cdlemd4 - Mathbox for Norm Megill

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Theorem cdlemd4 39706

Description: Part of proof of Lemma D in [Crawley] p. 113. (Contributed by NM, 30-May-2012.)

**Hypotheses**
Ref	Expression
cdlemd4.l	⊢ ≤ = (le‘𝐾)
cdlemd4.j	⊢ ∨ = (join‘𝐾)
cdlemd4.a	⊢ 𝐴 = (Atoms‘𝐾)
cdlemd4.h	⊢ 𝐻 = (LHyp‘𝐾)
cdlemd4.t	⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)

**Assertion**
Ref	Expression
cdlemd4	⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑅 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≠ 𝑃)) ∧ ((𝐹‘𝑃) = (𝐺‘𝑃) ∧ (𝐹‘𝑄) = (𝐺‘𝑄))) → (𝐹‘𝑅) = (𝐺‘𝑅))

Proof of Theorem cdlemd4 Dummy variable 𝑠 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp11l 1281	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑅 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≠ 𝑃)) ∧ ((𝐹‘𝑃) = (𝐺‘𝑃) ∧ (𝐹‘𝑄) = (𝐺‘𝑄))) → 𝐾 ∈ HL)
2		simp11r 1282	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑅 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≠ 𝑃)) ∧ ((𝐹‘𝑃) = (𝐺‘𝑃) ∧ (𝐹‘𝑄) = (𝐺‘𝑄))) → 𝑊 ∈ 𝐻)
3		simp21 1203	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑅 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≠ 𝑃)) ∧ ((𝐹‘𝑃) = (𝐺‘𝑃) ∧ (𝐹‘𝑄) = (𝐺‘𝑄))) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
4		simp22 1204	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑅 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≠ 𝑃)) ∧ ((𝐹‘𝑃) = (𝐺‘𝑃) ∧ (𝐹‘𝑄) = (𝐺‘𝑄))) → (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))
5		simp231 1314	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑅 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≠ 𝑃)) ∧ ((𝐹‘𝑃) = (𝐺‘𝑃) ∧ (𝐹‘𝑄) = (𝐺‘𝑄))) → 𝑃 ≠ 𝑄)
6		cdlemd4.l	. . . 4 ⊢ ≤ = (le‘𝐾)
7		cdlemd4.j	. . . 4 ⊢ ∨ = (join‘𝐾)
8		cdlemd4.a	. . . 4 ⊢ 𝐴 = (Atoms‘𝐾)
9		cdlemd4.h	. . . 4 ⊢ 𝐻 = (LHyp‘𝐾)
10	6, 7, 8, 9	cdlemb2 39546	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 ≠ 𝑄) → ∃𝑠 ∈ 𝐴 (¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ (𝑃 ∨ 𝑄)))
11	1, 2, 3, 4, 5, 10	syl221anc 1378	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑅 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≠ 𝑃)) ∧ ((𝐹‘𝑃) = (𝐺‘𝑃) ∧ (𝐹‘𝑄) = (𝐺‘𝑄))) → ∃𝑠 ∈ 𝐴 (¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ (𝑃 ∨ 𝑄)))
12		simpl11 1245	. . 3 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑅 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≠ 𝑃)) ∧ ((𝐹‘𝑃) = (𝐺‘𝑃) ∧ (𝐹‘𝑄) = (𝐺‘𝑄))) ∧ (𝑠 ∈ 𝐴 ∧ (¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ (𝑃 ∨ 𝑄)))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
13		simpl12 1246	. . 3 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑅 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≠ 𝑃)) ∧ ((𝐹‘𝑃) = (𝐺‘𝑃) ∧ (𝐹‘𝑄) = (𝐺‘𝑄))) ∧ (𝑠 ∈ 𝐴 ∧ (¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ (𝑃 ∨ 𝑄)))) → (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇))
14		simpl13 1247	. . 3 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑅 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≠ 𝑃)) ∧ ((𝐹‘𝑃) = (𝐺‘𝑃) ∧ (𝐹‘𝑄) = (𝐺‘𝑄))) ∧ (𝑠 ∈ 𝐴 ∧ (¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ (𝑃 ∨ 𝑄)))) → 𝑅 ∈ 𝐴)
15		simpl21 1248	. . 3 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑅 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≠ 𝑃)) ∧ ((𝐹‘𝑃) = (𝐺‘𝑃) ∧ (𝐹‘𝑄) = (𝐺‘𝑄))) ∧ (𝑠 ∈ 𝐴 ∧ (¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ (𝑃 ∨ 𝑄)))) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
16		simprl 769	. . . 4 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑅 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≠ 𝑃)) ∧ ((𝐹‘𝑃) = (𝐺‘𝑃) ∧ (𝐹‘𝑄) = (𝐺‘𝑄))) ∧ (𝑠 ∈ 𝐴 ∧ (¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ (𝑃 ∨ 𝑄)))) → 𝑠 ∈ 𝐴)
17		simprrl 779	. . . 4 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑅 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≠ 𝑃)) ∧ ((𝐹‘𝑃) = (𝐺‘𝑃) ∧ (𝐹‘𝑄) = (𝐺‘𝑄))) ∧ (𝑠 ∈ 𝐴 ∧ (¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ (𝑃 ∨ 𝑄)))) → ¬ 𝑠 ≤ 𝑊)
18	16, 17	jca 510	. . 3 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑅 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≠ 𝑃)) ∧ ((𝐹‘𝑃) = (𝐺‘𝑃) ∧ (𝐹‘𝑄) = (𝐺‘𝑄))) ∧ (𝑠 ∈ 𝐴 ∧ (¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ (𝑃 ∨ 𝑄)))) → (𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊))
19	1	hllatd 38868	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑅 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≠ 𝑃)) ∧ ((𝐹‘𝑃) = (𝐺‘𝑃) ∧ (𝐹‘𝑄) = (𝐺‘𝑄))) → 𝐾 ∈ Lat)
20	19	adantr 479	. . . . 5 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑅 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≠ 𝑃)) ∧ ((𝐹‘𝑃) = (𝐺‘𝑃) ∧ (𝐹‘𝑄) = (𝐺‘𝑄))) ∧ (𝑠 ∈ 𝐴 ∧ (¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ (𝑃 ∨ 𝑄)))) → 𝐾 ∈ Lat)
21		eqid 2728	. . . . . . 7 ⊢ (Base‘𝐾) = (Base‘𝐾)
22	21, 8	atbase 38793	. . . . . 6 ⊢ (𝑠 ∈ 𝐴 → 𝑠 ∈ (Base‘𝐾))
23	22	ad2antrl 726	. . . . 5 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑅 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≠ 𝑃)) ∧ ((𝐹‘𝑃) = (𝐺‘𝑃) ∧ (𝐹‘𝑄) = (𝐺‘𝑄))) ∧ (𝑠 ∈ 𝐴 ∧ (¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ (𝑃 ∨ 𝑄)))) → 𝑠 ∈ (Base‘𝐾))
24		simp21l 1287	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑅 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≠ 𝑃)) ∧ ((𝐹‘𝑃) = (𝐺‘𝑃) ∧ (𝐹‘𝑄) = (𝐺‘𝑄))) → 𝑃 ∈ 𝐴)
25	21, 8	atbase 38793	. . . . . . 7 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ (Base‘𝐾))
26	24, 25	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) ∧ 𝑅 ≠ 𝑃)) ∧ ((𝐹‘𝑃) = (𝐺‘𝑃) ∧ (𝐹‘𝑄) = (𝐺‘𝑄))) → 𝑃 ∈ (Base‘𝐾))
27	26	adantr 479	. . . . 5 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑅 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≠ 𝑃)) ∧ ((𝐹‘𝑃) = (𝐺‘𝑃) ∧ (𝐹‘𝑄) = (𝐺‘𝑄))) ∧ (𝑠 ∈ 𝐴 ∧ (¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ (𝑃 ∨ 𝑄)))) → 𝑃 ∈ (Base‘𝐾))
28		simp22l 1289	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑅 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≠ 𝑃)) ∧ ((𝐹‘𝑃) = (𝐺‘𝑃) ∧ (𝐹‘𝑄) = (𝐺‘𝑄))) → 𝑄 ∈ 𝐴)
29	21, 8	atbase 38793	. . . . . . 7 ⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ (Base‘𝐾))
30	28, 29	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) ∧ 𝑅 ≠ 𝑃)) ∧ ((𝐹‘𝑃) = (𝐺‘𝑃) ∧ (𝐹‘𝑄) = (𝐺‘𝑄))) → 𝑄 ∈ (Base‘𝐾))
31	30	adantr 479	. . . . 5 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑅 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≠ 𝑃)) ∧ ((𝐹‘𝑃) = (𝐺‘𝑃) ∧ (𝐹‘𝑄) = (𝐺‘𝑄))) ∧ (𝑠 ∈ 𝐴 ∧ (¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ (𝑃 ∨ 𝑄)))) → 𝑄 ∈ (Base‘𝐾))
32		simprrr 780	. . . . 5 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑅 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≠ 𝑃)) ∧ ((𝐹‘𝑃) = (𝐺‘𝑃) ∧ (𝐹‘𝑄) = (𝐺‘𝑄))) ∧ (𝑠 ∈ 𝐴 ∧ (¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ (𝑃 ∨ 𝑄)))) → ¬ 𝑠 ≤ (𝑃 ∨ 𝑄))
33	21, 6, 7	latnlej1l 18456	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ (𝑠 ∈ (Base‘𝐾) ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾)) ∧ ¬ 𝑠 ≤ (𝑃 ∨ 𝑄)) → 𝑠 ≠ 𝑃)
34	33	necomd 2993	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ (𝑠 ∈ (Base‘𝐾) ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾)) ∧ ¬ 𝑠 ≤ (𝑃 ∨ 𝑄)) → 𝑃 ≠ 𝑠)
35	20, 23, 27, 31, 32, 34	syl131anc 1380	. . . 4 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑅 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≠ 𝑃)) ∧ ((𝐹‘𝑃) = (𝐺‘𝑃) ∧ (𝐹‘𝑄) = (𝐺‘𝑄))) ∧ (𝑠 ∈ 𝐴 ∧ (¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ (𝑃 ∨ 𝑄)))) → 𝑃 ≠ 𝑠)
36		simpl22 1249	. . . . 5 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑅 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≠ 𝑃)) ∧ ((𝐹‘𝑃) = (𝐺‘𝑃) ∧ (𝐹‘𝑄) = (𝐺‘𝑄))) ∧ (𝑠 ∈ 𝐴 ∧ (¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ (𝑃 ∨ 𝑄)))) → (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))
37		simpl23 1250	. . . . 5 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑅 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≠ 𝑃)) ∧ ((𝐹‘𝑃) = (𝐺‘𝑃) ∧ (𝐹‘𝑄) = (𝐺‘𝑄))) ∧ (𝑠 ∈ 𝐴 ∧ (¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ (𝑃 ∨ 𝑄)))) → (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≠ 𝑃))
38	6, 7, 8, 9	cdlemd3 39705	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≠ 𝑃)) ∧ (𝑅 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ (𝑃 ∨ 𝑄))) → ¬ 𝑅 ≤ (𝑃 ∨ 𝑠))
39	12, 15, 36, 37, 14, 16, 32, 38	syl133anc 1390	. . . 4 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑅 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≠ 𝑃)) ∧ ((𝐹‘𝑃) = (𝐺‘𝑃) ∧ (𝐹‘𝑄) = (𝐺‘𝑄))) ∧ (𝑠 ∈ 𝐴 ∧ (¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ (𝑃 ∨ 𝑄)))) → ¬ 𝑅 ≤ (𝑃 ∨ 𝑠))
40	35, 39	jca 510	. . 3 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑅 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≠ 𝑃)) ∧ ((𝐹‘𝑃) = (𝐺‘𝑃) ∧ (𝐹‘𝑄) = (𝐺‘𝑄))) ∧ (𝑠 ∈ 𝐴 ∧ (¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ (𝑃 ∨ 𝑄)))) → (𝑃 ≠ 𝑠 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑠)))
41		simpl3l 1225	. . 3 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑅 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≠ 𝑃)) ∧ ((𝐹‘𝑃) = (𝐺‘𝑃) ∧ (𝐹‘𝑄) = (𝐺‘𝑄))) ∧ (𝑠 ∈ 𝐴 ∧ (¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ (𝑃 ∨ 𝑄)))) → (𝐹‘𝑃) = (𝐺‘𝑃))
42	5	adantr 479	. . . . 5 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑅 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≠ 𝑃)) ∧ ((𝐹‘𝑃) = (𝐺‘𝑃) ∧ (𝐹‘𝑄) = (𝐺‘𝑄))) ∧ (𝑠 ∈ 𝐴 ∧ (¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ (𝑃 ∨ 𝑄)))) → 𝑃 ≠ 𝑄)
43	42, 32	jca 510	. . . 4 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑅 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≠ 𝑃)) ∧ ((𝐹‘𝑃) = (𝐺‘𝑃) ∧ (𝐹‘𝑄) = (𝐺‘𝑄))) ∧ (𝑠 ∈ 𝐴 ∧ (¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ (𝑃 ∨ 𝑄)))) → (𝑃 ≠ 𝑄 ∧ ¬ 𝑠 ≤ (𝑃 ∨ 𝑄)))
44		simpl3 1190	. . . 4 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑅 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≠ 𝑃)) ∧ ((𝐹‘𝑃) = (𝐺‘𝑃) ∧ (𝐹‘𝑄) = (𝐺‘𝑄))) ∧ (𝑠 ∈ 𝐴 ∧ (¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ (𝑃 ∨ 𝑄)))) → ((𝐹‘𝑃) = (𝐺‘𝑃) ∧ (𝐹‘𝑄) = (𝐺‘𝑄)))
45		cdlemd4.t	. . . . 5 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
46	6, 7, 8, 9, 45	cdlemd2 39704	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑠 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑠 ≤ (𝑃 ∨ 𝑄))) ∧ ((𝐹‘𝑃) = (𝐺‘𝑃) ∧ (𝐹‘𝑄) = (𝐺‘𝑄))) → (𝐹‘𝑠) = (𝐺‘𝑠))
47	12, 13, 16, 15, 36, 43, 44, 46	syl331anc 1392	. . 3 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑅 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≠ 𝑃)) ∧ ((𝐹‘𝑃) = (𝐺‘𝑃) ∧ (𝐹‘𝑄) = (𝐺‘𝑄))) ∧ (𝑠 ∈ 𝐴 ∧ (¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ (𝑃 ∨ 𝑄)))) → (𝐹‘𝑠) = (𝐺‘𝑠))
48	6, 7, 8, 9, 45	cdlemd2 39704	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑅 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊) ∧ (𝑃 ≠ 𝑠 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑠))) ∧ ((𝐹‘𝑃) = (𝐺‘𝑃) ∧ (𝐹‘𝑠) = (𝐺‘𝑠))) → (𝐹‘𝑅) = (𝐺‘𝑅))
49	12, 13, 14, 15, 18, 40, 41, 47, 48	syl332anc 1398	. 2 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑅 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≠ 𝑃)) ∧ ((𝐹‘𝑃) = (𝐺‘𝑃) ∧ (𝐹‘𝑄) = (𝐺‘𝑄))) ∧ (𝑠 ∈ 𝐴 ∧ (¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ (𝑃 ∨ 𝑄)))) → (𝐹‘𝑅) = (𝐺‘𝑅))
50	11, 49	rexlimddv 3158	1 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑅 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≠ 𝑃)) ∧ ((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 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ≠ wne 2937 ∃wrex 3067 class class class wbr 5152 ‘cfv 6553 (class class class)co 7426 Basecbs 17187 lecple 17247 joincjn 18310 Latclat 18430 Atomscatm 38767 HLchlt 38854 LHypclh 39489 LTrncltrn 39606

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 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-iin 5003 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-1st 7999 df-2nd 8000 df-map 8853 df-proset 18294 df-poset 18312 df-plt 18329 df-lub 18345 df-glb 18346 df-join 18347 df-meet 18348 df-p0 18424 df-p1 18425 df-lat 18431 df-clat 18498 df-oposet 38680 df-ol 38682 df-oml 38683 df-covers 38770 df-ats 38771 df-atl 38802 df-cvlat 38826 df-hlat 38855 df-psubsp 39008 df-pmap 39009 df-padd 39301 df-lhyp 39493 df-laut 39494 df-ldil 39609 df-ltrn 39610

This theorem is referenced by: cdlemd5 39707

W3C validator