cdlemk4 - Mathbox for Norm Megill

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Theorem cdlemk4 40795

Description: Part of proof of Lemma K of [Crawley] p. 118, last line. We use 𝑋 for their h, since 𝐻 is already used. (Contributed by NM, 24-Jun-2013.)

**Hypotheses**
Ref	Expression
cdlemk.b	⊢ 𝐵 = (Base‘𝐾)
cdlemk.l	⊢ ≤ = (le‘𝐾)
cdlemk.j	⊢ ∨ = (join‘𝐾)
cdlemk.a	⊢ 𝐴 = (Atoms‘𝐾)
cdlemk.h	⊢ 𝐻 = (LHyp‘𝐾)
cdlemk.t	⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
cdlemk.r	⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
cdlemk.m	⊢ ∧ = (meet‘𝐾)

**Assertion**
Ref	Expression
cdlemk4	⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝐹‘𝑃) ≤ ((𝑋‘𝑃) ∨ (𝑅‘(𝑋 ∘ ^◡𝐹))))

**Proof of Theorem cdlemk4**
Step	Hyp	Ref	Expression
1		simp1l 1197	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝐾 ∈ HL)
2		simp1 1136	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
3		simp2l 1199	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝐹 ∈ 𝑇)
4		simp3l 1201	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝑃 ∈ 𝐴)
5		cdlemk.l	. . . . 5 ⊢ ≤ = (le‘𝐾)
6		cdlemk.a	. . . . 5 ⊢ 𝐴 = (Atoms‘𝐾)
7		cdlemk.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
8		cdlemk.t	. . . . 5 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
9	5, 6, 7, 8	ltrnat 40101	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝑃 ∈ 𝐴) → (𝐹‘𝑃) ∈ 𝐴)
10	2, 3, 4, 9	syl3anc 1372	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝐹‘𝑃) ∈ 𝐴)
11		simp2r 1200	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝑋 ∈ 𝑇)
12	5, 6, 7, 8	ltrnat 40101	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑇 ∧ 𝑃 ∈ 𝐴) → (𝑋‘𝑃) ∈ 𝐴)
13	2, 11, 4, 12	syl3anc 1372	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑋‘𝑃) ∈ 𝐴)
14		cdlemk.j	. . . 4 ⊢ ∨ = (join‘𝐾)
15	5, 14, 6	hlatlej1 39335	. . 3 ⊢ ((𝐾 ∈ HL ∧ (𝐹‘𝑃) ∈ 𝐴 ∧ (𝑋‘𝑃) ∈ 𝐴) → (𝐹‘𝑃) ≤ ((𝐹‘𝑃) ∨ (𝑋‘𝑃)))
16	1, 10, 13, 15	syl3anc 1372	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝐹‘𝑃) ≤ ((𝐹‘𝑃) ∨ (𝑋‘𝑃)))
17	1	hllatd 39324	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝐾 ∈ Lat)
18		cdlemk.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝐾)
19	18, 6	atbase 39249	. . . . . 6 ⊢ ((𝐹‘𝑃) ∈ 𝐴 → (𝐹‘𝑃) ∈ 𝐵)
20	10, 19	syl 17	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝐹‘𝑃) ∈ 𝐵)
21	18, 6	atbase 39249	. . . . . 6 ⊢ ((𝑋‘𝑃) ∈ 𝐴 → (𝑋‘𝑃) ∈ 𝐵)
22	13, 21	syl 17	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑋‘𝑃) ∈ 𝐵)
23	18, 14	latjcl 18453	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ (𝐹‘𝑃) ∈ 𝐵 ∧ (𝑋‘𝑃) ∈ 𝐵) → ((𝐹‘𝑃) ∨ (𝑋‘𝑃)) ∈ 𝐵)
24	17, 20, 22, 23	syl3anc 1372	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ((𝐹‘𝑃) ∨ (𝑋‘𝑃)) ∈ 𝐵)
25		simp1r 1198	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝑊 ∈ 𝐻)
26	18, 7	lhpbase 39959	. . . . 5 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ 𝐵)
27	25, 26	syl 17	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝑊 ∈ 𝐵)
28	5, 14, 6	hlatlej2 39336	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ (𝐹‘𝑃) ∈ 𝐴 ∧ (𝑋‘𝑃) ∈ 𝐴) → (𝑋‘𝑃) ≤ ((𝐹‘𝑃) ∨ (𝑋‘𝑃)))
29	1, 10, 13, 28	syl3anc 1372	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑋‘𝑃) ≤ ((𝐹‘𝑃) ∨ (𝑋‘𝑃)))
30		cdlemk.m	. . . . 5 ⊢ ∧ = (meet‘𝐾)
31	18, 5, 14, 30, 6	atmod3i1 39825	. . . 4 ⊢ ((𝐾 ∈ HL ∧ ((𝑋‘𝑃) ∈ 𝐴 ∧ ((𝐹‘𝑃) ∨ (𝑋‘𝑃)) ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ (𝑋‘𝑃) ≤ ((𝐹‘𝑃) ∨ (𝑋‘𝑃))) → ((𝑋‘𝑃) ∨ (((𝐹‘𝑃) ∨ (𝑋‘𝑃)) ∧ 𝑊)) = (((𝐹‘𝑃) ∨ (𝑋‘𝑃)) ∧ ((𝑋‘𝑃) ∨ 𝑊)))
32	1, 13, 24, 27, 29, 31	syl131anc 1384	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ((𝑋‘𝑃) ∨ (((𝐹‘𝑃) ∨ (𝑋‘𝑃)) ∧ 𝑊)) = (((𝐹‘𝑃) ∨ (𝑋‘𝑃)) ∧ ((𝑋‘𝑃) ∨ 𝑊)))
33	7, 8	ltrncnv 40107	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → ^◡𝐹 ∈ 𝑇)
34	2, 3, 33	syl2anc 584	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ^◡𝐹 ∈ 𝑇)
35	7, 8	ltrnco 40680	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑇 ∧ ^◡𝐹 ∈ 𝑇) → (𝑋 ∘ ^◡𝐹) ∈ 𝑇)
36	2, 11, 34, 35	syl3anc 1372	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑋 ∘ ^◡𝐹) ∈ 𝑇)
37	5, 6, 7, 8	ltrnel 40100	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ((𝐹‘𝑃) ∈ 𝐴 ∧ ¬ (𝐹‘𝑃) ≤ 𝑊))
38	3, 37	syld3an2 1412	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ((𝐹‘𝑃) ∈ 𝐴 ∧ ¬ (𝐹‘𝑃) ≤ 𝑊))
39		cdlemk.r	. . . . . . 7 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
40	5, 14, 30, 6, 7, 8, 39	trlval2 40124	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∘ ^◡𝐹) ∈ 𝑇 ∧ ((𝐹‘𝑃) ∈ 𝐴 ∧ ¬ (𝐹‘𝑃) ≤ 𝑊)) → (𝑅‘(𝑋 ∘ ^◡𝐹)) = (((𝐹‘𝑃) ∨ ((𝑋 ∘ ^◡𝐹)‘(𝐹‘𝑃))) ∧ 𝑊))
41	2, 36, 38, 40	syl3anc 1372	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑅‘(𝑋 ∘ ^◡𝐹)) = (((𝐹‘𝑃) ∨ ((𝑋 ∘ ^◡𝐹)‘(𝐹‘𝑃))) ∧ 𝑊))
42	18, 7, 8	ltrn1o 40085	. . . . . . . . . . . . . . 15 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → 𝐹:𝐵–1-1-onto→𝐵)
43	2, 3, 42	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝐹:𝐵–1-1-onto→𝐵)
44		f1ococnv1 6857	. . . . . . . . . . . . . 14 ⊢ (𝐹:𝐵–1-1-onto→𝐵 → (^◡𝐹 ∘ 𝐹) = ( I ↾ 𝐵))
45	43, 44	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (^◡𝐹 ∘ 𝐹) = ( I ↾ 𝐵))
46	45	coeq2d 5853	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑋 ∘ (^◡𝐹 ∘ 𝐹)) = (𝑋 ∘ ( I ↾ 𝐵)))
47	18, 7, 8	ltrn1o 40085	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑇) → 𝑋:𝐵–1-1-onto→𝐵)
48	2, 11, 47	syl2anc 584	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝑋:𝐵–1-1-onto→𝐵)
49		f1of 6828	. . . . . . . . . . . . 13 ⊢ (𝑋:𝐵–1-1-onto→𝐵 → 𝑋:𝐵⟶𝐵)
50		fcoi1 6762	. . . . . . . . . . . . 13 ⊢ (𝑋:𝐵⟶𝐵 → (𝑋 ∘ ( I ↾ 𝐵)) = 𝑋)
51	48, 49, 50	3syl 18	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑋 ∘ ( I ↾ 𝐵)) = 𝑋)
52	46, 51	eqtr2d 2770	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝑋 = (𝑋 ∘ (^◡𝐹 ∘ 𝐹)))
53		coass 6265	. . . . . . . . . . 11 ⊢ ((𝑋 ∘ ^◡𝐹) ∘ 𝐹) = (𝑋 ∘ (^◡𝐹 ∘ 𝐹))
54	52, 53	eqtr4di 2787	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝑋 = ((𝑋 ∘ ^◡𝐹) ∘ 𝐹))
55	54	fveq1d 6888	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑋‘𝑃) = (((𝑋 ∘ ^◡𝐹) ∘ 𝐹)‘𝑃))
56	5, 6, 7, 8	ltrncoval 40106	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑋 ∘ ^◡𝐹) ∈ 𝑇 ∧ 𝐹 ∈ 𝑇) ∧ 𝑃 ∈ 𝐴) → (((𝑋 ∘ ^◡𝐹) ∘ 𝐹)‘𝑃) = ((𝑋 ∘ ^◡𝐹)‘(𝐹‘𝑃)))
57	2, 36, 3, 4, 56	syl121anc 1376	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (((𝑋 ∘ ^◡𝐹) ∘ 𝐹)‘𝑃) = ((𝑋 ∘ ^◡𝐹)‘(𝐹‘𝑃)))
58	55, 57	eqtrd 2769	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑋‘𝑃) = ((𝑋 ∘ ^◡𝐹)‘(𝐹‘𝑃)))
59	58	oveq2d 7429	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ((𝐹‘𝑃) ∨ (𝑋‘𝑃)) = ((𝐹‘𝑃) ∨ ((𝑋 ∘ ^◡𝐹)‘(𝐹‘𝑃))))
60	59	eqcomd 2740	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ((𝐹‘𝑃) ∨ ((𝑋 ∘ ^◡𝐹)‘(𝐹‘𝑃))) = ((𝐹‘𝑃) ∨ (𝑋‘𝑃)))
61	60	oveq1d 7428	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (((𝐹‘𝑃) ∨ ((𝑋 ∘ ^◡𝐹)‘(𝐹‘𝑃))) ∧ 𝑊) = (((𝐹‘𝑃) ∨ (𝑋‘𝑃)) ∧ 𝑊))
62	41, 61	eqtrd 2769	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑅‘(𝑋 ∘ ^◡𝐹)) = (((𝐹‘𝑃) ∨ (𝑋‘𝑃)) ∧ 𝑊))
63	62	oveq2d 7429	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ((𝑋‘𝑃) ∨ (𝑅‘(𝑋 ∘ ^◡𝐹))) = ((𝑋‘𝑃) ∨ (((𝐹‘𝑃) ∨ (𝑋‘𝑃)) ∧ 𝑊)))
64	5, 6, 7, 8	ltrnel 40100	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ((𝑋‘𝑃) ∈ 𝐴 ∧ ¬ (𝑋‘𝑃) ≤ 𝑊))
65	11, 64	syld3an2 1412	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ((𝑋‘𝑃) ∈ 𝐴 ∧ ¬ (𝑋‘𝑃) ≤ 𝑊))
66		eqid 2734	. . . . . . 7 ⊢ (1.‘𝐾) = (1.‘𝐾)
67	5, 14, 66, 6, 7	lhpjat2 39982	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑋‘𝑃) ∈ 𝐴 ∧ ¬ (𝑋‘𝑃) ≤ 𝑊)) → ((𝑋‘𝑃) ∨ 𝑊) = (1.‘𝐾))
68	2, 65, 67	syl2anc 584	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ((𝑋‘𝑃) ∨ 𝑊) = (1.‘𝐾))
69	68	oveq2d 7429	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (((𝐹‘𝑃) ∨ (𝑋‘𝑃)) ∧ ((𝑋‘𝑃) ∨ 𝑊)) = (((𝐹‘𝑃) ∨ (𝑋‘𝑃)) ∧ (1.‘𝐾)))
70		hlol 39321	. . . . . 6 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OL)
71	1, 70	syl 17	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝐾 ∈ OL)
72	18, 30, 66	olm11 39187	. . . . 5 ⊢ ((𝐾 ∈ OL ∧ ((𝐹‘𝑃) ∨ (𝑋‘𝑃)) ∈ 𝐵) → (((𝐹‘𝑃) ∨ (𝑋‘𝑃)) ∧ (1.‘𝐾)) = ((𝐹‘𝑃) ∨ (𝑋‘𝑃)))
73	71, 24, 72	syl2anc 584	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (((𝐹‘𝑃) ∨ (𝑋‘𝑃)) ∧ (1.‘𝐾)) = ((𝐹‘𝑃) ∨ (𝑋‘𝑃)))
74	69, 73	eqtr2d 2770	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ((𝐹‘𝑃) ∨ (𝑋‘𝑃)) = (((𝐹‘𝑃) ∨ (𝑋‘𝑃)) ∧ ((𝑋‘𝑃) ∨ 𝑊)))
75	32, 63, 74	3eqtr4rd 2780	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ((𝐹‘𝑃) ∨ (𝑋‘𝑃)) = ((𝑋‘𝑃) ∨ (𝑅‘(𝑋 ∘ ^◡𝐹))))
76	16, 75	breqtrd 5149	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝐹‘𝑃) ≤ ((𝑋‘𝑃) ∨ (𝑅‘(𝑋 ∘ ^◡𝐹))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1539 ∈ wcel 2107 class class class wbr 5123 I cid 5557 ^◡ccnv 5664 ↾ cres 5667 ∘ ccom 5669 ⟶wf 6537 –1-1-onto→wf1o 6540 ‘cfv 6541 (class class class)co 7413 Basecbs 17229 lecple 17280 joincjn 18327 meetcmee 18328 1.cp1 18438 Latclat 18445 OLcol 39134 Atomscatm 39223 HLchlt 39310 LHypclh 39945 LTrncltrn 40062 trLctrl 40119

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2706 ax-rep 5259 ax-sep 5276 ax-nul 5286 ax-pow 5345 ax-pr 5412 ax-un 7737 ax-riotaBAD 38913

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2808 df-nfc 2884 df-ne 2932 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3420 df-v 3465 df-sbc 3771 df-csb 3880 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-op 4613 df-uni 4888 df-iun 4973 df-iin 4974 df-br 5124 df-opab 5186 df-mpt 5206 df-id 5558 df-xp 5671 df-rel 5672 df-cnv 5673 df-co 5674 df-dm 5675 df-rn 5676 df-res 5677 df-ima 5678 df-iota 6494 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-riota 7370 df-ov 7416 df-oprab 7417 df-mpo 7418 df-1st 7996 df-2nd 7997 df-undef 8280 df-map 8850 df-proset 18310 df-poset 18329 df-plt 18344 df-lub 18360 df-glb 18361 df-join 18362 df-meet 18363 df-p0 18439 df-p1 18440 df-lat 18446 df-clat 18513 df-oposet 39136 df-ol 39138 df-oml 39139 df-covers 39226 df-ats 39227 df-atl 39258 df-cvlat 39282 df-hlat 39311 df-llines 39459 df-lplanes 39460 df-lvols 39461 df-lines 39462 df-psubsp 39464 df-pmap 39465 df-padd 39757 df-lhyp 39949 df-laut 39950 df-ldil 40065 df-ltrn 40066 df-trl 40120

This theorem is referenced by: cdlemk5a 40796

W3C validator