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Theorem dihmeetlem13N 39782
Description: Lemma for isomorphism H of a lattice meet. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)
Hypotheses
Ref Expression
dihmeetlem13.b 𝐵 = (Base‘𝐾)
dihmeetlem13.l = (le‘𝐾)
dihmeetlem13.j = (join‘𝐾)
dihmeetlem13.a 𝐴 = (Atoms‘𝐾)
dihmeetlem13.h 𝐻 = (LHyp‘𝐾)
dihmeetlem13.p 𝑃 = ((oc‘𝐾)‘𝑊)
dihmeetlem13.t 𝑇 = ((LTrn‘𝐾)‘𝑊)
dihmeetlem13.e 𝐸 = ((TEndo‘𝐾)‘𝑊)
dihmeetlem13.o 𝑂 = (𝑇 ↦ ( I ↾ 𝐵))
dihmeetlem13.i 𝐼 = ((DIsoH‘𝐾)‘𝑊)
dihmeetlem13.u 𝑈 = ((DVecH‘𝐾)‘𝑊)
dihmeetlem13.z 0 = (0g𝑈)
dihmeetlem13.f 𝐹 = (𝑇 (𝑃) = 𝑄)
dihmeetlem13.g 𝐺 = (𝑇 (𝑃) = 𝑅)
Assertion
Ref Expression
dihmeetlem13N (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) → ((𝐼𝑄) ∩ (𝐼𝑅)) = { 0 })
Distinct variable groups:   ,   𝐴,   𝐵,   ,𝐻   ,𝐾   𝑃,   𝑄,   𝑅,   𝑇,   ,𝑊
Allowed substitution hints:   𝑈()   𝐸()   𝐹()   𝐺()   𝐼()   ()   𝑂()   0 ()

Proof of Theorem dihmeetlem13N
Dummy variables 𝑓 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dihmeetlem13.h . . . . . 6 𝐻 = (LHyp‘𝐾)
2 dihmeetlem13.i . . . . . 6 𝐼 = ((DIsoH‘𝐾)‘𝑊)
31, 2dihvalrel 39742 . . . . 5 ((𝐾 ∈ HL ∧ 𝑊𝐻) → Rel (𝐼𝑄))
433ad2ant1 1133 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) → Rel (𝐼𝑄))
5 relin1 5768 . . . 4 (Rel (𝐼𝑄) → Rel ((𝐼𝑄) ∩ (𝐼𝑅)))
64, 5syl 17 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) → Rel ((𝐼𝑄) ∩ (𝐼𝑅)))
7 elin 3926 . . . . . 6 (⟨𝑓, 𝑠⟩ ∈ ((𝐼𝑄) ∩ (𝐼𝑅)) ↔ (⟨𝑓, 𝑠⟩ ∈ (𝐼𝑄) ∧ ⟨𝑓, 𝑠⟩ ∈ (𝐼𝑅)))
8 simp1 1136 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) → (𝐾 ∈ HL ∧ 𝑊𝐻))
9 simp2l 1199 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) → (𝑄𝐴 ∧ ¬ 𝑄 𝑊))
10 dihmeetlem13.l . . . . . . . . 9 = (le‘𝐾)
11 dihmeetlem13.a . . . . . . . . 9 𝐴 = (Atoms‘𝐾)
12 dihmeetlem13.p . . . . . . . . 9 𝑃 = ((oc‘𝐾)‘𝑊)
13 dihmeetlem13.t . . . . . . . . 9 𝑇 = ((LTrn‘𝐾)‘𝑊)
14 dihmeetlem13.e . . . . . . . . 9 𝐸 = ((TEndo‘𝐾)‘𝑊)
15 dihmeetlem13.f . . . . . . . . 9 𝐹 = (𝑇 (𝑃) = 𝑄)
16 vex 3449 . . . . . . . . 9 𝑓 ∈ V
17 vex 3449 . . . . . . . . 9 𝑠 ∈ V
1810, 11, 1, 12, 13, 14, 2, 15, 16, 17dihopelvalcqat 39709 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (⟨𝑓, 𝑠⟩ ∈ (𝐼𝑄) ↔ (𝑓 = (𝑠𝐹) ∧ 𝑠𝐸)))
198, 9, 18syl2anc 584 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) → (⟨𝑓, 𝑠⟩ ∈ (𝐼𝑄) ↔ (𝑓 = (𝑠𝐹) ∧ 𝑠𝐸)))
20 simp2r 1200 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) → (𝑅𝐴 ∧ ¬ 𝑅 𝑊))
21 dihmeetlem13.g . . . . . . . . 9 𝐺 = (𝑇 (𝑃) = 𝑅)
2210, 11, 1, 12, 13, 14, 2, 21, 16, 17dihopelvalcqat 39709 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) → (⟨𝑓, 𝑠⟩ ∈ (𝐼𝑅) ↔ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸)))
238, 20, 22syl2anc 584 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) → (⟨𝑓, 𝑠⟩ ∈ (𝐼𝑅) ↔ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸)))
2419, 23anbi12d 631 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) → ((⟨𝑓, 𝑠⟩ ∈ (𝐼𝑄) ∧ ⟨𝑓, 𝑠⟩ ∈ (𝐼𝑅)) ↔ ((𝑓 = (𝑠𝐹) ∧ 𝑠𝐸) ∧ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸))))
257, 24bitrid 282 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) → (⟨𝑓, 𝑠⟩ ∈ ((𝐼𝑄) ∩ (𝐼𝑅)) ↔ ((𝑓 = (𝑠𝐹) ∧ 𝑠𝐸) ∧ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸))))
26 simprll 777 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) ∧ ((𝑓 = (𝑠𝐹) ∧ 𝑠𝐸) ∧ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸))) → 𝑓 = (𝑠𝐹))
27 simpl3 1193 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) ∧ ((𝑓 = (𝑠𝐹) ∧ 𝑠𝐸) ∧ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸))) → 𝑄𝑅)
28 fveq1 6841 . . . . . . . . . . . . 13 (𝐹 = 𝐺 → (𝐹𝑃) = (𝐺𝑃))
29 simpl1 1191 . . . . . . . . . . . . . . 15 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) ∧ ((𝑓 = (𝑠𝐹) ∧ 𝑠𝐸) ∧ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸))) → (𝐾 ∈ HL ∧ 𝑊𝐻))
3010, 11, 1, 12lhpocnel2 38482 . . . . . . . . . . . . . . . 16 ((𝐾 ∈ HL ∧ 𝑊𝐻) → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
3129, 30syl 17 . . . . . . . . . . . . . . 15 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) ∧ ((𝑓 = (𝑠𝐹) ∧ 𝑠𝐸) ∧ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸))) → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
32 simpl2l 1226 . . . . . . . . . . . . . . 15 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) ∧ ((𝑓 = (𝑠𝐹) ∧ 𝑠𝐸) ∧ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸))) → (𝑄𝐴 ∧ ¬ 𝑄 𝑊))
3310, 11, 1, 13, 15ltrniotaval 39044 . . . . . . . . . . . . . . 15 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝐹𝑃) = 𝑄)
3429, 31, 32, 33syl3anc 1371 . . . . . . . . . . . . . 14 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) ∧ ((𝑓 = (𝑠𝐹) ∧ 𝑠𝐸) ∧ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸))) → (𝐹𝑃) = 𝑄)
35 simpl2r 1227 . . . . . . . . . . . . . . 15 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) ∧ ((𝑓 = (𝑠𝐹) ∧ 𝑠𝐸) ∧ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸))) → (𝑅𝐴 ∧ ¬ 𝑅 𝑊))
3610, 11, 1, 13, 21ltrniotaval 39044 . . . . . . . . . . . . . . 15 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) → (𝐺𝑃) = 𝑅)
3729, 31, 35, 36syl3anc 1371 . . . . . . . . . . . . . 14 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) ∧ ((𝑓 = (𝑠𝐹) ∧ 𝑠𝐸) ∧ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸))) → (𝐺𝑃) = 𝑅)
3834, 37eqeq12d 2752 . . . . . . . . . . . . 13 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) ∧ ((𝑓 = (𝑠𝐹) ∧ 𝑠𝐸) ∧ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸))) → ((𝐹𝑃) = (𝐺𝑃) ↔ 𝑄 = 𝑅))
3928, 38imbitrid 243 . . . . . . . . . . . 12 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) ∧ ((𝑓 = (𝑠𝐹) ∧ 𝑠𝐸) ∧ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸))) → (𝐹 = 𝐺𝑄 = 𝑅))
4039necon3d 2964 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) ∧ ((𝑓 = (𝑠𝐹) ∧ 𝑠𝐸) ∧ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸))) → (𝑄𝑅𝐹𝐺))
4127, 40mpd 15 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) ∧ ((𝑓 = (𝑠𝐹) ∧ 𝑠𝐸) ∧ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸))) → 𝐹𝐺)
42 simp2ll 1240 . . . . . . . . . . . . . 14 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) ∧ ((𝑓 = (𝑠𝐹) ∧ 𝑠𝐸) ∧ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸)) ∧ 𝑠𝑂) → 𝑓 = (𝑠𝐹))
43 simp2rl 1242 . . . . . . . . . . . . . 14 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) ∧ ((𝑓 = (𝑠𝐹) ∧ 𝑠𝐸) ∧ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸)) ∧ 𝑠𝑂) → 𝑓 = (𝑠𝐺))
4442, 43eqtr3d 2778 . . . . . . . . . . . . 13 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) ∧ ((𝑓 = (𝑠𝐹) ∧ 𝑠𝐸) ∧ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸)) ∧ 𝑠𝑂) → (𝑠𝐹) = (𝑠𝐺))
45 simp11 1203 . . . . . . . . . . . . . 14 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) ∧ ((𝑓 = (𝑠𝐹) ∧ 𝑠𝐸) ∧ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸)) ∧ 𝑠𝑂) → (𝐾 ∈ HL ∧ 𝑊𝐻))
46 simp2rr 1243 . . . . . . . . . . . . . 14 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) ∧ ((𝑓 = (𝑠𝐹) ∧ 𝑠𝐸) ∧ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸)) ∧ 𝑠𝑂) → 𝑠𝐸)
47 simp3 1138 . . . . . . . . . . . . . 14 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) ∧ ((𝑓 = (𝑠𝐹) ∧ 𝑠𝐸) ∧ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸)) ∧ 𝑠𝑂) → 𝑠𝑂)
4845, 30syl 17 . . . . . . . . . . . . . . 15 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) ∧ ((𝑓 = (𝑠𝐹) ∧ 𝑠𝐸) ∧ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸)) ∧ 𝑠𝑂) → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
49 simp12l 1286 . . . . . . . . . . . . . . 15 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) ∧ ((𝑓 = (𝑠𝐹) ∧ 𝑠𝐸) ∧ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸)) ∧ 𝑠𝑂) → (𝑄𝐴 ∧ ¬ 𝑄 𝑊))
5010, 11, 1, 13, 15ltrniotacl 39042 . . . . . . . . . . . . . . 15 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → 𝐹𝑇)
5145, 48, 49, 50syl3anc 1371 . . . . . . . . . . . . . 14 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) ∧ ((𝑓 = (𝑠𝐹) ∧ 𝑠𝐸) ∧ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸)) ∧ 𝑠𝑂) → 𝐹𝑇)
52 simp12r 1287 . . . . . . . . . . . . . . 15 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) ∧ ((𝑓 = (𝑠𝐹) ∧ 𝑠𝐸) ∧ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸)) ∧ 𝑠𝑂) → (𝑅𝐴 ∧ ¬ 𝑅 𝑊))
5310, 11, 1, 13, 21ltrniotacl 39042 . . . . . . . . . . . . . . 15 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) → 𝐺𝑇)
5445, 48, 52, 53syl3anc 1371 . . . . . . . . . . . . . 14 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) ∧ ((𝑓 = (𝑠𝐹) ∧ 𝑠𝐸) ∧ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸)) ∧ 𝑠𝑂) → 𝐺𝑇)
55 dihmeetlem13.b . . . . . . . . . . . . . . 15 𝐵 = (Base‘𝐾)
56 dihmeetlem13.o . . . . . . . . . . . . . . 15 𝑂 = (𝑇 ↦ ( I ↾ 𝐵))
5755, 1, 13, 14, 56tendospcanN 39486 . . . . . . . . . . . . . 14 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑠𝑂) ∧ (𝐹𝑇𝐺𝑇)) → ((𝑠𝐹) = (𝑠𝐺) ↔ 𝐹 = 𝐺))
5845, 46, 47, 51, 54, 57syl122anc 1379 . . . . . . . . . . . . 13 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) ∧ ((𝑓 = (𝑠𝐹) ∧ 𝑠𝐸) ∧ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸)) ∧ 𝑠𝑂) → ((𝑠𝐹) = (𝑠𝐺) ↔ 𝐹 = 𝐺))
5944, 58mpbid 231 . . . . . . . . . . . 12 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) ∧ ((𝑓 = (𝑠𝐹) ∧ 𝑠𝐸) ∧ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸)) ∧ 𝑠𝑂) → 𝐹 = 𝐺)
60593expia 1121 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) ∧ ((𝑓 = (𝑠𝐹) ∧ 𝑠𝐸) ∧ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸))) → (𝑠𝑂𝐹 = 𝐺))
6160necon1d 2965 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) ∧ ((𝑓 = (𝑠𝐹) ∧ 𝑠𝐸) ∧ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸))) → (𝐹𝐺𝑠 = 𝑂))
6241, 61mpd 15 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) ∧ ((𝑓 = (𝑠𝐹) ∧ 𝑠𝐸) ∧ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸))) → 𝑠 = 𝑂)
6362fveq1d 6844 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) ∧ ((𝑓 = (𝑠𝐹) ∧ 𝑠𝐸) ∧ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸))) → (𝑠𝐹) = (𝑂𝐹))
6429, 31, 32, 50syl3anc 1371 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) ∧ ((𝑓 = (𝑠𝐹) ∧ 𝑠𝐸) ∧ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸))) → 𝐹𝑇)
6556, 55tendo02 39250 . . . . . . . . 9 (𝐹𝑇 → (𝑂𝐹) = ( I ↾ 𝐵))
6664, 65syl 17 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) ∧ ((𝑓 = (𝑠𝐹) ∧ 𝑠𝐸) ∧ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸))) → (𝑂𝐹) = ( I ↾ 𝐵))
6726, 63, 663eqtrd 2780 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) ∧ ((𝑓 = (𝑠𝐹) ∧ 𝑠𝐸) ∧ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸))) → 𝑓 = ( I ↾ 𝐵))
6867, 62jca 512 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) ∧ ((𝑓 = (𝑠𝐹) ∧ 𝑠𝐸) ∧ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸))) → (𝑓 = ( I ↾ 𝐵) ∧ 𝑠 = 𝑂))
6968ex 413 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) → (((𝑓 = (𝑠𝐹) ∧ 𝑠𝐸) ∧ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸)) → (𝑓 = ( I ↾ 𝐵) ∧ 𝑠 = 𝑂)))
7025, 69sylbid 239 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) → (⟨𝑓, 𝑠⟩ ∈ ((𝐼𝑄) ∩ (𝐼𝑅)) → (𝑓 = ( I ↾ 𝐵) ∧ 𝑠 = 𝑂)))
71 opex 5421 . . . . . . 7 𝑓, 𝑠⟩ ∈ V
7271elsn 4601 . . . . . 6 (⟨𝑓, 𝑠⟩ ∈ {⟨( I ↾ 𝐵), 𝑂⟩} ↔ ⟨𝑓, 𝑠⟩ = ⟨( I ↾ 𝐵), 𝑂⟩)
7316, 17opth 5433 . . . . . 6 (⟨𝑓, 𝑠⟩ = ⟨( I ↾ 𝐵), 𝑂⟩ ↔ (𝑓 = ( I ↾ 𝐵) ∧ 𝑠 = 𝑂))
7472, 73bitr2i 275 . . . . 5 ((𝑓 = ( I ↾ 𝐵) ∧ 𝑠 = 𝑂) ↔ ⟨𝑓, 𝑠⟩ ∈ {⟨( I ↾ 𝐵), 𝑂⟩})
75 dihmeetlem13.u . . . . . . . . 9 𝑈 = ((DVecH‘𝐾)‘𝑊)
76 dihmeetlem13.z . . . . . . . . 9 0 = (0g𝑈)
7755, 1, 13, 75, 76, 56dvh0g 39574 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑊𝐻) → 0 = ⟨( I ↾ 𝐵), 𝑂⟩)
78773ad2ant1 1133 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) → 0 = ⟨( I ↾ 𝐵), 𝑂⟩)
7978sneqd 4598 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) → { 0 } = {⟨( I ↾ 𝐵), 𝑂⟩})
8079eleq2d 2823 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) → (⟨𝑓, 𝑠⟩ ∈ { 0 } ↔ ⟨𝑓, 𝑠⟩ ∈ {⟨( I ↾ 𝐵), 𝑂⟩}))
8174, 80bitr4id 289 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) → ((𝑓 = ( I ↾ 𝐵) ∧ 𝑠 = 𝑂) ↔ ⟨𝑓, 𝑠⟩ ∈ { 0 }))
8270, 81sylibd 238 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) → (⟨𝑓, 𝑠⟩ ∈ ((𝐼𝑄) ∩ (𝐼𝑅)) → ⟨𝑓, 𝑠⟩ ∈ { 0 }))
836, 82relssdv 5744 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) → ((𝐼𝑄) ∩ (𝐼𝑅)) ⊆ { 0 })
841, 75, 8dvhlmod 39573 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) → 𝑈 ∈ LMod)
85 simp2ll 1240 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) → 𝑄𝐴)
8655, 11atbase 37751 . . . . . 6 (𝑄𝐴𝑄𝐵)
8785, 86syl 17 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) → 𝑄𝐵)
88 eqid 2736 . . . . . 6 (LSubSp‘𝑈) = (LSubSp‘𝑈)
8955, 1, 2, 75, 88dihlss 39713 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑄𝐵) → (𝐼𝑄) ∈ (LSubSp‘𝑈))
908, 87, 89syl2anc 584 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) → (𝐼𝑄) ∈ (LSubSp‘𝑈))
91 simp2rl 1242 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) → 𝑅𝐴)
9255, 11atbase 37751 . . . . . 6 (𝑅𝐴𝑅𝐵)
9391, 92syl 17 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) → 𝑅𝐵)
9455, 1, 2, 75, 88dihlss 39713 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑅𝐵) → (𝐼𝑅) ∈ (LSubSp‘𝑈))
958, 93, 94syl2anc 584 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) → (𝐼𝑅) ∈ (LSubSp‘𝑈))
9688lssincl 20426 . . . 4 ((𝑈 ∈ LMod ∧ (𝐼𝑄) ∈ (LSubSp‘𝑈) ∧ (𝐼𝑅) ∈ (LSubSp‘𝑈)) → ((𝐼𝑄) ∩ (𝐼𝑅)) ∈ (LSubSp‘𝑈))
9784, 90, 95, 96syl3anc 1371 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) → ((𝐼𝑄) ∩ (𝐼𝑅)) ∈ (LSubSp‘𝑈))
9876, 88lss0ss 20409 . . 3 ((𝑈 ∈ LMod ∧ ((𝐼𝑄) ∩ (𝐼𝑅)) ∈ (LSubSp‘𝑈)) → { 0 } ⊆ ((𝐼𝑄) ∩ (𝐼𝑅)))
9984, 97, 98syl2anc 584 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) → { 0 } ⊆ ((𝐼𝑄) ∩ (𝐼𝑅)))
10083, 99eqssd 3961 1 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) → ((𝐼𝑄) ∩ (𝐼𝑅)) = { 0 })
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wne 2943  cin 3909  wss 3910  {csn 4586  cop 4592   class class class wbr 5105  cmpt 5188   I cid 5530  cres 5635  Rel wrel 5638  cfv 6496  crio 7312  Basecbs 17083  lecple 17140  occoc 17141  0gc0g 17321  joincjn 18200  LModclmod 20322  LSubSpclss 20392  Atomscatm 37725  HLchlt 37812  LHypclh 38447  LTrncltrn 38564  TEndoctendo 39215  DVecHcdvh 39541  DIsoHcdih 39691
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-riotaBAD 37415
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-iin 4957  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-tpos 8157  df-undef 8204  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-map 8767  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-3 12217  df-4 12218  df-5 12219  df-6 12220  df-n0 12414  df-z 12500  df-uz 12764  df-fz 13425  df-struct 17019  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-mulr 17147  df-sca 17149  df-vsca 17150  df-0g 17323  df-proset 18184  df-poset 18202  df-plt 18219  df-lub 18235  df-glb 18236  df-join 18237  df-meet 18238  df-p0 18314  df-p1 18315  df-lat 18321  df-clat 18388  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-submnd 18602  df-grp 18751  df-minusg 18752  df-sbg 18753  df-subg 18925  df-cntz 19097  df-lsm 19418  df-cmn 19564  df-abl 19565  df-mgp 19897  df-ur 19914  df-ring 19966  df-oppr 20049  df-dvdsr 20070  df-unit 20071  df-invr 20101  df-dvr 20112  df-drng 20187  df-lmod 20324  df-lss 20393  df-lsp 20433  df-lvec 20564  df-oposet 37638  df-ol 37640  df-oml 37641  df-covers 37728  df-ats 37729  df-atl 37760  df-cvlat 37784  df-hlat 37813  df-llines 37961  df-lplanes 37962  df-lvols 37963  df-lines 37964  df-psubsp 37966  df-pmap 37967  df-padd 38259  df-lhyp 38451  df-laut 38452  df-ldil 38567  df-ltrn 38568  df-trl 38622  df-tendo 39218  df-edring 39220  df-disoa 39492  df-dvech 39542  df-dib 39602  df-dic 39636  df-dih 39692
This theorem is referenced by:  dihmeetlem15N  39784
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