Theorem tendococl 40891

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 2733	. 2 ⊢ (le‘𝐾) = (le‘𝐾)
2		tendoco.h	. 2 ⊢ 𝐻 = (LHyp‘𝐾)
3		eqid 2733	. 2 ⊢ ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊)
4		eqid 2733	. 2 ⊢ ((trL‘𝐾)‘𝑊) = ((trL‘𝐾)‘𝑊)
5		tendoco.e	. 2 ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)
6		simp1 1136	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝑇 ∈ 𝐸) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
7		simp2 1137	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝑇 ∈ 𝐸) → 𝑆 ∈ 𝐸)
8	2, 3, 5	tendof 40882	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸) → 𝑆:((LTrn‘𝐾)‘𝑊)⟶((LTrn‘𝐾)‘𝑊))
9	6, 7, 8	syl2anc 584	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝑇 ∈ 𝐸) → 𝑆:((LTrn‘𝐾)‘𝑊)⟶((LTrn‘𝐾)‘𝑊))
10		simp3 1138	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝑇 ∈ 𝐸) → 𝑇 ∈ 𝐸)
11	2, 3, 5	tendof 40882	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑇 ∈ 𝐸) → 𝑇:((LTrn‘𝐾)‘𝑊)⟶((LTrn‘𝐾)‘𝑊))
12	6, 10, 11	syl2anc 584	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝑇 ∈ 𝐸) → 𝑇:((LTrn‘𝐾)‘𝑊)⟶((LTrn‘𝐾)‘𝑊))
13		fco 6680	. . 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 584	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝑇 ∈ 𝐸) → (𝑆 ∘ 𝑇):((LTrn‘𝐾)‘𝑊)⟶((LTrn‘𝐾)‘𝑊))
15		simp11l 1285	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝑇 ∈ 𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∧ 𝑔 ∈ ((LTrn‘𝐾)‘𝑊)) → 𝐾 ∈ HL)
16		simp11r 1286	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝑇 ∈ 𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∧ 𝑔 ∈ ((LTrn‘𝐾)‘𝑊)) → 𝑊 ∈ 𝐻)
17		simp13 1206	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝑇 ∈ 𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∧ 𝑔 ∈ ((LTrn‘𝐾)‘𝑊)) → 𝑇 ∈ 𝐸)
18		simp2 1137	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝑇 ∈ 𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∧ 𝑔 ∈ ((LTrn‘𝐾)‘𝑊)) → 𝑓 ∈ ((LTrn‘𝐾)‘𝑊))
19		simp3 1138	. . . . . 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 40884	. . . . . 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 1380	. . . . 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 6832	. . . 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 1205	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝑇 ∈ 𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∧ 𝑔 ∈ ((LTrn‘𝐾)‘𝑊)) → 𝑆 ∈ 𝐸)
24		simp11 1204	. . . . . 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 40886	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑇 ∈ 𝐸 ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊)) → (𝑇‘𝑓) ∈ ((LTrn‘𝐾)‘𝑊))
26	24, 17, 18, 25	syl3anc 1373	. . . . 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 40886	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑇 ∈ 𝐸 ∧ 𝑔 ∈ ((LTrn‘𝐾)‘𝑊)) → (𝑇‘𝑔) ∈ ((LTrn‘𝐾)‘𝑊))
28	24, 17, 19, 27	syl3anc 1373	. . . . 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 40884	. . . . 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 1380	. . . 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 2768	. . 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 40838	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∧ 𝑔 ∈ ((LTrn‘𝐾)‘𝑊)) → (𝑓 ∘ 𝑔) ∈ ((LTrn‘𝐾)‘𝑊))
33	24, 18, 19, 32	syl3anc 1373	. . . 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 40885	. . . 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 1377	. . 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 40885	. . . . 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 1383	. . . 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 40885	. . . . 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 1383	. . . 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 5808	. . 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 2778	. 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 2733	. . 3 ⊢ (Base‘𝐾) = (Base‘𝐾)
43		simpl1l 1225	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝑇 ∈ 𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊)) → 𝐾 ∈ HL)
44	43	hllatd 39483	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝑇 ∈ 𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊)) → 𝐾 ∈ Lat)
45		simpl1 1192	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝑇 ∈ 𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
46		simpl2 1193	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝑇 ∈ 𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊)) → 𝑆 ∈ 𝐸)
47		simpl3 1194	. . . . . 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 1377	. . . . 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 1373	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝑇 ∈ 𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊)) → (𝑇‘𝑓) ∈ ((LTrn‘𝐾)‘𝑊))
51	2, 3, 5	tendocl 40886	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ (𝑇‘𝑓) ∈ ((LTrn‘𝐾)‘𝑊)) → (𝑆‘(𝑇‘𝑓)) ∈ ((LTrn‘𝐾)‘𝑊))
52	45, 46, 50, 51	syl3anc 1373	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝑇 ∈ 𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊)) → (𝑆‘(𝑇‘𝑓)) ∈ ((LTrn‘𝐾)‘𝑊))
53	49, 52	eqeltrd 2833	. . . 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 40283	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑆 ∘ 𝑇)‘𝑓) ∈ ((LTrn‘𝐾)‘𝑊)) → (((trL‘𝐾)‘𝑊)‘((𝑆 ∘ 𝑇)‘𝑓)) ∈ (Base‘𝐾))
55	45, 53, 54	syl2anc 584	. . 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 40283	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑇‘𝑓) ∈ ((LTrn‘𝐾)‘𝑊)) → (((trL‘𝐾)‘𝑊)‘(𝑇‘𝑓)) ∈ (Base‘𝐾))
57	45, 50, 56	syl2anc 584	. . 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 40283	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊)) → (((trL‘𝐾)‘𝑊)‘𝑓) ∈ (Base‘𝐾))
59	45, 48, 58	syl2anc 584	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝑇 ∈ 𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊)) → (((trL‘𝐾)‘𝑊)‘𝑓) ∈ (Base‘𝐾))
60		simpl1r 1226	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝑇 ∈ 𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊)) → 𝑊 ∈ 𝐻)
61	43, 60, 46, 47, 48, 36	syl221anc 1383	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝑇 ∈ 𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊)) → ((𝑆 ∘ 𝑇)‘𝑓) = (𝑆‘(𝑇‘𝑓)))
62	61	fveq2d 6832	. . . 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 40880	. . . . 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 1373	. . . 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 5115	. . 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 40880	. . . 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 1373	. . 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 18354	. 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 40881	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝑇 ∈ 𝐸) → (𝑆 ∘ 𝑇) ∈ 𝐸)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113   class class class wbr 5093   ∘ ccom 5623  ⟶wf 6482  ‘cfv 6486  Basecbs 17122  lecple 17170  HLchlt 39469  LHypclh 40103  LTrncltrn 40220  trLctrl 40277  TEndoctendo 40871

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5219  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674  ax-riotaBAD 39072

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rmo 3347  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-iun 4943  df-iin 4944  df-br 5094  df-opab 5156  df-mpt 5175  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-riota 7309  df-ov 7355  df-oprab 7356  df-mpo 7357  df-1st 7927  df-2nd 7928  df-undef 8209  df-map 8758  df-proset 18202  df-poset 18221  df-plt 18236  df-lub 18252  df-glb 18253  df-join 18254  df-meet 18255  df-p0 18331  df-p1 18332  df-lat 18340  df-clat 18407  df-oposet 39295  df-ol 39297  df-oml 39298  df-covers 39385  df-ats 39386  df-atl 39417  df-cvlat 39441  df-hlat 39470  df-llines 39617  df-lplanes 39618  df-lvols 39619  df-lines 39620  df-psubsp 39622  df-pmap 39623  df-padd 39915  df-lhyp 40107  df-laut 40108  df-ldil 40223  df-ltrn 40224  df-trl 40278  df-tendo 40874

This theorem is referenced by:  tendodi1  40903  tendodi2  40904  tendo0mul  40945  tendo0mulr  40946  tendoconid  40948  cdleml3N  41097  cdleml8  41102  erngdvlem3  41109  erngdvlem3-rN  41117  dvalveclem  41144  dvhvscacl  41222  dvhlveclem  41227  diblss  41289  dicvscacl  41310  dih1dimatlem0  41447