cdlemh1 - Mathbox for Norm Megill

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Theorem cdlemh1 36103

Description: Part of proof of 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
cdlemh1	⊢ ((((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))))

**Proof of Theorem cdlemh1**
Step	Hyp	Ref	Expression
1		cdlemh.s	. . 3 ⊢ 𝑆 = ((𝑃 ∨ (𝑅‘𝐺)) ∧ (𝑄 ∨ (𝑅‘(𝐺 ∘ ^◡𝐹))))
2	1	oveq1i 6660	. 2 ⊢ (𝑆 ∨ (𝑅‘(𝐺 ∘ ^◡𝐹))) = (((𝑃 ∨ (𝑅‘𝐺)) ∧ (𝑄 ∨ (𝑅‘(𝐺 ∘ ^◡𝐹)))) ∨ (𝑅‘(𝐺 ∘ ^◡𝐹)))
3		simp11l 1172	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑄 ≤ (𝑃 ∨ (𝑅‘𝐹)) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺))) → 𝐾 ∈ HL)
4		simp11 1091	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑄 ≤ (𝑃 ∨ (𝑅‘𝐹)) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
5		simp13 1093	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑄 ≤ (𝑃 ∨ (𝑅‘𝐹)) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺))) → 𝐺 ∈ 𝑇)
6		simp12 1092	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑄 ≤ (𝑃 ∨ (𝑅‘𝐹)) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺))) → 𝐹 ∈ 𝑇)
7		simp3r 1090	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑄 ≤ (𝑃 ∨ (𝑅‘𝐹)) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺))) → (𝑅‘𝐹) ≠ (𝑅‘𝐺))
8	7	necomd 2849	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑄 ≤ (𝑃 ∨ (𝑅‘𝐹)) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺))) → (𝑅‘𝐺) ≠ (𝑅‘𝐹))
9		cdlemh.a	. . . . . 6 ⊢ 𝐴 = (Atoms‘𝐾)
10		cdlemh.h	. . . . . 6 ⊢ 𝐻 = (LHyp‘𝐾)
11		cdlemh.t	. . . . . 6 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
12		cdlemh.r	. . . . . 6 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
13	9, 10, 11, 12	trlcocnvat 36012	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐺 ∈ 𝑇 ∧ 𝐹 ∈ 𝑇) ∧ (𝑅‘𝐺) ≠ (𝑅‘𝐹)) → (𝑅‘(𝐺 ∘ ^◡𝐹)) ∈ 𝐴)
14	4, 5, 6, 8, 13	syl121anc 1331	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑄 ≤ (𝑃 ∨ (𝑅‘𝐹)) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺))) → (𝑅‘(𝐺 ∘ ^◡𝐹)) ∈ 𝐴)
15		hllat 34650	. . . . . 6 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat)
16	3, 15	syl 17	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑄 ≤ (𝑃 ∨ (𝑅‘𝐹)) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺))) → 𝐾 ∈ Lat)
17		simp2l 1087	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑄 ≤ (𝑃 ∨ (𝑅‘𝐹)) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺))) → 𝑃 ∈ 𝐴)
18		cdlemh.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝐾)
19	18, 9	atbase 34576	. . . . . 6 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ 𝐵)
20	17, 19	syl 17	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑄 ≤ (𝑃 ∨ (𝑅‘𝐹)) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺))) → 𝑃 ∈ 𝐵)
21	18, 10, 11, 12	trlcl 35451	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇) → (𝑅‘𝐺) ∈ 𝐵)
22	4, 5, 21	syl2anc 693	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑄 ≤ (𝑃 ∨ (𝑅‘𝐹)) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺))) → (𝑅‘𝐺) ∈ 𝐵)
23		cdlemh.j	. . . . . 6 ⊢ ∨ = (join‘𝐾)
24	18, 23	latjcl 17051	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐵 ∧ (𝑅‘𝐺) ∈ 𝐵) → (𝑃 ∨ (𝑅‘𝐺)) ∈ 𝐵)
25	16, 20, 22, 24	syl3anc 1326	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑄 ≤ (𝑃 ∨ (𝑅‘𝐹)) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺))) → (𝑃 ∨ (𝑅‘𝐺)) ∈ 𝐵)
26		simp2r 1088	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑄 ≤ (𝑃 ∨ (𝑅‘𝐹)) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺))) → 𝑄 ∈ 𝐴)
27	18, 23, 9	hlatjcl 34653	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ (𝑅‘(𝐺 ∘ ^◡𝐹)) ∈ 𝐴) → (𝑄 ∨ (𝑅‘(𝐺 ∘ ^◡𝐹))) ∈ 𝐵)
28	3, 26, 14, 27	syl3anc 1326	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑄 ≤ (𝑃 ∨ (𝑅‘𝐹)) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺))) → (𝑄 ∨ (𝑅‘(𝐺 ∘ ^◡𝐹))) ∈ 𝐵)
29		cdlemh.l	. . . . . 6 ⊢ ≤ = (le‘𝐾)
30	29, 23, 9	hlatlej2 34662	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ (𝑅‘(𝐺 ∘ ^◡𝐹)) ∈ 𝐴) → (𝑅‘(𝐺 ∘ ^◡𝐹)) ≤ (𝑄 ∨ (𝑅‘(𝐺 ∘ ^◡𝐹))))
31	3, 26, 14, 30	syl3anc 1326	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑄 ≤ (𝑃 ∨ (𝑅‘𝐹)) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺))) → (𝑅‘(𝐺 ∘ ^◡𝐹)) ≤ (𝑄 ∨ (𝑅‘(𝐺 ∘ ^◡𝐹))))
32		cdlemh.m	. . . . 5 ⊢ ∧ = (meet‘𝐾)
33	18, 29, 23, 32, 9	atmod4i1 35152	. . . 4 ⊢ ((𝐾 ∈ HL ∧ ((𝑅‘(𝐺 ∘ ^◡𝐹)) ∈ 𝐴 ∧ (𝑃 ∨ (𝑅‘𝐺)) ∈ 𝐵 ∧ (𝑄 ∨ (𝑅‘(𝐺 ∘ ^◡𝐹))) ∈ 𝐵) ∧ (𝑅‘(𝐺 ∘ ^◡𝐹)) ≤ (𝑄 ∨ (𝑅‘(𝐺 ∘ ^◡𝐹)))) → (((𝑃 ∨ (𝑅‘𝐺)) ∧ (𝑄 ∨ (𝑅‘(𝐺 ∘ ^◡𝐹)))) ∨ (𝑅‘(𝐺 ∘ ^◡𝐹))) = (((𝑃 ∨ (𝑅‘𝐺)) ∨ (𝑅‘(𝐺 ∘ ^◡𝐹))) ∧ (𝑄 ∨ (𝑅‘(𝐺 ∘ ^◡𝐹)))))
34	3, 14, 25, 28, 31, 33	syl131anc 1339	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑄 ≤ (𝑃 ∨ (𝑅‘𝐹)) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺))) → (((𝑃 ∨ (𝑅‘𝐺)) ∧ (𝑄 ∨ (𝑅‘(𝐺 ∘ ^◡𝐹)))) ∨ (𝑅‘(𝐺 ∘ ^◡𝐹))) = (((𝑃 ∨ (𝑅‘𝐺)) ∨ (𝑅‘(𝐺 ∘ ^◡𝐹))) ∧ (𝑄 ∨ (𝑅‘(𝐺 ∘ ^◡𝐹)))))
35	10, 11	ltrncnv 35432	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → ^◡𝐹 ∈ 𝑇)
36	4, 6, 35	syl2anc 693	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑄 ≤ (𝑃 ∨ (𝑅‘𝐹)) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺))) → ^◡𝐹 ∈ 𝑇)
37	23, 10, 11, 12	trljco2 36029	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇 ∧ ^◡𝐹 ∈ 𝑇) → ((𝑅‘𝐺) ∨ (𝑅‘(𝐺 ∘ ^◡𝐹))) = ((𝑅‘^◡𝐹) ∨ (𝑅‘(𝐺 ∘ ^◡𝐹))))
38	4, 5, 36, 37	syl3anc 1326	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑄 ≤ (𝑃 ∨ (𝑅‘𝐹)) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺))) → ((𝑅‘𝐺) ∨ (𝑅‘(𝐺 ∘ ^◡𝐹))) = ((𝑅‘^◡𝐹) ∨ (𝑅‘(𝐺 ∘ ^◡𝐹))))
39	10, 11, 12	trlcnv 35452	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (𝑅‘^◡𝐹) = (𝑅‘𝐹))
40	4, 6, 39	syl2anc 693	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑄 ≤ (𝑃 ∨ (𝑅‘𝐹)) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺))) → (𝑅‘^◡𝐹) = (𝑅‘𝐹))
41	40	oveq1d 6665	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑄 ≤ (𝑃 ∨ (𝑅‘𝐹)) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺))) → ((𝑅‘^◡𝐹) ∨ (𝑅‘(𝐺 ∘ ^◡𝐹))) = ((𝑅‘𝐹) ∨ (𝑅‘(𝐺 ∘ ^◡𝐹))))
42	38, 41	eqtrd 2656	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑄 ≤ (𝑃 ∨ (𝑅‘𝐹)) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺))) → ((𝑅‘𝐺) ∨ (𝑅‘(𝐺 ∘ ^◡𝐹))) = ((𝑅‘𝐹) ∨ (𝑅‘(𝐺 ∘ ^◡𝐹))))
43	42	oveq2d 6666	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑄 ≤ (𝑃 ∨ (𝑅‘𝐹)) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺))) → (𝑃 ∨ ((𝑅‘𝐺) ∨ (𝑅‘(𝐺 ∘ ^◡𝐹)))) = (𝑃 ∨ ((𝑅‘𝐹) ∨ (𝑅‘(𝐺 ∘ ^◡𝐹)))))
44	10, 11	ltrnco 36007	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇 ∧ ^◡𝐹 ∈ 𝑇) → (𝐺 ∘ ^◡𝐹) ∈ 𝑇)
45	4, 5, 36, 44	syl3anc 1326	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑄 ≤ (𝑃 ∨ (𝑅‘𝐹)) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺))) → (𝐺 ∘ ^◡𝐹) ∈ 𝑇)
46	18, 10, 11, 12	trlcl 35451	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐺 ∘ ^◡𝐹) ∈ 𝑇) → (𝑅‘(𝐺 ∘ ^◡𝐹)) ∈ 𝐵)
47	4, 45, 46	syl2anc 693	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑄 ≤ (𝑃 ∨ (𝑅‘𝐹)) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺))) → (𝑅‘(𝐺 ∘ ^◡𝐹)) ∈ 𝐵)
48	18, 23	latjass 17095	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∈ 𝐵 ∧ (𝑅‘𝐺) ∈ 𝐵 ∧ (𝑅‘(𝐺 ∘ ^◡𝐹)) ∈ 𝐵)) → ((𝑃 ∨ (𝑅‘𝐺)) ∨ (𝑅‘(𝐺 ∘ ^◡𝐹))) = (𝑃 ∨ ((𝑅‘𝐺) ∨ (𝑅‘(𝐺 ∘ ^◡𝐹)))))
49	16, 20, 22, 47, 48	syl13anc 1328	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑄 ≤ (𝑃 ∨ (𝑅‘𝐹)) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺))) → ((𝑃 ∨ (𝑅‘𝐺)) ∨ (𝑅‘(𝐺 ∘ ^◡𝐹))) = (𝑃 ∨ ((𝑅‘𝐺) ∨ (𝑅‘(𝐺 ∘ ^◡𝐹)))))
50	18, 10, 11, 12	trlcl 35451	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (𝑅‘𝐹) ∈ 𝐵)
51	4, 6, 50	syl2anc 693	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑄 ≤ (𝑃 ∨ (𝑅‘𝐹)) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺))) → (𝑅‘𝐹) ∈ 𝐵)
52	18, 23	latjass 17095	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∈ 𝐵 ∧ (𝑅‘𝐹) ∈ 𝐵 ∧ (𝑅‘(𝐺 ∘ ^◡𝐹)) ∈ 𝐵)) → ((𝑃 ∨ (𝑅‘𝐹)) ∨ (𝑅‘(𝐺 ∘ ^◡𝐹))) = (𝑃 ∨ ((𝑅‘𝐹) ∨ (𝑅‘(𝐺 ∘ ^◡𝐹)))))
53	16, 20, 51, 47, 52	syl13anc 1328	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑄 ≤ (𝑃 ∨ (𝑅‘𝐹)) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺))) → ((𝑃 ∨ (𝑅‘𝐹)) ∨ (𝑅‘(𝐺 ∘ ^◡𝐹))) = (𝑃 ∨ ((𝑅‘𝐹) ∨ (𝑅‘(𝐺 ∘ ^◡𝐹)))))
54	43, 49, 53	3eqtr4d 2666	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑄 ≤ (𝑃 ∨ (𝑅‘𝐹)) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺))) → ((𝑃 ∨ (𝑅‘𝐺)) ∨ (𝑅‘(𝐺 ∘ ^◡𝐹))) = ((𝑃 ∨ (𝑅‘𝐹)) ∨ (𝑅‘(𝐺 ∘ ^◡𝐹))))
55	54	oveq1d 6665	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑄 ≤ (𝑃 ∨ (𝑅‘𝐹)) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺))) → (((𝑃 ∨ (𝑅‘𝐺)) ∨ (𝑅‘(𝐺 ∘ ^◡𝐹))) ∧ (𝑄 ∨ (𝑅‘(𝐺 ∘ ^◡𝐹)))) = (((𝑃 ∨ (𝑅‘𝐹)) ∨ (𝑅‘(𝐺 ∘ ^◡𝐹))) ∧ (𝑄 ∨ (𝑅‘(𝐺 ∘ ^◡𝐹)))))
56		simp3l 1089	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑄 ≤ (𝑃 ∨ (𝑅‘𝐹)) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺))) → 𝑄 ≤ (𝑃 ∨ (𝑅‘𝐹)))
57	18, 9	atbase 34576	. . . . . . 7 ⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ 𝐵)
58	26, 57	syl 17	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑄 ≤ (𝑃 ∨ (𝑅‘𝐹)) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺))) → 𝑄 ∈ 𝐵)
59	18, 23	latjcl 17051	. . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐵 ∧ (𝑅‘𝐹) ∈ 𝐵) → (𝑃 ∨ (𝑅‘𝐹)) ∈ 𝐵)
60	16, 20, 51, 59	syl3anc 1326	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑄 ≤ (𝑃 ∨ (𝑅‘𝐹)) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺))) → (𝑃 ∨ (𝑅‘𝐹)) ∈ 𝐵)
61	18, 29, 23	latjlej1 17065	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ (𝑄 ∈ 𝐵 ∧ (𝑃 ∨ (𝑅‘𝐹)) ∈ 𝐵 ∧ (𝑅‘(𝐺 ∘ ^◡𝐹)) ∈ 𝐵)) → (𝑄 ≤ (𝑃 ∨ (𝑅‘𝐹)) → (𝑄 ∨ (𝑅‘(𝐺 ∘ ^◡𝐹))) ≤ ((𝑃 ∨ (𝑅‘𝐹)) ∨ (𝑅‘(𝐺 ∘ ^◡𝐹)))))
62	16, 58, 60, 47, 61	syl13anc 1328	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑄 ≤ (𝑃 ∨ (𝑅‘𝐹)) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺))) → (𝑄 ≤ (𝑃 ∨ (𝑅‘𝐹)) → (𝑄 ∨ (𝑅‘(𝐺 ∘ ^◡𝐹))) ≤ ((𝑃 ∨ (𝑅‘𝐹)) ∨ (𝑅‘(𝐺 ∘ ^◡𝐹)))))
63	56, 62	mpd 15	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑄 ≤ (𝑃 ∨ (𝑅‘𝐹)) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺))) → (𝑄 ∨ (𝑅‘(𝐺 ∘ ^◡𝐹))) ≤ ((𝑃 ∨ (𝑅‘𝐹)) ∨ (𝑅‘(𝐺 ∘ ^◡𝐹))))
64	18, 23	latjcl 17051	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ (𝑅‘𝐹)) ∈ 𝐵 ∧ (𝑅‘(𝐺 ∘ ^◡𝐹)) ∈ 𝐵) → ((𝑃 ∨ (𝑅‘𝐹)) ∨ (𝑅‘(𝐺 ∘ ^◡𝐹))) ∈ 𝐵)
65	16, 60, 47, 64	syl3anc 1326	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑄 ≤ (𝑃 ∨ (𝑅‘𝐹)) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺))) → ((𝑃 ∨ (𝑅‘𝐹)) ∨ (𝑅‘(𝐺 ∘ ^◡𝐹))) ∈ 𝐵)
66	18, 29, 32	latleeqm2 17080	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ (𝑄 ∨ (𝑅‘(𝐺 ∘ ^◡𝐹))) ∈ 𝐵 ∧ ((𝑃 ∨ (𝑅‘𝐹)) ∨ (𝑅‘(𝐺 ∘ ^◡𝐹))) ∈ 𝐵) → ((𝑄 ∨ (𝑅‘(𝐺 ∘ ^◡𝐹))) ≤ ((𝑃 ∨ (𝑅‘𝐹)) ∨ (𝑅‘(𝐺 ∘ ^◡𝐹))) ↔ (((𝑃 ∨ (𝑅‘𝐹)) ∨ (𝑅‘(𝐺 ∘ ^◡𝐹))) ∧ (𝑄 ∨ (𝑅‘(𝐺 ∘ ^◡𝐹)))) = (𝑄 ∨ (𝑅‘(𝐺 ∘ ^◡𝐹)))))
67	16, 28, 65, 66	syl3anc 1326	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑄 ≤ (𝑃 ∨ (𝑅‘𝐹)) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺))) → ((𝑄 ∨ (𝑅‘(𝐺 ∘ ^◡𝐹))) ≤ ((𝑃 ∨ (𝑅‘𝐹)) ∨ (𝑅‘(𝐺 ∘ ^◡𝐹))) ↔ (((𝑃 ∨ (𝑅‘𝐹)) ∨ (𝑅‘(𝐺 ∘ ^◡𝐹))) ∧ (𝑄 ∨ (𝑅‘(𝐺 ∘ ^◡𝐹)))) = (𝑄 ∨ (𝑅‘(𝐺 ∘ ^◡𝐹)))))
68	63, 67	mpbid 222	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑄 ≤ (𝑃 ∨ (𝑅‘𝐹)) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺))) → (((𝑃 ∨ (𝑅‘𝐹)) ∨ (𝑅‘(𝐺 ∘ ^◡𝐹))) ∧ (𝑄 ∨ (𝑅‘(𝐺 ∘ ^◡𝐹)))) = (𝑄 ∨ (𝑅‘(𝐺 ∘ ^◡𝐹))))
69	34, 55, 68	3eqtrd 2660	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑄 ≤ (𝑃 ∨ (𝑅‘𝐹)) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺))) → (((𝑃 ∨ (𝑅‘𝐺)) ∧ (𝑄 ∨ (𝑅‘(𝐺 ∘ ^◡𝐹)))) ∨ (𝑅‘(𝐺 ∘ ^◡𝐹))) = (𝑄 ∨ (𝑅‘(𝐺 ∘ ^◡𝐹))))
70	2, 69	syl5eq 2668	1 ⊢ ((((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))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 class class class wbr 4653 ^◡ccnv 5113 ∘ ccom 5118 ‘cfv 5888 (class class class)co 6650 Basecbs 15857 lecple 15948 joincjn 16944 meetcmee 16945 Latclat 17045 Atomscatm 34550 HLchlt 34637 LHypclh 35270 LTrncltrn 35387 trLctrl 35445

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-riotaBAD 34239

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 df-undef 7399 df-map 7859 df-preset 16928 df-poset 16946 df-plt 16958 df-lub 16974 df-glb 16975 df-join 16976 df-meet 16977 df-p0 17039 df-p1 17040 df-lat 17046 df-clat 17108 df-oposet 34463 df-ol 34465 df-oml 34466 df-covers 34553 df-ats 34554 df-atl 34585 df-cvlat 34609 df-hlat 34638 df-llines 34784 df-lplanes 34785 df-lvols 34786 df-lines 34787 df-psubsp 34789 df-pmap 34790 df-padd 35082 df-lhyp 35274 df-laut 35275 df-ldil 35390 df-ltrn 35391 df-trl 35446

This theorem is referenced by: cdlemh 36105

W3C validator