cdlemk4 - Mathbox for Norm Megill

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Theorem cdlemk4 40828

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 1198	. . 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 1200	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝐹 ∈ 𝑇)
4		simp3l 1202	. . . 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 40134	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝑃 ∈ 𝐴) → (𝐹‘𝑃) ∈ 𝐴)
10	2, 3, 4, 9	syl3anc 1373	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝐹‘𝑃) ∈ 𝐴)
11		simp2r 1201	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝑋 ∈ 𝑇)
12	5, 6, 7, 8	ltrnat 40134	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑇 ∧ 𝑃 ∈ 𝐴) → (𝑋‘𝑃) ∈ 𝐴)
13	2, 11, 4, 12	syl3anc 1373	. . 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 39368	. . 3 ⊢ ((𝐾 ∈ HL ∧ (𝐹‘𝑃) ∈ 𝐴 ∧ (𝑋‘𝑃) ∈ 𝐴) → (𝐹‘𝑃) ≤ ((𝐹‘𝑃) ∨ (𝑋‘𝑃)))
16	1, 10, 13, 15	syl3anc 1373	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝐹‘𝑃) ≤ ((𝐹‘𝑃) ∨ (𝑋‘𝑃)))
17	1	hllatd 39357	. . . . 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 39282	. . . . . 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 39282	. . . . . 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 18398	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ (𝐹‘𝑃) ∈ 𝐵 ∧ (𝑋‘𝑃) ∈ 𝐵) → ((𝐹‘𝑃) ∨ (𝑋‘𝑃)) ∈ 𝐵)
24	17, 20, 22, 23	syl3anc 1373	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ((𝐹‘𝑃) ∨ (𝑋‘𝑃)) ∈ 𝐵)
25		simp1r 1199	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝑊 ∈ 𝐻)
26	18, 7	lhpbase 39992	. . . . 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 39369	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ (𝐹‘𝑃) ∈ 𝐴 ∧ (𝑋‘𝑃) ∈ 𝐴) → (𝑋‘𝑃) ≤ ((𝐹‘𝑃) ∨ (𝑋‘𝑃)))
29	1, 10, 13, 28	syl3anc 1373	. . . 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 39858	. . . 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 1385	. . 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 40140	. . . . . . . 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 40713	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑇 ∧ ^◡𝐹 ∈ 𝑇) → (𝑋 ∘ ^◡𝐹) ∈ 𝑇)
36	2, 11, 34, 35	syl3anc 1373	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑋 ∘ ^◡𝐹) ∈ 𝑇)
37	5, 6, 7, 8	ltrnel 40133	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ((𝐹‘𝑃) ∈ 𝐴 ∧ ¬ (𝐹‘𝑃) ≤ 𝑊))
38	3, 37	syld3an2 1413	. . . . . 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 40157	. . . . . 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 1373	. . . . 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 40118	. . . . . . . . . . . . . . 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 6829	. . . . . . . . . . . . . 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 5826	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑋 ∘ (^◡𝐹 ∘ 𝐹)) = (𝑋 ∘ ( I ↾ 𝐵)))
47	18, 7, 8	ltrn1o 40118	. . . . . . . . . . . . . 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 6800	. . . . . . . . . . . . 13 ⊢ (𝑋:𝐵–1-1-onto→𝐵 → 𝑋:𝐵⟶𝐵)
50		fcoi1 6734	. . . . . . . . . . . . 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 2765	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝑋 = (𝑋 ∘ (^◡𝐹 ∘ 𝐹)))
53		coass 6238	. . . . . . . . . . 11 ⊢ ((𝑋 ∘ ^◡𝐹) ∘ 𝐹) = (𝑋 ∘ (^◡𝐹 ∘ 𝐹))
54	52, 53	eqtr4di 2782	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝑋 = ((𝑋 ∘ ^◡𝐹) ∘ 𝐹))
55	54	fveq1d 6860	. . . . . . . . 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 40139	. . . . . . . . . 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 1377	. . . . . . . . 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 2764	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑋‘𝑃) = ((𝑋 ∘ ^◡𝐹)‘(𝐹‘𝑃)))
59	58	oveq2d 7403	. . . . . . 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 2735	. . . . . 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 7402	. . . . 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 2764	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑅‘(𝑋 ∘ ^◡𝐹)) = (((𝐹‘𝑃) ∨ (𝑋‘𝑃)) ∧ 𝑊))
63	62	oveq2d 7403	. . 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 40133	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ((𝑋‘𝑃) ∈ 𝐴 ∧ ¬ (𝑋‘𝑃) ≤ 𝑊))
65	11, 64	syld3an2 1413	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ((𝑋‘𝑃) ∈ 𝐴 ∧ ¬ (𝑋‘𝑃) ≤ 𝑊))
66		eqid 2729	. . . . . . 7 ⊢ (1.‘𝐾) = (1.‘𝐾)
67	5, 14, 66, 6, 7	lhpjat2 40015	. . . . . 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 7403	. . . 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 39354	. . . . . 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 39220	. . . . 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 2765	. . 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 2775	. 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 5133	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 1540 ∈ wcel 2109 class class class wbr 5107 I cid 5532 ^◡ccnv 5637 ↾ cres 5640 ∘ ccom 5642 ⟶wf 6507 –1-1-onto→wf1o 6510 ‘cfv 6511 (class class class)co 7387 Basecbs 17179 lecple 17227 joincjn 18272 meetcmee 18273 1.cp1 18383 Latclat 18390 OLcol 39167 Atomscatm 39256 HLchlt 39343 LHypclh 39978 LTrncltrn 40095 trLctrl 40152

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5234 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-riotaBAD 38946

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-ral 3045 df-rex 3054 df-rmo 3354 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4872 df-iun 4957 df-iin 4958 df-br 5108 df-opab 5170 df-mpt 5189 df-id 5533 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-1st 7968 df-2nd 7969 df-undef 8252 df-map 8801 df-proset 18255 df-poset 18274 df-plt 18289 df-lub 18305 df-glb 18306 df-join 18307 df-meet 18308 df-p0 18384 df-p1 18385 df-lat 18391 df-clat 18458 df-oposet 39169 df-ol 39171 df-oml 39172 df-covers 39259 df-ats 39260 df-atl 39291 df-cvlat 39315 df-hlat 39344 df-llines 39492 df-lplanes 39493 df-lvols 39494 df-lines 39495 df-psubsp 39497 df-pmap 39498 df-padd 39790 df-lhyp 39982 df-laut 39983 df-ldil 40098 df-ltrn 40099 df-trl 40153

This theorem is referenced by: cdlemk5a 40829

W3C validator