Theorem cdlemd4 40661

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 1286	. . 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 1287	. . 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 1208	. . 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 1209	. . 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 1319	. . 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 40501	. . 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 1384	. 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 1250	. . 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 1251	. . 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 1252	. . 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 1253	. . 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 771	. . . 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 781	. . . 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 511	. . 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 39824	. . . . . 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 480	. . . . 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 2737	. . . . . . 7 ⊢ (Base‘𝐾) = (Base‘𝐾)
22	21, 8	atbase 39749	. . . . . 6 ⊢ (𝑠 ∈ 𝐴 → 𝑠 ∈ (Base‘𝐾))
23	22	ad2antrl 729	. . . . 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 1292	. . . . . . 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 39749	. . . . . . 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 480	. . . . 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 1294	. . . . . . 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 39749	. . . . . . 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 480	. . . . 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 782	. . . . 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 18414	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ (𝑠 ∈ (Base‘𝐾) ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾)) ∧ ¬ 𝑠 ≤ (𝑃 ∨ 𝑄)) → 𝑠 ≠ 𝑃)
34	33	necomd 2988	. . . . 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 1386	. . . 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 1254	. . . . 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 1255	. . . . 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 40660	. . . . 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 1396	. . . 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 511	. . 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 1230	. . 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 480	. . . . 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 511	. . . 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 1195	. . . 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 40659	. . . 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 1398	. . 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 40659	. . 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 1404	. 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 3145	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))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ∃wrex 3062   class class class wbr 5086  ‘cfv 6492  (class class class)co 7360  Basecbs 17170  lecple 17218  joincjn 18268  Latclat 18388  Atomscatm 39723  HLchlt 39810  LHypclh 40444  LTrncltrn 40561

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-iin 4937  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-1st 7935  df-2nd 7936  df-map 8768  df-proset 18251  df-poset 18270  df-plt 18285  df-lub 18301  df-glb 18302  df-join 18303  df-meet 18304  df-p0 18380  df-p1 18381  df-lat 18389  df-clat 18456  df-oposet 39636  df-ol 39638  df-oml 39639  df-covers 39726  df-ats 39727  df-atl 39758  df-cvlat 39782  df-hlat 39811  df-psubsp 39963  df-pmap 39964  df-padd 40256  df-lhyp 40448  df-laut 40449  df-ldil 40564  df-ltrn 40565

This theorem is referenced by:  cdlemd5  40662