trljco - Mathbox for Norm Megill

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Theorem trljco 41200

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 5806	. . . . 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 40584	. . . . . . 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 6774	. . . . . 6 ⊢ (𝐺:(Base‘𝐾)–1-1-onto→(Base‘𝐾) → 𝐺:(Base‘𝐾)⟶(Base‘𝐾))
8		fcoi2 6709	. . . . . 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 6838	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝐹 = ( I ↾ (Base‘𝐾))) → (𝑅‘(𝐹 ∘ 𝐺)) = (𝑅‘𝐺))
12	11	oveq2d 7376	. 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 39824	. . . . . 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 40624	. . . . . . 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 18419	. . . . . 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 39821	. . . . . . 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 39685	. . . . . 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 5807	. . . . . 6 ⊢ (𝐺 = ( I ↾ (Base‘𝐾)) → (𝐹 ∘ 𝐺) = (𝐹 ∘ ( I ↾ (Base‘𝐾))))
29	2, 3, 4	ltrn1o 40584	. . . . . . . 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 6774	. . . . . . 7 ⊢ (𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾) → 𝐹:(Base‘𝐾)⟶(Base‘𝐾))
32		fcoi1 6708	. . . . . . 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 6838	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝐺 = ( I ↾ (Base‘𝐾))) → (𝑅‘(𝐹 ∘ 𝐺)) = (𝑅‘𝐹))
36	35	oveq2d 7376	. . 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 40638	. . . . . 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 7376	. . 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 41179	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → (𝐹 ∘ 𝐺) ∈ 𝑇)
46	2, 3, 4, 15	trlcl 40624	. . . . . 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 18396	. . . . 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 40624	. . . . . 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 18396	. . . . 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 18405	. . . . . 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 41187	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → (𝑅‘(𝐹 ∘ 𝐺))(le‘𝐾)((𝑅‘𝐹) ∨ (𝑅‘𝐺)))
59	2, 42, 18	latjle12 18407	. . . . . 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 7376	. . . 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 18405	. . . . . . 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 5108	. . . . 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 5110	. . 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 18402	. 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 40633	. . . . 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 41183	. . . . 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 41188	. . . . 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 40633	. . . . 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 39937	. . . 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 5086 I cid 5518 ↾ cres 5626 ∘ ccom 5628 ⟶wf 6488 –1-1-onto→wf1o 6491 ‘cfv 6492 (class class class)co 7360 Basecbs 17170 lecple 17218 joincjn 18268 0.cp0 18378 Latclat 18388 OLcol 39634 Atomscatm 39723 HLchlt 39810 LHypclh 40444 LTrncltrn 40561 trLctrl 40618

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 5212 ax-sep 5231 ax-nul 5241 ax-pow 5302 ax-pr 5370 ax-un 7682 ax-riotaBAD 39413

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 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-iun 4936 df-iin 4937 df-br 5087 df-opab 5149 df-mpt 5168 df-id 5519 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-1st 7935 df-2nd 7936 df-undef 8216 df-map 8768 df-proset 18251 df-poset 18270 df-plt 18285 df-lub 18301 df-glb 18302 df-join 18303 df-meet 18304 df-p0 18380 df-p1 18381 df-lat 18389 df-clat 18456 df-oposet 39636 df-ol 39638 df-oml 39639 df-covers 39726 df-ats 39727 df-atl 39758 df-cvlat 39782 df-hlat 39811 df-llines 39958 df-lplanes 39959 df-lvols 39960 df-lines 39961 df-psubsp 39963 df-pmap 39964 df-padd 40256 df-lhyp 40448 df-laut 40449 df-ldil 40564 df-ltrn 40565 df-trl 40619

This theorem is referenced by: trljco2 41201 cdlemkid1 41382

W3C validator