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Theorem tendococl 36580
 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 2760 . 2 (le‘𝐾) = (le‘𝐾)
2 tendoco.h . 2 𝐻 = (LHyp‘𝐾)
3 eqid 2760 . 2 ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊)
4 eqid 2760 . 2 ((trL‘𝐾)‘𝑊) = ((trL‘𝐾)‘𝑊)
5 tendoco.e . 2 𝐸 = ((TEndo‘𝐾)‘𝑊)
6 simp1 1131 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) → (𝐾 ∈ HL ∧ 𝑊𝐻))
7 simp2 1132 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) → 𝑆𝐸)
82, 3, 5tendof 36571 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸) → 𝑆:((LTrn‘𝐾)‘𝑊)⟶((LTrn‘𝐾)‘𝑊))
96, 7, 8syl2anc 696 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) → 𝑆:((LTrn‘𝐾)‘𝑊)⟶((LTrn‘𝐾)‘𝑊))
10 simp3 1133 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) → 𝑇𝐸)
112, 3, 5tendof 36571 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑇𝐸) → 𝑇:((LTrn‘𝐾)‘𝑊)⟶((LTrn‘𝐾)‘𝑊))
126, 10, 11syl2anc 696 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) → 𝑇:((LTrn‘𝐾)‘𝑊)⟶((LTrn‘𝐾)‘𝑊))
13 fco 6219 . . 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 696 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) → (𝑆𝑇):((LTrn‘𝐾)‘𝑊)⟶((LTrn‘𝐾)‘𝑊))
15 simp11l 1369 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∧ 𝑔 ∈ ((LTrn‘𝐾)‘𝑊)) → 𝐾 ∈ HL)
16 simp11r 1370 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∧ 𝑔 ∈ ((LTrn‘𝐾)‘𝑊)) → 𝑊𝐻)
17 simp13 1248 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∧ 𝑔 ∈ ((LTrn‘𝐾)‘𝑊)) → 𝑇𝐸)
18 simp2 1132 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∧ 𝑔 ∈ ((LTrn‘𝐾)‘𝑊)) → 𝑓 ∈ ((LTrn‘𝐾)‘𝑊))
19 simp3 1133 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∧ 𝑔 ∈ ((LTrn‘𝐾)‘𝑊)) → 𝑔 ∈ ((LTrn‘𝐾)‘𝑊))
202, 3, 5tendovalco 36573 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻𝑇𝐸) ∧ (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∧ 𝑔 ∈ ((LTrn‘𝐾)‘𝑊))) → (𝑇‘(𝑓𝑔)) = ((𝑇𝑓) ∘ (𝑇𝑔)))
2115, 16, 17, 18, 19, 20syl32anc 1485 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∧ 𝑔 ∈ ((LTrn‘𝐾)‘𝑊)) → (𝑇‘(𝑓𝑔)) = ((𝑇𝑓) ∘ (𝑇𝑔)))
2221fveq2d 6357 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∧ 𝑔 ∈ ((LTrn‘𝐾)‘𝑊)) → (𝑆‘(𝑇‘(𝑓𝑔))) = (𝑆‘((𝑇𝑓) ∘ (𝑇𝑔))))
23 simp12 1247 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∧ 𝑔 ∈ ((LTrn‘𝐾)‘𝑊)) → 𝑆𝐸)
24 simp11 1246 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∧ 𝑔 ∈ ((LTrn‘𝐾)‘𝑊)) → (𝐾 ∈ HL ∧ 𝑊𝐻))
252, 3, 5tendocl 36575 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑇𝐸𝑓 ∈ ((LTrn‘𝐾)‘𝑊)) → (𝑇𝑓) ∈ ((LTrn‘𝐾)‘𝑊))
2624, 17, 18, 25syl3anc 1477 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∧ 𝑔 ∈ ((LTrn‘𝐾)‘𝑊)) → (𝑇𝑓) ∈ ((LTrn‘𝐾)‘𝑊))
272, 3, 5tendocl 36575 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑇𝐸𝑔 ∈ ((LTrn‘𝐾)‘𝑊)) → (𝑇𝑔) ∈ ((LTrn‘𝐾)‘𝑊))
2824, 17, 19, 27syl3anc 1477 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∧ 𝑔 ∈ ((LTrn‘𝐾)‘𝑊)) → (𝑇𝑔) ∈ ((LTrn‘𝐾)‘𝑊))
292, 3, 5tendovalco 36573 . . . . 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 1485 . . . 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 36527 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∧ 𝑔 ∈ ((LTrn‘𝐾)‘𝑊)) → (𝑓𝑔) ∈ ((LTrn‘𝐾)‘𝑊))
3324, 18, 19, 32syl3anc 1477 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∧ 𝑔 ∈ ((LTrn‘𝐾)‘𝑊)) → (𝑓𝑔) ∈ ((LTrn‘𝐾)‘𝑊))
342, 3, 5tendocoval 36574 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑆𝐸𝑇𝐸) ∧ (𝑓𝑔) ∈ ((LTrn‘𝐾)‘𝑊)) → ((𝑆𝑇)‘(𝑓𝑔)) = (𝑆‘(𝑇‘(𝑓𝑔))))
3524, 23, 17, 33, 34syl121anc 1482 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∧ 𝑔 ∈ ((LTrn‘𝐾)‘𝑊)) → ((𝑆𝑇)‘(𝑓𝑔)) = (𝑆‘(𝑇‘(𝑓𝑔))))
362, 3, 5tendocoval 36574 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊)) → ((𝑆𝑇)‘𝑓) = (𝑆‘(𝑇𝑓)))
3715, 16, 23, 17, 18, 36syl221anc 1488 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∧ 𝑔 ∈ ((LTrn‘𝐾)‘𝑊)) → ((𝑆𝑇)‘𝑓) = (𝑆‘(𝑇𝑓)))
382, 3, 5tendocoval 36574 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑆𝐸𝑇𝐸) ∧ 𝑔 ∈ ((LTrn‘𝐾)‘𝑊)) → ((𝑆𝑇)‘𝑔) = (𝑆‘(𝑇𝑔)))
3915, 16, 23, 17, 19, 38syl221anc 1488 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∧ 𝑔 ∈ ((LTrn‘𝐾)‘𝑊)) → ((𝑆𝑇)‘𝑔) = (𝑆‘(𝑇𝑔)))
4037, 39coeq12d 5442 . . 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 2760 . . 3 (Base‘𝐾) = (Base‘𝐾)
43 simpl1l 1279 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊)) → 𝐾 ∈ HL)
44 hllat 35171 . . . 4 (𝐾 ∈ HL → 𝐾 ∈ Lat)
4543, 44syl 17 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊)) → 𝐾 ∈ Lat)
46 simpl1 1228 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊)) → (𝐾 ∈ HL ∧ 𝑊𝐻))
47 simpl2 1230 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊)) → 𝑆𝐸)
48 simpl3 1232 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊)) → 𝑇𝐸)
49 simpr 479 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊)) → 𝑓 ∈ ((LTrn‘𝐾)‘𝑊))
5046, 47, 48, 49, 36syl121anc 1482 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊)) → ((𝑆𝑇)‘𝑓) = (𝑆‘(𝑇𝑓)))
5146, 48, 49, 25syl3anc 1477 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊)) → (𝑇𝑓) ∈ ((LTrn‘𝐾)‘𝑊))
522, 3, 5tendocl 36575 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸 ∧ (𝑇𝑓) ∈ ((LTrn‘𝐾)‘𝑊)) → (𝑆‘(𝑇𝑓)) ∈ ((LTrn‘𝐾)‘𝑊))
5346, 47, 51, 52syl3anc 1477 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊)) → (𝑆‘(𝑇𝑓)) ∈ ((LTrn‘𝐾)‘𝑊))
5450, 53eqeltrd 2839 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊)) → ((𝑆𝑇)‘𝑓) ∈ ((LTrn‘𝐾)‘𝑊))
5542, 2, 3, 4trlcl 35972 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑆𝑇)‘𝑓) ∈ ((LTrn‘𝐾)‘𝑊)) → (((trL‘𝐾)‘𝑊)‘((𝑆𝑇)‘𝑓)) ∈ (Base‘𝐾))
5646, 54, 55syl2anc 696 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊)) → (((trL‘𝐾)‘𝑊)‘((𝑆𝑇)‘𝑓)) ∈ (Base‘𝐾))
5742, 2, 3, 4trlcl 35972 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑇𝑓) ∈ ((LTrn‘𝐾)‘𝑊)) → (((trL‘𝐾)‘𝑊)‘(𝑇𝑓)) ∈ (Base‘𝐾))
5846, 51, 57syl2anc 696 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊)) → (((trL‘𝐾)‘𝑊)‘(𝑇𝑓)) ∈ (Base‘𝐾))
5942, 2, 3, 4trlcl 35972 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊)) → (((trL‘𝐾)‘𝑊)‘𝑓) ∈ (Base‘𝐾))
6046, 49, 59syl2anc 696 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊)) → (((trL‘𝐾)‘𝑊)‘𝑓) ∈ (Base‘𝐾))
61 simpl1r 1281 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊)) → 𝑊𝐻)
6243, 61, 47, 48, 49, 36syl221anc 1488 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊)) → ((𝑆𝑇)‘𝑓) = (𝑆‘(𝑇𝑓)))
6362fveq2d 6357 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊)) → (((trL‘𝐾)‘𝑊)‘((𝑆𝑇)‘𝑓)) = (((trL‘𝐾)‘𝑊)‘(𝑆‘(𝑇𝑓))))
641, 2, 3, 4, 5tendotp 36569 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸 ∧ (𝑇𝑓) ∈ ((LTrn‘𝐾)‘𝑊)) → (((trL‘𝐾)‘𝑊)‘(𝑆‘(𝑇𝑓)))(le‘𝐾)(((trL‘𝐾)‘𝑊)‘(𝑇𝑓)))
6546, 47, 51, 64syl3anc 1477 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊)) → (((trL‘𝐾)‘𝑊)‘(𝑆‘(𝑇𝑓)))(le‘𝐾)(((trL‘𝐾)‘𝑊)‘(𝑇𝑓)))
6663, 65eqbrtrd 4826 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊)) → (((trL‘𝐾)‘𝑊)‘((𝑆𝑇)‘𝑓))(le‘𝐾)(((trL‘𝐾)‘𝑊)‘(𝑇𝑓)))
671, 2, 3, 4, 5tendotp 36569 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑇𝐸𝑓 ∈ ((LTrn‘𝐾)‘𝑊)) → (((trL‘𝐾)‘𝑊)‘(𝑇𝑓))(le‘𝐾)(((trL‘𝐾)‘𝑊)‘𝑓))
6846, 48, 49, 67syl3anc 1477 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊)) → (((trL‘𝐾)‘𝑊)‘(𝑇𝑓))(le‘𝐾)(((trL‘𝐾)‘𝑊)‘𝑓))
6942, 1, 45, 56, 58, 60, 66, 68lattrd 17279 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊)) → (((trL‘𝐾)‘𝑊)‘((𝑆𝑇)‘𝑓))(le‘𝐾)(((trL‘𝐾)‘𝑊)‘𝑓))
701, 2, 3, 4, 5, 6, 14, 41, 69istendod 36570 1 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) → (𝑆𝑇) ∈ 𝐸)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1072   = wceq 1632   ∈ wcel 2139   class class class wbr 4804   ∘ ccom 5270  ⟶wf 6045  ‘cfv 6049  Basecbs 16079  lecple 16170  Latclat 17266  HLchlt 35158  LHypclh 35791  LTrncltrn 35908  trLctrl 35966  TEndoctendo 36560 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115  ax-riotaBAD 34760 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-iin 4675  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6775  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-1st 7334  df-2nd 7335  df-undef 7569  df-map 8027  df-preset 17149  df-poset 17167  df-plt 17179  df-lub 17195  df-glb 17196  df-join 17197  df-meet 17198  df-p0 17260  df-p1 17261  df-lat 17267  df-clat 17329  df-oposet 34984  df-ol 34986  df-oml 34987  df-covers 35074  df-ats 35075  df-atl 35106  df-cvlat 35130  df-hlat 35159  df-llines 35305  df-lplanes 35306  df-lvols 35307  df-lines 35308  df-psubsp 35310  df-pmap 35311  df-padd 35603  df-lhyp 35795  df-laut 35796  df-ldil 35911  df-ltrn 35912  df-trl 35967  df-tendo 36563 This theorem is referenced by:  tendodi1  36592  tendodi2  36593  tendo0mul  36634  tendo0mulr  36635  tendoconid  36637  cdleml3N  36786  cdleml8  36791  erngdvlem3  36798  erngdvlem3-rN  36806  dvalveclem  36834  dvhvscacl  36912  dvhlveclem  36917  diblss  36979  dicvscacl  37000  dih1dimatlem0  37137
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