cdlemh1 - Mathbox for Norm Megill

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Theorem cdlemh1 41014

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 7366	. 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 2985	. . . . 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 40923	. . . . 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 39563	. . . . 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 39488	. . . . . 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 40363	. . . . . 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 18360	. . . . 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 39566	. . . . 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 39575	. . . . 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 40065	. . . 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 40345	. . . . . . . . 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 40940	. . . . . . . 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 40364	. . . . . . . . 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 7371	. . . . . . 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 2769	. . . . . 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 7372	. . . . 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 40918	. . . . . . . 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 40363	. . . . . . 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 18404	. . . . . 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 40363	. . . . . . 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 18404	. . . . . 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 2779	. . . 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 7371	. . 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 39488	. . . . . . 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 18360	. . . . . . 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 18374	. . . . . 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 18360	. . . . . 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 18389	. . . . 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 2773	. 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 2781	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 2113 ≠ wne 2930 class class class wbr 5096 ^◡ccnv 5621 ∘ ccom 5626 ‘cfv 6490 (class class class)co 7356 Basecbs 17134 lecple 17182 joincjn 18232 meetcmee 18233 Latclat 18352 Atomscatm 39462 HLchlt 39549 LHypclh 40183 LTrncltrn 40300 trLctrl 40357

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 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2706 ax-rep 5222 ax-sep 5239 ax-nul 5249 ax-pow 5308 ax-pr 5375 ax-un 7678 ax-riotaBAD 39152

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 2537 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2809 df-nfc 2883 df-ne 2931 df-ral 3050 df-rex 3059 df-rmo 3348 df-reu 3349 df-rab 3398 df-v 3440 df-sbc 3739 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4579 df-pr 4581 df-op 4585 df-uni 4862 df-iun 4946 df-iin 4947 df-br 5097 df-opab 5159 df-mpt 5178 df-id 5517 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-f1 6495 df-fo 6496 df-f1o 6497 df-fv 6498 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-1st 7931 df-2nd 7932 df-undef 8213 df-map 8763 df-proset 18215 df-poset 18234 df-plt 18249 df-lub 18265 df-glb 18266 df-join 18267 df-meet 18268 df-p0 18344 df-p1 18345 df-lat 18353 df-clat 18420 df-oposet 39375 df-ol 39377 df-oml 39378 df-covers 39465 df-ats 39466 df-atl 39497 df-cvlat 39521 df-hlat 39550 df-llines 39697 df-lplanes 39698 df-lvols 39699 df-lines 39700 df-psubsp 39702 df-pmap 39703 df-padd 39995 df-lhyp 40187 df-laut 40188 df-ldil 40303 df-ltrn 40304 df-trl 40358

This theorem is referenced by: cdlemh 41016

W3C validator