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Theorem tendococl 36581
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 2771 . 2 (le‘𝐾) = (le‘𝐾)
2 tendoco.h . 2 𝐻 = (LHyp‘𝐾)
3 eqid 2771 . 2 ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊)
4 eqid 2771 . 2 ((trL‘𝐾)‘𝑊) = ((trL‘𝐾)‘𝑊)
5 tendoco.e . 2 𝐸 = ((TEndo‘𝐾)‘𝑊)
6 simp1 1130 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) → (𝐾 ∈ HL ∧ 𝑊𝐻))
7 simp2 1131 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) → 𝑆𝐸)
82, 3, 5tendof 36572 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸) → 𝑆:((LTrn‘𝐾)‘𝑊)⟶((LTrn‘𝐾)‘𝑊))
96, 7, 8syl2anc 573 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) → 𝑆:((LTrn‘𝐾)‘𝑊)⟶((LTrn‘𝐾)‘𝑊))
10 simp3 1132 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) → 𝑇𝐸)
112, 3, 5tendof 36572 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑇𝐸) → 𝑇:((LTrn‘𝐾)‘𝑊)⟶((LTrn‘𝐾)‘𝑊))
126, 10, 11syl2anc 573 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) → 𝑇:((LTrn‘𝐾)‘𝑊)⟶((LTrn‘𝐾)‘𝑊))
13 fco 6198 . . 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 573 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) → (𝑆𝑇):((LTrn‘𝐾)‘𝑊)⟶((LTrn‘𝐾)‘𝑊))
15 simp11l 1368 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∧ 𝑔 ∈ ((LTrn‘𝐾)‘𝑊)) → 𝐾 ∈ HL)
16 simp11r 1369 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∧ 𝑔 ∈ ((LTrn‘𝐾)‘𝑊)) → 𝑊𝐻)
17 simp13 1247 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∧ 𝑔 ∈ ((LTrn‘𝐾)‘𝑊)) → 𝑇𝐸)
18 simp2 1131 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∧ 𝑔 ∈ ((LTrn‘𝐾)‘𝑊)) → 𝑓 ∈ ((LTrn‘𝐾)‘𝑊))
19 simp3 1132 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∧ 𝑔 ∈ ((LTrn‘𝐾)‘𝑊)) → 𝑔 ∈ ((LTrn‘𝐾)‘𝑊))
202, 3, 5tendovalco 36574 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻𝑇𝐸) ∧ (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∧ 𝑔 ∈ ((LTrn‘𝐾)‘𝑊))) → (𝑇‘(𝑓𝑔)) = ((𝑇𝑓) ∘ (𝑇𝑔)))
2115, 16, 17, 18, 19, 20syl32anc 1484 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∧ 𝑔 ∈ ((LTrn‘𝐾)‘𝑊)) → (𝑇‘(𝑓𝑔)) = ((𝑇𝑓) ∘ (𝑇𝑔)))
2221fveq2d 6336 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∧ 𝑔 ∈ ((LTrn‘𝐾)‘𝑊)) → (𝑆‘(𝑇‘(𝑓𝑔))) = (𝑆‘((𝑇𝑓) ∘ (𝑇𝑔))))
23 simp12 1246 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∧ 𝑔 ∈ ((LTrn‘𝐾)‘𝑊)) → 𝑆𝐸)
24 simp11 1245 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∧ 𝑔 ∈ ((LTrn‘𝐾)‘𝑊)) → (𝐾 ∈ HL ∧ 𝑊𝐻))
252, 3, 5tendocl 36576 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑇𝐸𝑓 ∈ ((LTrn‘𝐾)‘𝑊)) → (𝑇𝑓) ∈ ((LTrn‘𝐾)‘𝑊))
2624, 17, 18, 25syl3anc 1476 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∧ 𝑔 ∈ ((LTrn‘𝐾)‘𝑊)) → (𝑇𝑓) ∈ ((LTrn‘𝐾)‘𝑊))
272, 3, 5tendocl 36576 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑇𝐸𝑔 ∈ ((LTrn‘𝐾)‘𝑊)) → (𝑇𝑔) ∈ ((LTrn‘𝐾)‘𝑊))
2824, 17, 19, 27syl3anc 1476 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∧ 𝑔 ∈ ((LTrn‘𝐾)‘𝑊)) → (𝑇𝑔) ∈ ((LTrn‘𝐾)‘𝑊))
292, 3, 5tendovalco 36574 . . . . 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 1484 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∧ 𝑔 ∈ ((LTrn‘𝐾)‘𝑊)) → (𝑆‘((𝑇𝑓) ∘ (𝑇𝑔))) = ((𝑆‘(𝑇𝑓)) ∘ (𝑆‘(𝑇𝑔))))
3122, 30eqtrd 2805 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∧ 𝑔 ∈ ((LTrn‘𝐾)‘𝑊)) → (𝑆‘(𝑇‘(𝑓𝑔))) = ((𝑆‘(𝑇𝑓)) ∘ (𝑆‘(𝑇𝑔))))
322, 3ltrnco 36528 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∧ 𝑔 ∈ ((LTrn‘𝐾)‘𝑊)) → (𝑓𝑔) ∈ ((LTrn‘𝐾)‘𝑊))
3324, 18, 19, 32syl3anc 1476 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∧ 𝑔 ∈ ((LTrn‘𝐾)‘𝑊)) → (𝑓𝑔) ∈ ((LTrn‘𝐾)‘𝑊))
342, 3, 5tendocoval 36575 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑆𝐸𝑇𝐸) ∧ (𝑓𝑔) ∈ ((LTrn‘𝐾)‘𝑊)) → ((𝑆𝑇)‘(𝑓𝑔)) = (𝑆‘(𝑇‘(𝑓𝑔))))
3524, 23, 17, 33, 34syl121anc 1481 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∧ 𝑔 ∈ ((LTrn‘𝐾)‘𝑊)) → ((𝑆𝑇)‘(𝑓𝑔)) = (𝑆‘(𝑇‘(𝑓𝑔))))
362, 3, 5tendocoval 36575 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊)) → ((𝑆𝑇)‘𝑓) = (𝑆‘(𝑇𝑓)))
3715, 16, 23, 17, 18, 36syl221anc 1487 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∧ 𝑔 ∈ ((LTrn‘𝐾)‘𝑊)) → ((𝑆𝑇)‘𝑓) = (𝑆‘(𝑇𝑓)))
382, 3, 5tendocoval 36575 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑆𝐸𝑇𝐸) ∧ 𝑔 ∈ ((LTrn‘𝐾)‘𝑊)) → ((𝑆𝑇)‘𝑔) = (𝑆‘(𝑇𝑔)))
3915, 16, 23, 17, 19, 38syl221anc 1487 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∧ 𝑔 ∈ ((LTrn‘𝐾)‘𝑊)) → ((𝑆𝑇)‘𝑔) = (𝑆‘(𝑇𝑔)))
4037, 39coeq12d 5425 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∧ 𝑔 ∈ ((LTrn‘𝐾)‘𝑊)) → (((𝑆𝑇)‘𝑓) ∘ ((𝑆𝑇)‘𝑔)) = ((𝑆‘(𝑇𝑓)) ∘ (𝑆‘(𝑇𝑔))))
4131, 35, 403eqtr4d 2815 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∧ 𝑔 ∈ ((LTrn‘𝐾)‘𝑊)) → ((𝑆𝑇)‘(𝑓𝑔)) = (((𝑆𝑇)‘𝑓) ∘ ((𝑆𝑇)‘𝑔)))
42 eqid 2771 . . 3 (Base‘𝐾) = (Base‘𝐾)
43 simpl1l 1278 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊)) → 𝐾 ∈ HL)
44 hllat 35172 . . . 4 (𝐾 ∈ HL → 𝐾 ∈ Lat)
4543, 44syl 17 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊)) → 𝐾 ∈ Lat)
46 simpl1 1227 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊)) → (𝐾 ∈ HL ∧ 𝑊𝐻))
47 simpl2 1229 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊)) → 𝑆𝐸)
48 simpl3 1231 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊)) → 𝑇𝐸)
49 simpr 471 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊)) → 𝑓 ∈ ((LTrn‘𝐾)‘𝑊))
5046, 47, 48, 49, 36syl121anc 1481 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊)) → ((𝑆𝑇)‘𝑓) = (𝑆‘(𝑇𝑓)))
5146, 48, 49, 25syl3anc 1476 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊)) → (𝑇𝑓) ∈ ((LTrn‘𝐾)‘𝑊))
522, 3, 5tendocl 36576 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸 ∧ (𝑇𝑓) ∈ ((LTrn‘𝐾)‘𝑊)) → (𝑆‘(𝑇𝑓)) ∈ ((LTrn‘𝐾)‘𝑊))
5346, 47, 51, 52syl3anc 1476 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊)) → (𝑆‘(𝑇𝑓)) ∈ ((LTrn‘𝐾)‘𝑊))
5450, 53eqeltrd 2850 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊)) → ((𝑆𝑇)‘𝑓) ∈ ((LTrn‘𝐾)‘𝑊))
5542, 2, 3, 4trlcl 35973 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑆𝑇)‘𝑓) ∈ ((LTrn‘𝐾)‘𝑊)) → (((trL‘𝐾)‘𝑊)‘((𝑆𝑇)‘𝑓)) ∈ (Base‘𝐾))
5646, 54, 55syl2anc 573 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊)) → (((trL‘𝐾)‘𝑊)‘((𝑆𝑇)‘𝑓)) ∈ (Base‘𝐾))
5742, 2, 3, 4trlcl 35973 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑇𝑓) ∈ ((LTrn‘𝐾)‘𝑊)) → (((trL‘𝐾)‘𝑊)‘(𝑇𝑓)) ∈ (Base‘𝐾))
5846, 51, 57syl2anc 573 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊)) → (((trL‘𝐾)‘𝑊)‘(𝑇𝑓)) ∈ (Base‘𝐾))
5942, 2, 3, 4trlcl 35973 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊)) → (((trL‘𝐾)‘𝑊)‘𝑓) ∈ (Base‘𝐾))
6046, 49, 59syl2anc 573 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊)) → (((trL‘𝐾)‘𝑊)‘𝑓) ∈ (Base‘𝐾))
61 simpl1r 1280 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊)) → 𝑊𝐻)
6243, 61, 47, 48, 49, 36syl221anc 1487 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊)) → ((𝑆𝑇)‘𝑓) = (𝑆‘(𝑇𝑓)))
6362fveq2d 6336 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊)) → (((trL‘𝐾)‘𝑊)‘((𝑆𝑇)‘𝑓)) = (((trL‘𝐾)‘𝑊)‘(𝑆‘(𝑇𝑓))))
641, 2, 3, 4, 5tendotp 36570 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸 ∧ (𝑇𝑓) ∈ ((LTrn‘𝐾)‘𝑊)) → (((trL‘𝐾)‘𝑊)‘(𝑆‘(𝑇𝑓)))(le‘𝐾)(((trL‘𝐾)‘𝑊)‘(𝑇𝑓)))
6546, 47, 51, 64syl3anc 1476 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊)) → (((trL‘𝐾)‘𝑊)‘(𝑆‘(𝑇𝑓)))(le‘𝐾)(((trL‘𝐾)‘𝑊)‘(𝑇𝑓)))
6663, 65eqbrtrd 4808 . . 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 36570 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑇𝐸𝑓 ∈ ((LTrn‘𝐾)‘𝑊)) → (((trL‘𝐾)‘𝑊)‘(𝑇𝑓))(le‘𝐾)(((trL‘𝐾)‘𝑊)‘𝑓))
6846, 48, 49, 67syl3anc 1476 . . 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 17266 . 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 36571 1 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) → (𝑆𝑇) ∈ 𝐸)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3a 1071   = wceq 1631  wcel 2145   class class class wbr 4786  ccom 5253  wf 6027  cfv 6031  Basecbs 16064  lecple 16156  Latclat 17253  HLchlt 35159  LHypclh 35792  LTrncltrn 35909  trLctrl 35967  TEndoctendo 36561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-riotaBAD 34761
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-iin 4657  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6754  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-1st 7315  df-2nd 7316  df-undef 7551  df-map 8011  df-preset 17136  df-poset 17154  df-plt 17166  df-lub 17182  df-glb 17183  df-join 17184  df-meet 17185  df-p0 17247  df-p1 17248  df-lat 17254  df-clat 17316  df-oposet 34985  df-ol 34987  df-oml 34988  df-covers 35075  df-ats 35076  df-atl 35107  df-cvlat 35131  df-hlat 35160  df-llines 35306  df-lplanes 35307  df-lvols 35308  df-lines 35309  df-psubsp 35311  df-pmap 35312  df-padd 35604  df-lhyp 35796  df-laut 35797  df-ldil 35912  df-ltrn 35913  df-trl 35968  df-tendo 36564
This theorem is referenced by:  tendodi1  36593  tendodi2  36594  tendo0mul  36635  tendo0mulr  36636  tendoconid  36638  cdleml3N  36787  cdleml8  36792  erngdvlem3  36799  erngdvlem3-rN  36807  dvalveclem  36835  dvhvscacl  36913  dvhlveclem  36918  diblss  36980  dicvscacl  37001  dih1dimatlem0  37138
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