cdlemd2 - Mathbox for Norm Megill

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Theorem cdlemd2 40237

Description: Part of proof of Lemma D in [Crawley] p. 113. (Contributed by NM, 29-May-2012.)

**Hypotheses**
Ref	Expression
cdlemd2.l	⊢ ≤ = (le‘𝐾)
cdlemd2.j	⊢ ∨ = (join‘𝐾)
cdlemd2.a	⊢ 𝐴 = (Atoms‘𝐾)
cdlemd2.h	⊢ 𝐻 = (LHyp‘𝐾)
cdlemd2.t	⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)

**Assertion**
Ref	Expression
cdlemd2	⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑅 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄))) ∧ ((𝐹‘𝑃) = (𝐺‘𝑃) ∧ (𝐹‘𝑄) = (𝐺‘𝑄))) → (𝐹‘𝑅) = (𝐺‘𝑅))

**Proof of Theorem cdlemd2**
Step	Hyp	Ref	Expression
1		simp3l 1202	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑅 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄))) ∧ ((𝐹‘𝑃) = (𝐺‘𝑃) ∧ (𝐹‘𝑄) = (𝐺‘𝑄))) → (𝐹‘𝑃) = (𝐺‘𝑃))
2		simp11 1204	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑅 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄))) ∧ ((𝐹‘𝑃) = (𝐺‘𝑃) ∧ (𝐹‘𝑄) = (𝐺‘𝑄))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
3		simp12l 1287	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑅 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄))) ∧ ((𝐹‘𝑃) = (𝐺‘𝑃) ∧ (𝐹‘𝑄) = (𝐺‘𝑄))) → 𝐹 ∈ 𝑇)
4		simp11l 1285	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑅 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄))) ∧ ((𝐹‘𝑃) = (𝐺‘𝑃) ∧ (𝐹‘𝑄) = (𝐺‘𝑄))) → 𝐾 ∈ HL)
5	4	hllatd 39402	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑅 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄))) ∧ ((𝐹‘𝑃) = (𝐺‘𝑃) ∧ (𝐹‘𝑄) = (𝐺‘𝑄))) → 𝐾 ∈ Lat)
6		simp21l 1291	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑅 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄))) ∧ ((𝐹‘𝑃) = (𝐺‘𝑃) ∧ (𝐹‘𝑄) = (𝐺‘𝑄))) → 𝑃 ∈ 𝐴)
7		simp13 1206	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑅 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄))) ∧ ((𝐹‘𝑃) = (𝐺‘𝑃) ∧ (𝐹‘𝑄) = (𝐺‘𝑄))) → 𝑅 ∈ 𝐴)
8		eqid 2731	. . . . . . . . . . 11 ⊢ (Base‘𝐾) = (Base‘𝐾)
9		cdlemd2.j	. . . . . . . . . . 11 ⊢ ∨ = (join‘𝐾)
10		cdlemd2.a	. . . . . . . . . . 11 ⊢ 𝐴 = (Atoms‘𝐾)
11	8, 9, 10	hlatjcl 39405	. . . . . . . . . 10 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) → (𝑃 ∨ 𝑅) ∈ (Base‘𝐾))
12	4, 6, 7, 11	syl3anc 1373	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑅 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄))) ∧ ((𝐹‘𝑃) = (𝐺‘𝑃) ∧ (𝐹‘𝑄) = (𝐺‘𝑄))) → (𝑃 ∨ 𝑅) ∈ (Base‘𝐾))
13		simp11r 1286	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑅 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄))) ∧ ((𝐹‘𝑃) = (𝐺‘𝑃) ∧ (𝐹‘𝑄) = (𝐺‘𝑄))) → 𝑊 ∈ 𝐻)
14		cdlemd2.h	. . . . . . . . . . 11 ⊢ 𝐻 = (LHyp‘𝐾)
15	8, 14	lhpbase 40036	. . . . . . . . . 10 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ (Base‘𝐾))
16	13, 15	syl 17	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑅 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄))) ∧ ((𝐹‘𝑃) = (𝐺‘𝑃) ∧ (𝐹‘𝑄) = (𝐺‘𝑄))) → 𝑊 ∈ (Base‘𝐾))
17		eqid 2731	. . . . . . . . . 10 ⊢ (meet‘𝐾) = (meet‘𝐾)
18	8, 17	latmcl 18343	. . . . . . . . 9 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑅) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 ∨ 𝑅)(meet‘𝐾)𝑊) ∈ (Base‘𝐾))
19	5, 12, 16, 18	syl3anc 1373	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑅 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄))) ∧ ((𝐹‘𝑃) = (𝐺‘𝑃) ∧ (𝐹‘𝑄) = (𝐺‘𝑄))) → ((𝑃 ∨ 𝑅)(meet‘𝐾)𝑊) ∈ (Base‘𝐾))
20		cdlemd2.l	. . . . . . . . . 10 ⊢ ≤ = (le‘𝐾)
21	8, 20, 17	latmle2 18368	. . . . . . . . 9 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑅) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 ∨ 𝑅)(meet‘𝐾)𝑊) ≤ 𝑊)
22	5, 12, 16, 21	syl3anc 1373	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑅 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄))) ∧ ((𝐹‘𝑃) = (𝐺‘𝑃) ∧ (𝐹‘𝑄) = (𝐺‘𝑄))) → ((𝑃 ∨ 𝑅)(meet‘𝐾)𝑊) ≤ 𝑊)
23		cdlemd2.t	. . . . . . . . 9 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
24	8, 20, 14, 23	ltrnval1 40172	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (((𝑃 ∨ 𝑅)(meet‘𝐾)𝑊) ∈ (Base‘𝐾) ∧ ((𝑃 ∨ 𝑅)(meet‘𝐾)𝑊) ≤ 𝑊)) → (𝐹‘((𝑃 ∨ 𝑅)(meet‘𝐾)𝑊)) = ((𝑃 ∨ 𝑅)(meet‘𝐾)𝑊))
25	2, 3, 19, 22, 24	syl112anc 1376	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑅 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄))) ∧ ((𝐹‘𝑃) = (𝐺‘𝑃) ∧ (𝐹‘𝑄) = (𝐺‘𝑄))) → (𝐹‘((𝑃 ∨ 𝑅)(meet‘𝐾)𝑊)) = ((𝑃 ∨ 𝑅)(meet‘𝐾)𝑊))
26		simp12r 1288	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑅 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄))) ∧ ((𝐹‘𝑃) = (𝐺‘𝑃) ∧ (𝐹‘𝑄) = (𝐺‘𝑄))) → 𝐺 ∈ 𝑇)
27	8, 20, 14, 23	ltrnval1 40172	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇 ∧ (((𝑃 ∨ 𝑅)(meet‘𝐾)𝑊) ∈ (Base‘𝐾) ∧ ((𝑃 ∨ 𝑅)(meet‘𝐾)𝑊) ≤ 𝑊)) → (𝐺‘((𝑃 ∨ 𝑅)(meet‘𝐾)𝑊)) = ((𝑃 ∨ 𝑅)(meet‘𝐾)𝑊))
28	2, 26, 19, 22, 27	syl112anc 1376	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑅 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄))) ∧ ((𝐹‘𝑃) = (𝐺‘𝑃) ∧ (𝐹‘𝑄) = (𝐺‘𝑄))) → (𝐺‘((𝑃 ∨ 𝑅)(meet‘𝐾)𝑊)) = ((𝑃 ∨ 𝑅)(meet‘𝐾)𝑊))
29	25, 28	eqtr4d 2769	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑅 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄))) ∧ ((𝐹‘𝑃) = (𝐺‘𝑃) ∧ (𝐹‘𝑄) = (𝐺‘𝑄))) → (𝐹‘((𝑃 ∨ 𝑅)(meet‘𝐾)𝑊)) = (𝐺‘((𝑃 ∨ 𝑅)(meet‘𝐾)𝑊)))
30	1, 29	oveq12d 7364	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑅 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄))) ∧ ((𝐹‘𝑃) = (𝐺‘𝑃) ∧ (𝐹‘𝑄) = (𝐺‘𝑄))) → ((𝐹‘𝑃) ∨ (𝐹‘((𝑃 ∨ 𝑅)(meet‘𝐾)𝑊))) = ((𝐺‘𝑃) ∨ (𝐺‘((𝑃 ∨ 𝑅)(meet‘𝐾)𝑊))))
31	8, 10	atbase 39327	. . . . . . 7 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ (Base‘𝐾))
32	6, 31	syl 17	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑅 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄))) ∧ ((𝐹‘𝑃) = (𝐺‘𝑃) ∧ (𝐹‘𝑄) = (𝐺‘𝑄))) → 𝑃 ∈ (Base‘𝐾))
33	8, 9, 14, 23	ltrnj 40170	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ (Base‘𝐾) ∧ ((𝑃 ∨ 𝑅)(meet‘𝐾)𝑊) ∈ (Base‘𝐾))) → (𝐹‘(𝑃 ∨ ((𝑃 ∨ 𝑅)(meet‘𝐾)𝑊))) = ((𝐹‘𝑃) ∨ (𝐹‘((𝑃 ∨ 𝑅)(meet‘𝐾)𝑊))))
34	2, 3, 32, 19, 33	syl112anc 1376	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑅 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄))) ∧ ((𝐹‘𝑃) = (𝐺‘𝑃) ∧ (𝐹‘𝑄) = (𝐺‘𝑄))) → (𝐹‘(𝑃 ∨ ((𝑃 ∨ 𝑅)(meet‘𝐾)𝑊))) = ((𝐹‘𝑃) ∨ (𝐹‘((𝑃 ∨ 𝑅)(meet‘𝐾)𝑊))))
35	8, 9, 14, 23	ltrnj 40170	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇 ∧ (𝑃 ∈ (Base‘𝐾) ∧ ((𝑃 ∨ 𝑅)(meet‘𝐾)𝑊) ∈ (Base‘𝐾))) → (𝐺‘(𝑃 ∨ ((𝑃 ∨ 𝑅)(meet‘𝐾)𝑊))) = ((𝐺‘𝑃) ∨ (𝐺‘((𝑃 ∨ 𝑅)(meet‘𝐾)𝑊))))
36	2, 26, 32, 19, 35	syl112anc 1376	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑅 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄))) ∧ ((𝐹‘𝑃) = (𝐺‘𝑃) ∧ (𝐹‘𝑄) = (𝐺‘𝑄))) → (𝐺‘(𝑃 ∨ ((𝑃 ∨ 𝑅)(meet‘𝐾)𝑊))) = ((𝐺‘𝑃) ∨ (𝐺‘((𝑃 ∨ 𝑅)(meet‘𝐾)𝑊))))
37	30, 34, 36	3eqtr4d 2776	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑅 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄))) ∧ ((𝐹‘𝑃) = (𝐺‘𝑃) ∧ (𝐹‘𝑄) = (𝐺‘𝑄))) → (𝐹‘(𝑃 ∨ ((𝑃 ∨ 𝑅)(meet‘𝐾)𝑊))) = (𝐺‘(𝑃 ∨ ((𝑃 ∨ 𝑅)(meet‘𝐾)𝑊))))
38		simp3r 1203	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑅 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄))) ∧ ((𝐹‘𝑃) = (𝐺‘𝑃) ∧ (𝐹‘𝑄) = (𝐺‘𝑄))) → (𝐹‘𝑄) = (𝐺‘𝑄))
39		simp22l 1293	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑅 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄))) ∧ ((𝐹‘𝑃) = (𝐺‘𝑃) ∧ (𝐹‘𝑄) = (𝐺‘𝑄))) → 𝑄 ∈ 𝐴)
40	8, 9, 10	hlatjcl 39405	. . . . . . . . . 10 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) → (𝑄 ∨ 𝑅) ∈ (Base‘𝐾))
41	4, 39, 7, 40	syl3anc 1373	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑅 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄))) ∧ ((𝐹‘𝑃) = (𝐺‘𝑃) ∧ (𝐹‘𝑄) = (𝐺‘𝑄))) → (𝑄 ∨ 𝑅) ∈ (Base‘𝐾))
42	8, 17	latmcl 18343	. . . . . . . . 9 ⊢ ((𝐾 ∈ Lat ∧ (𝑄 ∨ 𝑅) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑄 ∨ 𝑅)(meet‘𝐾)𝑊) ∈ (Base‘𝐾))
43	5, 41, 16, 42	syl3anc 1373	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑅 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄))) ∧ ((𝐹‘𝑃) = (𝐺‘𝑃) ∧ (𝐹‘𝑄) = (𝐺‘𝑄))) → ((𝑄 ∨ 𝑅)(meet‘𝐾)𝑊) ∈ (Base‘𝐾))
44	8, 20, 17	latmle2 18368	. . . . . . . . 9 ⊢ ((𝐾 ∈ Lat ∧ (𝑄 ∨ 𝑅) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑄 ∨ 𝑅)(meet‘𝐾)𝑊) ≤ 𝑊)
45	5, 41, 16, 44	syl3anc 1373	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑅 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄))) ∧ ((𝐹‘𝑃) = (𝐺‘𝑃) ∧ (𝐹‘𝑄) = (𝐺‘𝑄))) → ((𝑄 ∨ 𝑅)(meet‘𝐾)𝑊) ≤ 𝑊)
46	8, 20, 14, 23	ltrnval1 40172	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (((𝑄 ∨ 𝑅)(meet‘𝐾)𝑊) ∈ (Base‘𝐾) ∧ ((𝑄 ∨ 𝑅)(meet‘𝐾)𝑊) ≤ 𝑊)) → (𝐹‘((𝑄 ∨ 𝑅)(meet‘𝐾)𝑊)) = ((𝑄 ∨ 𝑅)(meet‘𝐾)𝑊))
47	2, 3, 43, 45, 46	syl112anc 1376	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑅 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄))) ∧ ((𝐹‘𝑃) = (𝐺‘𝑃) ∧ (𝐹‘𝑄) = (𝐺‘𝑄))) → (𝐹‘((𝑄 ∨ 𝑅)(meet‘𝐾)𝑊)) = ((𝑄 ∨ 𝑅)(meet‘𝐾)𝑊))
48	8, 20, 14, 23	ltrnval1 40172	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇 ∧ (((𝑄 ∨ 𝑅)(meet‘𝐾)𝑊) ∈ (Base‘𝐾) ∧ ((𝑄 ∨ 𝑅)(meet‘𝐾)𝑊) ≤ 𝑊)) → (𝐺‘((𝑄 ∨ 𝑅)(meet‘𝐾)𝑊)) = ((𝑄 ∨ 𝑅)(meet‘𝐾)𝑊))
49	2, 26, 43, 45, 48	syl112anc 1376	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑅 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄))) ∧ ((𝐹‘𝑃) = (𝐺‘𝑃) ∧ (𝐹‘𝑄) = (𝐺‘𝑄))) → (𝐺‘((𝑄 ∨ 𝑅)(meet‘𝐾)𝑊)) = ((𝑄 ∨ 𝑅)(meet‘𝐾)𝑊))
50	47, 49	eqtr4d 2769	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑅 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄))) ∧ ((𝐹‘𝑃) = (𝐺‘𝑃) ∧ (𝐹‘𝑄) = (𝐺‘𝑄))) → (𝐹‘((𝑄 ∨ 𝑅)(meet‘𝐾)𝑊)) = (𝐺‘((𝑄 ∨ 𝑅)(meet‘𝐾)𝑊)))
51	38, 50	oveq12d 7364	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑅 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄))) ∧ ((𝐹‘𝑃) = (𝐺‘𝑃) ∧ (𝐹‘𝑄) = (𝐺‘𝑄))) → ((𝐹‘𝑄) ∨ (𝐹‘((𝑄 ∨ 𝑅)(meet‘𝐾)𝑊))) = ((𝐺‘𝑄) ∨ (𝐺‘((𝑄 ∨ 𝑅)(meet‘𝐾)𝑊))))
52	8, 10	atbase 39327	. . . . . . 7 ⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ (Base‘𝐾))
53	39, 52	syl 17	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑅 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄))) ∧ ((𝐹‘𝑃) = (𝐺‘𝑃) ∧ (𝐹‘𝑄) = (𝐺‘𝑄))) → 𝑄 ∈ (Base‘𝐾))
54	8, 9, 14, 23	ltrnj 40170	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑄 ∈ (Base‘𝐾) ∧ ((𝑄 ∨ 𝑅)(meet‘𝐾)𝑊) ∈ (Base‘𝐾))) → (𝐹‘(𝑄 ∨ ((𝑄 ∨ 𝑅)(meet‘𝐾)𝑊))) = ((𝐹‘𝑄) ∨ (𝐹‘((𝑄 ∨ 𝑅)(meet‘𝐾)𝑊))))
55	2, 3, 53, 43, 54	syl112anc 1376	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑅 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄))) ∧ ((𝐹‘𝑃) = (𝐺‘𝑃) ∧ (𝐹‘𝑄) = (𝐺‘𝑄))) → (𝐹‘(𝑄 ∨ ((𝑄 ∨ 𝑅)(meet‘𝐾)𝑊))) = ((𝐹‘𝑄) ∨ (𝐹‘((𝑄 ∨ 𝑅)(meet‘𝐾)𝑊))))
56	8, 9, 14, 23	ltrnj 40170	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇 ∧ (𝑄 ∈ (Base‘𝐾) ∧ ((𝑄 ∨ 𝑅)(meet‘𝐾)𝑊) ∈ (Base‘𝐾))) → (𝐺‘(𝑄 ∨ ((𝑄 ∨ 𝑅)(meet‘𝐾)𝑊))) = ((𝐺‘𝑄) ∨ (𝐺‘((𝑄 ∨ 𝑅)(meet‘𝐾)𝑊))))
57	2, 26, 53, 43, 56	syl112anc 1376	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑅 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄))) ∧ ((𝐹‘𝑃) = (𝐺‘𝑃) ∧ (𝐹‘𝑄) = (𝐺‘𝑄))) → (𝐺‘(𝑄 ∨ ((𝑄 ∨ 𝑅)(meet‘𝐾)𝑊))) = ((𝐺‘𝑄) ∨ (𝐺‘((𝑄 ∨ 𝑅)(meet‘𝐾)𝑊))))
58	51, 55, 57	3eqtr4d 2776	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑅 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄))) ∧ ((𝐹‘𝑃) = (𝐺‘𝑃) ∧ (𝐹‘𝑄) = (𝐺‘𝑄))) → (𝐹‘(𝑄 ∨ ((𝑄 ∨ 𝑅)(meet‘𝐾)𝑊))) = (𝐺‘(𝑄 ∨ ((𝑄 ∨ 𝑅)(meet‘𝐾)𝑊))))
59	37, 58	oveq12d 7364	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑅 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄))) ∧ ((𝐹‘𝑃) = (𝐺‘𝑃) ∧ (𝐹‘𝑄) = (𝐺‘𝑄))) → ((𝐹‘(𝑃 ∨ ((𝑃 ∨ 𝑅)(meet‘𝐾)𝑊)))(meet‘𝐾)(𝐹‘(𝑄 ∨ ((𝑄 ∨ 𝑅)(meet‘𝐾)𝑊)))) = ((𝐺‘(𝑃 ∨ ((𝑃 ∨ 𝑅)(meet‘𝐾)𝑊)))(meet‘𝐾)(𝐺‘(𝑄 ∨ ((𝑄 ∨ 𝑅)(meet‘𝐾)𝑊)))))
60	8, 9	latjcl 18342	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ ((𝑃 ∨ 𝑅)(meet‘𝐾)𝑊) ∈ (Base‘𝐾)) → (𝑃 ∨ ((𝑃 ∨ 𝑅)(meet‘𝐾)𝑊)) ∈ (Base‘𝐾))
61	5, 32, 19, 60	syl3anc 1373	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑅 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄))) ∧ ((𝐹‘𝑃) = (𝐺‘𝑃) ∧ (𝐹‘𝑄) = (𝐺‘𝑄))) → (𝑃 ∨ ((𝑃 ∨ 𝑅)(meet‘𝐾)𝑊)) ∈ (Base‘𝐾))
62	8, 9	latjcl 18342	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ 𝑄 ∈ (Base‘𝐾) ∧ ((𝑄 ∨ 𝑅)(meet‘𝐾)𝑊) ∈ (Base‘𝐾)) → (𝑄 ∨ ((𝑄 ∨ 𝑅)(meet‘𝐾)𝑊)) ∈ (Base‘𝐾))
63	5, 53, 43, 62	syl3anc 1373	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑅 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄))) ∧ ((𝐹‘𝑃) = (𝐺‘𝑃) ∧ (𝐹‘𝑄) = (𝐺‘𝑄))) → (𝑄 ∨ ((𝑄 ∨ 𝑅)(meet‘𝐾)𝑊)) ∈ (Base‘𝐾))
64	8, 17, 14, 23	ltrnm 40169	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ ((𝑃 ∨ ((𝑃 ∨ 𝑅)(meet‘𝐾)𝑊)) ∈ (Base‘𝐾) ∧ (𝑄 ∨ ((𝑄 ∨ 𝑅)(meet‘𝐾)𝑊)) ∈ (Base‘𝐾))) → (𝐹‘((𝑃 ∨ ((𝑃 ∨ 𝑅)(meet‘𝐾)𝑊))(meet‘𝐾)(𝑄 ∨ ((𝑄 ∨ 𝑅)(meet‘𝐾)𝑊)))) = ((𝐹‘(𝑃 ∨ ((𝑃 ∨ 𝑅)(meet‘𝐾)𝑊)))(meet‘𝐾)(𝐹‘(𝑄 ∨ ((𝑄 ∨ 𝑅)(meet‘𝐾)𝑊)))))
65	2, 3, 61, 63, 64	syl112anc 1376	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑅 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄))) ∧ ((𝐹‘𝑃) = (𝐺‘𝑃) ∧ (𝐹‘𝑄) = (𝐺‘𝑄))) → (𝐹‘((𝑃 ∨ ((𝑃 ∨ 𝑅)(meet‘𝐾)𝑊))(meet‘𝐾)(𝑄 ∨ ((𝑄 ∨ 𝑅)(meet‘𝐾)𝑊)))) = ((𝐹‘(𝑃 ∨ ((𝑃 ∨ 𝑅)(meet‘𝐾)𝑊)))(meet‘𝐾)(𝐹‘(𝑄 ∨ ((𝑄 ∨ 𝑅)(meet‘𝐾)𝑊)))))
66	8, 17, 14, 23	ltrnm 40169	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇 ∧ ((𝑃 ∨ ((𝑃 ∨ 𝑅)(meet‘𝐾)𝑊)) ∈ (Base‘𝐾) ∧ (𝑄 ∨ ((𝑄 ∨ 𝑅)(meet‘𝐾)𝑊)) ∈ (Base‘𝐾))) → (𝐺‘((𝑃 ∨ ((𝑃 ∨ 𝑅)(meet‘𝐾)𝑊))(meet‘𝐾)(𝑄 ∨ ((𝑄 ∨ 𝑅)(meet‘𝐾)𝑊)))) = ((𝐺‘(𝑃 ∨ ((𝑃 ∨ 𝑅)(meet‘𝐾)𝑊)))(meet‘𝐾)(𝐺‘(𝑄 ∨ ((𝑄 ∨ 𝑅)(meet‘𝐾)𝑊)))))
67	2, 26, 61, 63, 66	syl112anc 1376	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑅 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄))) ∧ ((𝐹‘𝑃) = (𝐺‘𝑃) ∧ (𝐹‘𝑄) = (𝐺‘𝑄))) → (𝐺‘((𝑃 ∨ ((𝑃 ∨ 𝑅)(meet‘𝐾)𝑊))(meet‘𝐾)(𝑄 ∨ ((𝑄 ∨ 𝑅)(meet‘𝐾)𝑊)))) = ((𝐺‘(𝑃 ∨ ((𝑃 ∨ 𝑅)(meet‘𝐾)𝑊)))(meet‘𝐾)(𝐺‘(𝑄 ∨ ((𝑄 ∨ 𝑅)(meet‘𝐾)𝑊)))))
68	59, 65, 67	3eqtr4d 2776	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑅 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄))) ∧ ((𝐹‘𝑃) = (𝐺‘𝑃) ∧ (𝐹‘𝑄) = (𝐺‘𝑄))) → (𝐹‘((𝑃 ∨ ((𝑃 ∨ 𝑅)(meet‘𝐾)𝑊))(meet‘𝐾)(𝑄 ∨ ((𝑄 ∨ 𝑅)(meet‘𝐾)𝑊)))) = (𝐺‘((𝑃 ∨ ((𝑃 ∨ 𝑅)(meet‘𝐾)𝑊))(meet‘𝐾)(𝑄 ∨ ((𝑄 ∨ 𝑅)(meet‘𝐾)𝑊)))))
69		simp21 1207	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑅 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄))) ∧ ((𝐹‘𝑃) = (𝐺‘𝑃) ∧ (𝐹‘𝑄) = (𝐺‘𝑄))) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
70		simp22 1208	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑅 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄))) ∧ ((𝐹‘𝑃) = (𝐺‘𝑃) ∧ (𝐹‘𝑄) = (𝐺‘𝑄))) → (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))
71		simp23l 1295	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑅 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄))) ∧ ((𝐹‘𝑃) = (𝐺‘𝑃) ∧ (𝐹‘𝑄) = (𝐺‘𝑄))) → 𝑃 ≠ 𝑄)
72		simp23r 1296	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑅 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄))) ∧ ((𝐹‘𝑃) = (𝐺‘𝑃) ∧ (𝐹‘𝑄) = (𝐺‘𝑄))) → ¬ 𝑅 ≤ (𝑃 ∨ 𝑄))
73	7, 71, 72	3jca 1128	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑅 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄))) ∧ ((𝐹‘𝑃) = (𝐺‘𝑃) ∧ (𝐹‘𝑄) = (𝐺‘𝑄))) → (𝑅 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄)))
74	20, 9, 17, 10, 14	cdlemd1 40236	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄)))) → 𝑅 = ((𝑃 ∨ ((𝑃 ∨ 𝑅)(meet‘𝐾)𝑊))(meet‘𝐾)(𝑄 ∨ ((𝑄 ∨ 𝑅)(meet‘𝐾)𝑊))))
75	2, 69, 70, 73, 74	syl13anc 1374	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑅 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄))) ∧ ((𝐹‘𝑃) = (𝐺‘𝑃) ∧ (𝐹‘𝑄) = (𝐺‘𝑄))) → 𝑅 = ((𝑃 ∨ ((𝑃 ∨ 𝑅)(meet‘𝐾)𝑊))(meet‘𝐾)(𝑄 ∨ ((𝑄 ∨ 𝑅)(meet‘𝐾)𝑊))))
76	75	fveq2d 6826	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑅 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄))) ∧ ((𝐹‘𝑃) = (𝐺‘𝑃) ∧ (𝐹‘𝑄) = (𝐺‘𝑄))) → (𝐹‘𝑅) = (𝐹‘((𝑃 ∨ ((𝑃 ∨ 𝑅)(meet‘𝐾)𝑊))(meet‘𝐾)(𝑄 ∨ ((𝑄 ∨ 𝑅)(meet‘𝐾)𝑊)))))
77	75	fveq2d 6826	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑅 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄))) ∧ ((𝐹‘𝑃) = (𝐺‘𝑃) ∧ (𝐹‘𝑄) = (𝐺‘𝑄))) → (𝐺‘𝑅) = (𝐺‘((𝑃 ∨ ((𝑃 ∨ 𝑅)(meet‘𝐾)𝑊))(meet‘𝐾)(𝑄 ∨ ((𝑄 ∨ 𝑅)(meet‘𝐾)𝑊)))))
78	68, 76, 77	3eqtr4d 2776	1 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑅 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄))) ∧ ((𝐹‘𝑃) = (𝐺‘𝑃) ∧ (𝐹‘𝑄) = (𝐺‘𝑄))) → (𝐹‘𝑅) = (𝐺‘𝑅))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1541 ∈ wcel 2111 ≠ wne 2928 class class class wbr 5091 ‘cfv 6481 (class class class)co 7346 Basecbs 17117 lecple 17165 joincjn 18214 meetcmee 18215 Latclat 18334 Atomscatm 39301 HLchlt 39388 LHypclh 40022 LTrncltrn 40139

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5217 ax-sep 5234 ax-nul 5244 ax-pow 5303 ax-pr 5370 ax-un 7668

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3742 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-nul 4284 df-if 4476 df-pw 4552 df-sn 4577 df-pr 4579 df-op 4583 df-uni 4860 df-iun 4943 df-iin 4944 df-br 5092 df-opab 5154 df-mpt 5173 df-id 5511 df-xp 5622 df-rel 5623 df-cnv 5624 df-co 5625 df-dm 5626 df-rn 5627 df-res 5628 df-ima 5629 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-riota 7303 df-ov 7349 df-oprab 7350 df-mpo 7351 df-1st 7921 df-2nd 7922 df-map 8752 df-proset 18197 df-poset 18216 df-plt 18231 df-lub 18247 df-glb 18248 df-join 18249 df-meet 18250 df-p0 18326 df-p1 18327 df-lat 18335 df-clat 18402 df-oposet 39214 df-ol 39216 df-oml 39217 df-covers 39304 df-ats 39305 df-atl 39336 df-cvlat 39360 df-hlat 39389 df-psubsp 39541 df-pmap 39542 df-padd 39834 df-lhyp 40026 df-laut 40027 df-ldil 40142 df-ltrn 40143

This theorem is referenced by: cdlemd4 40239 cdlemd5 40240

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