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Theorem tendococl 38333
 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 ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) → (𝑆𝑇) ∈ 𝐸)

Proof of Theorem tendococl
Dummy variables 𝑓 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2759 . 2 (le‘𝐾) = (le‘𝐾)
2 tendoco.h . 2 𝐻 = (LHyp‘𝐾)
3 eqid 2759 . 2 ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊)
4 eqid 2759 . 2 ((trL‘𝐾)‘𝑊) = ((trL‘𝐾)‘𝑊)
5 tendoco.e . 2 𝐸 = ((TEndo‘𝐾)‘𝑊)
6 simp1 1134 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) → (𝐾 ∈ HL ∧ 𝑊𝐻))
7 simp2 1135 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) → 𝑆𝐸)
82, 3, 5tendof 38324 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸) → 𝑆:((LTrn‘𝐾)‘𝑊)⟶((LTrn‘𝐾)‘𝑊))
96, 7, 8syl2anc 588 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) → 𝑆:((LTrn‘𝐾)‘𝑊)⟶((LTrn‘𝐾)‘𝑊))
10 simp3 1136 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) → 𝑇𝐸)
112, 3, 5tendof 38324 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑇𝐸) → 𝑇:((LTrn‘𝐾)‘𝑊)⟶((LTrn‘𝐾)‘𝑊))
126, 10, 11syl2anc 588 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) → 𝑇:((LTrn‘𝐾)‘𝑊)⟶((LTrn‘𝐾)‘𝑊))
13 fco 6509 . . 3 ((𝑆:((LTrn‘𝐾)‘𝑊)⟶((LTrn‘𝐾)‘𝑊) ∧ 𝑇:((LTrn‘𝐾)‘𝑊)⟶((LTrn‘𝐾)‘𝑊)) → (𝑆𝑇):((LTrn‘𝐾)‘𝑊)⟶((LTrn‘𝐾)‘𝑊))
149, 12, 13syl2anc 588 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) → (𝑆𝑇):((LTrn‘𝐾)‘𝑊)⟶((LTrn‘𝐾)‘𝑊))
15 simp11l 1282 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∧ 𝑔 ∈ ((LTrn‘𝐾)‘𝑊)) → 𝐾 ∈ HL)
16 simp11r 1283 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∧ 𝑔 ∈ ((LTrn‘𝐾)‘𝑊)) → 𝑊𝐻)
17 simp13 1203 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∧ 𝑔 ∈ ((LTrn‘𝐾)‘𝑊)) → 𝑇𝐸)
18 simp2 1135 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∧ 𝑔 ∈ ((LTrn‘𝐾)‘𝑊)) → 𝑓 ∈ ((LTrn‘𝐾)‘𝑊))
19 simp3 1136 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∧ 𝑔 ∈ ((LTrn‘𝐾)‘𝑊)) → 𝑔 ∈ ((LTrn‘𝐾)‘𝑊))
202, 3, 5tendovalco 38326 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻𝑇𝐸) ∧ (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∧ 𝑔 ∈ ((LTrn‘𝐾)‘𝑊))) → (𝑇‘(𝑓𝑔)) = ((𝑇𝑓) ∘ (𝑇𝑔)))
2115, 16, 17, 18, 19, 20syl32anc 1376 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∧ 𝑔 ∈ ((LTrn‘𝐾)‘𝑊)) → (𝑇‘(𝑓𝑔)) = ((𝑇𝑓) ∘ (𝑇𝑔)))
2221fveq2d 6655 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∧ 𝑔 ∈ ((LTrn‘𝐾)‘𝑊)) → (𝑆‘(𝑇‘(𝑓𝑔))) = (𝑆‘((𝑇𝑓) ∘ (𝑇𝑔))))
23 simp12 1202 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∧ 𝑔 ∈ ((LTrn‘𝐾)‘𝑊)) → 𝑆𝐸)
24 simp11 1201 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∧ 𝑔 ∈ ((LTrn‘𝐾)‘𝑊)) → (𝐾 ∈ HL ∧ 𝑊𝐻))
252, 3, 5tendocl 38328 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑇𝐸𝑓 ∈ ((LTrn‘𝐾)‘𝑊)) → (𝑇𝑓) ∈ ((LTrn‘𝐾)‘𝑊))
2624, 17, 18, 25syl3anc 1369 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∧ 𝑔 ∈ ((LTrn‘𝐾)‘𝑊)) → (𝑇𝑓) ∈ ((LTrn‘𝐾)‘𝑊))
272, 3, 5tendocl 38328 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑇𝐸𝑔 ∈ ((LTrn‘𝐾)‘𝑊)) → (𝑇𝑔) ∈ ((LTrn‘𝐾)‘𝑊))
2824, 17, 19, 27syl3anc 1369 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∧ 𝑔 ∈ ((LTrn‘𝐾)‘𝑊)) → (𝑇𝑔) ∈ ((LTrn‘𝐾)‘𝑊))
292, 3, 5tendovalco 38326 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻𝑆𝐸) ∧ ((𝑇𝑓) ∈ ((LTrn‘𝐾)‘𝑊) ∧ (𝑇𝑔) ∈ ((LTrn‘𝐾)‘𝑊))) → (𝑆‘((𝑇𝑓) ∘ (𝑇𝑔))) = ((𝑆‘(𝑇𝑓)) ∘ (𝑆‘(𝑇𝑔))))
3015, 16, 23, 26, 28, 29syl32anc 1376 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∧ 𝑔 ∈ ((LTrn‘𝐾)‘𝑊)) → (𝑆‘((𝑇𝑓) ∘ (𝑇𝑔))) = ((𝑆‘(𝑇𝑓)) ∘ (𝑆‘(𝑇𝑔))))
3122, 30eqtrd 2794 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∧ 𝑔 ∈ ((LTrn‘𝐾)‘𝑊)) → (𝑆‘(𝑇‘(𝑓𝑔))) = ((𝑆‘(𝑇𝑓)) ∘ (𝑆‘(𝑇𝑔))))
322, 3ltrnco 38280 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∧ 𝑔 ∈ ((LTrn‘𝐾)‘𝑊)) → (𝑓𝑔) ∈ ((LTrn‘𝐾)‘𝑊))
3324, 18, 19, 32syl3anc 1369 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∧ 𝑔 ∈ ((LTrn‘𝐾)‘𝑊)) → (𝑓𝑔) ∈ ((LTrn‘𝐾)‘𝑊))
342, 3, 5tendocoval 38327 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑆𝐸𝑇𝐸) ∧ (𝑓𝑔) ∈ ((LTrn‘𝐾)‘𝑊)) → ((𝑆𝑇)‘(𝑓𝑔)) = (𝑆‘(𝑇‘(𝑓𝑔))))
3524, 23, 17, 33, 34syl121anc 1373 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∧ 𝑔 ∈ ((LTrn‘𝐾)‘𝑊)) → ((𝑆𝑇)‘(𝑓𝑔)) = (𝑆‘(𝑇‘(𝑓𝑔))))
362, 3, 5tendocoval 38327 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊)) → ((𝑆𝑇)‘𝑓) = (𝑆‘(𝑇𝑓)))
3715, 16, 23, 17, 18, 36syl221anc 1379 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∧ 𝑔 ∈ ((LTrn‘𝐾)‘𝑊)) → ((𝑆𝑇)‘𝑓) = (𝑆‘(𝑇𝑓)))
382, 3, 5tendocoval 38327 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑆𝐸𝑇𝐸) ∧ 𝑔 ∈ ((LTrn‘𝐾)‘𝑊)) → ((𝑆𝑇)‘𝑔) = (𝑆‘(𝑇𝑔)))
3915, 16, 23, 17, 19, 38syl221anc 1379 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∧ 𝑔 ∈ ((LTrn‘𝐾)‘𝑊)) → ((𝑆𝑇)‘𝑔) = (𝑆‘(𝑇𝑔)))
4037, 39coeq12d 5697 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∧ 𝑔 ∈ ((LTrn‘𝐾)‘𝑊)) → (((𝑆𝑇)‘𝑓) ∘ ((𝑆𝑇)‘𝑔)) = ((𝑆‘(𝑇𝑓)) ∘ (𝑆‘(𝑇𝑔))))
4131, 35, 403eqtr4d 2804 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∧ 𝑔 ∈ ((LTrn‘𝐾)‘𝑊)) → ((𝑆𝑇)‘(𝑓𝑔)) = (((𝑆𝑇)‘𝑓) ∘ ((𝑆𝑇)‘𝑔)))
42 eqid 2759 . . 3 (Base‘𝐾) = (Base‘𝐾)
43 simpl1l 1222 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊)) → 𝐾 ∈ HL)
4443hllatd 36925 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊)) → 𝐾 ∈ Lat)
45 simpl1 1189 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊)) → (𝐾 ∈ HL ∧ 𝑊𝐻))
46 simpl2 1190 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊)) → 𝑆𝐸)
47 simpl3 1191 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊)) → 𝑇𝐸)
48 simpr 489 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊)) → 𝑓 ∈ ((LTrn‘𝐾)‘𝑊))
4945, 46, 47, 48, 36syl121anc 1373 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊)) → ((𝑆𝑇)‘𝑓) = (𝑆‘(𝑇𝑓)))
5045, 47, 48, 25syl3anc 1369 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊)) → (𝑇𝑓) ∈ ((LTrn‘𝐾)‘𝑊))
512, 3, 5tendocl 38328 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸 ∧ (𝑇𝑓) ∈ ((LTrn‘𝐾)‘𝑊)) → (𝑆‘(𝑇𝑓)) ∈ ((LTrn‘𝐾)‘𝑊))
5245, 46, 50, 51syl3anc 1369 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊)) → (𝑆‘(𝑇𝑓)) ∈ ((LTrn‘𝐾)‘𝑊))
5349, 52eqeltrd 2851 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊)) → ((𝑆𝑇)‘𝑓) ∈ ((LTrn‘𝐾)‘𝑊))
5442, 2, 3, 4trlcl 37725 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑆𝑇)‘𝑓) ∈ ((LTrn‘𝐾)‘𝑊)) → (((trL‘𝐾)‘𝑊)‘((𝑆𝑇)‘𝑓)) ∈ (Base‘𝐾))
5545, 53, 54syl2anc 588 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊)) → (((trL‘𝐾)‘𝑊)‘((𝑆𝑇)‘𝑓)) ∈ (Base‘𝐾))
5642, 2, 3, 4trlcl 37725 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑇𝑓) ∈ ((LTrn‘𝐾)‘𝑊)) → (((trL‘𝐾)‘𝑊)‘(𝑇𝑓)) ∈ (Base‘𝐾))
5745, 50, 56syl2anc 588 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊)) → (((trL‘𝐾)‘𝑊)‘(𝑇𝑓)) ∈ (Base‘𝐾))
5842, 2, 3, 4trlcl 37725 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊)) → (((trL‘𝐾)‘𝑊)‘𝑓) ∈ (Base‘𝐾))
5945, 48, 58syl2anc 588 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊)) → (((trL‘𝐾)‘𝑊)‘𝑓) ∈ (Base‘𝐾))
60 simpl1r 1223 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊)) → 𝑊𝐻)
6143, 60, 46, 47, 48, 36syl221anc 1379 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊)) → ((𝑆𝑇)‘𝑓) = (𝑆‘(𝑇𝑓)))
6261fveq2d 6655 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊)) → (((trL‘𝐾)‘𝑊)‘((𝑆𝑇)‘𝑓)) = (((trL‘𝐾)‘𝑊)‘(𝑆‘(𝑇𝑓))))
631, 2, 3, 4, 5tendotp 38322 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸 ∧ (𝑇𝑓) ∈ ((LTrn‘𝐾)‘𝑊)) → (((trL‘𝐾)‘𝑊)‘(𝑆‘(𝑇𝑓)))(le‘𝐾)(((trL‘𝐾)‘𝑊)‘(𝑇𝑓)))
6445, 46, 50, 63syl3anc 1369 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊)) → (((trL‘𝐾)‘𝑊)‘(𝑆‘(𝑇𝑓)))(le‘𝐾)(((trL‘𝐾)‘𝑊)‘(𝑇𝑓)))
6562, 64eqbrtrd 5047 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊)) → (((trL‘𝐾)‘𝑊)‘((𝑆𝑇)‘𝑓))(le‘𝐾)(((trL‘𝐾)‘𝑊)‘(𝑇𝑓)))
661, 2, 3, 4, 5tendotp 38322 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑇𝐸𝑓 ∈ ((LTrn‘𝐾)‘𝑊)) → (((trL‘𝐾)‘𝑊)‘(𝑇𝑓))(le‘𝐾)(((trL‘𝐾)‘𝑊)‘𝑓))
6745, 47, 48, 66syl3anc 1369 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊)) → (((trL‘𝐾)‘𝑊)‘(𝑇𝑓))(le‘𝐾)(((trL‘𝐾)‘𝑊)‘𝑓))
6842, 1, 44, 55, 57, 59, 65, 67lattrd 17719 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊)) → (((trL‘𝐾)‘𝑊)‘((𝑆𝑇)‘𝑓))(le‘𝐾)(((trL‘𝐾)‘𝑊)‘𝑓))
691, 2, 3, 4, 5, 6, 14, 41, 68istendod 38323 1 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) → (𝑆𝑇) ∈ 𝐸)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1085   = wceq 1539   ∈ wcel 2112   class class class wbr 5025   ∘ ccom 5521  ⟶wf 6324  ‘cfv 6328  Basecbs 16526  lecple 16615  HLchlt 36911  LHypclh 37545  LTrncltrn 37662  trLctrl 37719  TEndoctendo 38313 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-rep 5149  ax-sep 5162  ax-nul 5169  ax-pow 5227  ax-pr 5291  ax-un 7452  ax-riotaBAD 36514 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2899  df-ne 2950  df-ral 3073  df-rex 3074  df-reu 3075  df-rmo 3076  df-rab 3077  df-v 3409  df-sbc 3694  df-csb 3802  df-dif 3857  df-un 3859  df-in 3861  df-ss 3871  df-nul 4222  df-if 4414  df-pw 4489  df-sn 4516  df-pr 4518  df-op 4522  df-uni 4792  df-iun 4878  df-iin 4879  df-br 5026  df-opab 5088  df-mpt 5106  df-id 5423  df-xp 5523  df-rel 5524  df-cnv 5525  df-co 5526  df-dm 5527  df-rn 5528  df-res 5529  df-ima 5530  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7101  df-ov 7146  df-oprab 7147  df-mpo 7148  df-1st 7686  df-2nd 7687  df-undef 7942  df-map 8411  df-proset 17589  df-poset 17607  df-plt 17619  df-lub 17635  df-glb 17636  df-join 17637  df-meet 17638  df-p0 17700  df-p1 17701  df-lat 17707  df-clat 17769  df-oposet 36737  df-ol 36739  df-oml 36740  df-covers 36827  df-ats 36828  df-atl 36859  df-cvlat 36883  df-hlat 36912  df-llines 37059  df-lplanes 37060  df-lvols 37061  df-lines 37062  df-psubsp 37064  df-pmap 37065  df-padd 37357  df-lhyp 37549  df-laut 37550  df-ldil 37665  df-ltrn 37666  df-trl 37720  df-tendo 38316 This theorem is referenced by:  tendodi1  38345  tendodi2  38346  tendo0mul  38387  tendo0mulr  38388  tendoconid  38390  cdleml3N  38539  cdleml8  38544  erngdvlem3  38551  erngdvlem3-rN  38559  dvalveclem  38586  dvhvscacl  38664  dvhlveclem  38669  diblss  38731  dicvscacl  38752  dih1dimatlem0  38889
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