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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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