trlcoabs2N - Mathbox for Norm Megill

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Theorem trlcoabs2N 40431

Description: Absorption of the trace of a composition. (Contributed by NM, 29-Jul-2013.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
trlcoabs.l	⊢ ≤ = (le‘𝐾)
trlcoabs.j	⊢ ∨ = (join‘𝐾)
trlcoabs.a	⊢ 𝐴 = (Atoms‘𝐾)
trlcoabs.h	⊢ 𝐻 = (LHyp‘𝐾)
trlcoabs.t	⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
trlcoabs.r	⊢ 𝑅 = ((trL‘𝐾)‘𝑊)

**Assertion**
Ref	Expression
trlcoabs2N	⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ((𝐹‘𝑃) ∨ (𝑅‘(𝐺 ∘ ^◡𝐹))) = ((𝐹‘𝑃) ∨ (𝐺‘𝑃)))

**Proof of Theorem trlcoabs2N**
Step	Hyp	Ref	Expression
1		simp1 1133	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
2		simp2r 1197	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝐺 ∈ 𝑇)
3		simp2l 1196	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝐹 ∈ 𝑇)
4		trlcoabs.h	. . . . . . 7 ⊢ 𝐻 = (LHyp‘𝐾)
5		trlcoabs.t	. . . . . . 7 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
6	4, 5	ltrncnv 39855	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → ^◡𝐹 ∈ 𝑇)
7	1, 3, 6	syl2anc 582	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ^◡𝐹 ∈ 𝑇)
8	4, 5	ltrnco 40428	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇 ∧ ^◡𝐹 ∈ 𝑇) → (𝐺 ∘ ^◡𝐹) ∈ 𝑇)
9	1, 2, 7, 8	syl3anc 1368	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝐺 ∘ ^◡𝐹) ∈ 𝑇)
10		trlcoabs.l	. . . . . 6 ⊢ ≤ = (le‘𝐾)
11		trlcoabs.a	. . . . . 6 ⊢ 𝐴 = (Atoms‘𝐾)
12	10, 11, 4, 5	ltrnel 39848	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ((𝐹‘𝑃) ∈ 𝐴 ∧ ¬ (𝐹‘𝑃) ≤ 𝑊))
13	12	3adant2r 1176	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ((𝐹‘𝑃) ∈ 𝐴 ∧ ¬ (𝐹‘𝑃) ≤ 𝑊))
14		trlcoabs.j	. . . . 5 ⊢ ∨ = (join‘𝐾)
15		eqid 2726	. . . . 5 ⊢ (meet‘𝐾) = (meet‘𝐾)
16		trlcoabs.r	. . . . 5 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
17	10, 14, 15, 11, 4, 5, 16	trlval2 39872	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐺 ∘ ^◡𝐹) ∈ 𝑇 ∧ ((𝐹‘𝑃) ∈ 𝐴 ∧ ¬ (𝐹‘𝑃) ≤ 𝑊)) → (𝑅‘(𝐺 ∘ ^◡𝐹)) = (((𝐹‘𝑃) ∨ ((𝐺 ∘ ^◡𝐹)‘(𝐹‘𝑃)))(meet‘𝐾)𝑊))
18	1, 9, 13, 17	syl3anc 1368	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑅‘(𝐺 ∘ ^◡𝐹)) = (((𝐹‘𝑃) ∨ ((𝐺 ∘ ^◡𝐹)‘(𝐹‘𝑃)))(meet‘𝐾)𝑊))
19	18	oveq2d 7429	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ((𝐹‘𝑃) ∨ (𝑅‘(𝐺 ∘ ^◡𝐹))) = ((𝐹‘𝑃) ∨ (((𝐹‘𝑃) ∨ ((𝐺 ∘ ^◡𝐹)‘(𝐹‘𝑃)))(meet‘𝐾)𝑊)))
20		simp1l 1194	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝐾 ∈ HL)
21		simp3l 1198	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝑃 ∈ 𝐴)
22	10, 11, 4, 5	ltrnat 39849	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝑃 ∈ 𝐴) → (𝐹‘𝑃) ∈ 𝐴)
23	1, 3, 21, 22	syl3anc 1368	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝐹‘𝑃) ∈ 𝐴)
24	10, 11, 4, 5	ltrnat 39849	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐺 ∘ ^◡𝐹) ∈ 𝑇 ∧ (𝐹‘𝑃) ∈ 𝐴) → ((𝐺 ∘ ^◡𝐹)‘(𝐹‘𝑃)) ∈ 𝐴)
25	1, 9, 23, 24	syl3anc 1368	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ((𝐺 ∘ ^◡𝐹)‘(𝐹‘𝑃)) ∈ 𝐴)
26		eqid 2726	. . . . 5 ⊢ (Base‘𝐾) = (Base‘𝐾)
27	26, 14, 11	hlatjcl 39075	. . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝐹‘𝑃) ∈ 𝐴 ∧ ((𝐺 ∘ ^◡𝐹)‘(𝐹‘𝑃)) ∈ 𝐴) → ((𝐹‘𝑃) ∨ ((𝐺 ∘ ^◡𝐹)‘(𝐹‘𝑃))) ∈ (Base‘𝐾))
28	20, 23, 25, 27	syl3anc 1368	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ((𝐹‘𝑃) ∨ ((𝐺 ∘ ^◡𝐹)‘(𝐹‘𝑃))) ∈ (Base‘𝐾))
29		simp1r 1195	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝑊 ∈ 𝐻)
30	26, 4	lhpbase 39707	. . . 4 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ (Base‘𝐾))
31	29, 30	syl 17	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝑊 ∈ (Base‘𝐾))
32	10, 14, 11	hlatlej1 39083	. . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝐹‘𝑃) ∈ 𝐴 ∧ ((𝐺 ∘ ^◡𝐹)‘(𝐹‘𝑃)) ∈ 𝐴) → (𝐹‘𝑃) ≤ ((𝐹‘𝑃) ∨ ((𝐺 ∘ ^◡𝐹)‘(𝐹‘𝑃))))
33	20, 23, 25, 32	syl3anc 1368	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝐹‘𝑃) ≤ ((𝐹‘𝑃) ∨ ((𝐺 ∘ ^◡𝐹)‘(𝐹‘𝑃))))
34	26, 10, 14, 15, 11	atmod3i1 39573	. . 3 ⊢ ((𝐾 ∈ HL ∧ ((𝐹‘𝑃) ∈ 𝐴 ∧ ((𝐹‘𝑃) ∨ ((𝐺 ∘ ^◡𝐹)‘(𝐹‘𝑃))) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) ∧ (𝐹‘𝑃) ≤ ((𝐹‘𝑃) ∨ ((𝐺 ∘ ^◡𝐹)‘(𝐹‘𝑃)))) → ((𝐹‘𝑃) ∨ (((𝐹‘𝑃) ∨ ((𝐺 ∘ ^◡𝐹)‘(𝐹‘𝑃)))(meet‘𝐾)𝑊)) = (((𝐹‘𝑃) ∨ ((𝐺 ∘ ^◡𝐹)‘(𝐹‘𝑃)))(meet‘𝐾)((𝐹‘𝑃) ∨ 𝑊)))
35	20, 23, 28, 31, 33, 34	syl131anc 1380	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ((𝐹‘𝑃) ∨ (((𝐹‘𝑃) ∨ ((𝐺 ∘ ^◡𝐹)‘(𝐹‘𝑃)))(meet‘𝐾)𝑊)) = (((𝐹‘𝑃) ∨ ((𝐺 ∘ ^◡𝐹)‘(𝐹‘𝑃)))(meet‘𝐾)((𝐹‘𝑃) ∨ 𝑊)))
36	10, 11, 4, 5	ltrncoval 39854	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝐺 ∘ ^◡𝐹) ∈ 𝑇 ∧ 𝐹 ∈ 𝑇) ∧ 𝑃 ∈ 𝐴) → (((𝐺 ∘ ^◡𝐹) ∘ 𝐹)‘𝑃) = ((𝐺 ∘ ^◡𝐹)‘(𝐹‘𝑃)))
37	1, 9, 3, 21, 36	syl121anc 1372	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (((𝐺 ∘ ^◡𝐹) ∘ 𝐹)‘𝑃) = ((𝐺 ∘ ^◡𝐹)‘(𝐹‘𝑃)))
38		coass 6266	. . . . . . . 8 ⊢ ((𝐺 ∘ ^◡𝐹) ∘ 𝐹) = (𝐺 ∘ (^◡𝐹 ∘ 𝐹))
39	26, 4, 5	ltrn1o 39833	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → 𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
40	1, 3, 39	syl2anc 582	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
41		f1ococnv1 6861	. . . . . . . . . . 11 ⊢ (𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾) → (^◡𝐹 ∘ 𝐹) = ( I ↾ (Base‘𝐾)))
42	40, 41	syl 17	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (^◡𝐹 ∘ 𝐹) = ( I ↾ (Base‘𝐾)))
43	42	coeq2d 5859	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝐺 ∘ (^◡𝐹 ∘ 𝐹)) = (𝐺 ∘ ( I ↾ (Base‘𝐾))))
44	26, 4, 5	ltrn1o 39833	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇) → 𝐺:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
45	1, 2, 44	syl2anc 582	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝐺:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
46		f1of 6832	. . . . . . . . . 10 ⊢ (𝐺:(Base‘𝐾)–1-1-onto→(Base‘𝐾) → 𝐺:(Base‘𝐾)⟶(Base‘𝐾))
47		fcoi1 6765	. . . . . . . . . 10 ⊢ (𝐺:(Base‘𝐾)⟶(Base‘𝐾) → (𝐺 ∘ ( I ↾ (Base‘𝐾))) = 𝐺)
48	45, 46, 47	3syl 18	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝐺 ∘ ( I ↾ (Base‘𝐾))) = 𝐺)
49	43, 48	eqtrd 2766	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝐺 ∘ (^◡𝐹 ∘ 𝐹)) = 𝐺)
50	38, 49	eqtrid 2778	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ((𝐺 ∘ ^◡𝐹) ∘ 𝐹) = 𝐺)
51	50	fveq1d 6892	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (((𝐺 ∘ ^◡𝐹) ∘ 𝐹)‘𝑃) = (𝐺‘𝑃))
52	37, 51	eqtr3d 2768	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ((𝐺 ∘ ^◡𝐹)‘(𝐹‘𝑃)) = (𝐺‘𝑃))
53	52	oveq2d 7429	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ((𝐹‘𝑃) ∨ ((𝐺 ∘ ^◡𝐹)‘(𝐹‘𝑃))) = ((𝐹‘𝑃) ∨ (𝐺‘𝑃)))
54		eqid 2726	. . . . . 6 ⊢ (1.‘𝐾) = (1.‘𝐾)
55	10, 14, 54, 11, 4	lhpjat2 39730	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝐹‘𝑃) ∈ 𝐴 ∧ ¬ (𝐹‘𝑃) ≤ 𝑊)) → ((𝐹‘𝑃) ∨ 𝑊) = (1.‘𝐾))
56	1, 13, 55	syl2anc 582	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ((𝐹‘𝑃) ∨ 𝑊) = (1.‘𝐾))
57	53, 56	oveq12d 7431	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (((𝐹‘𝑃) ∨ ((𝐺 ∘ ^◡𝐹)‘(𝐹‘𝑃)))(meet‘𝐾)((𝐹‘𝑃) ∨ 𝑊)) = (((𝐹‘𝑃) ∨ (𝐺‘𝑃))(meet‘𝐾)(1.‘𝐾)))
58		hlol 39069	. . . . 5 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OL)
59	20, 58	syl 17	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝐾 ∈ OL)
60	10, 11, 4, 5	ltrnat 39849	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ∈ 𝐴) → (𝐺‘𝑃) ∈ 𝐴)
61	1, 2, 21, 60	syl3anc 1368	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝐺‘𝑃) ∈ 𝐴)
62	26, 14, 11	hlatjcl 39075	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ (𝐹‘𝑃) ∈ 𝐴 ∧ (𝐺‘𝑃) ∈ 𝐴) → ((𝐹‘𝑃) ∨ (𝐺‘𝑃)) ∈ (Base‘𝐾))
63	20, 23, 61, 62	syl3anc 1368	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ((𝐹‘𝑃) ∨ (𝐺‘𝑃)) ∈ (Base‘𝐾))
64	26, 15, 54	olm11 38935	. . . 4 ⊢ ((𝐾 ∈ OL ∧ ((𝐹‘𝑃) ∨ (𝐺‘𝑃)) ∈ (Base‘𝐾)) → (((𝐹‘𝑃) ∨ (𝐺‘𝑃))(meet‘𝐾)(1.‘𝐾)) = ((𝐹‘𝑃) ∨ (𝐺‘𝑃)))
65	59, 63, 64	syl2anc 582	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (((𝐹‘𝑃) ∨ (𝐺‘𝑃))(meet‘𝐾)(1.‘𝐾)) = ((𝐹‘𝑃) ∨ (𝐺‘𝑃)))
66	57, 65	eqtrd 2766	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (((𝐹‘𝑃) ∨ ((𝐺 ∘ ^◡𝐹)‘(𝐹‘𝑃)))(meet‘𝐾)((𝐹‘𝑃) ∨ 𝑊)) = ((𝐹‘𝑃) ∨ (𝐺‘𝑃)))
67	19, 35, 66	3eqtrd 2770	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ((𝐹‘𝑃) ∨ (𝑅‘(𝐺 ∘ ^◡𝐹))) = ((𝐹‘𝑃) ∨ (𝐺‘𝑃)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 394 ∧ w3a 1084 = wceq 1534 ∈ wcel 2099 class class class wbr 5143 I cid 5569 ^◡ccnv 5671 ↾ cres 5674 ∘ ccom 5676 ⟶wf 6539 –1-1-onto→wf1o 6542 ‘cfv 6543 (class class class)co 7413 Basecbs 17205 lecple 17265 joincjn 18328 meetcmee 18329 1.cp1 18441 OLcol 38882 Atomscatm 38971 HLchlt 39058 LHypclh 39693 LTrncltrn 39810 trLctrl 39867

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7735 ax-riotaBAD 38661

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-iun 4995 df-iin 4996 df-br 5144 df-opab 5206 df-mpt 5227 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-1st 7992 df-2nd 7993 df-undef 8277 df-map 8846 df-proset 18312 df-poset 18330 df-plt 18347 df-lub 18363 df-glb 18364 df-join 18365 df-meet 18366 df-p0 18442 df-p1 18443 df-lat 18449 df-clat 18516 df-oposet 38884 df-ol 38886 df-oml 38887 df-covers 38974 df-ats 38975 df-atl 39006 df-cvlat 39030 df-hlat 39059 df-llines 39207 df-lplanes 39208 df-lvols 39209 df-lines 39210 df-psubsp 39212 df-pmap 39213 df-padd 39505 df-lhyp 39697 df-laut 39698 df-ldil 39813 df-ltrn 39814 df-trl 39868

This theorem is referenced by: cdlemkfid1N 40630

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