trljco - Mathbox for Norm Megill

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Theorem trljco 41116

Description: Trace joined with trace of composition. (Contributed by NM, 15-Jun-2013.)

**Hypotheses**
Ref	Expression
trljco.j	⊢ ∨ = (join‘𝐾)
trljco.h	⊢ 𝐻 = (LHyp‘𝐾)
trljco.t	⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
trljco.r	⊢ 𝑅 = ((trL‘𝐾)‘𝑊)

**Assertion**
Ref	Expression
trljco	⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → ((𝑅‘𝐹) ∨ (𝑅‘(𝐹 ∘ 𝐺))) = ((𝑅‘𝐹) ∨ (𝑅‘𝐺)))

**Proof of Theorem trljco**
Step	Hyp	Ref	Expression
1		coeq1 5814	. . . . 5 ⊢ (𝐹 = ( I ↾ (Base‘𝐾)) → (𝐹 ∘ 𝐺) = (( I ↾ (Base‘𝐾)) ∘ 𝐺))
2		eqid 2737	. . . . . . . 8 ⊢ (Base‘𝐾) = (Base‘𝐾)
3		trljco.h	. . . . . . . 8 ⊢ 𝐻 = (LHyp‘𝐾)
4		trljco.t	. . . . . . . 8 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
5	2, 3, 4	ltrn1o 40500	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇) → 𝐺:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
6	5	3adant2 1132	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → 𝐺:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
7		f1of 6782	. . . . . 6 ⊢ (𝐺:(Base‘𝐾)–1-1-onto→(Base‘𝐾) → 𝐺:(Base‘𝐾)⟶(Base‘𝐾))
8		fcoi2 6717	. . . . . 6 ⊢ (𝐺:(Base‘𝐾)⟶(Base‘𝐾) → (( I ↾ (Base‘𝐾)) ∘ 𝐺) = 𝐺)
9	6, 7, 8	3syl 18	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → (( I ↾ (Base‘𝐾)) ∘ 𝐺) = 𝐺)
10	1, 9	sylan9eqr 2794	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝐹 = ( I ↾ (Base‘𝐾))) → (𝐹 ∘ 𝐺) = 𝐺)
11	10	fveq2d 6846	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝐹 = ( I ↾ (Base‘𝐾))) → (𝑅‘(𝐹 ∘ 𝐺)) = (𝑅‘𝐺))
12	11	oveq2d 7384	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝐹 = ( I ↾ (Base‘𝐾))) → ((𝑅‘𝐹) ∨ (𝑅‘(𝐹 ∘ 𝐺))) = ((𝑅‘𝐹) ∨ (𝑅‘𝐺)))
13		simp1l 1199	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → 𝐾 ∈ HL)
14	13	hllatd 39740	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → 𝐾 ∈ Lat)
15		trljco.r	. . . . . . . 8 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
16	2, 3, 4, 15	trlcl 40540	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (𝑅‘𝐹) ∈ (Base‘𝐾))
17	16	3adant3 1133	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → (𝑅‘𝐹) ∈ (Base‘𝐾))
18		trljco.j	. . . . . . 7 ⊢ ∨ = (join‘𝐾)
19	2, 18	latjidm 18397	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ (𝑅‘𝐹) ∈ (Base‘𝐾)) → ((𝑅‘𝐹) ∨ (𝑅‘𝐹)) = (𝑅‘𝐹))
20	14, 17, 19	syl2anc 585	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → ((𝑅‘𝐹) ∨ (𝑅‘𝐹)) = (𝑅‘𝐹))
21		hlol 39737	. . . . . . 7 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OL)
22	13, 21	syl 17	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → 𝐾 ∈ OL)
23		eqid 2737	. . . . . . 7 ⊢ (0.‘𝐾) = (0.‘𝐾)
24	2, 18, 23	olj01 39601	. . . . . 6 ⊢ ((𝐾 ∈ OL ∧ (𝑅‘𝐹) ∈ (Base‘𝐾)) → ((𝑅‘𝐹) ∨ (0.‘𝐾)) = (𝑅‘𝐹))
25	22, 17, 24	syl2anc 585	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → ((𝑅‘𝐹) ∨ (0.‘𝐾)) = (𝑅‘𝐹))
26	20, 25	eqtr4d 2775	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → ((𝑅‘𝐹) ∨ (𝑅‘𝐹)) = ((𝑅‘𝐹) ∨ (0.‘𝐾)))
27	26	adantr 480	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝐺 = ( I ↾ (Base‘𝐾))) → ((𝑅‘𝐹) ∨ (𝑅‘𝐹)) = ((𝑅‘𝐹) ∨ (0.‘𝐾)))
28		coeq2 5815	. . . . . 6 ⊢ (𝐺 = ( I ↾ (Base‘𝐾)) → (𝐹 ∘ 𝐺) = (𝐹 ∘ ( I ↾ (Base‘𝐾))))
29	2, 3, 4	ltrn1o 40500	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → 𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
30	29	3adant3 1133	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → 𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
31		f1of 6782	. . . . . . 7 ⊢ (𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾) → 𝐹:(Base‘𝐾)⟶(Base‘𝐾))
32		fcoi1 6716	. . . . . . 7 ⊢ (𝐹:(Base‘𝐾)⟶(Base‘𝐾) → (𝐹 ∘ ( I ↾ (Base‘𝐾))) = 𝐹)
33	30, 31, 32	3syl 18	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → (𝐹 ∘ ( I ↾ (Base‘𝐾))) = 𝐹)
34	28, 33	sylan9eqr 2794	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝐺 = ( I ↾ (Base‘𝐾))) → (𝐹 ∘ 𝐺) = 𝐹)
35	34	fveq2d 6846	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝐺 = ( I ↾ (Base‘𝐾))) → (𝑅‘(𝐹 ∘ 𝐺)) = (𝑅‘𝐹))
36	35	oveq2d 7384	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝐺 = ( I ↾ (Base‘𝐾))) → ((𝑅‘𝐹) ∨ (𝑅‘(𝐹 ∘ 𝐺))) = ((𝑅‘𝐹) ∨ (𝑅‘𝐹)))
37	2, 23, 3, 4, 15	trlid0b 40554	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇) → (𝐺 = ( I ↾ (Base‘𝐾)) ↔ (𝑅‘𝐺) = (0.‘𝐾)))
38	37	3adant2 1132	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → (𝐺 = ( I ↾ (Base‘𝐾)) ↔ (𝑅‘𝐺) = (0.‘𝐾)))
39	38	biimpa 476	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝐺 = ( I ↾ (Base‘𝐾))) → (𝑅‘𝐺) = (0.‘𝐾))
40	39	oveq2d 7384	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝐺 = ( I ↾ (Base‘𝐾))) → ((𝑅‘𝐹) ∨ (𝑅‘𝐺)) = ((𝑅‘𝐹) ∨ (0.‘𝐾)))
41	27, 36, 40	3eqtr4d 2782	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝐺 = ( I ↾ (Base‘𝐾))) → ((𝑅‘𝐹) ∨ (𝑅‘(𝐹 ∘ 𝐺))) = ((𝑅‘𝐹) ∨ (𝑅‘𝐺)))
42		eqid 2737	. . 3 ⊢ (le‘𝐾) = (le‘𝐾)
43	14	adantr 480	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑅‘𝐹) = (𝑅‘𝐺)) → 𝐾 ∈ Lat)
44		simp1 1137	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
45	3, 4	ltrnco 41095	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → (𝐹 ∘ 𝐺) ∈ 𝑇)
46	2, 3, 4, 15	trlcl 40540	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∘ 𝐺) ∈ 𝑇) → (𝑅‘(𝐹 ∘ 𝐺)) ∈ (Base‘𝐾))
47	44, 45, 46	syl2anc 585	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → (𝑅‘(𝐹 ∘ 𝐺)) ∈ (Base‘𝐾))
48	2, 18	latjcl 18374	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ (𝑅‘𝐹) ∈ (Base‘𝐾) ∧ (𝑅‘(𝐹 ∘ 𝐺)) ∈ (Base‘𝐾)) → ((𝑅‘𝐹) ∨ (𝑅‘(𝐹 ∘ 𝐺))) ∈ (Base‘𝐾))
49	14, 17, 47, 48	syl3anc 1374	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → ((𝑅‘𝐹) ∨ (𝑅‘(𝐹 ∘ 𝐺))) ∈ (Base‘𝐾))
50	49	adantr 480	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑅‘𝐹) = (𝑅‘𝐺)) → ((𝑅‘𝐹) ∨ (𝑅‘(𝐹 ∘ 𝐺))) ∈ (Base‘𝐾))
51	2, 3, 4, 15	trlcl 40540	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇) → (𝑅‘𝐺) ∈ (Base‘𝐾))
52	51	3adant2 1132	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → (𝑅‘𝐺) ∈ (Base‘𝐾))
53	2, 18	latjcl 18374	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ (𝑅‘𝐹) ∈ (Base‘𝐾) ∧ (𝑅‘𝐺) ∈ (Base‘𝐾)) → ((𝑅‘𝐹) ∨ (𝑅‘𝐺)) ∈ (Base‘𝐾))
54	14, 17, 52, 53	syl3anc 1374	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → ((𝑅‘𝐹) ∨ (𝑅‘𝐺)) ∈ (Base‘𝐾))
55	54	adantr 480	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑅‘𝐹) = (𝑅‘𝐺)) → ((𝑅‘𝐹) ∨ (𝑅‘𝐺)) ∈ (Base‘𝐾))
56	2, 42, 18	latlej1 18383	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ (𝑅‘𝐹) ∈ (Base‘𝐾) ∧ (𝑅‘𝐺) ∈ (Base‘𝐾)) → (𝑅‘𝐹)(le‘𝐾)((𝑅‘𝐹) ∨ (𝑅‘𝐺)))
57	14, 17, 52, 56	syl3anc 1374	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → (𝑅‘𝐹)(le‘𝐾)((𝑅‘𝐹) ∨ (𝑅‘𝐺)))
58	42, 18, 3, 4, 15	trlco 41103	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → (𝑅‘(𝐹 ∘ 𝐺))(le‘𝐾)((𝑅‘𝐹) ∨ (𝑅‘𝐺)))
59	2, 42, 18	latjle12 18385	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ ((𝑅‘𝐹) ∈ (Base‘𝐾) ∧ (𝑅‘(𝐹 ∘ 𝐺)) ∈ (Base‘𝐾) ∧ ((𝑅‘𝐹) ∨ (𝑅‘𝐺)) ∈ (Base‘𝐾))) → (((𝑅‘𝐹)(le‘𝐾)((𝑅‘𝐹) ∨ (𝑅‘𝐺)) ∧ (𝑅‘(𝐹 ∘ 𝐺))(le‘𝐾)((𝑅‘𝐹) ∨ (𝑅‘𝐺))) ↔ ((𝑅‘𝐹) ∨ (𝑅‘(𝐹 ∘ 𝐺)))(le‘𝐾)((𝑅‘𝐹) ∨ (𝑅‘𝐺))))
60	14, 17, 47, 54, 59	syl13anc 1375	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → (((𝑅‘𝐹)(le‘𝐾)((𝑅‘𝐹) ∨ (𝑅‘𝐺)) ∧ (𝑅‘(𝐹 ∘ 𝐺))(le‘𝐾)((𝑅‘𝐹) ∨ (𝑅‘𝐺))) ↔ ((𝑅‘𝐹) ∨ (𝑅‘(𝐹 ∘ 𝐺)))(le‘𝐾)((𝑅‘𝐹) ∨ (𝑅‘𝐺))))
61	57, 58, 60	mpbi2and 713	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → ((𝑅‘𝐹) ∨ (𝑅‘(𝐹 ∘ 𝐺)))(le‘𝐾)((𝑅‘𝐹) ∨ (𝑅‘𝐺)))
62	61	adantr 480	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑅‘𝐹) = (𝑅‘𝐺)) → ((𝑅‘𝐹) ∨ (𝑅‘(𝐹 ∘ 𝐺)))(le‘𝐾)((𝑅‘𝐹) ∨ (𝑅‘𝐺)))
63		simpr 484	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑅‘𝐹) = (𝑅‘𝐺)) → (𝑅‘𝐹) = (𝑅‘𝐺))
64	63	oveq2d 7384	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑅‘𝐹) = (𝑅‘𝐺)) → ((𝑅‘𝐹) ∨ (𝑅‘𝐹)) = ((𝑅‘𝐹) ∨ (𝑅‘𝐺)))
65	2, 42, 18	latlej1 18383	. . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ (𝑅‘𝐹) ∈ (Base‘𝐾) ∧ (𝑅‘(𝐹 ∘ 𝐺)) ∈ (Base‘𝐾)) → (𝑅‘𝐹)(le‘𝐾)((𝑅‘𝐹) ∨ (𝑅‘(𝐹 ∘ 𝐺))))
66	14, 17, 47, 65	syl3anc 1374	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → (𝑅‘𝐹)(le‘𝐾)((𝑅‘𝐹) ∨ (𝑅‘(𝐹 ∘ 𝐺))))
67	20, 66	eqbrtrd 5122	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → ((𝑅‘𝐹) ∨ (𝑅‘𝐹))(le‘𝐾)((𝑅‘𝐹) ∨ (𝑅‘(𝐹 ∘ 𝐺))))
68	67	adantr 480	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑅‘𝐹) = (𝑅‘𝐺)) → ((𝑅‘𝐹) ∨ (𝑅‘𝐹))(le‘𝐾)((𝑅‘𝐹) ∨ (𝑅‘(𝐹 ∘ 𝐺))))
69	64, 68	eqbrtrrd 5124	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑅‘𝐹) = (𝑅‘𝐺)) → ((𝑅‘𝐹) ∨ (𝑅‘𝐺))(le‘𝐾)((𝑅‘𝐹) ∨ (𝑅‘(𝐹 ∘ 𝐺))))
70	2, 42, 43, 50, 55, 62, 69	latasymd 18380	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑅‘𝐹) = (𝑅‘𝐺)) → ((𝑅‘𝐹) ∨ (𝑅‘(𝐹 ∘ 𝐺))) = ((𝑅‘𝐹) ∨ (𝑅‘𝐺)))
71	61	adantr 480	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝐹 ≠ ( I ↾ (Base‘𝐾)) ∧ 𝐺 ≠ ( I ↾ (Base‘𝐾)) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺))) → ((𝑅‘𝐹) ∨ (𝑅‘(𝐹 ∘ 𝐺)))(le‘𝐾)((𝑅‘𝐹) ∨ (𝑅‘𝐺)))
72		simpl1l 1226	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝐹 ≠ ( I ↾ (Base‘𝐾)) ∧ 𝐺 ≠ ( I ↾ (Base‘𝐾)) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺))) → 𝐾 ∈ HL)
73		simpl1 1193	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝐹 ≠ ( I ↾ (Base‘𝐾)) ∧ 𝐺 ≠ ( I ↾ (Base‘𝐾)) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
74		simpl2 1194	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝐹 ≠ ( I ↾ (Base‘𝐾)) ∧ 𝐺 ≠ ( I ↾ (Base‘𝐾)) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺))) → 𝐹 ∈ 𝑇)
75		simpr1 1196	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝐹 ≠ ( I ↾ (Base‘𝐾)) ∧ 𝐺 ≠ ( I ↾ (Base‘𝐾)) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺))) → 𝐹 ≠ ( I ↾ (Base‘𝐾)))
76		eqid 2737	. . . . . 6 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
77	2, 76, 3, 4, 15	trlnidat 40549	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ (Base‘𝐾))) → (𝑅‘𝐹) ∈ (Atoms‘𝐾))
78	73, 74, 75, 77	syl3anc 1374	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝐹 ≠ ( I ↾ (Base‘𝐾)) ∧ 𝐺 ≠ ( I ↾ (Base‘𝐾)) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺))) → (𝑅‘𝐹) ∈ (Atoms‘𝐾))
79		simpl3 1195	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝐹 ≠ ( I ↾ (Base‘𝐾)) ∧ 𝐺 ≠ ( I ↾ (Base‘𝐾)) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺))) → 𝐺 ∈ 𝑇)
80	74, 79	jca 511	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝐹 ≠ ( I ↾ (Base‘𝐾)) ∧ 𝐺 ≠ ( I ↾ (Base‘𝐾)) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺))) → (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇))
81		simpr3 1198	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝐹 ≠ ( I ↾ (Base‘𝐾)) ∧ 𝐺 ≠ ( I ↾ (Base‘𝐾)) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺))) → (𝑅‘𝐹) ≠ (𝑅‘𝐺))
82	76, 3, 4, 15	trlcoat 41099	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺)) → (𝑅‘(𝐹 ∘ 𝐺)) ∈ (Atoms‘𝐾))
83	73, 80, 81, 82	syl3anc 1374	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝐹 ≠ ( I ↾ (Base‘𝐾)) ∧ 𝐺 ≠ ( I ↾ (Base‘𝐾)) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺))) → (𝑅‘(𝐹 ∘ 𝐺)) ∈ (Atoms‘𝐾))
84		simpr2 1197	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝐹 ≠ ( I ↾ (Base‘𝐾)) ∧ 𝐺 ≠ ( I ↾ (Base‘𝐾)) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺))) → 𝐺 ≠ ( I ↾ (Base‘𝐾)))
85	2, 3, 4, 15	trlcone 41104	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑅‘𝐹) ≠ (𝑅‘𝐺) ∧ 𝐺 ≠ ( I ↾ (Base‘𝐾)))) → (𝑅‘𝐹) ≠ (𝑅‘(𝐹 ∘ 𝐺)))
86	73, 80, 81, 84, 85	syl112anc 1377	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝐹 ≠ ( I ↾ (Base‘𝐾)) ∧ 𝐺 ≠ ( I ↾ (Base‘𝐾)) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺))) → (𝑅‘𝐹) ≠ (𝑅‘(𝐹 ∘ 𝐺)))
87	2, 76, 3, 4, 15	trlnidat 40549	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇 ∧ 𝐺 ≠ ( I ↾ (Base‘𝐾))) → (𝑅‘𝐺) ∈ (Atoms‘𝐾))
88	73, 79, 84, 87	syl3anc 1374	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝐹 ≠ ( I ↾ (Base‘𝐾)) ∧ 𝐺 ≠ ( I ↾ (Base‘𝐾)) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺))) → (𝑅‘𝐺) ∈ (Atoms‘𝐾))
89	42, 18, 76	ps-1 39853	. . . 4 ⊢ ((𝐾 ∈ HL ∧ ((𝑅‘𝐹) ∈ (Atoms‘𝐾) ∧ (𝑅‘(𝐹 ∘ 𝐺)) ∈ (Atoms‘𝐾) ∧ (𝑅‘𝐹) ≠ (𝑅‘(𝐹 ∘ 𝐺))) ∧ ((𝑅‘𝐹) ∈ (Atoms‘𝐾) ∧ (𝑅‘𝐺) ∈ (Atoms‘𝐾))) → (((𝑅‘𝐹) ∨ (𝑅‘(𝐹 ∘ 𝐺)))(le‘𝐾)((𝑅‘𝐹) ∨ (𝑅‘𝐺)) ↔ ((𝑅‘𝐹) ∨ (𝑅‘(𝐹 ∘ 𝐺))) = ((𝑅‘𝐹) ∨ (𝑅‘𝐺))))
90	72, 78, 83, 86, 78, 88, 89	syl132anc 1391	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝐹 ≠ ( I ↾ (Base‘𝐾)) ∧ 𝐺 ≠ ( I ↾ (Base‘𝐾)) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺))) → (((𝑅‘𝐹) ∨ (𝑅‘(𝐹 ∘ 𝐺)))(le‘𝐾)((𝑅‘𝐹) ∨ (𝑅‘𝐺)) ↔ ((𝑅‘𝐹) ∨ (𝑅‘(𝐹 ∘ 𝐺))) = ((𝑅‘𝐹) ∨ (𝑅‘𝐺))))
91	71, 90	mpbid 232	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝐹 ≠ ( I ↾ (Base‘𝐾)) ∧ 𝐺 ≠ ( I ↾ (Base‘𝐾)) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺))) → ((𝑅‘𝐹) ∨ (𝑅‘(𝐹 ∘ 𝐺))) = ((𝑅‘𝐹) ∨ (𝑅‘𝐺)))
92	12, 41, 70, 91	pm2.61da3ne 3022	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → ((𝑅‘𝐹) ∨ (𝑅‘(𝐹 ∘ 𝐺))) = ((𝑅‘𝐹) ∨ (𝑅‘𝐺)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ≠ wne 2933 class class class wbr 5100 I cid 5526 ↾ cres 5634 ∘ ccom 5636 ⟶wf 6496 –1-1-onto→wf1o 6499 ‘cfv 6500 (class class class)co 7368 Basecbs 17148 lecple 17196 joincjn 18246 0.cp0 18356 Latclat 18366 OLcol 39550 Atomscatm 39639 HLchlt 39726 LHypclh 40360 LTrncltrn 40477 trLctrl 40534

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5226 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 ax-riotaBAD 39329

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-ral 3053 df-rex 3063 df-rmo 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-iun 4950 df-iin 4951 df-br 5101 df-opab 5163 df-mpt 5182 df-id 5527 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-1st 7943 df-2nd 7944 df-undef 8225 df-map 8777 df-proset 18229 df-poset 18248 df-plt 18263 df-lub 18279 df-glb 18280 df-join 18281 df-meet 18282 df-p0 18358 df-p1 18359 df-lat 18367 df-clat 18434 df-oposet 39552 df-ol 39554 df-oml 39555 df-covers 39642 df-ats 39643 df-atl 39674 df-cvlat 39698 df-hlat 39727 df-llines 39874 df-lplanes 39875 df-lvols 39876 df-lines 39877 df-psubsp 39879 df-pmap 39880 df-padd 40172 df-lhyp 40364 df-laut 40365 df-ldil 40480 df-ltrn 40481 df-trl 40535

This theorem is referenced by: trljco2 41117 cdlemkid1 41298

W3C validator