trlcoabs2N - Mathbox for Norm Megill

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Theorem trlcoabs2N 41529

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 1154	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
2		simp2r 1219	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝐺 ∈ 𝑇)
3		simp2l 1218	. . . . . 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 40953	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → ^◡𝐹 ∈ 𝑇)
7	1, 3, 6	syl2anc 596	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ^◡𝐹 ∈ 𝑇)
8	4, 5	ltrnco 41526	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇 ∧ ^◡𝐹 ∈ 𝑇) → (𝐺 ∘ ^◡𝐹) ∈ 𝑇)
9	1, 2, 7, 8	syl3anc 1398	. . . 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 40946	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ((𝐹‘𝑃) ∈ 𝐴 ∧ ¬ (𝐹‘𝑃) ≤ 𝑊))
13	12	3adant2r 1198	. . . 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 2765	. . . . 5 ⊢ (meet‘𝐾) = (meet‘𝐾)
16		trlcoabs.r	. . . . 5 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
17	10, 14, 15, 11, 4, 5, 16	trlval2 40970	. . . 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 1398	. . 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 7432	. 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 1216	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝐾 ∈ HL)
21		simp3l 1220	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝑃 ∈ 𝐴)
22	10, 11, 4, 5	ltrnat 40947	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝑃 ∈ 𝐴) → (𝐹‘𝑃) ∈ 𝐴)
23	1, 3, 21, 22	syl3anc 1398	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝐹‘𝑃) ∈ 𝐴)
24	10, 11, 4, 5	ltrnat 40947	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐺 ∘ ^◡𝐹) ∈ 𝑇 ∧ (𝐹‘𝑃) ∈ 𝐴) → ((𝐺 ∘ ^◡𝐹)‘(𝐹‘𝑃)) ∈ 𝐴)
25	1, 9, 23, 24	syl3anc 1398	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ((𝐺 ∘ ^◡𝐹)‘(𝐹‘𝑃)) ∈ 𝐴)
26		eqid 2765	. . . . 5 ⊢ (Base‘𝐾) = (Base‘𝐾)
27	26, 14, 11	hlatjcl 40174	. . . 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 1398	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ((𝐹‘𝑃) ∨ ((𝐺 ∘ ^◡𝐹)‘(𝐹‘𝑃))) ∈ (Base‘𝐾))
29		simp1r 1217	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝑊 ∈ 𝐻)
30	26, 4	lhpbase 40805	. . . 4 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ (Base‘𝐾))
31	29, 30	syl 18	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝑊 ∈ (Base‘𝐾))
32	10, 14, 11	hlatlej1 40182	. . . 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 1398	. . 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 40671	. . 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 1410	. 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 40952	. . . . . . 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 1402	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (((𝐺 ∘ ^◡𝐹) ∘ 𝐹)‘𝑃) = ((𝐺 ∘ ^◡𝐹)‘(𝐹‘𝑃)))
38		coass 6269	. . . . . . . 8 ⊢ ((𝐺 ∘ ^◡𝐹) ∘ 𝐹) = (𝐺 ∘ (^◡𝐹 ∘ 𝐹))
39	26, 4, 5	ltrn1o 40931	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → 𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
40	1, 3, 39	syl2anc 596	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
41		f1ococnv1 6854	. . . . . . . . . . 11 ⊢ (𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾) → (^◡𝐹 ∘ 𝐹) = ( I ↾ (Base‘𝐾)))
42	40, 41	syl 18	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (^◡𝐹 ∘ 𝐹) = ( I ↾ (Base‘𝐾)))
43	42	coeq2d 5850	. . . . . . . . 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 40931	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇) → 𝐺:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
45	1, 2, 44	syl2anc 596	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝐺:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
46		f1of 6824	. . . . . . . . . 10 ⊢ (𝐺:(Base‘𝐾)–1-1-onto→(Base‘𝐾) → 𝐺:(Base‘𝐾)⟶(Base‘𝐾))
47		fcoi1 6756	. . . . . . . . . 10 ⊢ (𝐺:(Base‘𝐾)⟶(Base‘𝐾) → (𝐺 ∘ ( I ↾ (Base‘𝐾))) = 𝐺)
48	45, 46, 47	3syl 19	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝐺 ∘ ( I ↾ (Base‘𝐾))) = 𝐺)
49	43, 48	eqtrd 2800	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝐺 ∘ (^◡𝐹 ∘ 𝐹)) = 𝐺)
50	38, 49	eqtrid 2812	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ((𝐺 ∘ ^◡𝐹) ∘ 𝐹) = 𝐺)
51	50	fveq1d 6887	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (((𝐺 ∘ ^◡𝐹) ∘ 𝐹)‘𝑃) = (𝐺‘𝑃))
52	37, 51	eqtr3d 2802	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ((𝐺 ∘ ^◡𝐹)‘(𝐹‘𝑃)) = (𝐺‘𝑃))
53	52	oveq2d 7432	. . . 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 2765	. . . . . 6 ⊢ (1.‘𝐾) = (1.‘𝐾)
55	10, 14, 54, 11, 4	lhpjat2 40828	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝐹‘𝑃) ∈ 𝐴 ∧ ¬ (𝐹‘𝑃) ≤ 𝑊)) → ((𝐹‘𝑃) ∨ 𝑊) = (1.‘𝐾))
56	1, 13, 55	syl2anc 596	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ((𝐹‘𝑃) ∨ 𝑊) = (1.‘𝐾))
57	53, 56	oveq12d 7434	. . 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 40168	. . . . 5 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OL)
59	20, 58	syl 18	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝐾 ∈ OL)
60	10, 11, 4, 5	ltrnat 40947	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ∈ 𝐴) → (𝐺‘𝑃) ∈ 𝐴)
61	1, 2, 21, 60	syl3anc 1398	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝐺‘𝑃) ∈ 𝐴)
62	26, 14, 11	hlatjcl 40174	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ (𝐹‘𝑃) ∈ 𝐴 ∧ (𝐺‘𝑃) ∈ 𝐴) → ((𝐹‘𝑃) ∨ (𝐺‘𝑃)) ∈ (Base‘𝐾))
63	20, 23, 61, 62	syl3anc 1398	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ((𝐹‘𝑃) ∨ (𝐺‘𝑃)) ∈ (Base‘𝐾))
64	26, 15, 54	olm11 40034	. . . 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 596	. . 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 2800	. 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 2804	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

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2146 class class class wbr 5111 I cid 5557 ^◡ccnv 5662 ↾ cres 5665 ∘ ccom 5667 ⟶wf 6536 –1-1-onto→wf1o 6539 ‘cfv 6540 (class class class)co 7416 Basecbs 17286 lecple 17334 joincjn 18384 meetcmee 18385 1.cp1 18495 OLcol 39981 Atomscatm 40070 HLchlt 40157 LHypclh 40791 LTrncltrn 40908 trLctrl 40965

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-rep 5240 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7738 ax-riotaBAD 39760

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-ral 3082 df-rex 3092 df-rmo 3371 df-reu 3372 df-rab 3419 df-v 3459 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4875 df-iun 4960 df-iin 4961 df-br 5112 df-opab 5176 df-mpt 5195 df-id 5558 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7373 df-ov 7419 df-oprab 7420 df-mpo 7421 df-1st 7988 df-2nd 7989 df-undef 8271 df-map 8828 df-proset 18367 df-poset 18386 df-plt 18401 df-lub 18417 df-glb 18418 df-join 18419 df-meet 18420 df-p0 18496 df-p1 18497 df-lat 18505 df-clat 18572 df-oposet 39983 df-ol 39985 df-oml 39986 df-covers 40073 df-ats 40074 df-atl 40105 df-cvlat 40129 df-hlat 40158 df-llines 40305 df-lplanes 40306 df-lvols 40307 df-lines 40308 df-psubsp 40310 df-pmap 40311 df-padd 40603 df-lhyp 40795 df-laut 40796 df-ldil 40911 df-ltrn 40912 df-trl 40966

This theorem is used by: cdlemkfid1N 41728

W3C validator