trlcoabs2N - Mathbox for Norm Megill

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Theorem trlcoabs2N 40723

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 1136	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
2		simp2r 1201	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝐺 ∈ 𝑇)
3		simp2l 1200	. . . . . 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 40147	. . . . . 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 40720	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇 ∧ ^◡𝐹 ∈ 𝑇) → (𝐺 ∘ ^◡𝐹) ∈ 𝑇)
9	1, 2, 7, 8	syl3anc 1373	. . . 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 40140	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ((𝐹‘𝑃) ∈ 𝐴 ∧ ¬ (𝐹‘𝑃) ≤ 𝑊))
13	12	3adant2r 1180	. . . 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 2730	. . . . 5 ⊢ (meet‘𝐾) = (meet‘𝐾)
16		trlcoabs.r	. . . . 5 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
17	10, 14, 15, 11, 4, 5, 16	trlval2 40164	. . . 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 1373	. . 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 7406	. 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 1198	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝐾 ∈ HL)
21		simp3l 1202	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝑃 ∈ 𝐴)
22	10, 11, 4, 5	ltrnat 40141	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝑃 ∈ 𝐴) → (𝐹‘𝑃) ∈ 𝐴)
23	1, 3, 21, 22	syl3anc 1373	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝐹‘𝑃) ∈ 𝐴)
24	10, 11, 4, 5	ltrnat 40141	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐺 ∘ ^◡𝐹) ∈ 𝑇 ∧ (𝐹‘𝑃) ∈ 𝐴) → ((𝐺 ∘ ^◡𝐹)‘(𝐹‘𝑃)) ∈ 𝐴)
25	1, 9, 23, 24	syl3anc 1373	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ((𝐺 ∘ ^◡𝐹)‘(𝐹‘𝑃)) ∈ 𝐴)
26		eqid 2730	. . . . 5 ⊢ (Base‘𝐾) = (Base‘𝐾)
27	26, 14, 11	hlatjcl 39367	. . . 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 1373	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ((𝐹‘𝑃) ∨ ((𝐺 ∘ ^◡𝐹)‘(𝐹‘𝑃))) ∈ (Base‘𝐾))
29		simp1r 1199	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝑊 ∈ 𝐻)
30	26, 4	lhpbase 39999	. . . 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 39375	. . . 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 1373	. . 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 39865	. . 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 1385	. 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 40146	. . . . . . 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 1377	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (((𝐺 ∘ ^◡𝐹) ∘ 𝐹)‘𝑃) = ((𝐺 ∘ ^◡𝐹)‘(𝐹‘𝑃)))
38		coass 6241	. . . . . . . 8 ⊢ ((𝐺 ∘ ^◡𝐹) ∘ 𝐹) = (𝐺 ∘ (^◡𝐹 ∘ 𝐹))
39	26, 4, 5	ltrn1o 40125	. . . . . . . . . . . 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 6832	. . . . . . . . . . 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 5829	. . . . . . . . 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 40125	. . . . . . . . . . 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 6803	. . . . . . . . . 10 ⊢ (𝐺:(Base‘𝐾)–1-1-onto→(Base‘𝐾) → 𝐺:(Base‘𝐾)⟶(Base‘𝐾))
47		fcoi1 6737	. . . . . . . . . 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 2765	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝐺 ∘ (^◡𝐹 ∘ 𝐹)) = 𝐺)
50	38, 49	eqtrid 2777	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ((𝐺 ∘ ^◡𝐹) ∘ 𝐹) = 𝐺)
51	50	fveq1d 6863	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (((𝐺 ∘ ^◡𝐹) ∘ 𝐹)‘𝑃) = (𝐺‘𝑃))
52	37, 51	eqtr3d 2767	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ((𝐺 ∘ ^◡𝐹)‘(𝐹‘𝑃)) = (𝐺‘𝑃))
53	52	oveq2d 7406	. . . 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 2730	. . . . . 6 ⊢ (1.‘𝐾) = (1.‘𝐾)
55	10, 14, 54, 11, 4	lhpjat2 40022	. . . . 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 7408	. . 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 39361	. . . . 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 40141	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ∈ 𝐴) → (𝐺‘𝑃) ∈ 𝐴)
61	1, 2, 21, 60	syl3anc 1373	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝐺‘𝑃) ∈ 𝐴)
62	26, 14, 11	hlatjcl 39367	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ (𝐹‘𝑃) ∈ 𝐴 ∧ (𝐺‘𝑃) ∈ 𝐴) → ((𝐹‘𝑃) ∨ (𝐺‘𝑃)) ∈ (Base‘𝐾))
63	20, 23, 61, 62	syl3anc 1373	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ((𝐹‘𝑃) ∨ (𝐺‘𝑃)) ∈ (Base‘𝐾))
64	26, 15, 54	olm11 39227	. . . 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 2765	. 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 2769	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 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 class class class wbr 5110 I cid 5535 ^◡ccnv 5640 ↾ cres 5643 ∘ ccom 5645 ⟶wf 6510 –1-1-onto→wf1o 6513 ‘cfv 6514 (class class class)co 7390 Basecbs 17186 lecple 17234 joincjn 18279 meetcmee 18280 1.cp1 18390 OLcol 39174 Atomscatm 39263 HLchlt 39350 LHypclh 39985 LTrncltrn 40102 trLctrl 40159

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-rep 5237 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 ax-riotaBAD 38953

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-ral 3046 df-rex 3055 df-rmo 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4875 df-iun 4960 df-iin 4961 df-br 5111 df-opab 5173 df-mpt 5192 df-id 5536 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-1st 7971 df-2nd 7972 df-undef 8255 df-map 8804 df-proset 18262 df-poset 18281 df-plt 18296 df-lub 18312 df-glb 18313 df-join 18314 df-meet 18315 df-p0 18391 df-p1 18392 df-lat 18398 df-clat 18465 df-oposet 39176 df-ol 39178 df-oml 39179 df-covers 39266 df-ats 39267 df-atl 39298 df-cvlat 39322 df-hlat 39351 df-llines 39499 df-lplanes 39500 df-lvols 39501 df-lines 39502 df-psubsp 39504 df-pmap 39505 df-padd 39797 df-lhyp 39989 df-laut 39990 df-ldil 40105 df-ltrn 40106 df-trl 40160

This theorem is referenced by: cdlemkfid1N 40922

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