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Theorem cdlemc6 39579
Description: Lemma for cdlemc 39580. (Contributed by NM, 26-May-2012.)
Hypotheses
Ref Expression
cdlemc3.l ≀ = (leβ€˜πΎ)
cdlemc3.j ∨ = (joinβ€˜πΎ)
cdlemc3.m ∧ = (meetβ€˜πΎ)
cdlemc3.a 𝐴 = (Atomsβ€˜πΎ)
cdlemc3.h 𝐻 = (LHypβ€˜πΎ)
cdlemc3.t 𝑇 = ((LTrnβ€˜πΎ)β€˜π‘Š)
cdlemc3.r 𝑅 = ((trLβ€˜πΎ)β€˜π‘Š)
Assertion
Ref Expression
cdlemc6 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (πΉβ€˜π‘ƒ) = 𝑃) β†’ (πΉβ€˜π‘„) = ((𝑄 ∨ (π‘…β€˜πΉ)) ∧ ((πΉβ€˜π‘ƒ) ∨ ((𝑃 ∨ 𝑄) ∧ π‘Š))))

Proof of Theorem cdlemc6
StepHypRef Expression
1 simp1l 1194 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (πΉβ€˜π‘ƒ) = 𝑃) β†’ 𝐾 ∈ HL)
2 simp22l 1289 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (πΉβ€˜π‘ƒ) = 𝑃) β†’ 𝑃 ∈ 𝐴)
3 simp23l 1291 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (πΉβ€˜π‘ƒ) = 𝑃) β†’ 𝑄 ∈ 𝐴)
4 cdlemc3.j . . . . . 6 ∨ = (joinβ€˜πΎ)
5 cdlemc3.a . . . . . 6 𝐴 = (Atomsβ€˜πΎ)
64, 5hlatjcom 38750 . . . . 5 ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) β†’ (𝑃 ∨ 𝑄) = (𝑄 ∨ 𝑃))
71, 2, 3, 6syl3anc 1368 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (πΉβ€˜π‘ƒ) = 𝑃) β†’ (𝑃 ∨ 𝑄) = (𝑄 ∨ 𝑃))
87oveq2d 7420 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (πΉβ€˜π‘ƒ) = 𝑃) β†’ (𝑄 ∧ (𝑃 ∨ 𝑄)) = (𝑄 ∧ (𝑄 ∨ 𝑃)))
91hllatd 38746 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (πΉβ€˜π‘ƒ) = 𝑃) β†’ 𝐾 ∈ Lat)
10 eqid 2726 . . . . . 6 (Baseβ€˜πΎ) = (Baseβ€˜πΎ)
1110, 5atbase 38671 . . . . 5 (𝑄 ∈ 𝐴 β†’ 𝑄 ∈ (Baseβ€˜πΎ))
123, 11syl 17 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (πΉβ€˜π‘ƒ) = 𝑃) β†’ 𝑄 ∈ (Baseβ€˜πΎ))
1310, 5atbase 38671 . . . . 5 (𝑃 ∈ 𝐴 β†’ 𝑃 ∈ (Baseβ€˜πΎ))
142, 13syl 17 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (πΉβ€˜π‘ƒ) = 𝑃) β†’ 𝑃 ∈ (Baseβ€˜πΎ))
15 cdlemc3.m . . . . 5 ∧ = (meetβ€˜πΎ)
1610, 4, 15latabs2 18438 . . . 4 ((𝐾 ∈ Lat ∧ 𝑄 ∈ (Baseβ€˜πΎ) ∧ 𝑃 ∈ (Baseβ€˜πΎ)) β†’ (𝑄 ∧ (𝑄 ∨ 𝑃)) = 𝑄)
179, 12, 14, 16syl3anc 1368 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (πΉβ€˜π‘ƒ) = 𝑃) β†’ (𝑄 ∧ (𝑄 ∨ 𝑃)) = 𝑄)
188, 17eqtrd 2766 . 2 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (πΉβ€˜π‘ƒ) = 𝑃) β†’ (𝑄 ∧ (𝑃 ∨ 𝑄)) = 𝑄)
19 simp1 1133 . . . . . 6 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (πΉβ€˜π‘ƒ) = 𝑃) β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))
20 simp22 1204 . . . . . 6 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (πΉβ€˜π‘ƒ) = 𝑃) β†’ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š))
21 simp21 1203 . . . . . 6 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (πΉβ€˜π‘ƒ) = 𝑃) β†’ 𝐹 ∈ 𝑇)
22 simp3 1135 . . . . . 6 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (πΉβ€˜π‘ƒ) = 𝑃) β†’ (πΉβ€˜π‘ƒ) = 𝑃)
23 cdlemc3.l . . . . . . 7 ≀ = (leβ€˜πΎ)
24 eqid 2726 . . . . . . 7 (0.β€˜πΎ) = (0.β€˜πΎ)
25 cdlemc3.h . . . . . . 7 𝐻 = (LHypβ€˜πΎ)
26 cdlemc3.t . . . . . . 7 𝑇 = ((LTrnβ€˜πΎ)β€˜π‘Š)
27 cdlemc3.r . . . . . . 7 𝑅 = ((trLβ€˜πΎ)β€˜π‘Š)
2823, 24, 5, 25, 26, 27trl0 39553 . . . . . 6 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝐹 ∈ 𝑇 ∧ (πΉβ€˜π‘ƒ) = 𝑃)) β†’ (π‘…β€˜πΉ) = (0.β€˜πΎ))
2919, 20, 21, 22, 28syl112anc 1371 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (πΉβ€˜π‘ƒ) = 𝑃) β†’ (π‘…β€˜πΉ) = (0.β€˜πΎ))
3029oveq2d 7420 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (πΉβ€˜π‘ƒ) = 𝑃) β†’ (𝑄 ∨ (π‘…β€˜πΉ)) = (𝑄 ∨ (0.β€˜πΎ)))
31 hlol 38743 . . . . . 6 (𝐾 ∈ HL β†’ 𝐾 ∈ OL)
321, 31syl 17 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (πΉβ€˜π‘ƒ) = 𝑃) β†’ 𝐾 ∈ OL)
3310, 4, 24olj01 38607 . . . . 5 ((𝐾 ∈ OL ∧ 𝑄 ∈ (Baseβ€˜πΎ)) β†’ (𝑄 ∨ (0.β€˜πΎ)) = 𝑄)
3432, 12, 33syl2anc 583 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (πΉβ€˜π‘ƒ) = 𝑃) β†’ (𝑄 ∨ (0.β€˜πΎ)) = 𝑄)
3530, 34eqtrd 2766 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (πΉβ€˜π‘ƒ) = 𝑃) β†’ (𝑄 ∨ (π‘…β€˜πΉ)) = 𝑄)
3622oveq1d 7419 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (πΉβ€˜π‘ƒ) = 𝑃) β†’ ((πΉβ€˜π‘ƒ) ∨ ((𝑃 ∨ 𝑄) ∧ π‘Š)) = (𝑃 ∨ ((𝑃 ∨ 𝑄) ∧ π‘Š)))
3710, 4, 5hlatjcl 38749 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) β†’ (𝑃 ∨ 𝑄) ∈ (Baseβ€˜πΎ))
381, 2, 3, 37syl3anc 1368 . . . . . 6 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (πΉβ€˜π‘ƒ) = 𝑃) β†’ (𝑃 ∨ 𝑄) ∈ (Baseβ€˜πΎ))
39 simp1r 1195 . . . . . . 7 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (πΉβ€˜π‘ƒ) = 𝑃) β†’ π‘Š ∈ 𝐻)
4010, 25lhpbase 39381 . . . . . . 7 (π‘Š ∈ 𝐻 β†’ π‘Š ∈ (Baseβ€˜πΎ))
4139, 40syl 17 . . . . . 6 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (πΉβ€˜π‘ƒ) = 𝑃) β†’ π‘Š ∈ (Baseβ€˜πΎ))
4210, 15latmcl 18402 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑄) ∈ (Baseβ€˜πΎ) ∧ π‘Š ∈ (Baseβ€˜πΎ)) β†’ ((𝑃 ∨ 𝑄) ∧ π‘Š) ∈ (Baseβ€˜πΎ))
439, 38, 41, 42syl3anc 1368 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (πΉβ€˜π‘ƒ) = 𝑃) β†’ ((𝑃 ∨ 𝑄) ∧ π‘Š) ∈ (Baseβ€˜πΎ))
4410, 4latjcom 18409 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Baseβ€˜πΎ) ∧ ((𝑃 ∨ 𝑄) ∧ π‘Š) ∈ (Baseβ€˜πΎ)) β†’ (𝑃 ∨ ((𝑃 ∨ 𝑄) ∧ π‘Š)) = (((𝑃 ∨ 𝑄) ∧ π‘Š) ∨ 𝑃))
459, 14, 43, 44syl3anc 1368 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (πΉβ€˜π‘ƒ) = 𝑃) β†’ (𝑃 ∨ ((𝑃 ∨ 𝑄) ∧ π‘Š)) = (((𝑃 ∨ 𝑄) ∧ π‘Š) ∨ 𝑃))
4623, 4, 5hlatlej1 38757 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) β†’ 𝑃 ≀ (𝑃 ∨ 𝑄))
471, 2, 3, 46syl3anc 1368 . . . . . 6 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (πΉβ€˜π‘ƒ) = 𝑃) β†’ 𝑃 ≀ (𝑃 ∨ 𝑄))
4810, 23, 4, 15, 5atmod2i1 39244 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ (𝑃 ∨ 𝑄) ∈ (Baseβ€˜πΎ) ∧ π‘Š ∈ (Baseβ€˜πΎ)) ∧ 𝑃 ≀ (𝑃 ∨ 𝑄)) β†’ (((𝑃 ∨ 𝑄) ∧ π‘Š) ∨ 𝑃) = ((𝑃 ∨ 𝑄) ∧ (π‘Š ∨ 𝑃)))
491, 2, 38, 41, 47, 48syl131anc 1380 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (πΉβ€˜π‘ƒ) = 𝑃) β†’ (((𝑃 ∨ 𝑄) ∧ π‘Š) ∨ 𝑃) = ((𝑃 ∨ 𝑄) ∧ (π‘Š ∨ 𝑃)))
50 eqid 2726 . . . . . . . 8 (1.β€˜πΎ) = (1.β€˜πΎ)
5123, 4, 50, 5, 25lhpjat1 39403 . . . . . . 7 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š)) β†’ (π‘Š ∨ 𝑃) = (1.β€˜πΎ))
521, 39, 20, 51syl21anc 835 . . . . . 6 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (πΉβ€˜π‘ƒ) = 𝑃) β†’ (π‘Š ∨ 𝑃) = (1.β€˜πΎ))
5352oveq2d 7420 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (πΉβ€˜π‘ƒ) = 𝑃) β†’ ((𝑃 ∨ 𝑄) ∧ (π‘Š ∨ 𝑃)) = ((𝑃 ∨ 𝑄) ∧ (1.β€˜πΎ)))
5410, 15, 50olm11 38609 . . . . . 6 ((𝐾 ∈ OL ∧ (𝑃 ∨ 𝑄) ∈ (Baseβ€˜πΎ)) β†’ ((𝑃 ∨ 𝑄) ∧ (1.β€˜πΎ)) = (𝑃 ∨ 𝑄))
5532, 38, 54syl2anc 583 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (πΉβ€˜π‘ƒ) = 𝑃) β†’ ((𝑃 ∨ 𝑄) ∧ (1.β€˜πΎ)) = (𝑃 ∨ 𝑄))
5649, 53, 553eqtrd 2770 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (πΉβ€˜π‘ƒ) = 𝑃) β†’ (((𝑃 ∨ 𝑄) ∧ π‘Š) ∨ 𝑃) = (𝑃 ∨ 𝑄))
5736, 45, 563eqtrd 2770 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (πΉβ€˜π‘ƒ) = 𝑃) β†’ ((πΉβ€˜π‘ƒ) ∨ ((𝑃 ∨ 𝑄) ∧ π‘Š)) = (𝑃 ∨ 𝑄))
5835, 57oveq12d 7422 . 2 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (πΉβ€˜π‘ƒ) = 𝑃) β†’ ((𝑄 ∨ (π‘…β€˜πΉ)) ∧ ((πΉβ€˜π‘ƒ) ∨ ((𝑃 ∨ 𝑄) ∧ π‘Š))) = (𝑄 ∧ (𝑃 ∨ 𝑄)))
5923, 5, 25, 26ltrnateq 39564 . 2 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (πΉβ€˜π‘ƒ) = 𝑃) β†’ (πΉβ€˜π‘„) = 𝑄)
6018, 58, 593eqtr4rd 2777 1 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (πΉβ€˜π‘ƒ) = 𝑃) β†’ (πΉβ€˜π‘„) = ((𝑄 ∨ (π‘…β€˜πΉ)) ∧ ((πΉβ€˜π‘ƒ) ∨ ((𝑃 ∨ 𝑄) ∧ π‘Š))))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   class class class wbr 5141  β€˜cfv 6536  (class class class)co 7404  Basecbs 17150  lecple 17210  joincjn 18273  meetcmee 18274  0.cp0 18385  1.cp1 18386  Latclat 18393  OLcol 38556  Atomscatm 38645  HLchlt 38732  LHypclh 39367  LTrncltrn 39484  trLctrl 39541
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 7721
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-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 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7360  df-ov 7407  df-oprab 7408  df-mpo 7409  df-1st 7971  df-2nd 7972  df-map 8821  df-proset 18257  df-poset 18275  df-plt 18292  df-lub 18308  df-glb 18309  df-join 18310  df-meet 18311  df-p0 18387  df-p1 18388  df-lat 18394  df-clat 18461  df-oposet 38558  df-ol 38560  df-oml 38561  df-covers 38648  df-ats 38649  df-atl 38680  df-cvlat 38704  df-hlat 38733  df-psubsp 38886  df-pmap 38887  df-padd 39179  df-lhyp 39371  df-laut 39372  df-ldil 39487  df-ltrn 39488  df-trl 39542
This theorem is referenced by:  cdlemc  39580
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