Theorem tendococl 40099

Description: The composition of two trace-preserving endomorphisms (multiplication in the endormorphism ring) is a trace-preserving endomorphism. (Contributed by NM, 9-Jun-2013.)

Hypotheses

Ref	Expression
tendoco.h	⊢ 𝐻 = (LHyp‘𝐾)
tendoco.e	⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)

Assertion

Ref	Expression
tendococl	⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝑇 ∈ 𝐸) → (𝑆 ∘ 𝑇) ∈ 𝐸)

Proof of Theorem tendococl

Dummy variables 𝑓 𝑔 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2724	. 2 ⊢ (le‘𝐾) = (le‘𝐾)
2		tendoco.h	. 2 ⊢ 𝐻 = (LHyp‘𝐾)
3		eqid 2724	. 2 ⊢ ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊)
4		eqid 2724	. 2 ⊢ ((trL‘𝐾)‘𝑊) = ((trL‘𝐾)‘𝑊)
5		tendoco.e	. 2 ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)
6		simp1 1133	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝑇 ∈ 𝐸) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
7		simp2 1134	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝑇 ∈ 𝐸) → 𝑆 ∈ 𝐸)
8	2, 3, 5	tendof 40090	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸) → 𝑆:((LTrn‘𝐾)‘𝑊)⟶((LTrn‘𝐾)‘𝑊))
9	6, 7, 8	syl2anc 583	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝑇 ∈ 𝐸) → 𝑆:((LTrn‘𝐾)‘𝑊)⟶((LTrn‘𝐾)‘𝑊))
10		simp3 1135	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝑇 ∈ 𝐸) → 𝑇 ∈ 𝐸)
11	2, 3, 5	tendof 40090	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑇 ∈ 𝐸) → 𝑇:((LTrn‘𝐾)‘𝑊)⟶((LTrn‘𝐾)‘𝑊))
12	6, 10, 11	syl2anc 583	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝑇 ∈ 𝐸) → 𝑇:((LTrn‘𝐾)‘𝑊)⟶((LTrn‘𝐾)‘𝑊))
13		fco 6731	. . 3 ⊢ ((𝑆:((LTrn‘𝐾)‘𝑊)⟶((LTrn‘𝐾)‘𝑊) ∧ 𝑇:((LTrn‘𝐾)‘𝑊)⟶((LTrn‘𝐾)‘𝑊)) → (𝑆 ∘ 𝑇):((LTrn‘𝐾)‘𝑊)⟶((LTrn‘𝐾)‘𝑊))
14	9, 12, 13	syl2anc 583	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝑇 ∈ 𝐸) → (𝑆 ∘ 𝑇):((LTrn‘𝐾)‘𝑊)⟶((LTrn‘𝐾)‘𝑊))
15		simp11l 1281	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝑇 ∈ 𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∧ 𝑔 ∈ ((LTrn‘𝐾)‘𝑊)) → 𝐾 ∈ HL)
16		simp11r 1282	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝑇 ∈ 𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∧ 𝑔 ∈ ((LTrn‘𝐾)‘𝑊)) → 𝑊 ∈ 𝐻)
17		simp13 1202	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝑇 ∈ 𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∧ 𝑔 ∈ ((LTrn‘𝐾)‘𝑊)) → 𝑇 ∈ 𝐸)
18		simp2 1134	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝑇 ∈ 𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∧ 𝑔 ∈ ((LTrn‘𝐾)‘𝑊)) → 𝑓 ∈ ((LTrn‘𝐾)‘𝑊))
19		simp3 1135	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝑇 ∈ 𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∧ 𝑔 ∈ ((LTrn‘𝐾)‘𝑊)) → 𝑔 ∈ ((LTrn‘𝐾)‘𝑊))
20	2, 3, 5	tendovalco 40092	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝑇 ∈ 𝐸) ∧ (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∧ 𝑔 ∈ ((LTrn‘𝐾)‘𝑊))) → (𝑇‘(𝑓 ∘ 𝑔)) = ((𝑇‘𝑓) ∘ (𝑇‘𝑔)))
21	15, 16, 17, 18, 19, 20	syl32anc 1375	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝑇 ∈ 𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∧ 𝑔 ∈ ((LTrn‘𝐾)‘𝑊)) → (𝑇‘(𝑓 ∘ 𝑔)) = ((𝑇‘𝑓) ∘ (𝑇‘𝑔)))
22	21	fveq2d 6885	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝑇 ∈ 𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∧ 𝑔 ∈ ((LTrn‘𝐾)‘𝑊)) → (𝑆‘(𝑇‘(𝑓 ∘ 𝑔))) = (𝑆‘((𝑇‘𝑓) ∘ (𝑇‘𝑔))))
23		simp12 1201	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝑇 ∈ 𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∧ 𝑔 ∈ ((LTrn‘𝐾)‘𝑊)) → 𝑆 ∈ 𝐸)
24		simp11 1200	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝑇 ∈ 𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∧ 𝑔 ∈ ((LTrn‘𝐾)‘𝑊)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
25	2, 3, 5	tendocl 40094	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑇 ∈ 𝐸 ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊)) → (𝑇‘𝑓) ∈ ((LTrn‘𝐾)‘𝑊))
26	24, 17, 18, 25	syl3anc 1368	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝑇 ∈ 𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∧ 𝑔 ∈ ((LTrn‘𝐾)‘𝑊)) → (𝑇‘𝑓) ∈ ((LTrn‘𝐾)‘𝑊))
27	2, 3, 5	tendocl 40094	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑇 ∈ 𝐸 ∧ 𝑔 ∈ ((LTrn‘𝐾)‘𝑊)) → (𝑇‘𝑔) ∈ ((LTrn‘𝐾)‘𝑊))
28	24, 17, 19, 27	syl3anc 1368	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝑇 ∈ 𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∧ 𝑔 ∈ ((LTrn‘𝐾)‘𝑊)) → (𝑇‘𝑔) ∈ ((LTrn‘𝐾)‘𝑊))
29	2, 3, 5	tendovalco 40092	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝑆 ∈ 𝐸) ∧ ((𝑇‘𝑓) ∈ ((LTrn‘𝐾)‘𝑊) ∧ (𝑇‘𝑔) ∈ ((LTrn‘𝐾)‘𝑊))) → (𝑆‘((𝑇‘𝑓) ∘ (𝑇‘𝑔))) = ((𝑆‘(𝑇‘𝑓)) ∘ (𝑆‘(𝑇‘𝑔))))
30	15, 16, 23, 26, 28, 29	syl32anc 1375	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝑇 ∈ 𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∧ 𝑔 ∈ ((LTrn‘𝐾)‘𝑊)) → (𝑆‘((𝑇‘𝑓) ∘ (𝑇‘𝑔))) = ((𝑆‘(𝑇‘𝑓)) ∘ (𝑆‘(𝑇‘𝑔))))
31	22, 30	eqtrd 2764	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝑇 ∈ 𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∧ 𝑔 ∈ ((LTrn‘𝐾)‘𝑊)) → (𝑆‘(𝑇‘(𝑓 ∘ 𝑔))) = ((𝑆‘(𝑇‘𝑓)) ∘ (𝑆‘(𝑇‘𝑔))))
32	2, 3	ltrnco 40046	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∧ 𝑔 ∈ ((LTrn‘𝐾)‘𝑊)) → (𝑓 ∘ 𝑔) ∈ ((LTrn‘𝐾)‘𝑊))
33	24, 18, 19, 32	syl3anc 1368	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝑇 ∈ 𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∧ 𝑔 ∈ ((LTrn‘𝐾)‘𝑊)) → (𝑓 ∘ 𝑔) ∈ ((LTrn‘𝐾)‘𝑊))
34	2, 3, 5	tendocoval 40093	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ∈ 𝐸 ∧ 𝑇 ∈ 𝐸) ∧ (𝑓 ∘ 𝑔) ∈ ((LTrn‘𝐾)‘𝑊)) → ((𝑆 ∘ 𝑇)‘(𝑓 ∘ 𝑔)) = (𝑆‘(𝑇‘(𝑓 ∘ 𝑔))))
35	24, 23, 17, 33, 34	syl121anc 1372	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝑇 ∈ 𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∧ 𝑔 ∈ ((LTrn‘𝐾)‘𝑊)) → ((𝑆 ∘ 𝑇)‘(𝑓 ∘ 𝑔)) = (𝑆‘(𝑇‘(𝑓 ∘ 𝑔))))
36	2, 3, 5	tendocoval 40093	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ∈ 𝐸 ∧ 𝑇 ∈ 𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊)) → ((𝑆 ∘ 𝑇)‘𝑓) = (𝑆‘(𝑇‘𝑓)))
37	15, 16, 23, 17, 18, 36	syl221anc 1378	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝑇 ∈ 𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∧ 𝑔 ∈ ((LTrn‘𝐾)‘𝑊)) → ((𝑆 ∘ 𝑇)‘𝑓) = (𝑆‘(𝑇‘𝑓)))
38	2, 3, 5	tendocoval 40093	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ∈ 𝐸 ∧ 𝑇 ∈ 𝐸) ∧ 𝑔 ∈ ((LTrn‘𝐾)‘𝑊)) → ((𝑆 ∘ 𝑇)‘𝑔) = (𝑆‘(𝑇‘𝑔)))
39	15, 16, 23, 17, 19, 38	syl221anc 1378	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝑇 ∈ 𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∧ 𝑔 ∈ ((LTrn‘𝐾)‘𝑊)) → ((𝑆 ∘ 𝑇)‘𝑔) = (𝑆‘(𝑇‘𝑔)))
40	37, 39	coeq12d 5854	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝑇 ∈ 𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∧ 𝑔 ∈ ((LTrn‘𝐾)‘𝑊)) → (((𝑆 ∘ 𝑇)‘𝑓) ∘ ((𝑆 ∘ 𝑇)‘𝑔)) = ((𝑆‘(𝑇‘𝑓)) ∘ (𝑆‘(𝑇‘𝑔))))
41	31, 35, 40	3eqtr4d 2774	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝑇 ∈ 𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∧ 𝑔 ∈ ((LTrn‘𝐾)‘𝑊)) → ((𝑆 ∘ 𝑇)‘(𝑓 ∘ 𝑔)) = (((𝑆 ∘ 𝑇)‘𝑓) ∘ ((𝑆 ∘ 𝑇)‘𝑔)))
42		eqid 2724	. . 3 ⊢ (Base‘𝐾) = (Base‘𝐾)
43		simpl1l 1221	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝑇 ∈ 𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊)) → 𝐾 ∈ HL)
44	43	hllatd 38690	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝑇 ∈ 𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊)) → 𝐾 ∈ Lat)
45		simpl1 1188	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝑇 ∈ 𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
46		simpl2 1189	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝑇 ∈ 𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊)) → 𝑆 ∈ 𝐸)
47		simpl3 1190	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝑇 ∈ 𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊)) → 𝑇 ∈ 𝐸)
48		simpr 484	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝑇 ∈ 𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊)) → 𝑓 ∈ ((LTrn‘𝐾)‘𝑊))
49	45, 46, 47, 48, 36	syl121anc 1372	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝑇 ∈ 𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊)) → ((𝑆 ∘ 𝑇)‘𝑓) = (𝑆‘(𝑇‘𝑓)))
50	45, 47, 48, 25	syl3anc 1368	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝑇 ∈ 𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊)) → (𝑇‘𝑓) ∈ ((LTrn‘𝐾)‘𝑊))
51	2, 3, 5	tendocl 40094	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ (𝑇‘𝑓) ∈ ((LTrn‘𝐾)‘𝑊)) → (𝑆‘(𝑇‘𝑓)) ∈ ((LTrn‘𝐾)‘𝑊))
52	45, 46, 50, 51	syl3anc 1368	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝑇 ∈ 𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊)) → (𝑆‘(𝑇‘𝑓)) ∈ ((LTrn‘𝐾)‘𝑊))
53	49, 52	eqeltrd 2825	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝑇 ∈ 𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊)) → ((𝑆 ∘ 𝑇)‘𝑓) ∈ ((LTrn‘𝐾)‘𝑊))
54	42, 2, 3, 4	trlcl 39491	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑆 ∘ 𝑇)‘𝑓) ∈ ((LTrn‘𝐾)‘𝑊)) → (((trL‘𝐾)‘𝑊)‘((𝑆 ∘ 𝑇)‘𝑓)) ∈ (Base‘𝐾))
55	45, 53, 54	syl2anc 583	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝑇 ∈ 𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊)) → (((trL‘𝐾)‘𝑊)‘((𝑆 ∘ 𝑇)‘𝑓)) ∈ (Base‘𝐾))
56	42, 2, 3, 4	trlcl 39491	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑇‘𝑓) ∈ ((LTrn‘𝐾)‘𝑊)) → (((trL‘𝐾)‘𝑊)‘(𝑇‘𝑓)) ∈ (Base‘𝐾))
57	45, 50, 56	syl2anc 583	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝑇 ∈ 𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊)) → (((trL‘𝐾)‘𝑊)‘(𝑇‘𝑓)) ∈ (Base‘𝐾))
58	42, 2, 3, 4	trlcl 39491	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊)) → (((trL‘𝐾)‘𝑊)‘𝑓) ∈ (Base‘𝐾))
59	45, 48, 58	syl2anc 583	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝑇 ∈ 𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊)) → (((trL‘𝐾)‘𝑊)‘𝑓) ∈ (Base‘𝐾))
60		simpl1r 1222	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝑇 ∈ 𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊)) → 𝑊 ∈ 𝐻)
61	43, 60, 46, 47, 48, 36	syl221anc 1378	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝑇 ∈ 𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊)) → ((𝑆 ∘ 𝑇)‘𝑓) = (𝑆‘(𝑇‘𝑓)))
62	61	fveq2d 6885	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝑇 ∈ 𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊)) → (((trL‘𝐾)‘𝑊)‘((𝑆 ∘ 𝑇)‘𝑓)) = (((trL‘𝐾)‘𝑊)‘(𝑆‘(𝑇‘𝑓))))
63	1, 2, 3, 4, 5	tendotp 40088	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ (𝑇‘𝑓) ∈ ((LTrn‘𝐾)‘𝑊)) → (((trL‘𝐾)‘𝑊)‘(𝑆‘(𝑇‘𝑓)))(le‘𝐾)(((trL‘𝐾)‘𝑊)‘(𝑇‘𝑓)))
64	45, 46, 50, 63	syl3anc 1368	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝑇 ∈ 𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊)) → (((trL‘𝐾)‘𝑊)‘(𝑆‘(𝑇‘𝑓)))(le‘𝐾)(((trL‘𝐾)‘𝑊)‘(𝑇‘𝑓)))
65	62, 64	eqbrtrd 5160	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝑇 ∈ 𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊)) → (((trL‘𝐾)‘𝑊)‘((𝑆 ∘ 𝑇)‘𝑓))(le‘𝐾)(((trL‘𝐾)‘𝑊)‘(𝑇‘𝑓)))
66	1, 2, 3, 4, 5	tendotp 40088	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑇 ∈ 𝐸 ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊)) → (((trL‘𝐾)‘𝑊)‘(𝑇‘𝑓))(le‘𝐾)(((trL‘𝐾)‘𝑊)‘𝑓))
67	45, 47, 48, 66	syl3anc 1368	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝑇 ∈ 𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊)) → (((trL‘𝐾)‘𝑊)‘(𝑇‘𝑓))(le‘𝐾)(((trL‘𝐾)‘𝑊)‘𝑓))
68	42, 1, 44, 55, 57, 59, 65, 67	lattrd 18398	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝑇 ∈ 𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊)) → (((trL‘𝐾)‘𝑊)‘((𝑆 ∘ 𝑇)‘𝑓))(le‘𝐾)(((trL‘𝐾)‘𝑊)‘𝑓))
69	1, 2, 3, 4, 5, 6, 14, 41, 68	istendod 40089	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝑇 ∈ 𝐸) → (𝑆 ∘ 𝑇) ∈ 𝐸)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   class class class wbr 5138   ∘ ccom 5670  ⟶wf 6529  ‘cfv 6533  Basecbs 17140  lecple 17200  HLchlt 38676  LHypclh 39311  LTrncltrn 39428  trLctrl 39485  TEndoctendo 40079

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718  ax-riotaBAD 38279

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rmo 3368  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-iun 4989  df-iin 4990  df-br 5139  df-opab 5201  df-mpt 5222  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7357  df-ov 7404  df-oprab 7405  df-mpo 7406  df-1st 7968  df-2nd 7969  df-undef 8253  df-map 8817  df-proset 18247  df-poset 18265  df-plt 18282  df-lub 18298  df-glb 18299  df-join 18300  df-meet 18301  df-p0 18377  df-p1 18378  df-lat 18384  df-clat 18451  df-oposet 38502  df-ol 38504  df-oml 38505  df-covers 38592  df-ats 38593  df-atl 38624  df-cvlat 38648  df-hlat 38677  df-llines 38825  df-lplanes 38826  df-lvols 38827  df-lines 38828  df-psubsp 38830  df-pmap 38831  df-padd 39123  df-lhyp 39315  df-laut 39316  df-ldil 39431  df-ltrn 39432  df-trl 39486  df-tendo 40082

This theorem is referenced by:  tendodi1  40111  tendodi2  40112  tendo0mul  40153  tendo0mulr  40154  tendoconid  40156  cdleml3N  40305  cdleml8  40310  erngdvlem3  40317  erngdvlem3-rN  40325  dvalveclem  40352  dvhvscacl  40430  dvhlveclem  40435  diblss  40497  dicvscacl  40518  dih1dimatlem0  40655