cdlemh2 - Mathbox for Norm Megill

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Theorem cdlemh2 40199

Description: Part of proof of Lemma H of [Crawley] p. 118. (Contributed by NM, 16-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))))
cdlemh.z	⊢ 0 = (0.‘𝐾)

**Assertion**
Ref	Expression
cdlemh2	⊢ ((((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 )

**Proof of Theorem cdlemh2**
Step	Hyp	Ref	Expression
1		cdlemh.s	. . 3 ⊢ 𝑆 = ((𝑃 ∨ (𝑅‘𝐺)) ∧ (𝑄 ∨ (𝑅‘(𝐺 ∘ ^◡𝐹))))
2	1	oveq1i 7414	. 2 ⊢ (𝑆 ∧ 𝑊) = (((𝑃 ∨ (𝑅‘𝐺)) ∧ (𝑄 ∨ (𝑅‘(𝐺 ∘ ^◡𝐹)))) ∧ 𝑊)
3		simp11l 1281	. . . . 5 ⊢ ((((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))) → 𝐾 ∈ HL)
4		hlol 38743	. . . . 5 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OL)
5	3, 4	syl 17	. . . 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))) → 𝐾 ∈ OL)
6	3	hllatd 38746	. . . . 5 ⊢ ((((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))) → 𝐾 ∈ Lat)
7		simp2ll 1237	. . . . . 6 ⊢ ((((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))) → 𝑃 ∈ 𝐴)
8		cdlemh.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝐾)
9		cdlemh.a	. . . . . . 7 ⊢ 𝐴 = (Atoms‘𝐾)
10	8, 9	atbase 38671	. . . . . 6 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ 𝐵)
11	7, 10	syl 17	. . . . 5 ⊢ ((((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))) → 𝑃 ∈ 𝐵)
12		simp11r 1282	. . . . . . 7 ⊢ ((((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))) → 𝑊 ∈ 𝐻)
13	3, 12	jca 511	. . . . . 6 ⊢ ((((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))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
14		simp13 1202	. . . . . 6 ⊢ ((((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))) → 𝐺 ∈ 𝑇)
15		cdlemh.h	. . . . . . 7 ⊢ 𝐻 = (LHyp‘𝐾)
16		cdlemh.t	. . . . . . 7 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
17		cdlemh.r	. . . . . . 7 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
18	8, 15, 16, 17	trlcl 39547	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇) → (𝑅‘𝐺) ∈ 𝐵)
19	13, 14, 18	syl2anc 583	. . . . 5 ⊢ ((((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))) → (𝑅‘𝐺) ∈ 𝐵)
20		cdlemh.j	. . . . . 6 ⊢ ∨ = (join‘𝐾)
21	8, 20	latjcl 18401	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐵 ∧ (𝑅‘𝐺) ∈ 𝐵) → (𝑃 ∨ (𝑅‘𝐺)) ∈ 𝐵)
22	6, 11, 19, 21	syl3anc 1368	. . . 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))) → (𝑃 ∨ (𝑅‘𝐺)) ∈ 𝐵)
23		simp2rl 1239	. . . . . 6 ⊢ ((((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))) → 𝑄 ∈ 𝐴)
24	8, 9	atbase 38671	. . . . . 6 ⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ 𝐵)
25	23, 24	syl 17	. . . . 5 ⊢ ((((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))) → 𝑄 ∈ 𝐵)
26		simp12 1201	. . . . . . . 8 ⊢ ((((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))) → 𝐹 ∈ 𝑇)
27	15, 16	ltrncnv 39529	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → ^◡𝐹 ∈ 𝑇)
28	13, 26, 27	syl2anc 583	. . . . . . 7 ⊢ ((((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))) → ^◡𝐹 ∈ 𝑇)
29	15, 16	ltrnco 40102	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇 ∧ ^◡𝐹 ∈ 𝑇) → (𝐺 ∘ ^◡𝐹) ∈ 𝑇)
30	13, 14, 28, 29	syl3anc 1368	. . . . . 6 ⊢ ((((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))) → (𝐺 ∘ ^◡𝐹) ∈ 𝑇)
31	8, 15, 16, 17	trlcl 39547	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐺 ∘ ^◡𝐹) ∈ 𝑇) → (𝑅‘(𝐺 ∘ ^◡𝐹)) ∈ 𝐵)
32	13, 30, 31	syl2anc 583	. . . . 5 ⊢ ((((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))) → (𝑅‘(𝐺 ∘ ^◡𝐹)) ∈ 𝐵)
33	8, 20	latjcl 18401	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ 𝑄 ∈ 𝐵 ∧ (𝑅‘(𝐺 ∘ ^◡𝐹)) ∈ 𝐵) → (𝑄 ∨ (𝑅‘(𝐺 ∘ ^◡𝐹))) ∈ 𝐵)
34	6, 25, 32, 33	syl3anc 1368	. . . 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))) → (𝑄 ∨ (𝑅‘(𝐺 ∘ ^◡𝐹))) ∈ 𝐵)
35	8, 15	lhpbase 39381	. . . . 5 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ 𝐵)
36	12, 35	syl 17	. . . 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))) → 𝑊 ∈ 𝐵)
37		cdlemh.m	. . . . 5 ⊢ ∧ = (meet‘𝐾)
38	8, 37	latmassOLD 38611	. . . 4 ⊢ ((𝐾 ∈ OL ∧ ((𝑃 ∨ (𝑅‘𝐺)) ∈ 𝐵 ∧ (𝑄 ∨ (𝑅‘(𝐺 ∘ ^◡𝐹))) ∈ 𝐵 ∧ 𝑊 ∈ 𝐵)) → (((𝑃 ∨ (𝑅‘𝐺)) ∧ (𝑄 ∨ (𝑅‘(𝐺 ∘ ^◡𝐹)))) ∧ 𝑊) = ((𝑃 ∨ (𝑅‘𝐺)) ∧ ((𝑄 ∨ (𝑅‘(𝐺 ∘ ^◡𝐹))) ∧ 𝑊)))
39	5, 22, 34, 36, 38	syl13anc 1369	. . 3 ⊢ ((((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))) → (((𝑃 ∨ (𝑅‘𝐺)) ∧ (𝑄 ∨ (𝑅‘(𝐺 ∘ ^◡𝐹)))) ∧ 𝑊) = ((𝑃 ∨ (𝑅‘𝐺)) ∧ ((𝑄 ∨ (𝑅‘(𝐺 ∘ ^◡𝐹))) ∧ 𝑊)))
40		simp2r 1197	. . . . . . 7 ⊢ ((((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))) → (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))
41		cdlemh.l	. . . . . . . 8 ⊢ ≤ = (le‘𝐾)
42		cdlemh.z	. . . . . . . 8 ⊢ 0 = (0.‘𝐾)
43	41, 37, 42, 9, 15	lhpmat 39413	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝑄 ∧ 𝑊) = 0 )
44	13, 40, 43	syl2anc 583	. . . . . 6 ⊢ ((((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))) → (𝑄 ∧ 𝑊) = 0 )
45	44	oveq1d 7419	. . . . 5 ⊢ ((((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))) → ((𝑄 ∧ 𝑊) ∨ (𝑅‘(𝐺 ∘ ^◡𝐹))) = ( 0 ∨ (𝑅‘(𝐺 ∘ ^◡𝐹))))
46	41, 15, 16, 17	trlle 39567	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐺 ∘ ^◡𝐹) ∈ 𝑇) → (𝑅‘(𝐺 ∘ ^◡𝐹)) ≤ 𝑊)
47	13, 30, 46	syl2anc 583	. . . . . 6 ⊢ ((((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))) → (𝑅‘(𝐺 ∘ ^◡𝐹)) ≤ 𝑊)
48	8, 41, 20, 37, 9	atmod4i2 39250	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ (𝑄 ∈ 𝐴 ∧ (𝑅‘(𝐺 ∘ ^◡𝐹)) ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ (𝑅‘(𝐺 ∘ ^◡𝐹)) ≤ 𝑊) → ((𝑄 ∧ 𝑊) ∨ (𝑅‘(𝐺 ∘ ^◡𝐹))) = ((𝑄 ∨ (𝑅‘(𝐺 ∘ ^◡𝐹))) ∧ 𝑊))
49	3, 23, 32, 36, 47, 48	syl131anc 1380	. . . . 5 ⊢ ((((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))) → ((𝑄 ∧ 𝑊) ∨ (𝑅‘(𝐺 ∘ ^◡𝐹))) = ((𝑄 ∨ (𝑅‘(𝐺 ∘ ^◡𝐹))) ∧ 𝑊))
50	8, 20, 42	olj02 38608	. . . . . 6 ⊢ ((𝐾 ∈ OL ∧ (𝑅‘(𝐺 ∘ ^◡𝐹)) ∈ 𝐵) → ( 0 ∨ (𝑅‘(𝐺 ∘ ^◡𝐹))) = (𝑅‘(𝐺 ∘ ^◡𝐹)))
51	5, 32, 50	syl2anc 583	. . . . 5 ⊢ ((((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))) → ( 0 ∨ (𝑅‘(𝐺 ∘ ^◡𝐹))) = (𝑅‘(𝐺 ∘ ^◡𝐹)))
52	45, 49, 51	3eqtr3rd 2775	. . . 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))) → (𝑅‘(𝐺 ∘ ^◡𝐹)) = ((𝑄 ∨ (𝑅‘(𝐺 ∘ ^◡𝐹))) ∧ 𝑊))
53	52	oveq2d 7420	. . 3 ⊢ ((((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))) → ((𝑃 ∨ (𝑅‘𝐺)) ∧ (𝑅‘(𝐺 ∘ ^◡𝐹))) = ((𝑃 ∨ (𝑅‘𝐺)) ∧ ((𝑄 ∨ (𝑅‘(𝐺 ∘ ^◡𝐹))) ∧ 𝑊)))
54		simp2l 1196	. . . 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))) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
55	14, 28	jca 511	. . . . 5 ⊢ ((((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))) → (𝐺 ∈ 𝑇 ∧ ^◡𝐹 ∈ 𝑇))
56		simp33 1208	. . . . . . 7 ⊢ ((((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))) → (𝑅‘𝐹) ≠ (𝑅‘𝐺))
57	56	necomd 2990	. . . . . 6 ⊢ ((((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))) → (𝑅‘𝐺) ≠ (𝑅‘𝐹))
58	15, 16, 17	trlcnv 39548	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (𝑅‘^◡𝐹) = (𝑅‘𝐹))
59	13, 26, 58	syl2anc 583	. . . . . 6 ⊢ ((((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))) → (𝑅‘^◡𝐹) = (𝑅‘𝐹))
60	57, 59	neeqtrrd 3009	. . . . 5 ⊢ ((((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))) → (𝑅‘𝐺) ≠ (𝑅‘^◡𝐹))
61		simp31 1206	. . . . . 6 ⊢ ((((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))) → 𝐹 ≠ ( I ↾ 𝐵))
62	8, 15, 16	ltrncnvnid 39510	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵)) → ^◡𝐹 ≠ ( I ↾ 𝐵))
63	13, 26, 61, 62	syl3anc 1368	. . . . 5 ⊢ ((((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))) → ^◡𝐹 ≠ ( I ↾ 𝐵))
64	8, 15, 16, 17	trlcone 40111	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐺 ∈ 𝑇 ∧ ^◡𝐹 ∈ 𝑇) ∧ ((𝑅‘𝐺) ≠ (𝑅‘^◡𝐹) ∧ ^◡𝐹 ≠ ( I ↾ 𝐵))) → (𝑅‘𝐺) ≠ (𝑅‘(𝐺 ∘ ^◡𝐹)))
65	13, 55, 60, 63, 64	syl112anc 1371	. . . 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))) → (𝑅‘𝐺) ≠ (𝑅‘(𝐺 ∘ ^◡𝐹)))
66		simp32 1207	. . . . 5 ⊢ ((((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))) → 𝐺 ≠ ( I ↾ 𝐵))
67	8, 9, 15, 16, 17	trlnidat 39556	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇 ∧ 𝐺 ≠ ( I ↾ 𝐵)) → (𝑅‘𝐺) ∈ 𝐴)
68	13, 14, 66, 67	syl3anc 1368	. . . 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))) → (𝑅‘𝐺) ∈ 𝐴)
69	41, 15, 16, 17	trlle 39567	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇) → (𝑅‘𝐺) ≤ 𝑊)
70	13, 14, 69	syl2anc 583	. . . 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))) → (𝑅‘𝐺) ≤ 𝑊)
71	9, 15, 16, 17	trlcoat 40106	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐺 ∈ 𝑇 ∧ ^◡𝐹 ∈ 𝑇) ∧ (𝑅‘𝐺) ≠ (𝑅‘^◡𝐹)) → (𝑅‘(𝐺 ∘ ^◡𝐹)) ∈ 𝐴)
72	13, 55, 60, 71	syl3anc 1368	. . . 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))) → (𝑅‘(𝐺 ∘ ^◡𝐹)) ∈ 𝐴)
73	41, 20, 37, 42, 9, 15	lhp2at0 39415	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑅‘𝐺) ≠ (𝑅‘(𝐺 ∘ ^◡𝐹))) ∧ ((𝑅‘𝐺) ∈ 𝐴 ∧ (𝑅‘𝐺) ≤ 𝑊) ∧ ((𝑅‘(𝐺 ∘ ^◡𝐹)) ∈ 𝐴 ∧ (𝑅‘(𝐺 ∘ ^◡𝐹)) ≤ 𝑊)) → ((𝑃 ∨ (𝑅‘𝐺)) ∧ (𝑅‘(𝐺 ∘ ^◡𝐹))) = 0 )
74	13, 54, 65, 68, 70, 72, 47, 73	syl322anc 1395	. . 3 ⊢ ((((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))) → ((𝑃 ∨ (𝑅‘𝐺)) ∧ (𝑅‘(𝐺 ∘ ^◡𝐹))) = 0 )
75	39, 53, 74	3eqtr2rd 2773	. 2 ⊢ ((((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))) → 0 = (((𝑃 ∨ (𝑅‘𝐺)) ∧ (𝑄 ∨ (𝑅‘(𝐺 ∘ ^◡𝐹)))) ∧ 𝑊))
76	2, 75	eqtr4id 2785	1 ⊢ ((((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 )

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ≠ wne 2934 class class class wbr 5141 I cid 5566 ^◡ccnv 5668 ↾ cres 5671 ∘ ccom 5673 ‘cfv 6536 (class class class)co 7404 Basecbs 17150 lecple 17210 joincjn 18273 meetcmee 18274 0.cp0 18385 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 ax-riotaBAD 38335

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 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-undef 8256 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-llines 38881 df-lplanes 38882 df-lvols 38883 df-lines 38884 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: cdlemh 40200

W3C validator