trlcoabs2N - Mathbox for Norm Megill

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Theorem trlcoabs2N 38736

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 1135	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
2		simp2r 1199	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝐺 ∈ 𝑇)
3		simp2l 1198	. . . . . 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 38160	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → ^◡𝐹 ∈ 𝑇)
7	1, 3, 6	syl2anc 584	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ^◡𝐹 ∈ 𝑇)
8	4, 5	ltrnco 38733	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇 ∧ ^◡𝐹 ∈ 𝑇) → (𝐺 ∘ ^◡𝐹) ∈ 𝑇)
9	1, 2, 7, 8	syl3anc 1370	. . . 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 38153	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ((𝐹‘𝑃) ∈ 𝐴 ∧ ¬ (𝐹‘𝑃) ≤ 𝑊))
13	12	3adant2r 1178	. . . 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 2738	. . . . 5 ⊢ (meet‘𝐾) = (meet‘𝐾)
16		trlcoabs.r	. . . . 5 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
17	10, 14, 15, 11, 4, 5, 16	trlval2 38177	. . . 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 1370	. . 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 7291	. 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 1196	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝐾 ∈ HL)
21		simp3l 1200	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝑃 ∈ 𝐴)
22	10, 11, 4, 5	ltrnat 38154	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝑃 ∈ 𝐴) → (𝐹‘𝑃) ∈ 𝐴)
23	1, 3, 21, 22	syl3anc 1370	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝐹‘𝑃) ∈ 𝐴)
24	10, 11, 4, 5	ltrnat 38154	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐺 ∘ ^◡𝐹) ∈ 𝑇 ∧ (𝐹‘𝑃) ∈ 𝐴) → ((𝐺 ∘ ^◡𝐹)‘(𝐹‘𝑃)) ∈ 𝐴)
25	1, 9, 23, 24	syl3anc 1370	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ((𝐺 ∘ ^◡𝐹)‘(𝐹‘𝑃)) ∈ 𝐴)
26		eqid 2738	. . . . 5 ⊢ (Base‘𝐾) = (Base‘𝐾)
27	26, 14, 11	hlatjcl 37381	. . . 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 1370	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ((𝐹‘𝑃) ∨ ((𝐺 ∘ ^◡𝐹)‘(𝐹‘𝑃))) ∈ (Base‘𝐾))
29		simp1r 1197	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝑊 ∈ 𝐻)
30	26, 4	lhpbase 38012	. . . 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 37389	. . . 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 1370	. . 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 37878	. . 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 1382	. 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 38159	. . . . . . 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 1374	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (((𝐺 ∘ ^◡𝐹) ∘ 𝐹)‘𝑃) = ((𝐺 ∘ ^◡𝐹)‘(𝐹‘𝑃)))
38		coass 6169	. . . . . . . 8 ⊢ ((𝐺 ∘ ^◡𝐹) ∘ 𝐹) = (𝐺 ∘ (^◡𝐹 ∘ 𝐹))
39	26, 4, 5	ltrn1o 38138	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → 𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
40	1, 3, 39	syl2anc 584	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
41		f1ococnv1 6745	. . . . . . . . . . 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 5771	. . . . . . . . 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 38138	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇) → 𝐺:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
45	1, 2, 44	syl2anc 584	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝐺:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
46		f1of 6716	. . . . . . . . . 10 ⊢ (𝐺:(Base‘𝐾)–1-1-onto→(Base‘𝐾) → 𝐺:(Base‘𝐾)⟶(Base‘𝐾))
47		fcoi1 6648	. . . . . . . . . 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 2778	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝐺 ∘ (^◡𝐹 ∘ 𝐹)) = 𝐺)
50	38, 49	eqtrid 2790	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ((𝐺 ∘ ^◡𝐹) ∘ 𝐹) = 𝐺)
51	50	fveq1d 6776	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (((𝐺 ∘ ^◡𝐹) ∘ 𝐹)‘𝑃) = (𝐺‘𝑃))
52	37, 51	eqtr3d 2780	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ((𝐺 ∘ ^◡𝐹)‘(𝐹‘𝑃)) = (𝐺‘𝑃))
53	52	oveq2d 7291	. . . 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 2738	. . . . . 6 ⊢ (1.‘𝐾) = (1.‘𝐾)
55	10, 14, 54, 11, 4	lhpjat2 38035	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝐹‘𝑃) ∈ 𝐴 ∧ ¬ (𝐹‘𝑃) ≤ 𝑊)) → ((𝐹‘𝑃) ∨ 𝑊) = (1.‘𝐾))
56	1, 13, 55	syl2anc 584	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ((𝐹‘𝑃) ∨ 𝑊) = (1.‘𝐾))
57	53, 56	oveq12d 7293	. . 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 37375	. . . . 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 38154	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ∈ 𝐴) → (𝐺‘𝑃) ∈ 𝐴)
61	1, 2, 21, 60	syl3anc 1370	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝐺‘𝑃) ∈ 𝐴)
62	26, 14, 11	hlatjcl 37381	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ (𝐹‘𝑃) ∈ 𝐴 ∧ (𝐺‘𝑃) ∈ 𝐴) → ((𝐹‘𝑃) ∨ (𝐺‘𝑃)) ∈ (Base‘𝐾))
63	20, 23, 61, 62	syl3anc 1370	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ((𝐹‘𝑃) ∨ (𝐺‘𝑃)) ∈ (Base‘𝐾))
64	26, 15, 54	olm11 37241	. . . 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 584	. . 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 2778	. 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 2782	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 396 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 class class class wbr 5074 I cid 5488 ^◡ccnv 5588 ↾ cres 5591 ∘ ccom 5593 ⟶wf 6429 –1-1-onto→wf1o 6432 ‘cfv 6433 (class class class)co 7275 Basecbs 16912 lecple 16969 joincjn 18029 meetcmee 18030 1.cp1 18142 OLcol 37188 Atomscatm 37277 HLchlt 37364 LHypclh 37998 LTrncltrn 38115 trLctrl 38172

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-riotaBAD 36967

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-iin 4927 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5489 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-1st 7831 df-2nd 7832 df-undef 8089 df-map 8617 df-proset 18013 df-poset 18031 df-plt 18048 df-lub 18064 df-glb 18065 df-join 18066 df-meet 18067 df-p0 18143 df-p1 18144 df-lat 18150 df-clat 18217 df-oposet 37190 df-ol 37192 df-oml 37193 df-covers 37280 df-ats 37281 df-atl 37312 df-cvlat 37336 df-hlat 37365 df-llines 37512 df-lplanes 37513 df-lvols 37514 df-lines 37515 df-psubsp 37517 df-pmap 37518 df-padd 37810 df-lhyp 38002 df-laut 38003 df-ldil 38118 df-ltrn 38119 df-trl 38173

This theorem is referenced by: cdlemkfid1N 38935

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