Theorem tendococl 40773

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 2730	. 2 ⊢ (le‘𝐾) = (le‘𝐾)
2		tendoco.h	. 2 ⊢ 𝐻 = (LHyp‘𝐾)
3		eqid 2730	. 2 ⊢ ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊)
4		eqid 2730	. 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 40764	. . . 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 40764	. . . 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 6715	. . 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 40766	. . . . . 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 6865	. . . 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 40768	. . . . . 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 40768	. . . . . 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 40766	. . . . 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 2765	. . 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 40720	. . . . 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 40767	. . . 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 40767	. . . . 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 40767	. . . . 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 5831	. . 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 2775	. 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 2730	. . 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 39364	. . 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 40768	. . . . . 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 2829	. . . 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 40165	. . . 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 40165	. . . 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 40165	. . . 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 6865	. . . 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 40762	. . . . 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 5132	. . 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 40762	. . . 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 18412	. 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 40763	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝑇 ∈ 𝐸) → (𝑆 ∘ 𝑇) ∈ 𝐸)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   class class class wbr 5110   ∘ ccom 5645  ⟶wf 6510  ‘cfv 6514  Basecbs 17186  lecple 17234  HLchlt 39350  LHypclh 39985  LTrncltrn 40102  trLctrl 40159  TEndoctendo 40753

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-riotaBAD 38953

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-iin 4961  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-1st 7971  df-2nd 7972  df-undef 8255  df-map 8804  df-proset 18262  df-poset 18281  df-plt 18296  df-lub 18312  df-glb 18313  df-join 18314  df-meet 18315  df-p0 18391  df-p1 18392  df-lat 18398  df-clat 18465  df-oposet 39176  df-ol 39178  df-oml 39179  df-covers 39266  df-ats 39267  df-atl 39298  df-cvlat 39322  df-hlat 39351  df-llines 39499  df-lplanes 39500  df-lvols 39501  df-lines 39502  df-psubsp 39504  df-pmap 39505  df-padd 39797  df-lhyp 39989  df-laut 39990  df-ldil 40105  df-ltrn 40106  df-trl 40160  df-tendo 40756

This theorem is referenced by:  tendodi1  40785  tendodi2  40786  tendo0mul  40827  tendo0mulr  40828  tendoconid  40830  cdleml3N  40979  cdleml8  40984  erngdvlem3  40991  erngdvlem3-rN  40999  dvalveclem  41026  dvhvscacl  41104  dvhlveclem  41109  diblss  41171  dicvscacl  41192  dih1dimatlem0  41329