cdlemk15 - Mathbox for Norm Megill

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Theorem cdlemk15 40360

Description: Part of proof of Lemma K of [Crawley] p. 118. Line 21 on p. 119. 𝑂, 𝐷 are k₁, f₁. (Contributed by NM, 1-Jul-2013.)

**Hypotheses**
Ref	Expression
cdlemk1.b	⊢ 𝐵 = (Base‘𝐾)
cdlemk1.l	⊢ ≤ = (le‘𝐾)
cdlemk1.j	⊢ ∨ = (join‘𝐾)
cdlemk1.m	⊢ ∧ = (meet‘𝐾)
cdlemk1.a	⊢ 𝐴 = (Atoms‘𝐾)
cdlemk1.h	⊢ 𝐻 = (LHyp‘𝐾)
cdlemk1.t	⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
cdlemk1.r	⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
cdlemk1.s	⊢ 𝑆 = (𝑓 ∈ 𝑇 ↦ (℩𝑖 ∈ 𝑇 (𝑖‘𝑃) = ((𝑃 ∨ (𝑅‘𝑓)) ∧ ((𝑁‘𝑃) ∨ (𝑅‘(𝑓 ∘ ^◡𝐹))))))
cdlemk1.o	⊢ 𝑂 = (𝑆‘𝐷)

**Assertion**
Ref	Expression
cdlemk15	⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐷 ∈ 𝑇) ∧ (𝑁 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ (𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐷 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝐷) ≠ (𝑅‘𝐹))) → (𝑁‘𝑃) ≤ ((𝑃 ∨ (𝑅‘𝐹)) ∧ ((𝑂‘𝑃) ∨ (𝑅‘(𝐹 ∘ ^◡𝐷)))))

Distinct variable groups: 𝑓,𝑖, ∧ ≤ ,𝑖 ∨ ,𝑓,𝑖 𝐴,𝑖 𝐷,𝑓,𝑖 𝑓,𝐹,𝑖 𝑖,𝐻 𝑖,𝐾 𝑓,𝑁,𝑖 𝑃,𝑓,𝑖 𝑅,𝑓,𝑖 𝑇,𝑓,𝑖 𝑓,𝑊,𝑖 Allowed substitution hints: 𝐴(𝑓) 𝐵(𝑓,𝑖) 𝑆(𝑓,𝑖) 𝐻(𝑓) 𝐾(𝑓) ≤ (𝑓) 𝑂(𝑓,𝑖)

**Proof of Theorem cdlemk15**
Step	Hyp	Ref	Expression
1		simp11l 1281	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐷 ∈ 𝑇) ∧ (𝑁 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ (𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐷 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝐷) ≠ (𝑅‘𝐹))) → 𝐾 ∈ HL)
2		simp22l 1289	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐷 ∈ 𝑇) ∧ (𝑁 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ (𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐷 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝐷) ≠ (𝑅‘𝐹))) → 𝑃 ∈ 𝐴)
3		simp11 1200	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐷 ∈ 𝑇) ∧ (𝑁 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ (𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐷 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝐷) ≠ (𝑅‘𝐹))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
4		simp21 1203	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐷 ∈ 𝑇) ∧ (𝑁 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ (𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐷 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝐷) ≠ (𝑅‘𝐹))) → 𝑁 ∈ 𝑇)
5		cdlemk1.l	. . . . . 6 ⊢ ≤ = (le‘𝐾)
6		cdlemk1.a	. . . . . 6 ⊢ 𝐴 = (Atoms‘𝐾)
7		cdlemk1.h	. . . . . 6 ⊢ 𝐻 = (LHyp‘𝐾)
8		cdlemk1.t	. . . . . 6 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
9	5, 6, 7, 8	ltrnat 39645	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑁 ∈ 𝑇 ∧ 𝑃 ∈ 𝐴) → (𝑁‘𝑃) ∈ 𝐴)
10	3, 4, 2, 9	syl3anc 1368	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐷 ∈ 𝑇) ∧ (𝑁 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ (𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐷 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝐷) ≠ (𝑅‘𝐹))) → (𝑁‘𝑃) ∈ 𝐴)
11		cdlemk1.j	. . . . 5 ⊢ ∨ = (join‘𝐾)
12	5, 11, 6	hlatlej2 38880	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ (𝑁‘𝑃) ∈ 𝐴) → (𝑁‘𝑃) ≤ (𝑃 ∨ (𝑁‘𝑃)))
13	1, 2, 10, 12	syl3anc 1368	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐷 ∈ 𝑇) ∧ (𝑁 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ (𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐷 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝐷) ≠ (𝑅‘𝐹))) → (𝑁‘𝑃) ≤ (𝑃 ∨ (𝑁‘𝑃)))
14		simp23 1205	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐷 ∈ 𝑇) ∧ (𝑁 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ (𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐷 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝐷) ≠ (𝑅‘𝐹))) → (𝑅‘𝐹) = (𝑅‘𝑁))
15	14	oveq2d 7442	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐷 ∈ 𝑇) ∧ (𝑁 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ (𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐷 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝐷) ≠ (𝑅‘𝐹))) → (𝑃 ∨ (𝑅‘𝐹)) = (𝑃 ∨ (𝑅‘𝑁)))
16		simp22 1204	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐷 ∈ 𝑇) ∧ (𝑁 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ (𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐷 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝐷) ≠ (𝑅‘𝐹))) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
17		cdlemk1.r	. . . . . 6 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
18	5, 11, 6, 7, 8, 17	trljat1 39671	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑁 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑃 ∨ (𝑅‘𝑁)) = (𝑃 ∨ (𝑁‘𝑃)))
19	3, 4, 16, 18	syl3anc 1368	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐷 ∈ 𝑇) ∧ (𝑁 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ (𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐷 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝐷) ≠ (𝑅‘𝐹))) → (𝑃 ∨ (𝑅‘𝑁)) = (𝑃 ∨ (𝑁‘𝑃)))
20	15, 19	eqtr2d 2769	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐷 ∈ 𝑇) ∧ (𝑁 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ (𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐷 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝐷) ≠ (𝑅‘𝐹))) → (𝑃 ∨ (𝑁‘𝑃)) = (𝑃 ∨ (𝑅‘𝐹)))
21	13, 20	breqtrd 5178	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐷 ∈ 𝑇) ∧ (𝑁 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ (𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐷 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝐷) ≠ (𝑅‘𝐹))) → (𝑁‘𝑃) ≤ (𝑃 ∨ (𝑅‘𝐹)))
22		cdlemk1.b	. . 3 ⊢ 𝐵 = (Base‘𝐾)
23		cdlemk1.m	. . 3 ⊢ ∧ = (meet‘𝐾)
24		cdlemk1.s	. . 3 ⊢ 𝑆 = (𝑓 ∈ 𝑇 ↦ (℩𝑖 ∈ 𝑇 (𝑖‘𝑃) = ((𝑃 ∨ (𝑅‘𝑓)) ∧ ((𝑁‘𝑃) ∨ (𝑅‘(𝑓 ∘ ^◡𝐹))))))
25		cdlemk1.o	. . 3 ⊢ 𝑂 = (𝑆‘𝐷)
26	22, 5, 11, 23, 6, 7, 8, 17, 24, 25	cdlemk14 40359	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐷 ∈ 𝑇) ∧ (𝑁 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ (𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐷 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝐷) ≠ (𝑅‘𝐹))) → (𝑁‘𝑃) ≤ ((𝑂‘𝑃) ∨ (𝑅‘(𝐹 ∘ ^◡𝐷))))
27	1	hllatd 38868	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐷 ∈ 𝑇) ∧ (𝑁 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ (𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐷 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝐷) ≠ (𝑅‘𝐹))) → 𝐾 ∈ Lat)
28	22, 6	atbase 38793	. . . 4 ⊢ ((𝑁‘𝑃) ∈ 𝐴 → (𝑁‘𝑃) ∈ 𝐵)
29	10, 28	syl 17	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐷 ∈ 𝑇) ∧ (𝑁 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ (𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐷 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝐷) ≠ (𝑅‘𝐹))) → (𝑁‘𝑃) ∈ 𝐵)
30		simp12 1201	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐷 ∈ 𝑇) ∧ (𝑁 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ (𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐷 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝐷) ≠ (𝑅‘𝐹))) → 𝐹 ∈ 𝑇)
31		simp31 1206	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐷 ∈ 𝑇) ∧ (𝑁 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ (𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐷 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝐷) ≠ (𝑅‘𝐹))) → 𝐹 ≠ ( I ↾ 𝐵))
32	22, 6, 7, 8, 17	trlnidat 39678	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵)) → (𝑅‘𝐹) ∈ 𝐴)
33	3, 30, 31, 32	syl3anc 1368	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐷 ∈ 𝑇) ∧ (𝑁 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ (𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐷 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝐷) ≠ (𝑅‘𝐹))) → (𝑅‘𝐹) ∈ 𝐴)
34	22, 11, 6	hlatjcl 38871	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ (𝑅‘𝐹) ∈ 𝐴) → (𝑃 ∨ (𝑅‘𝐹)) ∈ 𝐵)
35	1, 2, 33, 34	syl3anc 1368	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐷 ∈ 𝑇) ∧ (𝑁 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ (𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐷 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝐷) ≠ (𝑅‘𝐹))) → (𝑃 ∨ (𝑅‘𝐹)) ∈ 𝐵)
36	25	fveq1i 6903	. . . . 5 ⊢ (𝑂‘𝑃) = ((𝑆‘𝐷)‘𝑃)
37	22, 5, 11, 6, 7, 8, 17, 23, 24	cdlemksat 40351	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐷 ∈ 𝑇) ∧ (𝑁 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ (𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐷 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝐷) ≠ (𝑅‘𝐹))) → ((𝑆‘𝐷)‘𝑃) ∈ 𝐴)
38	36, 37	eqeltrid 2833	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐷 ∈ 𝑇) ∧ (𝑁 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ (𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐷 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝐷) ≠ (𝑅‘𝐹))) → (𝑂‘𝑃) ∈ 𝐴)
39		simp13 1202	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐷 ∈ 𝑇) ∧ (𝑁 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ (𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐷 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝐷) ≠ (𝑅‘𝐹))) → 𝐷 ∈ 𝑇)
40		simp33 1208	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐷 ∈ 𝑇) ∧ (𝑁 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ (𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐷 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝐷) ≠ (𝑅‘𝐹))) → (𝑅‘𝐷) ≠ (𝑅‘𝐹))
41	40	necomd 2993	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐷 ∈ 𝑇) ∧ (𝑁 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ (𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐷 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝐷) ≠ (𝑅‘𝐹))) → (𝑅‘𝐹) ≠ (𝑅‘𝐷))
42	6, 7, 8, 17	trlcocnvat 40229	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐷 ∈ 𝑇) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐷)) → (𝑅‘(𝐹 ∘ ^◡𝐷)) ∈ 𝐴)
43	3, 30, 39, 41, 42	syl121anc 1372	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐷 ∈ 𝑇) ∧ (𝑁 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ (𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐷 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝐷) ≠ (𝑅‘𝐹))) → (𝑅‘(𝐹 ∘ ^◡𝐷)) ∈ 𝐴)
44	22, 11, 6	hlatjcl 38871	. . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑂‘𝑃) ∈ 𝐴 ∧ (𝑅‘(𝐹 ∘ ^◡𝐷)) ∈ 𝐴) → ((𝑂‘𝑃) ∨ (𝑅‘(𝐹 ∘ ^◡𝐷))) ∈ 𝐵)
45	1, 38, 43, 44	syl3anc 1368	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐷 ∈ 𝑇) ∧ (𝑁 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ (𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐷 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝐷) ≠ (𝑅‘𝐹))) → ((𝑂‘𝑃) ∨ (𝑅‘(𝐹 ∘ ^◡𝐷))) ∈ 𝐵)
46	22, 5, 23	latlem12 18465	. . 3 ⊢ ((𝐾 ∈ Lat ∧ ((𝑁‘𝑃) ∈ 𝐵 ∧ (𝑃 ∨ (𝑅‘𝐹)) ∈ 𝐵 ∧ ((𝑂‘𝑃) ∨ (𝑅‘(𝐹 ∘ ^◡𝐷))) ∈ 𝐵)) → (((𝑁‘𝑃) ≤ (𝑃 ∨ (𝑅‘𝐹)) ∧ (𝑁‘𝑃) ≤ ((𝑂‘𝑃) ∨ (𝑅‘(𝐹 ∘ ^◡𝐷)))) ↔ (𝑁‘𝑃) ≤ ((𝑃 ∨ (𝑅‘𝐹)) ∧ ((𝑂‘𝑃) ∨ (𝑅‘(𝐹 ∘ ^◡𝐷))))))
47	27, 29, 35, 45, 46	syl13anc 1369	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐷 ∈ 𝑇) ∧ (𝑁 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ (𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐷 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝐷) ≠ (𝑅‘𝐹))) → (((𝑁‘𝑃) ≤ (𝑃 ∨ (𝑅‘𝐹)) ∧ (𝑁‘𝑃) ≤ ((𝑂‘𝑃) ∨ (𝑅‘(𝐹 ∘ ^◡𝐷)))) ↔ (𝑁‘𝑃) ≤ ((𝑃 ∨ (𝑅‘𝐹)) ∧ ((𝑂‘𝑃) ∨ (𝑅‘(𝐹 ∘ ^◡𝐷))))))
48	21, 26, 47	mpbi2and 710	1 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐷 ∈ 𝑇) ∧ (𝑁 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ (𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐷 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝐷) ≠ (𝑅‘𝐹))) → (𝑁‘𝑃) ≤ ((𝑃 ∨ (𝑅‘𝐹)) ∧ ((𝑂‘𝑃) ∨ (𝑅‘(𝐹 ∘ ^◡𝐷)))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ≠ wne 2937 class class class wbr 5152 ↦ cmpt 5235 I cid 5579 ^◡ccnv 5681 ↾ cres 5684 ∘ ccom 5686 ‘cfv 6553 ℩crio 7381 (class class class)co 7426 Basecbs 17187 lecple 17247 joincjn 18310 meetcmee 18311 Latclat 18430 Atomscatm 38767 HLchlt 38854 LHypclh 39489 LTrncltrn 39606 trLctrl 39663

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 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-riotaBAD 38457

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-iin 5003 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-1st 7999 df-2nd 8000 df-undef 8285 df-map 8853 df-proset 18294 df-poset 18312 df-plt 18329 df-lub 18345 df-glb 18346 df-join 18347 df-meet 18348 df-p0 18424 df-p1 18425 df-lat 18431 df-clat 18498 df-oposet 38680 df-ol 38682 df-oml 38683 df-covers 38770 df-ats 38771 df-atl 38802 df-cvlat 38826 df-hlat 38855 df-llines 39003 df-lplanes 39004 df-lvols 39005 df-lines 39006 df-psubsp 39008 df-pmap 39009 df-padd 39301 df-lhyp 39493 df-laut 39494 df-ldil 39609 df-ltrn 39610 df-trl 39664

This theorem is referenced by: cdlemk17 40363 cdlemk15-2N 40384

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