trljco - Mathbox for Norm Megill

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Theorem trljco 41547

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 5845	. . . . 5 ⊢ (𝐹 = ( I ↾ (Base‘𝐾)) → (𝐹 ∘ 𝐺) = (( I ↾ (Base‘𝐾)) ∘ 𝐺))
2		eqid 2765	. . . . . . . 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 40931	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇) → 𝐺:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
6	5	3adant2 1149	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → 𝐺:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
7		f1of 6824	. . . . . 6 ⊢ (𝐺:(Base‘𝐾)–1-1-onto→(Base‘𝐾) → 𝐺:(Base‘𝐾)⟶(Base‘𝐾))
8		fcoi2 6757	. . . . . 6 ⊢ (𝐺:(Base‘𝐾)⟶(Base‘𝐾) → (( I ↾ (Base‘𝐾)) ∘ 𝐺) = 𝐺)
9	6, 7, 8	3syl 19	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → (( I ↾ (Base‘𝐾)) ∘ 𝐺) = 𝐺)
10	1, 9	sylan9eqr 2822	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝐹 = ( I ↾ (Base‘𝐾))) → (𝐹 ∘ 𝐺) = 𝐺)
11	10	fveq2d 6889	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝐹 = ( I ↾ (Base‘𝐾))) → (𝑅‘(𝐹 ∘ 𝐺)) = (𝑅‘𝐺))
12	11	oveq2d 7432	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝐹 = ( I ↾ (Base‘𝐾))) → ((𝑅‘𝐹) ∨ (𝑅‘(𝐹 ∘ 𝐺))) = ((𝑅‘𝐹) ∨ (𝑅‘𝐺)))
13		simp1l 1216	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → 𝐾 ∈ HL)
14	13	hllatd 40171	. . . . . 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 40971	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (𝑅‘𝐹) ∈ (Base‘𝐾))
17	16	3adant3 1150	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → (𝑅‘𝐹) ∈ (Base‘𝐾))
18		trljco.j	. . . . . . 7 ⊢ ∨ = (join‘𝐾)
19	2, 18	latjidm 18536	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ (𝑅‘𝐹) ∈ (Base‘𝐾)) → ((𝑅‘𝐹) ∨ (𝑅‘𝐹)) = (𝑅‘𝐹))
20	14, 17, 19	syl2anc 596	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → ((𝑅‘𝐹) ∨ (𝑅‘𝐹)) = (𝑅‘𝐹))
21		hlol 40168	. . . . . . 7 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OL)
22	13, 21	syl 18	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → 𝐾 ∈ OL)
23		eqid 2765	. . . . . . 7 ⊢ (0.‘𝐾) = (0.‘𝐾)
24	2, 18, 23	olj01 40032	. . . . . 6 ⊢ ((𝐾 ∈ OL ∧ (𝑅‘𝐹) ∈ (Base‘𝐾)) → ((𝑅‘𝐹) ∨ (0.‘𝐾)) = (𝑅‘𝐹))
25	22, 17, 24	syl2anc 596	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → ((𝑅‘𝐹) ∨ (0.‘𝐾)) = (𝑅‘𝐹))
26	20, 25	eqtr4d 2803	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → ((𝑅‘𝐹) ∨ (𝑅‘𝐹)) = ((𝑅‘𝐹) ∨ (0.‘𝐾)))
27	26	adantr 486	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝐺 = ( I ↾ (Base‘𝐾))) → ((𝑅‘𝐹) ∨ (𝑅‘𝐹)) = ((𝑅‘𝐹) ∨ (0.‘𝐾)))
28		coeq2 5846	. . . . . 6 ⊢ (𝐺 = ( I ↾ (Base‘𝐾)) → (𝐹 ∘ 𝐺) = (𝐹 ∘ ( I ↾ (Base‘𝐾))))
29	2, 3, 4	ltrn1o 40931	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → 𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
30	29	3adant3 1150	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → 𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
31		f1of 6824	. . . . . . 7 ⊢ (𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾) → 𝐹:(Base‘𝐾)⟶(Base‘𝐾))
32		fcoi1 6756	. . . . . . 7 ⊢ (𝐹:(Base‘𝐾)⟶(Base‘𝐾) → (𝐹 ∘ ( I ↾ (Base‘𝐾))) = 𝐹)
33	30, 31, 32	3syl 19	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → (𝐹 ∘ ( I ↾ (Base‘𝐾))) = 𝐹)
34	28, 33	sylan9eqr 2822	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝐺 = ( I ↾ (Base‘𝐾))) → (𝐹 ∘ 𝐺) = 𝐹)
35	34	fveq2d 6889	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝐺 = ( I ↾ (Base‘𝐾))) → (𝑅‘(𝐹 ∘ 𝐺)) = (𝑅‘𝐹))
36	35	oveq2d 7432	. . 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 40985	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇) → (𝐺 = ( I ↾ (Base‘𝐾)) ↔ (𝑅‘𝐺) = (0.‘𝐾)))
38	37	3adant2 1149	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → (𝐺 = ( I ↾ (Base‘𝐾)) ↔ (𝑅‘𝐺) = (0.‘𝐾)))
39	38	biimpa 482	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝐺 = ( I ↾ (Base‘𝐾))) → (𝑅‘𝐺) = (0.‘𝐾))
40	39	oveq2d 7432	. . 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 2810	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝐺 = ( I ↾ (Base‘𝐾))) → ((𝑅‘𝐹) ∨ (𝑅‘(𝐹 ∘ 𝐺))) = ((𝑅‘𝐹) ∨ (𝑅‘𝐺)))
42		eqid 2765	. . 3 ⊢ (le‘𝐾) = (le‘𝐾)
43	14	adantr 486	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑅‘𝐹) = (𝑅‘𝐺)) → 𝐾 ∈ Lat)
44		simp1 1154	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
45	3, 4	ltrnco 41526	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → (𝐹 ∘ 𝐺) ∈ 𝑇)
46	2, 3, 4, 15	trlcl 40971	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∘ 𝐺) ∈ 𝑇) → (𝑅‘(𝐹 ∘ 𝐺)) ∈ (Base‘𝐾))
47	44, 45, 46	syl2anc 596	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → (𝑅‘(𝐹 ∘ 𝐺)) ∈ (Base‘𝐾))
48	2, 18	latjcl 18513	. . . . 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 1398	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → ((𝑅‘𝐹) ∨ (𝑅‘(𝐹 ∘ 𝐺))) ∈ (Base‘𝐾))
50	49	adantr 486	. . 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 40971	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇) → (𝑅‘𝐺) ∈ (Base‘𝐾))
52	51	3adant2 1149	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → (𝑅‘𝐺) ∈ (Base‘𝐾))
53	2, 18	latjcl 18513	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ (𝑅‘𝐹) ∈ (Base‘𝐾) ∧ (𝑅‘𝐺) ∈ (Base‘𝐾)) → ((𝑅‘𝐹) ∨ (𝑅‘𝐺)) ∈ (Base‘𝐾))
54	14, 17, 52, 53	syl3anc 1398	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → ((𝑅‘𝐹) ∨ (𝑅‘𝐺)) ∈ (Base‘𝐾))
55	54	adantr 486	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑅‘𝐹) = (𝑅‘𝐺)) → ((𝑅‘𝐹) ∨ (𝑅‘𝐺)) ∈ (Base‘𝐾))
56	2, 42, 18	latlej1 18522	. . . . . 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 1398	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → (𝑅‘𝐹)(le‘𝐾)((𝑅‘𝐹) ∨ (𝑅‘𝐺)))
58	42, 18, 3, 4, 15	trlco 41534	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → (𝑅‘(𝐹 ∘ 𝐺))(le‘𝐾)((𝑅‘𝐹) ∨ (𝑅‘𝐺)))
59	2, 42, 18	latjle12 18524	. . . . . 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 1399	. . . . 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 725	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → ((𝑅‘𝐹) ∨ (𝑅‘(𝐹 ∘ 𝐺)))(le‘𝐾)((𝑅‘𝐹) ∨ (𝑅‘𝐺)))
62	61	adantr 486	. . 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 490	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑅‘𝐹) = (𝑅‘𝐺)) → (𝑅‘𝐹) = (𝑅‘𝐺))
64	63	oveq2d 7432	. . . 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 18522	. . . . . . 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 1398	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → (𝑅‘𝐹)(le‘𝐾)((𝑅‘𝐹) ∨ (𝑅‘(𝐹 ∘ 𝐺))))
67	20, 66	eqbrtrd 5135	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → ((𝑅‘𝐹) ∨ (𝑅‘𝐹))(le‘𝐾)((𝑅‘𝐹) ∨ (𝑅‘(𝐹 ∘ 𝐺))))
68	67	adantr 486	. . . 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 5137	. . 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 18519	. 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 486	. . 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 1243	. . . 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 1210	. . . . 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 1211	. . . . 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 1213	. . . . 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 2765	. . . . . 6 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
77	2, 76, 3, 4, 15	trlnidat 40980	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ (Base‘𝐾))) → (𝑅‘𝐹) ∈ (Atoms‘𝐾))
78	73, 74, 75, 77	syl3anc 1398	. . . 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 1212	. . . . . 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 521	. . . . 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 1215	. . . . 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 41530	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺)) → (𝑅‘(𝐹 ∘ 𝐺)) ∈ (Atoms‘𝐾))
83	73, 80, 81, 82	syl3anc 1398	. . . 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 1214	. . . . 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 41535	. . . . 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 1401	. . . 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 40980	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇 ∧ 𝐺 ≠ ( I ↾ (Base‘𝐾))) → (𝑅‘𝐺) ∈ (Atoms‘𝐾))
88	73, 79, 84, 87	syl3anc 1398	. . . 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 40284	. . . 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 1415	. . 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 235	. 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 3049	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → ((𝑅‘𝐹) ∨ (𝑅‘(𝐹 ∘ 𝐺))) = ((𝑅‘𝐹) ∨ (𝑅‘𝐺)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2146 ≠ wne 2960 class class class wbr 5111 I cid 5557 ↾ cres 5665 ∘ ccom 5667 ⟶wf 6536 –1-1-onto→wf1o 6539 ‘cfv 6540 (class class class)co 7416 Basecbs 17287 lecple 17335 joincjn 18385 0.cp0 18495 Latclat 18505 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 18368 df-poset 18387 df-plt 18402 df-lub 18418 df-glb 18419 df-join 18420 df-meet 18421 df-p0 18497 df-p1 18498 df-lat 18506 df-clat 18573 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: trljco2 41548 cdlemkid1 41729

W3C validator