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Theorem dihord1 36326
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 35130, 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 1089 . . 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 1091 . . 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 1090 . . 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 1170 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑋𝐵𝑌𝐵) ∧ ((𝑄 (𝑋 𝑊)) = 𝑋 ∧ (𝑅 (𝑌 𝑊)) = 𝑌𝑋 𝑌)) → 𝐾 ∈ HL)
5 hllat 34469 . . . . . 6 (𝐾 ∈ HL → 𝐾 ∈ Lat)
64, 5syl 17 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑋𝐵𝑌𝐵) ∧ ((𝑄 (𝑋 𝑊)) = 𝑋 ∧ (𝑅 (𝑌 𝑊)) = 𝑌𝑋 𝑌)) → 𝐾 ∈ Lat)
7 simp2r 1086 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑋𝐵𝑌𝐵) ∧ ((𝑄 (𝑋 𝑊)) = 𝑋 ∧ (𝑅 (𝑌 𝑊)) = 𝑌𝑋 𝑌)) → 𝑌𝐵)
8 simp11r 1171 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑋𝐵𝑌𝐵) ∧ ((𝑄 (𝑋 𝑊)) = 𝑋 ∧ (𝑅 (𝑌 𝑊)) = 𝑌𝑋 𝑌)) → 𝑊𝐻)
9 dihjust.b . . . . . . 7 𝐵 = (Base‘𝐾)
10 dihjust.h . . . . . . 7 𝐻 = (LHyp‘𝐾)
119, 10lhpbase 35103 . . . . . 6 (𝑊𝐻𝑊𝐵)
128, 11syl 17 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑋𝐵𝑌𝐵) ∧ ((𝑄 (𝑋 𝑊)) = 𝑋 ∧ (𝑅 (𝑌 𝑊)) = 𝑌𝑋 𝑌)) → 𝑊𝐵)
13 dihjust.m . . . . . 6 = (meet‘𝐾)
149, 13latmcl 17033 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑌𝐵𝑊𝐵) → (𝑌 𝑊) ∈ 𝐵)
156, 7, 12, 14syl3anc 1324 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑋𝐵𝑌𝐵) ∧ ((𝑄 (𝑋 𝑊)) = 𝑋 ∧ (𝑅 (𝑌 𝑊)) = 𝑌𝑋 𝑌)) → (𝑌 𝑊) ∈ 𝐵)
16 dihjust.l . . . . . 6 = (le‘𝐾)
179, 16, 13latmle2 17058 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑌𝐵𝑊𝐵) → (𝑌 𝑊) 𝑊)
186, 7, 12, 17syl3anc 1324 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑋𝐵𝑌𝐵) ∧ ((𝑄 (𝑋 𝑊)) = 𝑋 ∧ (𝑅 (𝑌 𝑊)) = 𝑌𝑋 𝑌)) → (𝑌 𝑊) 𝑊)
1915, 18jca 554 . . 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))
20 simp12l 1172 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑋𝐵𝑌𝐵) ∧ ((𝑄 (𝑋 𝑊)) = 𝑋 ∧ (𝑅 (𝑌 𝑊)) = 𝑌𝑋 𝑌)) → 𝑄𝐴)
21 dihjust.a . . . . . . 7 𝐴 = (Atoms‘𝐾)
229, 21atbase 34395 . . . . . 6 (𝑄𝐴𝑄𝐵)
2320, 22syl 17 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑋𝐵𝑌𝐵) ∧ ((𝑄 (𝑋 𝑊)) = 𝑋 ∧ (𝑅 (𝑌 𝑊)) = 𝑌𝑋 𝑌)) → 𝑄𝐵)
24 simp2l 1085 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑋𝐵𝑌𝐵) ∧ ((𝑄 (𝑋 𝑊)) = 𝑋 ∧ (𝑅 (𝑌 𝑊)) = 𝑌𝑋 𝑌)) → 𝑋𝐵)
259, 13latmcl 17033 . . . . . . 7 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑊𝐵) → (𝑋 𝑊) ∈ 𝐵)
266, 24, 12, 25syl3anc 1324 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑋𝐵𝑌𝐵) ∧ ((𝑄 (𝑋 𝑊)) = 𝑋 ∧ (𝑅 (𝑌 𝑊)) = 𝑌𝑋 𝑌)) → (𝑋 𝑊) ∈ 𝐵)
27 dihjust.j . . . . . . 7 = (join‘𝐾)
289, 27latjcl 17032 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑄𝐵 ∧ (𝑋 𝑊) ∈ 𝐵) → (𝑄 (𝑋 𝑊)) ∈ 𝐵)
296, 23, 26, 28syl3anc 1324 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑋𝐵𝑌𝐵) ∧ ((𝑄 (𝑋 𝑊)) = 𝑋 ∧ (𝑅 (𝑌 𝑊)) = 𝑌𝑋 𝑌)) → (𝑄 (𝑋 𝑊)) ∈ 𝐵)
309, 16, 27latlej1 17041 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑄𝐵 ∧ (𝑋 𝑊) ∈ 𝐵) → 𝑄 (𝑄 (𝑋 𝑊)))
316, 23, 26, 30syl3anc 1324 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑋𝐵𝑌𝐵) ∧ ((𝑄 (𝑋 𝑊)) = 𝑋 ∧ (𝑅 (𝑌 𝑊)) = 𝑌𝑋 𝑌)) → 𝑄 (𝑄 (𝑋 𝑊)))
32 simp31 1095 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑋𝐵𝑌𝐵) ∧ ((𝑄 (𝑋 𝑊)) = 𝑋 ∧ (𝑅 (𝑌 𝑊)) = 𝑌𝑋 𝑌)) → (𝑄 (𝑋 𝑊)) = 𝑋)
33 simp33 1097 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑋𝐵𝑌𝐵) ∧ ((𝑄 (𝑋 𝑊)) = 𝑋 ∧ (𝑅 (𝑌 𝑊)) = 𝑌𝑋 𝑌)) → 𝑋 𝑌)
3432, 33eqbrtrd 4666 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑋𝐵𝑌𝐵) ∧ ((𝑄 (𝑋 𝑊)) = 𝑋 ∧ (𝑅 (𝑌 𝑊)) = 𝑌𝑋 𝑌)) → (𝑄 (𝑋 𝑊)) 𝑌)
359, 16, 6, 23, 29, 7, 31, 34lattrd 17039 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑋𝐵𝑌𝐵) ∧ ((𝑄 (𝑋 𝑊)) = 𝑋 ∧ (𝑅 (𝑌 𝑊)) = 𝑌𝑋 𝑌)) → 𝑄 𝑌)
36 simp32 1096 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑋𝐵𝑌𝐵) ∧ ((𝑄 (𝑋 𝑊)) = 𝑋 ∧ (𝑅 (𝑌 𝑊)) = 𝑌𝑋 𝑌)) → (𝑅 (𝑌 𝑊)) = 𝑌)
3735, 36breqtrrd 4672 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑋𝐵𝑌𝐵) ∧ ((𝑄 (𝑋 𝑊)) = 𝑋 ∧ (𝑅 (𝑌 𝑊)) = 𝑌𝑋 𝑌)) → 𝑄 (𝑅 (𝑌 𝑊)))
38 dihjust.u . . . 4 𝑈 = ((DVecH‘𝐾)‘𝑊)
39 dihjust.s . . . 4 = (LSSum‘𝑈)
40 dihjust.i . . . 4 𝐼 = ((DIsoB‘𝐾)‘𝑊)
41 dihjust.J . . . 4 𝐽 = ((DIsoC‘𝐾)‘𝑊)
429, 16, 27, 21, 10, 38, 39, 40, 41cdlemn5 36309 . . 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))))
431, 2, 3, 19, 37, 42syl131anc 1337 . 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))))
449, 16, 13latmlem1 17062 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑊𝐵)) → (𝑋 𝑌 → (𝑋 𝑊) (𝑌 𝑊)))
456, 24, 7, 12, 44syl13anc 1326 . . . . 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)))
4633, 45mpd 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))
479, 16, 13latmle2 17058 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑊𝐵) → (𝑋 𝑊) 𝑊)
486, 24, 12, 47syl3anc 1324 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑋𝐵𝑌𝐵) ∧ ((𝑄 (𝑋 𝑊)) = 𝑋 ∧ (𝑅 (𝑌 𝑊)) = 𝑌𝑋 𝑌)) → (𝑋 𝑊) 𝑊)
499, 16, 10, 40dibord 36267 . . . . 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)))
501, 26, 48, 15, 18, 49syl122anc 1333 . . . 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)))
5146, 50mpbird 247 . . 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)))
5210, 38, 1dvhlmod 36218 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑋𝐵𝑌𝐵) ∧ ((𝑄 (𝑋 𝑊)) = 𝑋 ∧ (𝑅 (𝑌 𝑊)) = 𝑌𝑋 𝑌)) → 𝑈 ∈ LMod)
53 eqid 2620 . . . . . . 7 (LSubSp‘𝑈) = (LSubSp‘𝑈)
5453lsssssubg 18939 . . . . . 6 (𝑈 ∈ LMod → (LSubSp‘𝑈) ⊆ (SubGrp‘𝑈))
5552, 54syl 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))
5616, 21, 10, 38, 41, 53diclss 36301 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) → (𝐽𝑅) ∈ (LSubSp‘𝑈))
571, 2, 56syl2anc 692 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑋𝐵𝑌𝐵) ∧ ((𝑄 (𝑋 𝑊)) = 𝑋 ∧ (𝑅 (𝑌 𝑊)) = 𝑌𝑋 𝑌)) → (𝐽𝑅) ∈ (LSubSp‘𝑈))
5855, 57sseldd 3596 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑋𝐵𝑌𝐵) ∧ ((𝑄 (𝑋 𝑊)) = 𝑋 ∧ (𝑅 (𝑌 𝑊)) = 𝑌𝑋 𝑌)) → (𝐽𝑅) ∈ (SubGrp‘𝑈))
599, 16, 10, 38, 40, 53diblss 36278 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑌 𝑊) ∈ 𝐵 ∧ (𝑌 𝑊) 𝑊)) → (𝐼‘(𝑌 𝑊)) ∈ (LSubSp‘𝑈))
601, 15, 18, 59syl12anc 1322 . . . . 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))
6155, 60sseldd 3596 . . . 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))
6239lsmub2 18053 . . . 4 (((𝐽𝑅) ∈ (SubGrp‘𝑈) ∧ (𝐼‘(𝑌 𝑊)) ∈ (SubGrp‘𝑈)) → (𝐼‘(𝑌 𝑊)) ⊆ ((𝐽𝑅) (𝐼‘(𝑌 𝑊))))
6358, 61, 62syl2anc 692 . . 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))))
6451, 63sstrd 3605 . 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))))
6516, 21, 10, 38, 41, 53diclss 36301 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝐽𝑄) ∈ (LSubSp‘𝑈))
661, 3, 65syl2anc 692 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑋𝐵𝑌𝐵) ∧ ((𝑄 (𝑋 𝑊)) = 𝑋 ∧ (𝑅 (𝑌 𝑊)) = 𝑌𝑋 𝑌)) → (𝐽𝑄) ∈ (LSubSp‘𝑈))
6755, 66sseldd 3596 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑋𝐵𝑌𝐵) ∧ ((𝑄 (𝑋 𝑊)) = 𝑋 ∧ (𝑅 (𝑌 𝑊)) = 𝑌𝑋 𝑌)) → (𝐽𝑄) ∈ (SubGrp‘𝑈))
689, 16, 10, 38, 40, 53diblss 36278 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑋 𝑊) ∈ 𝐵 ∧ (𝑋 𝑊) 𝑊)) → (𝐼‘(𝑋 𝑊)) ∈ (LSubSp‘𝑈))
691, 26, 48, 68syl12anc 1322 . . . 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))
7055, 69sseldd 3596 . . 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))
7153, 39lsmcl 19064 . . . . 5 ((𝑈 ∈ LMod ∧ (𝐽𝑅) ∈ (LSubSp‘𝑈) ∧ (𝐼‘(𝑌 𝑊)) ∈ (LSubSp‘𝑈)) → ((𝐽𝑅) (𝐼‘(𝑌 𝑊))) ∈ (LSubSp‘𝑈))
7252, 57, 60, 71syl3anc 1324 . . . 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))
7355, 72sseldd 3596 . . 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))
7439lsmlub 18059 . . 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)))))
7567, 70, 73, 74syl3anc 1324 . 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)))))
7643, 64, 75mpbi2and 955 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 196  wa 384  w3a 1036   = wceq 1481  wcel 1988  wss 3567   class class class wbr 4644  cfv 5876  (class class class)co 6635  Basecbs 15838  lecple 15929  joincjn 16925  meetcmee 16926  Latclat 17026  SubGrpcsubg 17569  LSSumclsm 18030  LModclmod 18844  LSubSpclss 18913  Atomscatm 34369  HLchlt 34456  LHypclh 35089  DVecHcdvh 36186  DIsoBcdib 36246  DIsoCcdic 36280
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934  ax-cnex 9977  ax-resscn 9978  ax-1cn 9979  ax-icn 9980  ax-addcl 9981  ax-addrcl 9982  ax-mulcl 9983  ax-mulrcl 9984  ax-mulcom 9985  ax-addass 9986  ax-mulass 9987  ax-distr 9988  ax-i2m1 9989  ax-1ne0 9990  ax-1rid 9991  ax-rnegex 9992  ax-rrecex 9993  ax-cnre 9994  ax-pre-lttri 9995  ax-pre-lttrn 9996  ax-pre-ltadd 9997  ax-pre-mulgt0 9998  ax-riotaBAD 34058
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-fal 1487  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-nel 2895  df-ral 2914  df-rex 2915  df-reu 2916  df-rmo 2917  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-int 4467  df-iun 4513  df-iin 4514  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-riota 6596  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-om 7051  df-1st 7153  df-2nd 7154  df-tpos 7337  df-undef 7384  df-wrecs 7392  df-recs 7453  df-rdg 7491  df-1o 7545  df-oadd 7549  df-er 7727  df-map 7844  df-en 7941  df-dom 7942  df-sdom 7943  df-fin 7944  df-pnf 10061  df-mnf 10062  df-xr 10063  df-ltxr 10064  df-le 10065  df-sub 10253  df-neg 10254  df-nn 11006  df-2 11064  df-3 11065  df-4 11066  df-5 11067  df-6 11068  df-n0 11278  df-z 11363  df-uz 11673  df-fz 12312  df-struct 15840  df-ndx 15841  df-slot 15842  df-base 15844  df-sets 15845  df-ress 15846  df-plusg 15935  df-mulr 15936  df-sca 15938  df-vsca 15939  df-0g 16083  df-preset 16909  df-poset 16927  df-plt 16939  df-lub 16955  df-glb 16956  df-join 16957  df-meet 16958  df-p0 17020  df-p1 17021  df-lat 17027  df-clat 17089  df-mgm 17223  df-sgrp 17265  df-mnd 17276  df-submnd 17317  df-grp 17406  df-minusg 17407  df-sbg 17408  df-subg 17572  df-cntz 17731  df-lsm 18032  df-cmn 18176  df-abl 18177  df-mgp 18471  df-ur 18483  df-ring 18530  df-oppr 18604  df-dvdsr 18622  df-unit 18623  df-invr 18653  df-dvr 18664  df-drng 18730  df-lmod 18846  df-lss 18914  df-lsp 18953  df-lvec 19084  df-oposet 34282  df-ol 34284  df-oml 34285  df-covers 34372  df-ats 34373  df-atl 34404  df-cvlat 34428  df-hlat 34457  df-llines 34603  df-lplanes 34604  df-lvols 34605  df-lines 34606  df-psubsp 34608  df-pmap 34609  df-padd 34901  df-lhyp 35093  df-laut 35094  df-ldil 35209  df-ltrn 35210  df-trl 35265  df-tendo 35862  df-edring 35864  df-disoa 36137  df-dvech 36187  df-dib 36247  df-dic 36281
This theorem is referenced by:  dihord4  36366
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