cdlemh1 - Mathbox for Norm Megill

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Theorem cdlemh1 40920

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 7362	. 2 ⊢ (𝑆 ∨ (𝑅‘(𝐺 ∘ ^◡𝐹))) = (((𝑃 ∨ (𝑅‘𝐺)) ∧ (𝑄 ∨ (𝑅‘(𝐺 ∘ ^◡𝐹)))) ∨ (𝑅‘(𝐺 ∘ ^◡𝐹)))
3		simp11l 1285	. . . 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 1204	. . . . 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 1206	. . . . 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 1205	. . . . 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 1203	. . . . . 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 2983	. . . . 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 40829	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐺 ∈ 𝑇 ∧ 𝐹 ∈ 𝑇) ∧ (𝑅‘𝐺) ≠ (𝑅‘𝐹)) → (𝑅‘(𝐺 ∘ ^◡𝐹)) ∈ 𝐴)
14	4, 5, 6, 8, 13	syl121anc 1377	. . . 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	3	hllatd 39469	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑄 ≤ (𝑃 ∨ (𝑅‘𝐹)) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺))) → 𝐾 ∈ Lat)
16		simp2l 1200	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑄 ≤ (𝑃 ∨ (𝑅‘𝐹)) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺))) → 𝑃 ∈ 𝐴)
17		cdlemh.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝐾)
18	17, 9	atbase 39394	. . . . . 6 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ 𝐵)
19	16, 18	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)
20	17, 10, 11, 12	trlcl 40269	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇) → (𝑅‘𝐺) ∈ 𝐵)
21	4, 5, 20	syl2anc 584	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑄 ≤ (𝑃 ∨ (𝑅‘𝐹)) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺))) → (𝑅‘𝐺) ∈ 𝐵)
22		cdlemh.j	. . . . . 6 ⊢ ∨ = (join‘𝐾)
23	17, 22	latjcl 18351	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐵 ∧ (𝑅‘𝐺) ∈ 𝐵) → (𝑃 ∨ (𝑅‘𝐺)) ∈ 𝐵)
24	15, 19, 21, 23	syl3anc 1373	. . . 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)
25		simp2r 1201	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑄 ≤ (𝑃 ∨ (𝑅‘𝐹)) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺))) → 𝑄 ∈ 𝐴)
26	17, 22, 9	hlatjcl 39472	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ (𝑅‘(𝐺 ∘ ^◡𝐹)) ∈ 𝐴) → (𝑄 ∨ (𝑅‘(𝐺 ∘ ^◡𝐹))) ∈ 𝐵)
27	3, 25, 14, 26	syl3anc 1373	. . . 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)
28		cdlemh.l	. . . . . 6 ⊢ ≤ = (le‘𝐾)
29	28, 22, 9	hlatlej2 39481	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ (𝑅‘(𝐺 ∘ ^◡𝐹)) ∈ 𝐴) → (𝑅‘(𝐺 ∘ ^◡𝐹)) ≤ (𝑄 ∨ (𝑅‘(𝐺 ∘ ^◡𝐹))))
30	3, 25, 14, 29	syl3anc 1373	. . . 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))))
31		cdlemh.m	. . . . 5 ⊢ ∧ = (meet‘𝐾)
32	17, 28, 22, 31, 9	atmod4i1 39971	. . . 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)))))
33	3, 14, 24, 27, 30, 32	syl131anc 1385	. . 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)))))
34	10, 11	ltrncnv 40251	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → ^◡𝐹 ∈ 𝑇)
35	4, 6, 34	syl2anc 584	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑄 ≤ (𝑃 ∨ (𝑅‘𝐹)) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺))) → ^◡𝐹 ∈ 𝑇)
36	22, 10, 11, 12	trljco2 40846	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇 ∧ ^◡𝐹 ∈ 𝑇) → ((𝑅‘𝐺) ∨ (𝑅‘(𝐺 ∘ ^◡𝐹))) = ((𝑅‘^◡𝐹) ∨ (𝑅‘(𝐺 ∘ ^◡𝐹))))
37	4, 5, 35, 36	syl3anc 1373	. . . . . . 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))))
38	10, 11, 12	trlcnv 40270	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (𝑅‘^◡𝐹) = (𝑅‘𝐹))
39	4, 6, 38	syl2anc 584	. . . . . . . 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))
40	39	oveq1d 7367	. . . . . . 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))))
41	37, 40	eqtrd 2766	. . . . . 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))))
42	41	oveq2d 7368	. . . . 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)))))
43	10, 11	ltrnco 40824	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇 ∧ ^◡𝐹 ∈ 𝑇) → (𝐺 ∘ ^◡𝐹) ∈ 𝑇)
44	4, 5, 35, 43	syl3anc 1373	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑄 ≤ (𝑃 ∨ (𝑅‘𝐹)) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺))) → (𝐺 ∘ ^◡𝐹) ∈ 𝑇)
45	17, 10, 11, 12	trlcl 40269	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐺 ∘ ^◡𝐹) ∈ 𝑇) → (𝑅‘(𝐺 ∘ ^◡𝐹)) ∈ 𝐵)
46	4, 44, 45	syl2anc 584	. . . . . 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)
47	17, 22	latjass 18395	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∈ 𝐵 ∧ (𝑅‘𝐺) ∈ 𝐵 ∧ (𝑅‘(𝐺 ∘ ^◡𝐹)) ∈ 𝐵)) → ((𝑃 ∨ (𝑅‘𝐺)) ∨ (𝑅‘(𝐺 ∘ ^◡𝐹))) = (𝑃 ∨ ((𝑅‘𝐺) ∨ (𝑅‘(𝐺 ∘ ^◡𝐹)))))
48	15, 19, 21, 46, 47	syl13anc 1374	. . . . 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)))))
49	17, 10, 11, 12	trlcl 40269	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (𝑅‘𝐹) ∈ 𝐵)
50	4, 6, 49	syl2anc 584	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑄 ≤ (𝑃 ∨ (𝑅‘𝐹)) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺))) → (𝑅‘𝐹) ∈ 𝐵)
51	17, 22	latjass 18395	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∈ 𝐵 ∧ (𝑅‘𝐹) ∈ 𝐵 ∧ (𝑅‘(𝐺 ∘ ^◡𝐹)) ∈ 𝐵)) → ((𝑃 ∨ (𝑅‘𝐹)) ∨ (𝑅‘(𝐺 ∘ ^◡𝐹))) = (𝑃 ∨ ((𝑅‘𝐹) ∨ (𝑅‘(𝐺 ∘ ^◡𝐹)))))
52	15, 19, 50, 46, 51	syl13anc 1374	. . . . 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)))))
53	42, 48, 52	3eqtr4d 2776	. . . 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))))
54	53	oveq1d 7367	. . 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)))))
55		simp3l 1202	. . . . 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)))
56	17, 9	atbase 39394	. . . . . . 7 ⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ 𝐵)
57	25, 56	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)
58	17, 22	latjcl 18351	. . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐵 ∧ (𝑅‘𝐹) ∈ 𝐵) → (𝑃 ∨ (𝑅‘𝐹)) ∈ 𝐵)
59	15, 19, 50, 58	syl3anc 1373	. . . . . 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)
60	17, 28, 22	latjlej1 18365	. . . . . 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)))))
61	15, 57, 59, 46, 60	syl13anc 1374	. . . . 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)))))
62	55, 61	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))))
63	17, 22	latjcl 18351	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ (𝑅‘𝐹)) ∈ 𝐵 ∧ (𝑅‘(𝐺 ∘ ^◡𝐹)) ∈ 𝐵) → ((𝑃 ∨ (𝑅‘𝐹)) ∨ (𝑅‘(𝐺 ∘ ^◡𝐹))) ∈ 𝐵)
64	15, 59, 46, 63	syl3anc 1373	. . . . 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)
65	17, 28, 31	latleeqm2 18380	. . . . 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)))))
66	15, 27, 64, 65	syl3anc 1373	. . . 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)))))
67	62, 66	mpbid 232	. . 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))))
68	33, 54, 67	3eqtrd 2770	. 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))))
69	2, 68	eqtrid 2778	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 206 ∧ wa 395 ∧ w3a 1086 = wceq 1541 ∈ wcel 2111 ≠ wne 2928 class class class wbr 5093 ^◡ccnv 5618 ∘ ccom 5623 ‘cfv 6487 (class class class)co 7352 Basecbs 17126 lecple 17174 joincjn 18223 meetcmee 18224 Latclat 18343 Atomscatm 39368 HLchlt 39455 LHypclh 40089 LTrncltrn 40206 trLctrl 40263

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5219 ax-sep 5236 ax-nul 5246 ax-pow 5305 ax-pr 5372 ax-un 7674 ax-riotaBAD 39058

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-nul 4283 df-if 4475 df-pw 4551 df-sn 4576 df-pr 4578 df-op 4582 df-uni 4859 df-iun 4943 df-iin 4944 df-br 5094 df-opab 5156 df-mpt 5175 df-id 5514 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-iota 6443 df-fun 6489 df-fn 6490 df-f 6491 df-f1 6492 df-fo 6493 df-f1o 6494 df-fv 6495 df-riota 7309 df-ov 7355 df-oprab 7356 df-mpo 7357 df-1st 7927 df-2nd 7928 df-undef 8209 df-map 8758 df-proset 18206 df-poset 18225 df-plt 18240 df-lub 18256 df-glb 18257 df-join 18258 df-meet 18259 df-p0 18335 df-p1 18336 df-lat 18344 df-clat 18411 df-oposet 39281 df-ol 39283 df-oml 39284 df-covers 39371 df-ats 39372 df-atl 39403 df-cvlat 39427 df-hlat 39456 df-llines 39603 df-lplanes 39604 df-lvols 39605 df-lines 39606 df-psubsp 39608 df-pmap 39609 df-padd 39901 df-lhyp 40093 df-laut 40094 df-ldil 40209 df-ltrn 40210 df-trl 40264

This theorem is referenced by: cdlemh 40922

W3C validator