
131, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12cdleme29ex 39758 . 2 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ 𝑃 β‰  𝑄 ∧ (𝑋 ∈ 𝐡 ∧ Β¬ 𝑋 ≀ π‘Š)) β†’ βˆƒπ‘  ∈ 𝐴 ((Β¬ 𝑠 ≀ π‘Š ∧ (𝑠 ∨ (𝑋 ∧ π‘Š)) = 𝑋) ∧ (𝐢 ∨ (𝑋 ∧ π‘Š)) ∈ 𝐡))
14 eqid 2726 . . 3 ((𝑑 ∨ π‘ˆ) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑑) ∧ π‘Š))) = ((𝑑 ∨ π‘ˆ) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑑) ∧ π‘Š)))
15 eqid 2726 . . 3 ((𝑃 ∨ 𝑄) ∧ (𝑍 ∨ ((𝑑 ∨ 𝑧) ∧ π‘Š))) = ((𝑃 ∨ 𝑄) ∧ (𝑍 ∨ ((𝑑 ∨ 𝑧) ∧ π‘Š)))
16 eqid 2726 . . 3 (℩𝑒 ∈ 𝐡 βˆ€π‘§ ∈ 𝐴 ((Β¬ 𝑧 ≀ π‘Š ∧ Β¬ 𝑧 ≀ (𝑃 ∨ 𝑄)) β†’ 𝑒 = ((𝑃 ∨ 𝑄) ∧ (𝑍 ∨ ((𝑑 ∨ 𝑧) ∧ π‘Š))))) = (℩𝑒 ∈ 𝐡 βˆ€π‘§ ∈ 𝐴 ((Β¬ 𝑧 ≀ π‘Š ∧ Β¬ 𝑧 ≀ (𝑃 ∨ 𝑄)) β†’ 𝑒 = ((𝑃 ∨ 𝑄) ∧ (𝑍 ∨ ((𝑑 ∨ 𝑧) ∧ π‘Š)))))
17 eqid 2726 . . 3 if(𝑑 ≀ (𝑃 ∨ 𝑄), (℩𝑒 ∈ 𝐡 βˆ€π‘§ ∈ 𝐴 ((Β¬ 𝑧 ≀ π‘Š ∧ Β¬ 𝑧 ≀ (𝑃 ∨ 𝑄)) β†’ 𝑒 = ((𝑃 ∨ 𝑄) ∧ (𝑍 ∨ ((𝑑 ∨ 𝑧) ∧ π‘Š))))), ((𝑑 ∨ π‘ˆ) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑑) ∧ π‘Š)))) = if(𝑑 ≀ (𝑃 ∨ 𝑄), (℩𝑒 ∈ 𝐡 βˆ€π‘§ ∈ 𝐴 ((Β¬ 𝑧 ≀ π‘Š ∧ Β¬ 𝑧 ≀ (𝑃 ∨ 𝑄)) β†’ 𝑒 = ((𝑃 ∨ 𝑄) ∧ (𝑍 ∨ ((𝑑 ∨ 𝑧) ∧ π‘Š))))), ((𝑑 ∨ π‘ˆ) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑑) ∧ π‘Š))))
181, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 17cdleme28 39757 . 2 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ 𝑃 β‰  𝑄 ∧ (𝑋 ∈ 𝐡 ∧ Β¬ 𝑋 ≀ π‘Š)) β†’ βˆ€π‘  ∈ 𝐴 βˆ€π‘‘ ∈ 𝐴 (((Β¬ 𝑠 ≀ π‘Š ∧ (𝑠 ∨ (𝑋 ∧ π‘Š)) = 𝑋) ∧ (Β¬ 𝑑 ≀ π‘Š ∧ (𝑑 ∨ (𝑋 ∧ π‘Š)) = 𝑋)) β†’ (𝐢 ∨ (𝑋 ∧ π‘Š)) = (if(𝑑 ≀ (𝑃 ∨ 𝑄), (℩𝑒 ∈ 𝐡 βˆ€π‘§ ∈ 𝐴 ((Β¬ 𝑧 ≀ π‘Š ∧ Β¬ 𝑧 ≀ (𝑃 ∨ 𝑄)) β†’ 𝑒 = ((𝑃 ∨ 𝑄) ∧ (𝑍 ∨ ((𝑑 ∨ 𝑧) ∧ π‘Š))))), ((𝑑 ∨ π‘ˆ) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑑) ∧ π‘Š)))) ∨ (𝑋 ∧ π‘Š))))
19 breq1 5144 . . . . . 6 (𝑠 = 𝑑 β†’ (𝑠 ≀ π‘Š ↔ 𝑑 ≀ π‘Š))
2019notbid 318 . . . . 5 (𝑠 = 𝑑 β†’ (Β¬ 𝑠 ≀ π‘Š ↔ Β¬ 𝑑 ≀ π‘Š))
21 oveq1 7412 . . . . . 6 (𝑠 = 𝑑 β†’ (𝑠 ∨ (𝑋 ∧ π‘Š)) = (𝑑 ∨ (𝑋 ∧ π‘Š)))
2221eqeq1d 2728 . . . . 5 (𝑠 = 𝑑 β†’ ((𝑠 ∨ (𝑋 ∧ π‘Š)) = 𝑋 ↔ (𝑑 ∨ (𝑋 ∧ π‘Š)) = 𝑋))
2320, 22anbi12d 630 . . . 4 (𝑠 = 𝑑 β†’ ((Β¬ 𝑠 ≀ π‘Š ∧ (𝑠 ∨ (𝑋 ∧ π‘Š)) = 𝑋) ↔ (Β¬ 𝑑 ≀ π‘Š ∧ (𝑑 ∨ (𝑋 ∧ π‘Š)) = 𝑋)))
2412oveq1i 7415 . . . . 5 (𝐢 ∨ (𝑋 ∧ π‘Š)) = (if(𝑠 ≀ (𝑃 ∨ 𝑄), 𝐷, 𝐹) ∨ (𝑋 ∧ π‘Š))
25 breq1 5144 . . . . . . 7 (𝑠 = 𝑑 β†’ (𝑠 ≀ (𝑃 ∨ 𝑄) ↔ 𝑑 ≀ (𝑃 ∨ 𝑄)))
26 oveq1 7412 . . . . . . . . . . . . . . . 16 (𝑠 = 𝑑 β†’ (𝑠 ∨ 𝑧) = (𝑑 ∨ 𝑧))
2726oveq1d 7420 . . . . . . . . . . . . . . 15 (𝑠 = 𝑑 β†’ ((𝑠 ∨ 𝑧) ∧ π‘Š) = ((𝑑 ∨ 𝑧) ∧ π‘Š))
2827oveq2d 7421 . . . . . . . . . . . . . 14 (𝑠 = 𝑑 β†’ (𝑍 ∨ ((𝑠 ∨ 𝑧) ∧ π‘Š)) = (𝑍 ∨ ((𝑑 ∨ 𝑧) ∧ π‘Š)))
2928oveq2d 7421 . . . . . . . . . . . . 13 (𝑠 = 𝑑 β†’ ((𝑃 ∨ 𝑄) ∧ (𝑍 ∨ ((𝑠 ∨ 𝑧) ∧ π‘Š))) = ((𝑃 ∨ 𝑄) ∧ (𝑍 ∨ ((𝑑 ∨ 𝑧) ∧ π‘Š))))
3010, 29eqtrid 2778 . . . . . . . . . . . 12 (𝑠 = 𝑑 β†’ 𝑁 = ((𝑃 ∨ 𝑄) ∧ (𝑍 ∨ ((𝑑 ∨ 𝑧) ∧ π‘Š))))
3130eqeq2d 2737 . . . . . . . . . . 11 (𝑠 = 𝑑 β†’ (𝑒 = 𝑁 ↔ 𝑒 = ((𝑃 ∨ 𝑄) ∧ (𝑍 ∨ ((𝑑 ∨ 𝑧) ∧ π‘Š)))))
3231imbi2d 340 . . . . . . . . . 10 (𝑠 = 𝑑 β†’ (((Β¬ 𝑧 ≀ π‘Š ∧ Β¬ 𝑧 ≀ (𝑃 ∨ 𝑄)) β†’ 𝑒 = 𝑁) ↔ ((Β¬ 𝑧 ≀ π‘Š ∧ Β¬ 𝑧 ≀ (𝑃 ∨ 𝑄)) β†’ 𝑒 = ((𝑃 ∨ 𝑄) ∧ (𝑍 ∨ ((𝑑 ∨ 𝑧) ∧ π‘Š))))))
3332ralbidv 3171 . . . . . . . . 9 (𝑠 = 𝑑 β†’ (βˆ€π‘§ ∈ 𝐴 ((Β¬ 𝑧 ≀ π‘Š ∧ Β¬ 𝑧 ≀ (𝑃 ∨ 𝑄)) β†’ 𝑒 = 𝑁) ↔ βˆ€π‘§ ∈ 𝐴 ((Β¬ 𝑧 ≀ π‘Š ∧ Β¬ 𝑧 ≀ (𝑃 ∨ 𝑄)) β†’ 𝑒 = ((𝑃 ∨ 𝑄) ∧ (𝑍 ∨ ((𝑑 ∨ 𝑧) ∧ π‘Š))))))
3433riotabidv 7363 . . . . . . . 8 (𝑠 = 𝑑 β†’ (℩𝑒 ∈ 𝐡 βˆ€π‘§ ∈ 𝐴 ((Β¬ 𝑧 ≀ π‘Š ∧ Β¬ 𝑧 ≀ (𝑃 ∨ 𝑄)) β†’ 𝑒 = 𝑁)) = (℩𝑒 ∈ 𝐡 βˆ€π‘§ ∈ 𝐴 ((Β¬ 𝑧 ≀ π‘Š ∧ Β¬ 𝑧 ≀ (𝑃 ∨ 𝑄)) β†’ 𝑒 = ((𝑃 ∨ 𝑄) ∧ (𝑍 ∨ ((𝑑 ∨ 𝑧) ∧ π‘Š))))))
3511, 34eqtrid 2778 . . . . . . 7 (𝑠 = 𝑑 β†’ 𝐷 = (℩𝑒 ∈ 𝐡 βˆ€π‘§ ∈ 𝐴 ((Β¬ 𝑧 ≀ π‘Š ∧ Β¬ 𝑧 ≀ (𝑃 ∨ 𝑄)) β†’ 𝑒 = ((𝑃 ∨ 𝑄) ∧ (𝑍 ∨ ((𝑑 ∨ 𝑧) ∧ π‘Š))))))
36 oveq1 7412 . . . . . . . . 9 (𝑠 = 𝑑 β†’ (𝑠 ∨ π‘ˆ) = (𝑑 ∨ π‘ˆ))
37 oveq2 7413 . . . . . . . . . . 11 (𝑠 = 𝑑 β†’ (𝑃 ∨ 𝑠) = (𝑃 ∨ 𝑑))
3837oveq1d 7420 . . . . . . . . . 10 (𝑠 = 𝑑 β†’ ((𝑃 ∨ 𝑠) ∧ π‘Š) = ((𝑃 ∨ 𝑑) ∧ π‘Š))
3938oveq2d 7421 . . . . . . . . 9 (𝑠 = 𝑑 β†’ (𝑄 ∨ ((𝑃 ∨ 𝑠) ∧ π‘Š)) = (𝑄 ∨ ((𝑃 ∨ 𝑑) ∧ π‘Š)))
4036, 39oveq12d 7423 . . . . . . . 8 (𝑠 = 𝑑 β†’ ((𝑠 ∨ π‘ˆ) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑠) ∧ π‘Š))) = ((𝑑 ∨ π‘ˆ) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑑) ∧ π‘Š))))
418, 40eqtrid 2778 . . . . . . 7 (𝑠 = 𝑑 β†’ 𝐹 = ((𝑑 ∨ π‘ˆ) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑑) ∧ π‘Š))))
4225, 35, 41ifbieq12d 4551 . . . . . 6 (𝑠 = 𝑑 β†’ if(𝑠 ≀ (𝑃 ∨ 𝑄), 𝐷, 𝐹) = if(𝑑 ≀ (𝑃 ∨ 𝑄), (℩𝑒 ∈ 𝐡 βˆ€π‘§ ∈ 𝐴 ((Β¬ 𝑧 ≀ π‘Š ∧ Β¬ 𝑧 ≀ (𝑃 ∨ 𝑄)) β†’ 𝑒 = ((𝑃 ∨ 𝑄) ∧ (𝑍 ∨ ((𝑑 ∨ 𝑧) ∧ π‘Š))))), ((𝑑 ∨ π‘ˆ) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑑) ∧ π‘Š)))))
4342oveq1d 7420 . . . . 5 (𝑠 = 𝑑 β†’ (if(𝑠 ≀ (𝑃 ∨ 𝑄), 𝐷, 𝐹) ∨ (𝑋 ∧ π‘Š)) = (if(𝑑 ≀ (𝑃 ∨ 𝑄), (℩𝑒 ∈ 𝐡 βˆ€π‘§ ∈ 𝐴 ((Β¬ 𝑧 ≀ π‘Š ∧ Β¬ 𝑧 ≀ (𝑃 ∨ 𝑄)) β†’ 𝑒 = ((𝑃 ∨ 𝑄) ∧ (𝑍 ∨ ((𝑑 ∨ 𝑧) ∧ π‘Š))))), ((𝑑 ∨ π‘ˆ) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑑) ∧ π‘Š)))) ∨ (𝑋 ∧ π‘Š)))
4424, 43eqtrid 2778 . . . 4 (𝑠 = 𝑑 β†’ (𝐢 ∨ (𝑋 ∧ π‘Š)) = (if(𝑑 ≀ (𝑃 ∨ 𝑄), (℩𝑒 ∈ 𝐡 βˆ€π‘§ ∈ 𝐴 ((Β¬ 𝑧 ≀ π‘Š ∧ Β¬ 𝑧 ≀ (𝑃 ∨ 𝑄)) β†’ 𝑒 = ((𝑃 ∨ 𝑄) ∧ (𝑍 ∨ ((𝑑 ∨ 𝑧) ∧ π‘Š))))), ((𝑑 ∨ π‘ˆ) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑑) ∧ π‘Š)))) ∨ (𝑋 ∧ π‘Š)))
4523, 44reusv3 5396 . . 3 (βˆƒπ‘  ∈ 𝐴 ((Β¬ 𝑠 ≀ π‘Š ∧ (𝑠 ∨ (𝑋 ∧ π‘Š)) = 𝑋) ∧ (𝐢 ∨ (𝑋 ∧ π‘Š)) ∈ 𝐡) β†’ (βˆ€π‘  ∈ 𝐴 βˆ€π‘‘ ∈ 𝐴 (((Β¬ 𝑠 ≀ π‘Š ∧ (𝑠 ∨ (𝑋 ∧ π‘Š)) = 𝑋) ∧ (Β¬ 𝑑 ≀ π‘Š ∧ (𝑑 ∨ (𝑋 ∧ π‘Š)) = 𝑋)) β†’ (𝐢 ∨ (𝑋 ∧ π‘Š)) = (if(𝑑 ≀ (𝑃 ∨ 𝑄), (℩𝑒 ∈ 𝐡 βˆ€π‘§ ∈ 𝐴 ((Β¬ 𝑧 ≀ π‘Š ∧ Β¬ 𝑧 ≀ (𝑃 ∨ 𝑄)) β†’ 𝑒 = ((𝑃 ∨ 𝑄) ∧ (𝑍 ∨ ((𝑑 ∨ 𝑧) ∧ π‘Š))))), ((𝑑 ∨ π‘ˆ) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑑) ∧ π‘Š)))) ∨ (𝑋 ∧ π‘Š))) ↔ βˆƒπ‘£ ∈ 𝐡 βˆ€π‘  ∈ 𝐴 ((Β¬ 𝑠 ≀ π‘Š ∧ (𝑠 ∨ (𝑋 ∧ π‘Š)) = 𝑋) β†’ 𝑣 = (𝐢 ∨ (𝑋 ∧ π‘Š)))))
4645biimpd 228 . 2 (βˆƒπ‘  ∈ 𝐴 ((Β¬ 𝑠 ≀ π‘Š ∧ (𝑠 ∨ (𝑋 ∧ π‘Š)) = 𝑋) ∧ (𝐢 ∨ (𝑋 ∧ π‘Š)) ∈ 𝐡) β†’ (βˆ€π‘  ∈ 𝐴 βˆ€π‘‘ ∈ 𝐴 (((Β¬ 𝑠 ≀ π‘Š ∧ (𝑠 ∨ (𝑋 ∧ π‘Š)) = 𝑋) ∧ (Β¬ 𝑑 ≀ π‘Š ∧ (𝑑 ∨ (𝑋 ∧ π‘Š)) = 𝑋)) β†’ (𝐢 ∨ (𝑋 ∧ π‘Š)) = (if(𝑑 ≀ (𝑃 ∨ 𝑄), (℩𝑒 ∈ 𝐡 βˆ€π‘§ ∈ 𝐴 ((Β¬ 𝑧 ≀ π‘Š ∧ Β¬ 𝑧 ≀ (𝑃 ∨ 𝑄)) β†’ 𝑒 = ((𝑃 ∨ 𝑄) ∧ (𝑍 ∨ ((𝑑 ∨ 𝑧) ∧ π‘Š))))), ((𝑑 ∨ π‘ˆ) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑑) ∧ π‘Š)))) ∨ (𝑋 ∧ π‘Š))) β†’ βˆƒπ‘£ ∈ 𝐡 βˆ€π‘  ∈ 𝐴 ((Β¬ 𝑠 ≀ π‘Š ∧ (𝑠 ∨ (𝑋 ∧ π‘Š)) = 𝑋) β†’ 𝑣 = (𝐢 ∨ (𝑋 ∧ π‘Š)))))
4713, 18, 46sylc 65 1 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ 𝑃 β‰  𝑄 ∧ (𝑋 ∈ 𝐡 ∧ Β¬ 𝑋 ≀ π‘Š)) β†’ βˆƒπ‘£ ∈ 𝐡 βˆ€π‘  ∈ 𝐴 ((Β¬ 𝑠 ≀ π‘Š ∧ (𝑠 ∨ (𝑋 ∧ π‘Š)) = 𝑋) β†’ 𝑣 = (𝐢 ∨ (𝑋 ∧ π‘Š))))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   β‰  wne 2934  βˆ€wral 3055  βˆƒwrex 3064  ifcif 4523   class class class wbr 5141  β€˜cfv 6537  β„©crio 7360  (class class class)co 7405  Basecbs 17153  lecple 17213  joincjn 18276  meetcmee 18277  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  ax-riotaBAD 38336
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  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-iin 4993  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-mpo 7410  df-1st 7974  df-2nd 7975  df-undef 8259  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-llines 38882  df-lplanes 38883  df-lvols 38884  df-lines 38885  df-psubsp 38887  df-pmap 38888  df-padd 39180  df-lhyp 39372
This theorem is referenced by:  cdleme29c  39760
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