cdlemh - Mathbox for Norm Megill

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Theorem cdlemh 42066

Description: Lemma H of [Crawley] p. 118. (Contributed by NM, 17-Jun-2013.)

**Hypotheses**
Ref	Expression
cdlemh.b	⊢ 𝐵 = (Base‘𝐾)
cdlemh.l	⊢ ≤ = (le‘𝐾)
cdlemh.j	⊢ ∨ = (join‘𝐾)
cdlemh.m	⊢ ∧ = (meet‘𝐾)
cdlemh.a	⊢ 𝐴 = (Atoms‘𝐾)
cdlemh.h	⊢ 𝐻 = (LHyp‘𝐾)
cdlemh.t	⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
cdlemh.r	⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
cdlemh.s	⊢ 𝑆 = ((𝑃 ∨ (𝑅‘𝐺)) ∧ (𝑄 ∨ (𝑅‘(𝐺 ∘ ^◡𝐹))))

**Assertion**
Ref	Expression
cdlemh	⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑄 ≤ (𝑃 ∨ (𝑅‘𝐹))) ∧ (𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐺 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺))) → (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊))

**Proof of Theorem cdlemh**
Step	Hyp	Ref	Expression
1		simp1 1154	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑄 ≤ (𝑃 ∨ (𝑅‘𝐹))) ∧ (𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐺 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺))) → ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇))
2		simp21l 1309	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑄 ≤ (𝑃 ∨ (𝑅‘𝐹))) ∧ (𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐺 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺))) → 𝑃 ∈ 𝐴)
3		simp22l 1311	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑄 ≤ (𝑃 ∨ (𝑅‘𝐹))) ∧ (𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐺 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺))) → 𝑄 ∈ 𝐴)
4		simp23 1227	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑄 ≤ (𝑃 ∨ (𝑅‘𝐹))) ∧ (𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐺 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺))) → 𝑄 ≤ (𝑃 ∨ (𝑅‘𝐹)))
5		simp33 1230	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑄 ≤ (𝑃 ∨ (𝑅‘𝐹))) ∧ (𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐺 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺))) → (𝑅‘𝐹) ≠ (𝑅‘𝐺))
6		cdlemh.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐾)
7		cdlemh.l	. . . . . 6 ⊢ ≤ = (le‘𝐾)
8		cdlemh.j	. . . . . 6 ⊢ ∨ = (join‘𝐾)
9		cdlemh.m	. . . . . 6 ⊢ ∧ = (meet‘𝐾)
10		cdlemh.a	. . . . . 6 ⊢ 𝐴 = (Atoms‘𝐾)
11		cdlemh.h	. . . . . 6 ⊢ 𝐻 = (LHyp‘𝐾)
12		cdlemh.t	. . . . . 6 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
13		cdlemh.r	. . . . . 6 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
14		cdlemh.s	. . . . . 6 ⊢ 𝑆 = ((𝑃 ∨ (𝑅‘𝐺)) ∧ (𝑄 ∨ (𝑅‘(𝐺 ∘ ^◡𝐹))))
15	6, 7, 8, 9, 10, 11, 12, 13, 14	cdlemh1 42064	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑄 ≤ (𝑃 ∨ (𝑅‘𝐹)) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺))) → (𝑆 ∨ (𝑅‘(𝐺 ∘ ^◡𝐹))) = (𝑄 ∨ (𝑅‘(𝐺 ∘ ^◡𝐹))))
16	1, 2, 3, 4, 5, 15	syl122anc 1407	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑄 ≤ (𝑃 ∨ (𝑅‘𝐹))) ∧ (𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐺 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺))) → (𝑆 ∨ (𝑅‘(𝐺 ∘ ^◡𝐹))) = (𝑄 ∨ (𝑅‘(𝐺 ∘ ^◡𝐹))))
17		oveq1 7450	. . . . . . . 8 ⊢ (𝑆 = (0.‘𝐾) → (𝑆 ∨ (𝑅‘(𝐺 ∘ ^◡𝐹))) = ((0.‘𝐾) ∨ (𝑅‘(𝐺 ∘ ^◡𝐹))))
18		simp11l 1303	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑄 ≤ (𝑃 ∨ (𝑅‘𝐹))) ∧ (𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐺 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺))) → 𝐾 ∈ HL)
19		hlol 40610	. . . . . . . . . 10 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OL)
20	18, 19	syl 18	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑄 ≤ (𝑃 ∨ (𝑅‘𝐹))) ∧ (𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐺 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺))) → 𝐾 ∈ OL)
21		simp11r 1304	. . . . . . . . . . . 12 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑄 ≤ (𝑃 ∨ (𝑅‘𝐹))) ∧ (𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐺 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺))) → 𝑊 ∈ 𝐻)
22	18, 21	jca 521	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑄 ≤ (𝑃 ∨ (𝑅‘𝐹))) ∧ (𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐺 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
23		simp13 1224	. . . . . . . . . . . 12 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑄 ≤ (𝑃 ∨ (𝑅‘𝐹))) ∧ (𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐺 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺))) → 𝐺 ∈ 𝑇)
24		simp12 1223	. . . . . . . . . . . . 13 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑄 ≤ (𝑃 ∨ (𝑅‘𝐹))) ∧ (𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐺 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺))) → 𝐹 ∈ 𝑇)
25	11, 12	ltrncnv 41395	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → ^◡𝐹 ∈ 𝑇)
26	22, 24, 25	syl2anc 596	. . . . . . . . . . . 12 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑄 ≤ (𝑃 ∨ (𝑅‘𝐹))) ∧ (𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐺 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺))) → ^◡𝐹 ∈ 𝑇)
27	23, 26	jca 521	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑄 ≤ (𝑃 ∨ (𝑅‘𝐹))) ∧ (𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐺 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺))) → (𝐺 ∈ 𝑇 ∧ ^◡𝐹 ∈ 𝑇))
28	5	necomd 3014	. . . . . . . . . . . 12 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑄 ≤ (𝑃 ∨ (𝑅‘𝐹))) ∧ (𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐺 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺))) → (𝑅‘𝐺) ≠ (𝑅‘𝐹))
29	11, 12, 13	trlcnv 41414	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (𝑅‘^◡𝐹) = (𝑅‘𝐹))
30	22, 24, 29	syl2anc 596	. . . . . . . . . . . 12 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑄 ≤ (𝑃 ∨ (𝑅‘𝐹))) ∧ (𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐺 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺))) → (𝑅‘^◡𝐹) = (𝑅‘𝐹))
31	28, 30	neeqtrrd 3033	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑄 ≤ (𝑃 ∨ (𝑅‘𝐹))) ∧ (𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐺 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺))) → (𝑅‘𝐺) ≠ (𝑅‘^◡𝐹))
32	10, 11, 12, 13	trlcoat 41972	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐺 ∈ 𝑇 ∧ ^◡𝐹 ∈ 𝑇) ∧ (𝑅‘𝐺) ≠ (𝑅‘^◡𝐹)) → (𝑅‘(𝐺 ∘ ^◡𝐹)) ∈ 𝐴)
33	22, 27, 31, 32	syl3anc 1399	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑄 ≤ (𝑃 ∨ (𝑅‘𝐹))) ∧ (𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐺 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺))) → (𝑅‘(𝐺 ∘ ^◡𝐹)) ∈ 𝐴)
34	6, 10	atbase 40538	. . . . . . . . . 10 ⊢ ((𝑅‘(𝐺 ∘ ^◡𝐹)) ∈ 𝐴 → (𝑅‘(𝐺 ∘ ^◡𝐹)) ∈ 𝐵)
35	33, 34	syl 18	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑄 ≤ (𝑃 ∨ (𝑅‘𝐹))) ∧ (𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐺 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺))) → (𝑅‘(𝐺 ∘ ^◡𝐹)) ∈ 𝐵)
36		eqid 2763	. . . . . . . . . 10 ⊢ (0.‘𝐾) = (0.‘𝐾)
37	6, 8, 36	olj02 40475	. . . . . . . . 9 ⊢ ((𝐾 ∈ OL ∧ (𝑅‘(𝐺 ∘ ^◡𝐹)) ∈ 𝐵) → ((0.‘𝐾) ∨ (𝑅‘(𝐺 ∘ ^◡𝐹))) = (𝑅‘(𝐺 ∘ ^◡𝐹)))
38	20, 35, 37	syl2anc 596	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑄 ≤ (𝑃 ∨ (𝑅‘𝐹))) ∧ (𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐺 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺))) → ((0.‘𝐾) ∨ (𝑅‘(𝐺 ∘ ^◡𝐹))) = (𝑅‘(𝐺 ∘ ^◡𝐹)))
39	17, 38	sylan9eqr 2820	. . . . . . 7 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑄 ≤ (𝑃 ∨ (𝑅‘𝐹))) ∧ (𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐺 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺))) ∧ 𝑆 = (0.‘𝐾)) → (𝑆 ∨ (𝑅‘(𝐺 ∘ ^◡𝐹))) = (𝑅‘(𝐺 ∘ ^◡𝐹)))
40	11, 12	ltrnco 41968	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇 ∧ ^◡𝐹 ∈ 𝑇) → (𝐺 ∘ ^◡𝐹) ∈ 𝑇)
41	22, 23, 26, 40	syl3anc 1399	. . . . . . . . . . . . 13 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑄 ≤ (𝑃 ∨ (𝑅‘𝐹))) ∧ (𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐺 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺))) → (𝐺 ∘ ^◡𝐹) ∈ 𝑇)
42	7, 11, 12, 13	trlle 41433	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐺 ∘ ^◡𝐹) ∈ 𝑇) → (𝑅‘(𝐺 ∘ ^◡𝐹)) ≤ 𝑊)
43	22, 41, 42	syl2anc 596	. . . . . . . . . . . 12 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑄 ≤ (𝑃 ∨ (𝑅‘𝐹))) ∧ (𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐺 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺))) → (𝑅‘(𝐺 ∘ ^◡𝐹)) ≤ 𝑊)
44		simp22r 1312	. . . . . . . . . . . 12 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑄 ≤ (𝑃 ∨ (𝑅‘𝐹))) ∧ (𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐺 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺))) → ¬ 𝑄 ≤ 𝑊)
45		nbrne2 5128	. . . . . . . . . . . . 13 ⊢ (((𝑅‘(𝐺 ∘ ^◡𝐹)) ≤ 𝑊 ∧ ¬ 𝑄 ≤ 𝑊) → (𝑅‘(𝐺 ∘ ^◡𝐹)) ≠ 𝑄)
46	45	necomd 3014	. . . . . . . . . . . 12 ⊢ (((𝑅‘(𝐺 ∘ ^◡𝐹)) ≤ 𝑊 ∧ ¬ 𝑄 ≤ 𝑊) → 𝑄 ≠ (𝑅‘(𝐺 ∘ ^◡𝐹)))
47	43, 44, 46	syl2anc 596	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑄 ≤ (𝑃 ∨ (𝑅‘𝐹))) ∧ (𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐺 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺))) → 𝑄 ≠ (𝑅‘(𝐺 ∘ ^◡𝐹)))
48		eqid 2763	. . . . . . . . . . . 12 ⊢ (LLines‘𝐾) = (LLines‘𝐾)
49	8, 10, 48	llni2 40761	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ (𝑅‘(𝐺 ∘ ^◡𝐹)) ∈ 𝐴) ∧ 𝑄 ≠ (𝑅‘(𝐺 ∘ ^◡𝐹))) → (𝑄 ∨ (𝑅‘(𝐺 ∘ ^◡𝐹))) ∈ (LLines‘𝐾))
50	18, 3, 33, 47, 49	syl31anc 1401	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑄 ≤ (𝑃 ∨ (𝑅‘𝐹))) ∧ (𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐺 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺))) → (𝑄 ∨ (𝑅‘(𝐺 ∘ ^◡𝐹))) ∈ (LLines‘𝐾))
51	10, 48	llnneat 40763	. . . . . . . . . 10 ⊢ ((𝐾 ∈ HL ∧ (𝑄 ∨ (𝑅‘(𝐺 ∘ ^◡𝐹))) ∈ (LLines‘𝐾)) → ¬ (𝑄 ∨ (𝑅‘(𝐺 ∘ ^◡𝐹))) ∈ 𝐴)
52	18, 50, 51	syl2anc 596	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑄 ≤ (𝑃 ∨ (𝑅‘𝐹))) ∧ (𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐺 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺))) → ¬ (𝑄 ∨ (𝑅‘(𝐺 ∘ ^◡𝐹))) ∈ 𝐴)
53		nelne2 3057	. . . . . . . . 9 ⊢ (((𝑅‘(𝐺 ∘ ^◡𝐹)) ∈ 𝐴 ∧ ¬ (𝑄 ∨ (𝑅‘(𝐺 ∘ ^◡𝐹))) ∈ 𝐴) → (𝑅‘(𝐺 ∘ ^◡𝐹)) ≠ (𝑄 ∨ (𝑅‘(𝐺 ∘ ^◡𝐹))))
54	33, 52, 53	syl2anc 596	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑄 ≤ (𝑃 ∨ (𝑅‘𝐹))) ∧ (𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐺 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺))) → (𝑅‘(𝐺 ∘ ^◡𝐹)) ≠ (𝑄 ∨ (𝑅‘(𝐺 ∘ ^◡𝐹))))
55	54	adantr 486	. . . . . . 7 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑄 ≤ (𝑃 ∨ (𝑅‘𝐹))) ∧ (𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐺 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺))) ∧ 𝑆 = (0.‘𝐾)) → (𝑅‘(𝐺 ∘ ^◡𝐹)) ≠ (𝑄 ∨ (𝑅‘(𝐺 ∘ ^◡𝐹))))
56	39, 55	eqnetrd 3026	. . . . . 6 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑄 ≤ (𝑃 ∨ (𝑅‘𝐹))) ∧ (𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐺 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺))) ∧ 𝑆 = (0.‘𝐾)) → (𝑆 ∨ (𝑅‘(𝐺 ∘ ^◡𝐹))) ≠ (𝑄 ∨ (𝑅‘(𝐺 ∘ ^◡𝐹))))
57	56	ex 418	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑄 ≤ (𝑃 ∨ (𝑅‘𝐹))) ∧ (𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐺 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺))) → (𝑆 = (0.‘𝐾) → (𝑆 ∨ (𝑅‘(𝐺 ∘ ^◡𝐹))) ≠ (𝑄 ∨ (𝑅‘(𝐺 ∘ ^◡𝐹)))))
58	57	necon2d 2982	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑄 ≤ (𝑃 ∨ (𝑅‘𝐹))) ∧ (𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐺 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺))) → ((𝑆 ∨ (𝑅‘(𝐺 ∘ ^◡𝐹))) = (𝑄 ∨ (𝑅‘(𝐺 ∘ ^◡𝐹))) → 𝑆 ≠ (0.‘𝐾)))
59	16, 58	mpd 16	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑄 ≤ (𝑃 ∨ (𝑅‘𝐹))) ∧ (𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐺 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺))) → 𝑆 ≠ (0.‘𝐾))
60		simp32 1229	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑄 ≤ (𝑃 ∨ (𝑅‘𝐹))) ∧ (𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐺 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺))) → 𝐺 ≠ ( I ↾ 𝐵))
61	6, 10, 11, 12, 13	trlnidat 41422	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇 ∧ 𝐺 ≠ ( I ↾ 𝐵)) → (𝑅‘𝐺) ∈ 𝐴)
62	22, 23, 60, 61	syl3anc 1399	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑄 ≤ (𝑃 ∨ (𝑅‘𝐹))) ∧ (𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐺 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺))) → (𝑅‘𝐺) ∈ 𝐴)
63	7, 8, 10	hlatlej2 40625	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ (𝑅‘𝐺) ∈ 𝐴) → (𝑅‘𝐺) ≤ (𝑃 ∨ (𝑅‘𝐺)))
64	18, 2, 62, 63	syl3anc 1399	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑄 ≤ (𝑃 ∨ (𝑅‘𝐹))) ∧ (𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐺 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺))) → (𝑅‘𝐺) ≤ (𝑃 ∨ (𝑅‘𝐺)))
65		simp22 1226	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑄 ≤ (𝑃 ∨ (𝑅‘𝐹))) ∧ (𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐺 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺))) → (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))
66		simp31 1228	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑄 ≤ (𝑃 ∨ (𝑅‘𝐹))) ∧ (𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐺 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺))) → 𝐹 ≠ ( I ↾ 𝐵))
67	6, 11, 12	ltrncnvnid 41376	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵)) → ^◡𝐹 ≠ ( I ↾ 𝐵))
68	22, 24, 66, 67	syl3anc 1399	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑄 ≤ (𝑃 ∨ (𝑅‘𝐹))) ∧ (𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐺 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺))) → ^◡𝐹 ≠ ( I ↾ 𝐵))
69	6, 11, 12, 13	trlcone 41977	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐺 ∈ 𝑇 ∧ ^◡𝐹 ∈ 𝑇) ∧ ((𝑅‘𝐺) ≠ (𝑅‘^◡𝐹) ∧ ^◡𝐹 ≠ ( I ↾ 𝐵))) → (𝑅‘𝐺) ≠ (𝑅‘(𝐺 ∘ ^◡𝐹)))
70	69	necomd 3014	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐺 ∈ 𝑇 ∧ ^◡𝐹 ∈ 𝑇) ∧ ((𝑅‘𝐺) ≠ (𝑅‘^◡𝐹) ∧ ^◡𝐹 ≠ ( I ↾ 𝐵))) → (𝑅‘(𝐺 ∘ ^◡𝐹)) ≠ (𝑅‘𝐺))
71	22, 23, 26, 31, 68, 70	syl122anc 1407	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑄 ≤ (𝑃 ∨ (𝑅‘𝐹))) ∧ (𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐺 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺))) → (𝑅‘(𝐺 ∘ ^◡𝐹)) ≠ (𝑅‘𝐺))
72	7, 11, 12, 13	trlle 41433	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇) → (𝑅‘𝐺) ≤ 𝑊)
73	22, 23, 72	syl2anc 596	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑄 ≤ (𝑃 ∨ (𝑅‘𝐹))) ∧ (𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐺 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺))) → (𝑅‘𝐺) ≤ 𝑊)
74	7, 8, 10, 11	lhp2atnle 41282	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅‘(𝐺 ∘ ^◡𝐹)) ≠ (𝑅‘𝐺)) ∧ ((𝑅‘(𝐺 ∘ ^◡𝐹)) ∈ 𝐴 ∧ (𝑅‘(𝐺 ∘ ^◡𝐹)) ≤ 𝑊) ∧ ((𝑅‘𝐺) ∈ 𝐴 ∧ (𝑅‘𝐺) ≤ 𝑊)) → ¬ (𝑅‘𝐺) ≤ (𝑄 ∨ (𝑅‘(𝐺 ∘ ^◡𝐹))))
75	22, 65, 71, 33, 43, 62, 73, 74	syl322anc 1426	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑄 ≤ (𝑃 ∨ (𝑅‘𝐹))) ∧ (𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐺 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺))) → ¬ (𝑅‘𝐺) ≤ (𝑄 ∨ (𝑅‘(𝐺 ∘ ^◡𝐹))))
76		nbrne1 5127	. . . . . . . 8 ⊢ (((𝑅‘𝐺) ≤ (𝑃 ∨ (𝑅‘𝐺)) ∧ ¬ (𝑅‘𝐺) ≤ (𝑄 ∨ (𝑅‘(𝐺 ∘ ^◡𝐹)))) → (𝑃 ∨ (𝑅‘𝐺)) ≠ (𝑄 ∨ (𝑅‘(𝐺 ∘ ^◡𝐹))))
77	64, 75, 76	syl2anc 596	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑄 ≤ (𝑃 ∨ (𝑅‘𝐹))) ∧ (𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐺 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺))) → (𝑃 ∨ (𝑅‘𝐺)) ≠ (𝑄 ∨ (𝑅‘(𝐺 ∘ ^◡𝐹))))
78	8, 9, 36, 10	2atmat0 40775	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ (𝑅‘𝐺) ∈ 𝐴) ∧ (𝑄 ∈ 𝐴 ∧ (𝑅‘(𝐺 ∘ ^◡𝐹)) ∈ 𝐴 ∧ (𝑃 ∨ (𝑅‘𝐺)) ≠ (𝑄 ∨ (𝑅‘(𝐺 ∘ ^◡𝐹))))) → (((𝑃 ∨ (𝑅‘𝐺)) ∧ (𝑄 ∨ (𝑅‘(𝐺 ∘ ^◡𝐹)))) ∈ 𝐴 ∨ ((𝑃 ∨ (𝑅‘𝐺)) ∧ (𝑄 ∨ (𝑅‘(𝐺 ∘ ^◡𝐹)))) = (0.‘𝐾)))
79	18, 2, 62, 3, 33, 77, 78	syl33anc 1413	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑄 ≤ (𝑃 ∨ (𝑅‘𝐹))) ∧ (𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐺 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺))) → (((𝑃 ∨ (𝑅‘𝐺)) ∧ (𝑄 ∨ (𝑅‘(𝐺 ∘ ^◡𝐹)))) ∈ 𝐴 ∨ ((𝑃 ∨ (𝑅‘𝐺)) ∧ (𝑄 ∨ (𝑅‘(𝐺 ∘ ^◡𝐹)))) = (0.‘𝐾)))
80	14	eleq1i 2855	. . . . . . 7 ⊢ (𝑆 ∈ 𝐴 ↔ ((𝑃 ∨ (𝑅‘𝐺)) ∧ (𝑄 ∨ (𝑅‘(𝐺 ∘ ^◡𝐹)))) ∈ 𝐴)
81	14	eqeq1i 2768	. . . . . . 7 ⊢ (𝑆 = (0.‘𝐾) ↔ ((𝑃 ∨ (𝑅‘𝐺)) ∧ (𝑄 ∨ (𝑅‘(𝐺 ∘ ^◡𝐹)))) = (0.‘𝐾))
82	80, 81	orbi12i 928	. . . . . 6 ⊢ ((𝑆 ∈ 𝐴 ∨ 𝑆 = (0.‘𝐾)) ↔ (((𝑃 ∨ (𝑅‘𝐺)) ∧ (𝑄 ∨ (𝑅‘(𝐺 ∘ ^◡𝐹)))) ∈ 𝐴 ∨ ((𝑃 ∨ (𝑅‘𝐺)) ∧ (𝑄 ∨ (𝑅‘(𝐺 ∘ ^◡𝐹)))) = (0.‘𝐾)))
83	79, 82	sylibr 237	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑄 ≤ (𝑃 ∨ (𝑅‘𝐹))) ∧ (𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐺 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺))) → (𝑆 ∈ 𝐴 ∨ 𝑆 = (0.‘𝐾)))
84	83	ord 878	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑄 ≤ (𝑃 ∨ (𝑅‘𝐹))) ∧ (𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐺 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺))) → (¬ 𝑆 ∈ 𝐴 → 𝑆 = (0.‘𝐾)))
85	84	necon1ad 2976	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑄 ≤ (𝑃 ∨ (𝑅‘𝐹))) ∧ (𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐺 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺))) → (𝑆 ≠ (0.‘𝐾) → 𝑆 ∈ 𝐴))
86	59, 85	mpd 16	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑄 ≤ (𝑃 ∨ (𝑅‘𝐹))) ∧ (𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐺 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺))) → 𝑆 ∈ 𝐴)
87		simp21 1225	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑄 ≤ (𝑃 ∨ (𝑅‘𝐹))) ∧ (𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐺 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺))) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
88	87, 65	jca 521	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑄 ≤ (𝑃 ∨ (𝑅‘𝐹))) ∧ (𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐺 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺))) → ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)))
89	6, 7, 8, 9, 10, 11, 12, 13, 14, 36	cdlemh2 42065	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐺 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺))) → (𝑆 ∧ 𝑊) = (0.‘𝐾))
90	88, 89	syld3an2 1439	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑄 ≤ (𝑃 ∨ (𝑅‘𝐹))) ∧ (𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐺 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺))) → (𝑆 ∧ 𝑊) = (0.‘𝐾))
91	7, 9, 36, 10, 11	lhpmatb 41280	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐴) → (¬ 𝑆 ≤ 𝑊 ↔ (𝑆 ∧ 𝑊) = (0.‘𝐾)))
92	18, 21, 86, 91	syl21anc 851	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑄 ≤ (𝑃 ∨ (𝑅‘𝐹))) ∧ (𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐺 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺))) → (¬ 𝑆 ≤ 𝑊 ↔ (𝑆 ∧ 𝑊) = (0.‘𝐾)))
93	90, 92	mpbird 260	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑄 ≤ (𝑃 ∨ (𝑅‘𝐹))) ∧ (𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐺 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺))) → ¬ 𝑆 ≤ 𝑊)
94	86, 93	jca 521	1 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑄 ≤ (𝑃 ∨ (𝑅‘𝐹))) ∧ (𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐺 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺))) → (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 ∨ wo 861 ∧ w3a 1103 = wceq 1570 ∈ wcel 2147 ≠ wne 2959 class class class wbr 5106 I cid 5549 ^◡ccnv 5654 ↾ cres 5657 ∘ ccom 5659 ‘cfv 6558 (class class class)co 7443 Basecbs 17416 lecple 17464 joincjn 18514 meetcmee 18515 0.cp0 18624 OLcol 40423 Atomscatm 40512 HLchlt 40599 LLinesclln 40740 LHypclh 41233 LTrncltrn 41350 trLctrl 41407

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5235 ax-sep 5252 ax-nul 5263 ax-pow 5331 ax-pr 5395 ax-un 7775 ax-riotaBAD 40202

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-op 4839 df-uni 4871 df-iun 4956 df-iin 4957 df-br 5107 df-opab 5171 df-mpt 5190 df-id 5550 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-iota 6512 df-fun 6560 df-fn 6561 df-f 6562 df-f1 6563 df-fo 6564 df-f1o 6565 df-fv 6566 df-riota 7400 df-ov 7446 df-oprab 7447 df-mpo 7448 df-1st 8025 df-2nd 8026 df-undef 8318 df-map 8877 df-proset 18497 df-poset 18516 df-plt 18531 df-lub 18547 df-glb 18548 df-join 18549 df-meet 18550 df-p0 18626 df-p1 18627 df-lat 18635 df-clat 18702 df-oposet 40425 df-ol 40427 df-oml 40428 df-covers 40515 df-ats 40516 df-atl 40547 df-cvlat 40571 df-hlat 40600 df-llines 40747 df-lplanes 40748 df-lvols 40749 df-lines 40750 df-psubsp 40752 df-pmap 40753 df-padd 41045 df-lhyp 41237 df-laut 41238 df-ldil 41353 df-ltrn 41354 df-trl 41408

This theorem is used by: cdlemi 42069 cdlemki 42090 cdlemksv2 42096 cdlemk16a 42105

W3C validator