Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dihord1 Structured version   Visualization version   GIF version

Theorem dihord1 38221
Description: Part of proof after Lemma N of [Crawley] p. 122. Forward ordering property. TODO: change (𝑄 (𝑋 𝑊)) = 𝑋 to 𝑄 𝑋 using lhpmcvr3 37028, here and all theorems below. (Contributed by NM, 2-Mar-2014.)
Hypotheses
Ref Expression
dihjust.b 𝐵 = (Base‘𝐾)
dihjust.l = (le‘𝐾)
dihjust.j = (join‘𝐾)
dihjust.m = (meet‘𝐾)
dihjust.a 𝐴 = (Atoms‘𝐾)
dihjust.h 𝐻 = (LHyp‘𝐾)
dihjust.i 𝐼 = ((DIsoB‘𝐾)‘𝑊)
dihjust.J 𝐽 = ((DIsoC‘𝐾)‘𝑊)
dihjust.u 𝑈 = ((DVecH‘𝐾)‘𝑊)
dihjust.s = (LSSum‘𝑈)
Assertion
Ref Expression
dihord1 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑋𝐵𝑌𝐵) ∧ ((𝑄 (𝑋 𝑊)) = 𝑋 ∧ (𝑅 (𝑌 𝑊)) = 𝑌𝑋 𝑌)) → ((𝐽𝑄) (𝐼‘(𝑋 𝑊))) ⊆ ((𝐽𝑅) (𝐼‘(𝑌 𝑊))))

Proof of Theorem dihord1
StepHypRef Expression
1 simp11 1197 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑋𝐵𝑌𝐵) ∧ ((𝑄 (𝑋 𝑊)) = 𝑋 ∧ (𝑅 (𝑌 𝑊)) = 𝑌𝑋 𝑌)) → (𝐾 ∈ HL ∧ 𝑊𝐻))
2 simp13 1199 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑋𝐵𝑌𝐵) ∧ ((𝑄 (𝑋 𝑊)) = 𝑋 ∧ (𝑅 (𝑌 𝑊)) = 𝑌𝑋 𝑌)) → (𝑅𝐴 ∧ ¬ 𝑅 𝑊))
3 simp12 1198 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑋𝐵𝑌𝐵) ∧ ((𝑄 (𝑋 𝑊)) = 𝑋 ∧ (𝑅 (𝑌 𝑊)) = 𝑌𝑋 𝑌)) → (𝑄𝐴 ∧ ¬ 𝑄 𝑊))
4 simp11l 1278 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑋𝐵𝑌𝐵) ∧ ((𝑄 (𝑋 𝑊)) = 𝑋 ∧ (𝑅 (𝑌 𝑊)) = 𝑌𝑋 𝑌)) → 𝐾 ∈ HL)
54hllatd 36367 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑋𝐵𝑌𝐵) ∧ ((𝑄 (𝑋 𝑊)) = 𝑋 ∧ (𝑅 (𝑌 𝑊)) = 𝑌𝑋 𝑌)) → 𝐾 ∈ Lat)
6 simp2r 1194 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑋𝐵𝑌𝐵) ∧ ((𝑄 (𝑋 𝑊)) = 𝑋 ∧ (𝑅 (𝑌 𝑊)) = 𝑌𝑋 𝑌)) → 𝑌𝐵)
7 simp11r 1279 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑋𝐵𝑌𝐵) ∧ ((𝑄 (𝑋 𝑊)) = 𝑋 ∧ (𝑅 (𝑌 𝑊)) = 𝑌𝑋 𝑌)) → 𝑊𝐻)
8 dihjust.b . . . . . . 7 𝐵 = (Base‘𝐾)
9 dihjust.h . . . . . . 7 𝐻 = (LHyp‘𝐾)
108, 9lhpbase 37001 . . . . . 6 (𝑊𝐻𝑊𝐵)
117, 10syl 17 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑋𝐵𝑌𝐵) ∧ ((𝑄 (𝑋 𝑊)) = 𝑋 ∧ (𝑅 (𝑌 𝑊)) = 𝑌𝑋 𝑌)) → 𝑊𝐵)
12 dihjust.m . . . . . 6 = (meet‘𝐾)
138, 12latmcl 17652 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑌𝐵𝑊𝐵) → (𝑌 𝑊) ∈ 𝐵)
145, 6, 11, 13syl3anc 1365 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑋𝐵𝑌𝐵) ∧ ((𝑄 (𝑋 𝑊)) = 𝑋 ∧ (𝑅 (𝑌 𝑊)) = 𝑌𝑋 𝑌)) → (𝑌 𝑊) ∈ 𝐵)
15 dihjust.l . . . . . 6 = (le‘𝐾)
168, 15, 12latmle2 17677 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑌𝐵𝑊𝐵) → (𝑌 𝑊) 𝑊)
175, 6, 11, 16syl3anc 1365 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑋𝐵𝑌𝐵) ∧ ((𝑄 (𝑋 𝑊)) = 𝑋 ∧ (𝑅 (𝑌 𝑊)) = 𝑌𝑋 𝑌)) → (𝑌 𝑊) 𝑊)
1814, 17jca 512 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑋𝐵𝑌𝐵) ∧ ((𝑄 (𝑋 𝑊)) = 𝑋 ∧ (𝑅 (𝑌 𝑊)) = 𝑌𝑋 𝑌)) → ((𝑌 𝑊) ∈ 𝐵 ∧ (𝑌 𝑊) 𝑊))
19 simp12l 1280 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑋𝐵𝑌𝐵) ∧ ((𝑄 (𝑋 𝑊)) = 𝑋 ∧ (𝑅 (𝑌 𝑊)) = 𝑌𝑋 𝑌)) → 𝑄𝐴)
20 dihjust.a . . . . . . 7 𝐴 = (Atoms‘𝐾)
218, 20atbase 36292 . . . . . 6 (𝑄𝐴𝑄𝐵)
2219, 21syl 17 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑋𝐵𝑌𝐵) ∧ ((𝑄 (𝑋 𝑊)) = 𝑋 ∧ (𝑅 (𝑌 𝑊)) = 𝑌𝑋 𝑌)) → 𝑄𝐵)
23 simp2l 1193 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑋𝐵𝑌𝐵) ∧ ((𝑄 (𝑋 𝑊)) = 𝑋 ∧ (𝑅 (𝑌 𝑊)) = 𝑌𝑋 𝑌)) → 𝑋𝐵)
248, 12latmcl 17652 . . . . . . 7 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑊𝐵) → (𝑋 𝑊) ∈ 𝐵)
255, 23, 11, 24syl3anc 1365 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑋𝐵𝑌𝐵) ∧ ((𝑄 (𝑋 𝑊)) = 𝑋 ∧ (𝑅 (𝑌 𝑊)) = 𝑌𝑋 𝑌)) → (𝑋 𝑊) ∈ 𝐵)
26 dihjust.j . . . . . . 7 = (join‘𝐾)
278, 26latjcl 17651 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑄𝐵 ∧ (𝑋 𝑊) ∈ 𝐵) → (𝑄 (𝑋 𝑊)) ∈ 𝐵)
285, 22, 25, 27syl3anc 1365 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑋𝐵𝑌𝐵) ∧ ((𝑄 (𝑋 𝑊)) = 𝑋 ∧ (𝑅 (𝑌 𝑊)) = 𝑌𝑋 𝑌)) → (𝑄 (𝑋 𝑊)) ∈ 𝐵)
298, 15, 26latlej1 17660 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑄𝐵 ∧ (𝑋 𝑊) ∈ 𝐵) → 𝑄 (𝑄 (𝑋 𝑊)))
305, 22, 25, 29syl3anc 1365 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑋𝐵𝑌𝐵) ∧ ((𝑄 (𝑋 𝑊)) = 𝑋 ∧ (𝑅 (𝑌 𝑊)) = 𝑌𝑋 𝑌)) → 𝑄 (𝑄 (𝑋 𝑊)))
31 simp31 1203 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑋𝐵𝑌𝐵) ∧ ((𝑄 (𝑋 𝑊)) = 𝑋 ∧ (𝑅 (𝑌 𝑊)) = 𝑌𝑋 𝑌)) → (𝑄 (𝑋 𝑊)) = 𝑋)
32 simp33 1205 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑋𝐵𝑌𝐵) ∧ ((𝑄 (𝑋 𝑊)) = 𝑋 ∧ (𝑅 (𝑌 𝑊)) = 𝑌𝑋 𝑌)) → 𝑋 𝑌)
3331, 32eqbrtrd 5085 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑋𝐵𝑌𝐵) ∧ ((𝑄 (𝑋 𝑊)) = 𝑋 ∧ (𝑅 (𝑌 𝑊)) = 𝑌𝑋 𝑌)) → (𝑄 (𝑋 𝑊)) 𝑌)
348, 15, 5, 22, 28, 6, 30, 33lattrd 17658 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑋𝐵𝑌𝐵) ∧ ((𝑄 (𝑋 𝑊)) = 𝑋 ∧ (𝑅 (𝑌 𝑊)) = 𝑌𝑋 𝑌)) → 𝑄 𝑌)
35 simp32 1204 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑋𝐵𝑌𝐵) ∧ ((𝑄 (𝑋 𝑊)) = 𝑋 ∧ (𝑅 (𝑌 𝑊)) = 𝑌𝑋 𝑌)) → (𝑅 (𝑌 𝑊)) = 𝑌)
3634, 35breqtrrd 5091 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑋𝐵𝑌𝐵) ∧ ((𝑄 (𝑋 𝑊)) = 𝑋 ∧ (𝑅 (𝑌 𝑊)) = 𝑌𝑋 𝑌)) → 𝑄 (𝑅 (𝑌 𝑊)))
37 dihjust.u . . . 4 𝑈 = ((DVecH‘𝐾)‘𝑊)
38 dihjust.s . . . 4 = (LSSum‘𝑈)
39 dihjust.i . . . 4 𝐼 = ((DIsoB‘𝐾)‘𝑊)
40 dihjust.J . . . 4 𝐽 = ((DIsoC‘𝐾)‘𝑊)
418, 15, 26, 20, 9, 37, 38, 39, 40cdlemn5 38204 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ ((𝑌 𝑊) ∈ 𝐵 ∧ (𝑌 𝑊) 𝑊)) ∧ 𝑄 (𝑅 (𝑌 𝑊))) → (𝐽𝑄) ⊆ ((𝐽𝑅) (𝐼‘(𝑌 𝑊))))
421, 2, 3, 18, 36, 41syl131anc 1377 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑋𝐵𝑌𝐵) ∧ ((𝑄 (𝑋 𝑊)) = 𝑋 ∧ (𝑅 (𝑌 𝑊)) = 𝑌𝑋 𝑌)) → (𝐽𝑄) ⊆ ((𝐽𝑅) (𝐼‘(𝑌 𝑊))))
438, 15, 12latmlem1 17681 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑊𝐵)) → (𝑋 𝑌 → (𝑋 𝑊) (𝑌 𝑊)))
445, 23, 6, 11, 43syl13anc 1366 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑋𝐵𝑌𝐵) ∧ ((𝑄 (𝑋 𝑊)) = 𝑋 ∧ (𝑅 (𝑌 𝑊)) = 𝑌𝑋 𝑌)) → (𝑋 𝑌 → (𝑋 𝑊) (𝑌 𝑊)))
4532, 44mpd 15 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑋𝐵𝑌𝐵) ∧ ((𝑄 (𝑋 𝑊)) = 𝑋 ∧ (𝑅 (𝑌 𝑊)) = 𝑌𝑋 𝑌)) → (𝑋 𝑊) (𝑌 𝑊))
468, 15, 12latmle2 17677 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑊𝐵) → (𝑋 𝑊) 𝑊)
475, 23, 11, 46syl3anc 1365 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑋𝐵𝑌𝐵) ∧ ((𝑄 (𝑋 𝑊)) = 𝑋 ∧ (𝑅 (𝑌 𝑊)) = 𝑌𝑋 𝑌)) → (𝑋 𝑊) 𝑊)
488, 15, 9, 39dibord 38162 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑋 𝑊) ∈ 𝐵 ∧ (𝑋 𝑊) 𝑊) ∧ ((𝑌 𝑊) ∈ 𝐵 ∧ (𝑌 𝑊) 𝑊)) → ((𝐼‘(𝑋 𝑊)) ⊆ (𝐼‘(𝑌 𝑊)) ↔ (𝑋 𝑊) (𝑌 𝑊)))
491, 25, 47, 14, 17, 48syl122anc 1373 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑋𝐵𝑌𝐵) ∧ ((𝑄 (𝑋 𝑊)) = 𝑋 ∧ (𝑅 (𝑌 𝑊)) = 𝑌𝑋 𝑌)) → ((𝐼‘(𝑋 𝑊)) ⊆ (𝐼‘(𝑌 𝑊)) ↔ (𝑋 𝑊) (𝑌 𝑊)))
5045, 49mpbird 258 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑋𝐵𝑌𝐵) ∧ ((𝑄 (𝑋 𝑊)) = 𝑋 ∧ (𝑅 (𝑌 𝑊)) = 𝑌𝑋 𝑌)) → (𝐼‘(𝑋 𝑊)) ⊆ (𝐼‘(𝑌 𝑊)))
519, 37, 1dvhlmod 38113 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑋𝐵𝑌𝐵) ∧ ((𝑄 (𝑋 𝑊)) = 𝑋 ∧ (𝑅 (𝑌 𝑊)) = 𝑌𝑋 𝑌)) → 𝑈 ∈ LMod)
52 eqid 2826 . . . . . . 7 (LSubSp‘𝑈) = (LSubSp‘𝑈)
5352lsssssubg 19650 . . . . . 6 (𝑈 ∈ LMod → (LSubSp‘𝑈) ⊆ (SubGrp‘𝑈))
5451, 53syl 17 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑋𝐵𝑌𝐵) ∧ ((𝑄 (𝑋 𝑊)) = 𝑋 ∧ (𝑅 (𝑌 𝑊)) = 𝑌𝑋 𝑌)) → (LSubSp‘𝑈) ⊆ (SubGrp‘𝑈))
5515, 20, 9, 37, 40, 52diclss 38196 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) → (𝐽𝑅) ∈ (LSubSp‘𝑈))
561, 2, 55syl2anc 584 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑋𝐵𝑌𝐵) ∧ ((𝑄 (𝑋 𝑊)) = 𝑋 ∧ (𝑅 (𝑌 𝑊)) = 𝑌𝑋 𝑌)) → (𝐽𝑅) ∈ (LSubSp‘𝑈))
5754, 56sseldd 3972 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑋𝐵𝑌𝐵) ∧ ((𝑄 (𝑋 𝑊)) = 𝑋 ∧ (𝑅 (𝑌 𝑊)) = 𝑌𝑋 𝑌)) → (𝐽𝑅) ∈ (SubGrp‘𝑈))
588, 15, 9, 37, 39, 52diblss 38173 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑌 𝑊) ∈ 𝐵 ∧ (𝑌 𝑊) 𝑊)) → (𝐼‘(𝑌 𝑊)) ∈ (LSubSp‘𝑈))
591, 14, 17, 58syl12anc 834 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑋𝐵𝑌𝐵) ∧ ((𝑄 (𝑋 𝑊)) = 𝑋 ∧ (𝑅 (𝑌 𝑊)) = 𝑌𝑋 𝑌)) → (𝐼‘(𝑌 𝑊)) ∈ (LSubSp‘𝑈))
6054, 59sseldd 3972 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑋𝐵𝑌𝐵) ∧ ((𝑄 (𝑋 𝑊)) = 𝑋 ∧ (𝑅 (𝑌 𝑊)) = 𝑌𝑋 𝑌)) → (𝐼‘(𝑌 𝑊)) ∈ (SubGrp‘𝑈))
6138lsmub2 18703 . . . 4 (((𝐽𝑅) ∈ (SubGrp‘𝑈) ∧ (𝐼‘(𝑌 𝑊)) ∈ (SubGrp‘𝑈)) → (𝐼‘(𝑌 𝑊)) ⊆ ((𝐽𝑅) (𝐼‘(𝑌 𝑊))))
6257, 60, 61syl2anc 584 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑋𝐵𝑌𝐵) ∧ ((𝑄 (𝑋 𝑊)) = 𝑋 ∧ (𝑅 (𝑌 𝑊)) = 𝑌𝑋 𝑌)) → (𝐼‘(𝑌 𝑊)) ⊆ ((𝐽𝑅) (𝐼‘(𝑌 𝑊))))
6350, 62sstrd 3981 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑋𝐵𝑌𝐵) ∧ ((𝑄 (𝑋 𝑊)) = 𝑋 ∧ (𝑅 (𝑌 𝑊)) = 𝑌𝑋 𝑌)) → (𝐼‘(𝑋 𝑊)) ⊆ ((𝐽𝑅) (𝐼‘(𝑌 𝑊))))
6415, 20, 9, 37, 40, 52diclss 38196 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝐽𝑄) ∈ (LSubSp‘𝑈))
651, 3, 64syl2anc 584 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑋𝐵𝑌𝐵) ∧ ((𝑄 (𝑋 𝑊)) = 𝑋 ∧ (𝑅 (𝑌 𝑊)) = 𝑌𝑋 𝑌)) → (𝐽𝑄) ∈ (LSubSp‘𝑈))
6654, 65sseldd 3972 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑋𝐵𝑌𝐵) ∧ ((𝑄 (𝑋 𝑊)) = 𝑋 ∧ (𝑅 (𝑌 𝑊)) = 𝑌𝑋 𝑌)) → (𝐽𝑄) ∈ (SubGrp‘𝑈))
678, 15, 9, 37, 39, 52diblss 38173 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑋 𝑊) ∈ 𝐵 ∧ (𝑋 𝑊) 𝑊)) → (𝐼‘(𝑋 𝑊)) ∈ (LSubSp‘𝑈))
681, 25, 47, 67syl12anc 834 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑋𝐵𝑌𝐵) ∧ ((𝑄 (𝑋 𝑊)) = 𝑋 ∧ (𝑅 (𝑌 𝑊)) = 𝑌𝑋 𝑌)) → (𝐼‘(𝑋 𝑊)) ∈ (LSubSp‘𝑈))
6954, 68sseldd 3972 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑋𝐵𝑌𝐵) ∧ ((𝑄 (𝑋 𝑊)) = 𝑋 ∧ (𝑅 (𝑌 𝑊)) = 𝑌𝑋 𝑌)) → (𝐼‘(𝑋 𝑊)) ∈ (SubGrp‘𝑈))
7052, 38lsmcl 19775 . . . . 5 ((𝑈 ∈ LMod ∧ (𝐽𝑅) ∈ (LSubSp‘𝑈) ∧ (𝐼‘(𝑌 𝑊)) ∈ (LSubSp‘𝑈)) → ((𝐽𝑅) (𝐼‘(𝑌 𝑊))) ∈ (LSubSp‘𝑈))
7151, 56, 59, 70syl3anc 1365 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑋𝐵𝑌𝐵) ∧ ((𝑄 (𝑋 𝑊)) = 𝑋 ∧ (𝑅 (𝑌 𝑊)) = 𝑌𝑋 𝑌)) → ((𝐽𝑅) (𝐼‘(𝑌 𝑊))) ∈ (LSubSp‘𝑈))
7254, 71sseldd 3972 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑋𝐵𝑌𝐵) ∧ ((𝑄 (𝑋 𝑊)) = 𝑋 ∧ (𝑅 (𝑌 𝑊)) = 𝑌𝑋 𝑌)) → ((𝐽𝑅) (𝐼‘(𝑌 𝑊))) ∈ (SubGrp‘𝑈))
7338lsmlub 18710 . . 3 (((𝐽𝑄) ∈ (SubGrp‘𝑈) ∧ (𝐼‘(𝑋 𝑊)) ∈ (SubGrp‘𝑈) ∧ ((𝐽𝑅) (𝐼‘(𝑌 𝑊))) ∈ (SubGrp‘𝑈)) → (((𝐽𝑄) ⊆ ((𝐽𝑅) (𝐼‘(𝑌 𝑊))) ∧ (𝐼‘(𝑋 𝑊)) ⊆ ((𝐽𝑅) (𝐼‘(𝑌 𝑊)))) ↔ ((𝐽𝑄) (𝐼‘(𝑋 𝑊))) ⊆ ((𝐽𝑅) (𝐼‘(𝑌 𝑊)))))
7466, 69, 72, 73syl3anc 1365 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑋𝐵𝑌𝐵) ∧ ((𝑄 (𝑋 𝑊)) = 𝑋 ∧ (𝑅 (𝑌 𝑊)) = 𝑌𝑋 𝑌)) → (((𝐽𝑄) ⊆ ((𝐽𝑅) (𝐼‘(𝑌 𝑊))) ∧ (𝐼‘(𝑋 𝑊)) ⊆ ((𝐽𝑅) (𝐼‘(𝑌 𝑊)))) ↔ ((𝐽𝑄) (𝐼‘(𝑋 𝑊))) ⊆ ((𝐽𝑅) (𝐼‘(𝑌 𝑊)))))
7542, 63, 74mpbi2and 708 1 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑋𝐵𝑌𝐵) ∧ ((𝑄 (𝑋 𝑊)) = 𝑋 ∧ (𝑅 (𝑌 𝑊)) = 𝑌𝑋 𝑌)) → ((𝐽𝑄) (𝐼‘(𝑋 𝑊))) ⊆ ((𝐽𝑅) (𝐼‘(𝑌 𝑊))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  w3a 1081   = wceq 1530  wcel 2107  wss 3940   class class class wbr 5063  cfv 6352  (class class class)co 7148  Basecbs 16473  lecple 16562  joincjn 17544  meetcmee 17545  Latclat 17645  SubGrpcsubg 18203  LSSumclsm 18679  LModclmod 19554  LSubSpclss 19623  Atomscatm 36266  HLchlt 36353  LHypclh 36987  DVecHcdvh 38081  DIsoBcdib 38141  DIsoCcdic 38175
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-13 2385  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603  ax-riotaBAD 35956
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-fal 1543  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-nel 3129  df-ral 3148  df-rex 3149  df-reu 3150  df-rmo 3151  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-int 4875  df-iun 4919  df-iin 4920  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6146  df-ord 6192  df-on 6193  df-lim 6194  df-suc 6195  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-f1 6357  df-fo 6358  df-f1o 6359  df-fv 6360  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-om 7569  df-1st 7680  df-2nd 7681  df-tpos 7883  df-undef 7930  df-wrecs 7938  df-recs 7999  df-rdg 8037  df-1o 8093  df-oadd 8097  df-er 8279  df-map 8398  df-en 8499  df-dom 8500  df-sdom 8501  df-fin 8502  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-nn 11628  df-2 11689  df-3 11690  df-4 11691  df-5 11692  df-6 11693  df-n0 11887  df-z 11971  df-uz 12233  df-fz 12883  df-struct 16475  df-ndx 16476  df-slot 16477  df-base 16479  df-sets 16480  df-ress 16481  df-plusg 16568  df-mulr 16569  df-sca 16571  df-vsca 16572  df-0g 16705  df-proset 17528  df-poset 17546  df-plt 17558  df-lub 17574  df-glb 17575  df-join 17576  df-meet 17577  df-p0 17639  df-p1 17640  df-lat 17646  df-clat 17708  df-mgm 17842  df-sgrp 17890  df-mnd 17901  df-submnd 17945  df-grp 18036  df-minusg 18037  df-sbg 18038  df-subg 18206  df-cntz 18377  df-lsm 18681  df-cmn 18828  df-abl 18829  df-mgp 19160  df-ur 19172  df-ring 19219  df-oppr 19293  df-dvdsr 19311  df-unit 19312  df-invr 19342  df-dvr 19353  df-drng 19424  df-lmod 19556  df-lss 19624  df-lsp 19664  df-lvec 19795  df-oposet 36179  df-ol 36181  df-oml 36182  df-covers 36269  df-ats 36270  df-atl 36301  df-cvlat 36325  df-hlat 36354  df-llines 36501  df-lplanes 36502  df-lvols 36503  df-lines 36504  df-psubsp 36506  df-pmap 36507  df-padd 36799  df-lhyp 36991  df-laut 36992  df-ldil 37107  df-ltrn 37108  df-trl 37162  df-tendo 37758  df-edring 37760  df-disoa 38032  df-dvech 38082  df-dib 38142  df-dic 38176
This theorem is referenced by:  dihord4  38261
  Copyright terms: Public domain W3C validator