trljco - Mathbox for Norm Megill

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Theorem trljco 41325

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 5825	. . . . 5 ⊢ (𝐹 = ( I ↾ (Base‘𝐾)) → (𝐹 ∘ 𝐺) = (( I ↾ (Base‘𝐾)) ∘ 𝐺))
2		eqid 2761	. . . . . . . 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 40709	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇) → 𝐺:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
6	5	3adant2 1143	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → 𝐺:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
7		f1of 6801	. . . . . 6 ⊢ (𝐺:(Base‘𝐾)–1-1-onto→(Base‘𝐾) → 𝐺:(Base‘𝐾)⟶(Base‘𝐾))
8		fcoi2 6734	. . . . . 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 2818	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝐹 = ( I ↾ (Base‘𝐾))) → (𝐹 ∘ 𝐺) = 𝐺)
11	10	fveq2d 6866	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝐹 = ( I ↾ (Base‘𝐾))) → (𝑅‘(𝐹 ∘ 𝐺)) = (𝑅‘𝐺))
12	11	oveq2d 7407	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝐹 = ( I ↾ (Base‘𝐾))) → ((𝑅‘𝐹) ∨ (𝑅‘(𝐹 ∘ 𝐺))) = ((𝑅‘𝐹) ∨ (𝑅‘𝐺)))
13		simp1l 1210	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → 𝐾 ∈ HL)
14	13	hllatd 39949	. . . . . 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 40749	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (𝑅‘𝐹) ∈ (Base‘𝐾))
17	16	3adant3 1144	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → (𝑅‘𝐹) ∈ (Base‘𝐾))
18		trljco.j	. . . . . . 7 ⊢ ∨ = (join‘𝐾)
19	2, 18	latjidm 18485	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ (𝑅‘𝐹) ∈ (Base‘𝐾)) → ((𝑅‘𝐹) ∨ (𝑅‘𝐹)) = (𝑅‘𝐹))
20	14, 17, 19	syl2anc 593	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → ((𝑅‘𝐹) ∨ (𝑅‘𝐹)) = (𝑅‘𝐹))
21		hlol 39946	. . . . . . 7 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OL)
22	13, 21	syl 17	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → 𝐾 ∈ OL)
23		eqid 2761	. . . . . . 7 ⊢ (0.‘𝐾) = (0.‘𝐾)
24	2, 18, 23	olj01 39810	. . . . . 6 ⊢ ((𝐾 ∈ OL ∧ (𝑅‘𝐹) ∈ (Base‘𝐾)) → ((𝑅‘𝐹) ∨ (0.‘𝐾)) = (𝑅‘𝐹))
25	22, 17, 24	syl2anc 593	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → ((𝑅‘𝐹) ∨ (0.‘𝐾)) = (𝑅‘𝐹))
26	20, 25	eqtr4d 2799	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → ((𝑅‘𝐹) ∨ (𝑅‘𝐹)) = ((𝑅‘𝐹) ∨ (0.‘𝐾)))
27	26	adantr 484	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝐺 = ( I ↾ (Base‘𝐾))) → ((𝑅‘𝐹) ∨ (𝑅‘𝐹)) = ((𝑅‘𝐹) ∨ (0.‘𝐾)))
28		coeq2 5826	. . . . . 6 ⊢ (𝐺 = ( I ↾ (Base‘𝐾)) → (𝐹 ∘ 𝐺) = (𝐹 ∘ ( I ↾ (Base‘𝐾))))
29	2, 3, 4	ltrn1o 40709	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → 𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
30	29	3adant3 1144	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → 𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
31		f1of 6801	. . . . . . 7 ⊢ (𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾) → 𝐹:(Base‘𝐾)⟶(Base‘𝐾))
32		fcoi1 6733	. . . . . . 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 2818	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝐺 = ( I ↾ (Base‘𝐾))) → (𝐹 ∘ 𝐺) = 𝐹)
35	34	fveq2d 6866	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝐺 = ( I ↾ (Base‘𝐾))) → (𝑅‘(𝐹 ∘ 𝐺)) = (𝑅‘𝐹))
36	35	oveq2d 7407	. . 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 40763	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇) → (𝐺 = ( I ↾ (Base‘𝐾)) ↔ (𝑅‘𝐺) = (0.‘𝐾)))
38	37	3adant2 1143	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → (𝐺 = ( I ↾ (Base‘𝐾)) ↔ (𝑅‘𝐺) = (0.‘𝐾)))
39	38	biimpa 480	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝐺 = ( I ↾ (Base‘𝐾))) → (𝑅‘𝐺) = (0.‘𝐾))
40	39	oveq2d 7407	. . 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 2806	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝐺 = ( I ↾ (Base‘𝐾))) → ((𝑅‘𝐹) ∨ (𝑅‘(𝐹 ∘ 𝐺))) = ((𝑅‘𝐹) ∨ (𝑅‘𝐺)))
42		eqid 2761	. . 3 ⊢ (le‘𝐾) = (le‘𝐾)
43	14	adantr 484	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑅‘𝐹) = (𝑅‘𝐺)) → 𝐾 ∈ Lat)
44		simp1 1148	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
45	3, 4	ltrnco 41304	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → (𝐹 ∘ 𝐺) ∈ 𝑇)
46	2, 3, 4, 15	trlcl 40749	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∘ 𝐺) ∈ 𝑇) → (𝑅‘(𝐹 ∘ 𝐺)) ∈ (Base‘𝐾))
47	44, 45, 46	syl2anc 593	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → (𝑅‘(𝐹 ∘ 𝐺)) ∈ (Base‘𝐾))
48	2, 18	latjcl 18462	. . . . 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 1389	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → ((𝑅‘𝐹) ∨ (𝑅‘(𝐹 ∘ 𝐺))) ∈ (Base‘𝐾))
50	49	adantr 484	. . 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 40749	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇) → (𝑅‘𝐺) ∈ (Base‘𝐾))
52	51	3adant2 1143	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → (𝑅‘𝐺) ∈ (Base‘𝐾))
53	2, 18	latjcl 18462	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ (𝑅‘𝐹) ∈ (Base‘𝐾) ∧ (𝑅‘𝐺) ∈ (Base‘𝐾)) → ((𝑅‘𝐹) ∨ (𝑅‘𝐺)) ∈ (Base‘𝐾))
54	14, 17, 52, 53	syl3anc 1389	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → ((𝑅‘𝐹) ∨ (𝑅‘𝐺)) ∈ (Base‘𝐾))
55	54	adantr 484	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑅‘𝐹) = (𝑅‘𝐺)) → ((𝑅‘𝐹) ∨ (𝑅‘𝐺)) ∈ (Base‘𝐾))
56	2, 42, 18	latlej1 18471	. . . . . 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 1389	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → (𝑅‘𝐹)(le‘𝐾)((𝑅‘𝐹) ∨ (𝑅‘𝐺)))
58	42, 18, 3, 4, 15	trlco 41312	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → (𝑅‘(𝐹 ∘ 𝐺))(le‘𝐾)((𝑅‘𝐹) ∨ (𝑅‘𝐺)))
59	2, 42, 18	latjle12 18473	. . . . . 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 1390	. . . . 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 722	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → ((𝑅‘𝐹) ∨ (𝑅‘(𝐹 ∘ 𝐺)))(le‘𝐾)((𝑅‘𝐹) ∨ (𝑅‘𝐺)))
62	61	adantr 484	. . 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 488	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑅‘𝐹) = (𝑅‘𝐺)) → (𝑅‘𝐹) = (𝑅‘𝐺))
64	63	oveq2d 7407	. . . 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 18471	. . . . . . 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 1389	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → (𝑅‘𝐹)(le‘𝐾)((𝑅‘𝐹) ∨ (𝑅‘(𝐹 ∘ 𝐺))))
67	20, 66	eqbrtrd 5119	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → ((𝑅‘𝐹) ∨ (𝑅‘𝐹))(le‘𝐾)((𝑅‘𝐹) ∨ (𝑅‘(𝐹 ∘ 𝐺))))
68	67	adantr 484	. . . 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 5121	. . 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 18468	. 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 484	. . 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 1237	. . . 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 1204	. . . . 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 1205	. . . . 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 1207	. . . . 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 2761	. . . . . 6 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
77	2, 76, 3, 4, 15	trlnidat 40758	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ (Base‘𝐾))) → (𝑅‘𝐹) ∈ (Atoms‘𝐾))
78	73, 74, 75, 77	syl3anc 1389	. . . 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 1206	. . . . . 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 519	. . . . 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 1209	. . . . 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 41308	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺)) → (𝑅‘(𝐹 ∘ 𝐺)) ∈ (Atoms‘𝐾))
83	73, 80, 81, 82	syl3anc 1389	. . . 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 1208	. . . . 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 41313	. . . . 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 1392	. . . 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 40758	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇 ∧ 𝐺 ≠ ( I ↾ (Base‘𝐾))) → (𝑅‘𝐺) ∈ (Atoms‘𝐾))
88	73, 79, 84, 87	syl3anc 1389	. . . 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 40062	. . . 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 1406	. . 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 234	. 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 3045	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 208 ∧ wa 399 ∧ w3a 1097 = wceq 1559 ∈ wcel 2141 ≠ wne 2956 class class class wbr 5097 I cid 5537 ↾ cres 5645 ∘ ccom 5647 ⟶wf 6512 –1-1-onto→wf1o 6515 ‘cfv 6516 (class class class)co 7391 Basecbs 17236 lecple 17284 joincjn 18334 0.cp0 18444 Latclat 18454 OLcol 39759 Atomscatm 39848 HLchlt 39935 LHypclh 40569 LTrncltrn 40686 trLctrl 40743

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-rep 5224 ax-sep 5243 ax-nul 5253 ax-pow 5319 ax-pr 5387 ax-un 7713 ax-riotaBAD 39538

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1098 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-ral 3076 df-rex 3086 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4580 df-pr 4582 df-op 4586 df-uni 4863 df-iun 4948 df-iin 4949 df-br 5098 df-opab 5160 df-mpt 5179 df-id 5538 df-xp 5649 df-rel 5650 df-cnv 5651 df-co 5652 df-dm 5653 df-rn 5654 df-res 5655 df-ima 5656 df-iota 6472 df-fun 6518 df-fn 6519 df-f 6520 df-f1 6521 df-fo 6522 df-f1o 6523 df-fv 6524 df-riota 7348 df-ov 7394 df-oprab 7395 df-mpo 7396 df-1st 7965 df-2nd 7966 df-undef 8247 df-map 8804 df-proset 18317 df-poset 18336 df-plt 18351 df-lub 18367 df-glb 18368 df-join 18369 df-meet 18370 df-p0 18446 df-p1 18447 df-lat 18455 df-clat 18522 df-oposet 39761 df-ol 39763 df-oml 39764 df-covers 39851 df-ats 39852 df-atl 39883 df-cvlat 39907 df-hlat 39936 df-llines 40083 df-lplanes 40084 df-lvols 40085 df-lines 40086 df-psubsp 40088 df-pmap 40089 df-padd 40381 df-lhyp 40573 df-laut 40574 df-ldil 40689 df-ltrn 40690 df-trl 40744

This theorem is referenced by: trljco2 41326 cdlemkid1 41507

W3C validator