Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  tendococl Structured version   Visualization version   GIF version

Theorem tendococl 37946
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 2821 . 2 (le‘𝐾) = (le‘𝐾)
2 tendoco.h . 2 𝐻 = (LHyp‘𝐾)
3 eqid 2821 . 2 ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊)
4 eqid 2821 . 2 ((trL‘𝐾)‘𝑊) = ((trL‘𝐾)‘𝑊)
5 tendoco.e . 2 𝐸 = ((TEndo‘𝐾)‘𝑊)
6 simp1 1133 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) → (𝐾 ∈ HL ∧ 𝑊𝐻))
7 simp2 1134 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) → 𝑆𝐸)
82, 3, 5tendof 37937 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸) → 𝑆:((LTrn‘𝐾)‘𝑊)⟶((LTrn‘𝐾)‘𝑊))
96, 7, 8syl2anc 587 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) → 𝑆:((LTrn‘𝐾)‘𝑊)⟶((LTrn‘𝐾)‘𝑊))
10 simp3 1135 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) → 𝑇𝐸)
112, 3, 5tendof 37937 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑇𝐸) → 𝑇:((LTrn‘𝐾)‘𝑊)⟶((LTrn‘𝐾)‘𝑊))
126, 10, 11syl2anc 587 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) → 𝑇:((LTrn‘𝐾)‘𝑊)⟶((LTrn‘𝐾)‘𝑊))
13 fco 6504 . . 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 587 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) → (𝑆𝑇):((LTrn‘𝐾)‘𝑊)⟶((LTrn‘𝐾)‘𝑊))
15 simp11l 1281 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∧ 𝑔 ∈ ((LTrn‘𝐾)‘𝑊)) → 𝐾 ∈ HL)
16 simp11r 1282 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∧ 𝑔 ∈ ((LTrn‘𝐾)‘𝑊)) → 𝑊𝐻)
17 simp13 1202 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∧ 𝑔 ∈ ((LTrn‘𝐾)‘𝑊)) → 𝑇𝐸)
18 simp2 1134 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∧ 𝑔 ∈ ((LTrn‘𝐾)‘𝑊)) → 𝑓 ∈ ((LTrn‘𝐾)‘𝑊))
19 simp3 1135 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∧ 𝑔 ∈ ((LTrn‘𝐾)‘𝑊)) → 𝑔 ∈ ((LTrn‘𝐾)‘𝑊))
202, 3, 5tendovalco 37939 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻𝑇𝐸) ∧ (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∧ 𝑔 ∈ ((LTrn‘𝐾)‘𝑊))) → (𝑇‘(𝑓𝑔)) = ((𝑇𝑓) ∘ (𝑇𝑔)))
2115, 16, 17, 18, 19, 20syl32anc 1375 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∧ 𝑔 ∈ ((LTrn‘𝐾)‘𝑊)) → (𝑇‘(𝑓𝑔)) = ((𝑇𝑓) ∘ (𝑇𝑔)))
2221fveq2d 6647 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∧ 𝑔 ∈ ((LTrn‘𝐾)‘𝑊)) → (𝑆‘(𝑇‘(𝑓𝑔))) = (𝑆‘((𝑇𝑓) ∘ (𝑇𝑔))))
23 simp12 1201 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∧ 𝑔 ∈ ((LTrn‘𝐾)‘𝑊)) → 𝑆𝐸)
24 simp11 1200 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∧ 𝑔 ∈ ((LTrn‘𝐾)‘𝑊)) → (𝐾 ∈ HL ∧ 𝑊𝐻))
252, 3, 5tendocl 37941 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑇𝐸𝑓 ∈ ((LTrn‘𝐾)‘𝑊)) → (𝑇𝑓) ∈ ((LTrn‘𝐾)‘𝑊))
2624, 17, 18, 25syl3anc 1368 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∧ 𝑔 ∈ ((LTrn‘𝐾)‘𝑊)) → (𝑇𝑓) ∈ ((LTrn‘𝐾)‘𝑊))
272, 3, 5tendocl 37941 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑇𝐸𝑔 ∈ ((LTrn‘𝐾)‘𝑊)) → (𝑇𝑔) ∈ ((LTrn‘𝐾)‘𝑊))
2824, 17, 19, 27syl3anc 1368 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∧ 𝑔 ∈ ((LTrn‘𝐾)‘𝑊)) → (𝑇𝑔) ∈ ((LTrn‘𝐾)‘𝑊))
292, 3, 5tendovalco 37939 . . . . 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 1375 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∧ 𝑔 ∈ ((LTrn‘𝐾)‘𝑊)) → (𝑆‘((𝑇𝑓) ∘ (𝑇𝑔))) = ((𝑆‘(𝑇𝑓)) ∘ (𝑆‘(𝑇𝑔))))
3122, 30eqtrd 2856 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∧ 𝑔 ∈ ((LTrn‘𝐾)‘𝑊)) → (𝑆‘(𝑇‘(𝑓𝑔))) = ((𝑆‘(𝑇𝑓)) ∘ (𝑆‘(𝑇𝑔))))
322, 3ltrnco 37893 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∧ 𝑔 ∈ ((LTrn‘𝐾)‘𝑊)) → (𝑓𝑔) ∈ ((LTrn‘𝐾)‘𝑊))
3324, 18, 19, 32syl3anc 1368 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∧ 𝑔 ∈ ((LTrn‘𝐾)‘𝑊)) → (𝑓𝑔) ∈ ((LTrn‘𝐾)‘𝑊))
342, 3, 5tendocoval 37940 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑆𝐸𝑇𝐸) ∧ (𝑓𝑔) ∈ ((LTrn‘𝐾)‘𝑊)) → ((𝑆𝑇)‘(𝑓𝑔)) = (𝑆‘(𝑇‘(𝑓𝑔))))
3524, 23, 17, 33, 34syl121anc 1372 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∧ 𝑔 ∈ ((LTrn‘𝐾)‘𝑊)) → ((𝑆𝑇)‘(𝑓𝑔)) = (𝑆‘(𝑇‘(𝑓𝑔))))
362, 3, 5tendocoval 37940 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊)) → ((𝑆𝑇)‘𝑓) = (𝑆‘(𝑇𝑓)))
3715, 16, 23, 17, 18, 36syl221anc 1378 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∧ 𝑔 ∈ ((LTrn‘𝐾)‘𝑊)) → ((𝑆𝑇)‘𝑓) = (𝑆‘(𝑇𝑓)))
382, 3, 5tendocoval 37940 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑆𝐸𝑇𝐸) ∧ 𝑔 ∈ ((LTrn‘𝐾)‘𝑊)) → ((𝑆𝑇)‘𝑔) = (𝑆‘(𝑇𝑔)))
3915, 16, 23, 17, 19, 38syl221anc 1378 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∧ 𝑔 ∈ ((LTrn‘𝐾)‘𝑊)) → ((𝑆𝑇)‘𝑔) = (𝑆‘(𝑇𝑔)))
4037, 39coeq12d 5708 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∧ 𝑔 ∈ ((LTrn‘𝐾)‘𝑊)) → (((𝑆𝑇)‘𝑓) ∘ ((𝑆𝑇)‘𝑔)) = ((𝑆‘(𝑇𝑓)) ∘ (𝑆‘(𝑇𝑔))))
4131, 35, 403eqtr4d 2866 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ∧ 𝑔 ∈ ((LTrn‘𝐾)‘𝑊)) → ((𝑆𝑇)‘(𝑓𝑔)) = (((𝑆𝑇)‘𝑓) ∘ ((𝑆𝑇)‘𝑔)))
42 eqid 2821 . . 3 (Base‘𝐾) = (Base‘𝐾)
43 simpl1l 1221 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊)) → 𝐾 ∈ HL)
4443hllatd 36538 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊)) → 𝐾 ∈ Lat)
45 simpl1 1188 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊)) → (𝐾 ∈ HL ∧ 𝑊𝐻))
46 simpl2 1189 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊)) → 𝑆𝐸)
47 simpl3 1190 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊)) → 𝑇𝐸)
48 simpr 488 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊)) → 𝑓 ∈ ((LTrn‘𝐾)‘𝑊))
4945, 46, 47, 48, 36syl121anc 1372 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊)) → ((𝑆𝑇)‘𝑓) = (𝑆‘(𝑇𝑓)))
5045, 47, 48, 25syl3anc 1368 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊)) → (𝑇𝑓) ∈ ((LTrn‘𝐾)‘𝑊))
512, 3, 5tendocl 37941 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸 ∧ (𝑇𝑓) ∈ ((LTrn‘𝐾)‘𝑊)) → (𝑆‘(𝑇𝑓)) ∈ ((LTrn‘𝐾)‘𝑊))
5245, 46, 50, 51syl3anc 1368 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊)) → (𝑆‘(𝑇𝑓)) ∈ ((LTrn‘𝐾)‘𝑊))
5349, 52eqeltrd 2912 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊)) → ((𝑆𝑇)‘𝑓) ∈ ((LTrn‘𝐾)‘𝑊))
5442, 2, 3, 4trlcl 37338 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑆𝑇)‘𝑓) ∈ ((LTrn‘𝐾)‘𝑊)) → (((trL‘𝐾)‘𝑊)‘((𝑆𝑇)‘𝑓)) ∈ (Base‘𝐾))
5545, 53, 54syl2anc 587 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊)) → (((trL‘𝐾)‘𝑊)‘((𝑆𝑇)‘𝑓)) ∈ (Base‘𝐾))
5642, 2, 3, 4trlcl 37338 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑇𝑓) ∈ ((LTrn‘𝐾)‘𝑊)) → (((trL‘𝐾)‘𝑊)‘(𝑇𝑓)) ∈ (Base‘𝐾))
5745, 50, 56syl2anc 587 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊)) → (((trL‘𝐾)‘𝑊)‘(𝑇𝑓)) ∈ (Base‘𝐾))
5842, 2, 3, 4trlcl 37338 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊)) → (((trL‘𝐾)‘𝑊)‘𝑓) ∈ (Base‘𝐾))
5945, 48, 58syl2anc 587 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊)) → (((trL‘𝐾)‘𝑊)‘𝑓) ∈ (Base‘𝐾))
60 simpl1r 1222 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊)) → 𝑊𝐻)
6143, 60, 46, 47, 48, 36syl221anc 1378 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊)) → ((𝑆𝑇)‘𝑓) = (𝑆‘(𝑇𝑓)))
6261fveq2d 6647 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊)) → (((trL‘𝐾)‘𝑊)‘((𝑆𝑇)‘𝑓)) = (((trL‘𝐾)‘𝑊)‘(𝑆‘(𝑇𝑓))))
631, 2, 3, 4, 5tendotp 37935 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸 ∧ (𝑇𝑓) ∈ ((LTrn‘𝐾)‘𝑊)) → (((trL‘𝐾)‘𝑊)‘(𝑆‘(𝑇𝑓)))(le‘𝐾)(((trL‘𝐾)‘𝑊)‘(𝑇𝑓)))
6445, 46, 50, 63syl3anc 1368 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) ∧ 𝑓 ∈ ((LTrn‘𝐾)‘𝑊)) → (((trL‘𝐾)‘𝑊)‘(𝑆‘(𝑇𝑓)))(le‘𝐾)(((trL‘𝐾)‘𝑊)‘(𝑇𝑓)))
6562, 64eqbrtrd 5061 . . 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 37935 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑇𝐸𝑓 ∈ ((LTrn‘𝐾)‘𝑊)) → (((trL‘𝐾)‘𝑊)‘(𝑇𝑓))(le‘𝐾)(((trL‘𝐾)‘𝑊)‘𝑓))
6745, 47, 48, 66syl3anc 1368 . . 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 17646 . 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 37936 1 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑇𝐸) → (𝑆𝑇) ∈ 𝐸)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1084   = wceq 1538  wcel 2115   class class class wbr 5039  ccom 5532  wf 6324  cfv 6328  Basecbs 16461  lecple 16550  HLchlt 36524  LHypclh 37158  LTrncltrn 37275  trLctrl 37332  TEndoctendo 37926
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-rep 5163  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436  ax-riotaBAD 36127
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-ral 3131  df-rex 3132  df-reu 3133  df-rmo 3134  df-rab 3135  df-v 3473  df-sbc 3750  df-csb 3858  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-iun 4894  df-iin 4895  df-br 5040  df-opab 5102  df-mpt 5120  df-id 5433  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  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 7088  df-ov 7133  df-oprab 7134  df-mpo 7135  df-1st 7664  df-2nd 7665  df-undef 7914  df-map 8383  df-proset 17516  df-poset 17534  df-plt 17546  df-lub 17562  df-glb 17563  df-join 17564  df-meet 17565  df-p0 17627  df-p1 17628  df-lat 17634  df-clat 17696  df-oposet 36350  df-ol 36352  df-oml 36353  df-covers 36440  df-ats 36441  df-atl 36472  df-cvlat 36496  df-hlat 36525  df-llines 36672  df-lplanes 36673  df-lvols 36674  df-lines 36675  df-psubsp 36677  df-pmap 36678  df-padd 36970  df-lhyp 37162  df-laut 37163  df-ldil 37278  df-ltrn 37279  df-trl 37333  df-tendo 37929
This theorem is referenced by:  tendodi1  37958  tendodi2  37959  tendo0mul  38000  tendo0mulr  38001  tendoconid  38003  cdleml3N  38152  cdleml8  38157  erngdvlem3  38164  erngdvlem3-rN  38172  dvalveclem  38199  dvhvscacl  38277  dvhlveclem  38282  diblss  38344  dicvscacl  38365  dih1dimatlem0  38502
  Copyright terms: Public domain W3C validator