trljco - Mathbox for Norm Megill

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Theorem trljco 40717

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 5848	. . . . 5 ⊢ (𝐹 = ( I ↾ (Base‘𝐾)) → (𝐹 ∘ 𝐺) = (( I ↾ (Base‘𝐾)) ∘ 𝐺))
2		eqid 2734	. . . . . . . 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 40101	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇) → 𝐺:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
6	5	3adant2 1131	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → 𝐺:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
7		f1of 6828	. . . . . 6 ⊢ (𝐺:(Base‘𝐾)–1-1-onto→(Base‘𝐾) → 𝐺:(Base‘𝐾)⟶(Base‘𝐾))
8		fcoi2 6763	. . . . . 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 2791	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝐹 = ( I ↾ (Base‘𝐾))) → (𝐹 ∘ 𝐺) = 𝐺)
11	10	fveq2d 6890	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝐹 = ( I ↾ (Base‘𝐾))) → (𝑅‘(𝐹 ∘ 𝐺)) = (𝑅‘𝐺))
12	11	oveq2d 7429	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝐹 = ( I ↾ (Base‘𝐾))) → ((𝑅‘𝐹) ∨ (𝑅‘(𝐹 ∘ 𝐺))) = ((𝑅‘𝐹) ∨ (𝑅‘𝐺)))
13		simp1l 1197	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → 𝐾 ∈ HL)
14	13	hllatd 39340	. . . . . 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 40141	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (𝑅‘𝐹) ∈ (Base‘𝐾))
17	16	3adant3 1132	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → (𝑅‘𝐹) ∈ (Base‘𝐾))
18		trljco.j	. . . . . . 7 ⊢ ∨ = (join‘𝐾)
19	2, 18	latjidm 18477	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ (𝑅‘𝐹) ∈ (Base‘𝐾)) → ((𝑅‘𝐹) ∨ (𝑅‘𝐹)) = (𝑅‘𝐹))
20	14, 17, 19	syl2anc 584	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → ((𝑅‘𝐹) ∨ (𝑅‘𝐹)) = (𝑅‘𝐹))
21		hlol 39337	. . . . . . 7 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OL)
22	13, 21	syl 17	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → 𝐾 ∈ OL)
23		eqid 2734	. . . . . . 7 ⊢ (0.‘𝐾) = (0.‘𝐾)
24	2, 18, 23	olj01 39201	. . . . . 6 ⊢ ((𝐾 ∈ OL ∧ (𝑅‘𝐹) ∈ (Base‘𝐾)) → ((𝑅‘𝐹) ∨ (0.‘𝐾)) = (𝑅‘𝐹))
25	22, 17, 24	syl2anc 584	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → ((𝑅‘𝐹) ∨ (0.‘𝐾)) = (𝑅‘𝐹))
26	20, 25	eqtr4d 2772	. . . 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 5849	. . . . . 6 ⊢ (𝐺 = ( I ↾ (Base‘𝐾)) → (𝐹 ∘ 𝐺) = (𝐹 ∘ ( I ↾ (Base‘𝐾))))
29	2, 3, 4	ltrn1o 40101	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → 𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
30	29	3adant3 1132	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → 𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
31		f1of 6828	. . . . . . 7 ⊢ (𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾) → 𝐹:(Base‘𝐾)⟶(Base‘𝐾))
32		fcoi1 6762	. . . . . . 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 2791	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝐺 = ( I ↾ (Base‘𝐾))) → (𝐹 ∘ 𝐺) = 𝐹)
35	34	fveq2d 6890	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝐺 = ( I ↾ (Base‘𝐾))) → (𝑅‘(𝐹 ∘ 𝐺)) = (𝑅‘𝐹))
36	35	oveq2d 7429	. . 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 40155	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇) → (𝐺 = ( I ↾ (Base‘𝐾)) ↔ (𝑅‘𝐺) = (0.‘𝐾)))
38	37	3adant2 1131	. . . . 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 7429	. . 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 2779	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝐺 = ( I ↾ (Base‘𝐾))) → ((𝑅‘𝐹) ∨ (𝑅‘(𝐹 ∘ 𝐺))) = ((𝑅‘𝐹) ∨ (𝑅‘𝐺)))
42		eqid 2734	. . 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 1136	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
45	3, 4	ltrnco 40696	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → (𝐹 ∘ 𝐺) ∈ 𝑇)
46	2, 3, 4, 15	trlcl 40141	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∘ 𝐺) ∈ 𝑇) → (𝑅‘(𝐹 ∘ 𝐺)) ∈ (Base‘𝐾))
47	44, 45, 46	syl2anc 584	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → (𝑅‘(𝐹 ∘ 𝐺)) ∈ (Base‘𝐾))
48	2, 18	latjcl 18454	. . . . 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 1372	. . . 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 40141	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇) → (𝑅‘𝐺) ∈ (Base‘𝐾))
52	51	3adant2 1131	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → (𝑅‘𝐺) ∈ (Base‘𝐾))
53	2, 18	latjcl 18454	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ (𝑅‘𝐹) ∈ (Base‘𝐾) ∧ (𝑅‘𝐺) ∈ (Base‘𝐾)) → ((𝑅‘𝐹) ∨ (𝑅‘𝐺)) ∈ (Base‘𝐾))
54	14, 17, 52, 53	syl3anc 1372	. . . 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 18463	. . . . . 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 1372	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → (𝑅‘𝐹)(le‘𝐾)((𝑅‘𝐹) ∨ (𝑅‘𝐺)))
58	42, 18, 3, 4, 15	trlco 40704	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → (𝑅‘(𝐹 ∘ 𝐺))(le‘𝐾)((𝑅‘𝐹) ∨ (𝑅‘𝐺)))
59	2, 42, 18	latjle12 18465	. . . . . 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 1373	. . . . 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 712	. . . 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 7429	. . . 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 18463	. . . . . . 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 1372	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → (𝑅‘𝐹)(le‘𝐾)((𝑅‘𝐹) ∨ (𝑅‘(𝐹 ∘ 𝐺))))
67	20, 66	eqbrtrd 5145	. . . . 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 5147	. . 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 18460	. 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 1224	. . . 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 1191	. . . . 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 1192	. . . . 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 1194	. . . . 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 2734	. . . . . 6 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
77	2, 76, 3, 4, 15	trlnidat 40150	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ (Base‘𝐾))) → (𝑅‘𝐹) ∈ (Atoms‘𝐾))
78	73, 74, 75, 77	syl3anc 1372	. . . 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 1193	. . . . . 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 1196	. . . . 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 40700	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺)) → (𝑅‘(𝐹 ∘ 𝐺)) ∈ (Atoms‘𝐾))
83	73, 80, 81, 82	syl3anc 1372	. . . 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 1195	. . . . 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 40705	. . . . 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 1375	. . . 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 40150	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇 ∧ 𝐺 ≠ ( I ↾ (Base‘𝐾))) → (𝑅‘𝐺) ∈ (Atoms‘𝐾))
88	73, 79, 84, 87	syl3anc 1372	. . . 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 39454	. . . 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 1389	. . 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 3020	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 1086 = wceq 1539 ∈ wcel 2107 ≠ wne 2931 class class class wbr 5123 I cid 5557 ↾ cres 5667 ∘ ccom 5669 ⟶wf 6537 –1-1-onto→wf1o 6540 ‘cfv 6541 (class class class)co 7413 Basecbs 17230 lecple 17281 joincjn 18328 0.cp0 18438 Latclat 18446 OLcol 39150 Atomscatm 39239 HLchlt 39326 LHypclh 39961 LTrncltrn 40078 trLctrl 40135

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2706 ax-rep 5259 ax-sep 5276 ax-nul 5286 ax-pow 5345 ax-pr 5412 ax-un 7737 ax-riotaBAD 38929

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2808 df-nfc 2884 df-ne 2932 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3420 df-v 3465 df-sbc 3771 df-csb 3880 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-op 4613 df-uni 4888 df-iun 4973 df-iin 4974 df-br 5124 df-opab 5186 df-mpt 5206 df-id 5558 df-xp 5671 df-rel 5672 df-cnv 5673 df-co 5674 df-dm 5675 df-rn 5676 df-res 5677 df-ima 5678 df-iota 6494 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-riota 7370 df-ov 7416 df-oprab 7417 df-mpo 7418 df-1st 7996 df-2nd 7997 df-undef 8280 df-map 8850 df-proset 18311 df-poset 18330 df-plt 18345 df-lub 18361 df-glb 18362 df-join 18363 df-meet 18364 df-p0 18440 df-p1 18441 df-lat 18447 df-clat 18514 df-oposet 39152 df-ol 39154 df-oml 39155 df-covers 39242 df-ats 39243 df-atl 39274 df-cvlat 39298 df-hlat 39327 df-llines 39475 df-lplanes 39476 df-lvols 39477 df-lines 39478 df-psubsp 39480 df-pmap 39481 df-padd 39773 df-lhyp 39965 df-laut 39966 df-ldil 40081 df-ltrn 40082 df-trl 40136

This theorem is referenced by: trljco2 40718 cdlemkid1 40899

W3C validator