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Theorem dihmeetlem13N 38573
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 38533 . . . . 5 ((𝐾 ∈ HL ∧ 𝑊𝐻) → Rel (𝐼𝑄))
433ad2ant1 1130 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) → Rel (𝐼𝑄))
5 relin1 5662 . . . 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 3924 . . . . . 6 (⟨𝑓, 𝑠⟩ ∈ ((𝐼𝑄) ∩ (𝐼𝑅)) ↔ (⟨𝑓, 𝑠⟩ ∈ (𝐼𝑄) ∧ ⟨𝑓, 𝑠⟩ ∈ (𝐼𝑅)))
8 simp1 1133 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) → (𝐾 ∈ HL ∧ 𝑊𝐻))
9 simp2l 1196 . . . . . . . 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 3472 . . . . . . . . 9 𝑓 ∈ V
17 vex 3472 . . . . . . . . 9 𝑠 ∈ V
1810, 11, 1, 12, 13, 14, 2, 15, 16, 17dihopelvalcqat 38500 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (⟨𝑓, 𝑠⟩ ∈ (𝐼𝑄) ↔ (𝑓 = (𝑠𝐹) ∧ 𝑠𝐸)))
198, 9, 18syl2anc 587 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) → (⟨𝑓, 𝑠⟩ ∈ (𝐼𝑄) ↔ (𝑓 = (𝑠𝐹) ∧ 𝑠𝐸)))
20 simp2r 1197 . . . . . . . 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 38500 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) → (⟨𝑓, 𝑠⟩ ∈ (𝐼𝑅) ↔ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸)))
238, 20, 22syl2anc 587 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) → (⟨𝑓, 𝑠⟩ ∈ (𝐼𝑅) ↔ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸)))
2419, 23anbi12d 633 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) → ((⟨𝑓, 𝑠⟩ ∈ (𝐼𝑄) ∧ ⟨𝑓, 𝑠⟩ ∈ (𝐼𝑅)) ↔ ((𝑓 = (𝑠𝐹) ∧ 𝑠𝐸) ∧ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸))))
257, 24syl5bb 286 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) → (⟨𝑓, 𝑠⟩ ∈ ((𝐼𝑄) ∩ (𝐼𝑅)) ↔ ((𝑓 = (𝑠𝐹) ∧ 𝑠𝐸) ∧ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸))))
26 simprll 778 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) ∧ ((𝑓 = (𝑠𝐹) ∧ 𝑠𝐸) ∧ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸))) → 𝑓 = (𝑠𝐹))
27 simpl3 1190 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) ∧ ((𝑓 = (𝑠𝐹) ∧ 𝑠𝐸) ∧ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸))) → 𝑄𝑅)
28 fveq1 6651 . . . . . . . . . . . . 13 (𝐹 = 𝐺 → (𝐹𝑃) = (𝐺𝑃))
29 simpl1 1188 . . . . . . . . . . . . . . 15 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) ∧ ((𝑓 = (𝑠𝐹) ∧ 𝑠𝐸) ∧ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸))) → (𝐾 ∈ HL ∧ 𝑊𝐻))
3010, 11, 1, 12lhpocnel2 37273 . . . . . . . . . . . . . . . 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 1223 . . . . . . . . . . . . . . 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 37835 . . . . . . . . . . . . . . 15 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝐹𝑃) = 𝑄)
3429, 31, 32, 33syl3anc 1368 . . . . . . . . . . . . . 14 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) ∧ ((𝑓 = (𝑠𝐹) ∧ 𝑠𝐸) ∧ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸))) → (𝐹𝑃) = 𝑄)
35 simpl2r 1224 . . . . . . . . . . . . . . 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 37835 . . . . . . . . . . . . . . 15 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) → (𝐺𝑃) = 𝑅)
3729, 31, 35, 36syl3anc 1368 . . . . . . . . . . . . . 14 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) ∧ ((𝑓 = (𝑠𝐹) ∧ 𝑠𝐸) ∧ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸))) → (𝐺𝑃) = 𝑅)
3834, 37eqeq12d 2838 . . . . . . . . . . . . 13 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) ∧ ((𝑓 = (𝑠𝐹) ∧ 𝑠𝐸) ∧ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸))) → ((𝐹𝑃) = (𝐺𝑃) ↔ 𝑄 = 𝑅))
3928, 38syl5ib 247 . . . . . . . . . . . 12 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) ∧ ((𝑓 = (𝑠𝐹) ∧ 𝑠𝐸) ∧ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸))) → (𝐹 = 𝐺𝑄 = 𝑅))
4039necon3d 3032 . . . . . . . . . . 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 1237 . . . . . . . . . . . . . 14 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) ∧ ((𝑓 = (𝑠𝐹) ∧ 𝑠𝐸) ∧ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸)) ∧ 𝑠𝑂) → 𝑓 = (𝑠𝐹))
43 simp2rl 1239 . . . . . . . . . . . . . 14 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) ∧ ((𝑓 = (𝑠𝐹) ∧ 𝑠𝐸) ∧ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸)) ∧ 𝑠𝑂) → 𝑓 = (𝑠𝐺))
4442, 43eqtr3d 2859 . . . . . . . . . . . . 13 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) ∧ ((𝑓 = (𝑠𝐹) ∧ 𝑠𝐸) ∧ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸)) ∧ 𝑠𝑂) → (𝑠𝐹) = (𝑠𝐺))
45 simp11 1200 . . . . . . . . . . . . . 14 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) ∧ ((𝑓 = (𝑠𝐹) ∧ 𝑠𝐸) ∧ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸)) ∧ 𝑠𝑂) → (𝐾 ∈ HL ∧ 𝑊𝐻))
46 simp2rr 1240 . . . . . . . . . . . . . 14 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) ∧ ((𝑓 = (𝑠𝐹) ∧ 𝑠𝐸) ∧ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸)) ∧ 𝑠𝑂) → 𝑠𝐸)
47 simp3 1135 . . . . . . . . . . . . . 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 1283 . . . . . . . . . . . . . . 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 37833 . . . . . . . . . . . . . . 15 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → 𝐹𝑇)
5145, 48, 49, 50syl3anc 1368 . . . . . . . . . . . . . 14 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) ∧ ((𝑓 = (𝑠𝐹) ∧ 𝑠𝐸) ∧ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸)) ∧ 𝑠𝑂) → 𝐹𝑇)
52 simp12r 1284 . . . . . . . . . . . . . . 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 37833 . . . . . . . . . . . . . . 15 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) → 𝐺𝑇)
5445, 48, 52, 53syl3anc 1368 . . . . . . . . . . . . . 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 38277 . . . . . . . . . . . . . 14 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑠𝑂) ∧ (𝐹𝑇𝐺𝑇)) → ((𝑠𝐹) = (𝑠𝐺) ↔ 𝐹 = 𝐺))
5845, 46, 47, 51, 54, 57syl122anc 1376 . . . . . . . . . . . . 13 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) ∧ ((𝑓 = (𝑠𝐹) ∧ 𝑠𝐸) ∧ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸)) ∧ 𝑠𝑂) → ((𝑠𝐹) = (𝑠𝐺) ↔ 𝐹 = 𝐺))
5944, 58mpbid 235 . . . . . . . . . . . 12 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) ∧ ((𝑓 = (𝑠𝐹) ∧ 𝑠𝐸) ∧ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸)) ∧ 𝑠𝑂) → 𝐹 = 𝐺)
60593expia 1118 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) ∧ ((𝑓 = (𝑠𝐹) ∧ 𝑠𝐸) ∧ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸))) → (𝑠𝑂𝐹 = 𝐺))
6160necon1d 3033 . . . . . . . . . 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 6654 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) ∧ ((𝑓 = (𝑠𝐹) ∧ 𝑠𝐸) ∧ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸))) → (𝑠𝐹) = (𝑂𝐹))
6429, 31, 32, 50syl3anc 1368 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) ∧ ((𝑓 = (𝑠𝐹) ∧ 𝑠𝐸) ∧ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸))) → 𝐹𝑇)
6556, 55tendo02 38041 . . . . . . . . 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 2861 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) ∧ ((𝑓 = (𝑠𝐹) ∧ 𝑠𝐸) ∧ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸))) → 𝑓 = ( I ↾ 𝐵))
6867, 62jca 515 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) ∧ ((𝑓 = (𝑠𝐹) ∧ 𝑠𝐸) ∧ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸))) → (𝑓 = ( I ↾ 𝐵) ∧ 𝑠 = 𝑂))
6968ex 416 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) → (((𝑓 = (𝑠𝐹) ∧ 𝑠𝐸) ∧ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸)) → (𝑓 = ( I ↾ 𝐵) ∧ 𝑠 = 𝑂)))
7025, 69sylbid 243 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) → (⟨𝑓, 𝑠⟩ ∈ ((𝐼𝑄) ∩ (𝐼𝑅)) → (𝑓 = ( I ↾ 𝐵) ∧ 𝑠 = 𝑂)))
71 dihmeetlem13.u . . . . . . . . 9 𝑈 = ((DVecH‘𝐾)‘𝑊)
72 dihmeetlem13.z . . . . . . . . 9 0 = (0g𝑈)
7355, 1, 13, 71, 72, 56dvh0g 38365 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑊𝐻) → 0 = ⟨( I ↾ 𝐵), 𝑂⟩)
74733ad2ant1 1130 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) → 0 = ⟨( I ↾ 𝐵), 𝑂⟩)
7574sneqd 4551 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) → { 0 } = {⟨( I ↾ 𝐵), 𝑂⟩})
7675eleq2d 2899 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) → (⟨𝑓, 𝑠⟩ ∈ { 0 } ↔ ⟨𝑓, 𝑠⟩ ∈ {⟨( I ↾ 𝐵), 𝑂⟩}))
77 opex 5333 . . . . . . 7 𝑓, 𝑠⟩ ∈ V
7877elsn 4554 . . . . . 6 (⟨𝑓, 𝑠⟩ ∈ {⟨( I ↾ 𝐵), 𝑂⟩} ↔ ⟨𝑓, 𝑠⟩ = ⟨( I ↾ 𝐵), 𝑂⟩)
7916, 17opth 5345 . . . . . 6 (⟨𝑓, 𝑠⟩ = ⟨( I ↾ 𝐵), 𝑂⟩ ↔ (𝑓 = ( I ↾ 𝐵) ∧ 𝑠 = 𝑂))
8078, 79bitr2i 279 . . . . 5 ((𝑓 = ( I ↾ 𝐵) ∧ 𝑠 = 𝑂) ↔ ⟨𝑓, 𝑠⟩ ∈ {⟨( I ↾ 𝐵), 𝑂⟩})
8176, 80syl6rbbr 293 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) → ((𝑓 = ( I ↾ 𝐵) ∧ 𝑠 = 𝑂) ↔ ⟨𝑓, 𝑠⟩ ∈ { 0 }))
8270, 81sylibd 242 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) → (⟨𝑓, 𝑠⟩ ∈ ((𝐼𝑄) ∩ (𝐼𝑅)) → ⟨𝑓, 𝑠⟩ ∈ { 0 }))
836, 82relssdv 5638 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) → ((𝐼𝑄) ∩ (𝐼𝑅)) ⊆ { 0 })
841, 71, 8dvhlmod 38364 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) → 𝑈 ∈ LMod)
85 simp2ll 1237 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) → 𝑄𝐴)
8655, 11atbase 36543 . . . . . 6 (𝑄𝐴𝑄𝐵)
8785, 86syl 17 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) → 𝑄𝐵)
88 eqid 2822 . . . . . 6 (LSubSp‘𝑈) = (LSubSp‘𝑈)
8955, 1, 2, 71, 88dihlss 38504 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑄𝐵) → (𝐼𝑄) ∈ (LSubSp‘𝑈))
908, 87, 89syl2anc 587 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) → (𝐼𝑄) ∈ (LSubSp‘𝑈))
91 simp2rl 1239 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) → 𝑅𝐴)
9255, 11atbase 36543 . . . . . 6 (𝑅𝐴𝑅𝐵)
9391, 92syl 17 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) → 𝑅𝐵)
9455, 1, 2, 71, 88dihlss 38504 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑅𝐵) → (𝐼𝑅) ∈ (LSubSp‘𝑈))
958, 93, 94syl2anc 587 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) → (𝐼𝑅) ∈ (LSubSp‘𝑈))
9688lssincl 19728 . . . 4 ((𝑈 ∈ LMod ∧ (𝐼𝑄) ∈ (LSubSp‘𝑈) ∧ (𝐼𝑅) ∈ (LSubSp‘𝑈)) → ((𝐼𝑄) ∩ (𝐼𝑅)) ∈ (LSubSp‘𝑈))
9784, 90, 95, 96syl3anc 1368 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) → ((𝐼𝑄) ∩ (𝐼𝑅)) ∈ (LSubSp‘𝑈))
9872, 88lss0ss 19711 . . 3 ((𝑈 ∈ LMod ∧ ((𝐼𝑄) ∩ (𝐼𝑅)) ∈ (LSubSp‘𝑈)) → { 0 } ⊆ ((𝐼𝑄) ∩ (𝐼𝑅)))
9984, 97, 98syl2anc 587 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) → { 0 } ⊆ ((𝐼𝑄) ∩ (𝐼𝑅)))
10083, 99eqssd 3959 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 209  wa 399  w3a 1084   = wceq 1538  wcel 2114  wne 3011  cin 3907  wss 3908  {csn 4539  cop 4545   class class class wbr 5042  cmpt 5122   I cid 5436  cres 5534  Rel wrel 5537  cfv 6334  crio 7097  Basecbs 16474  lecple 16563  occoc 16564  0gc0g 16704  joincjn 17545  LModclmod 19625  LSubSpclss 19694  Atomscatm 36517  HLchlt 36604  LHypclh 37238  LTrncltrn 37355  TEndoctendo 38006  DVecHcdvh 38332  DIsoHcdih 38482
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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446  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 36207
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-nel 3116  df-ral 3135  df-rex 3136  df-reu 3137  df-rmo 3138  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-tp 4544  df-op 4546  df-uni 4814  df-int 4852  df-iun 4896  df-iin 4897  df-br 5043  df-opab 5105  df-mpt 5123  df-tr 5149  df-id 5437  df-eprel 5442  df-po 5451  df-so 5452  df-fr 5491  df-we 5493  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-pred 6126  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-riota 7098  df-ov 7143  df-oprab 7144  df-mpo 7145  df-om 7566  df-1st 7675  df-2nd 7676  df-tpos 7879  df-undef 7926  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-oadd 8093  df-er 8276  df-map 8395  df-en 8497  df-dom 8498  df-sdom 8499  df-fin 8500  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-nn 11626  df-2 11688  df-3 11689  df-4 11690  df-5 11691  df-6 11692  df-n0 11886  df-z 11970  df-uz 12232  df-fz 12886  df-struct 16476  df-ndx 16477  df-slot 16478  df-base 16480  df-sets 16481  df-ress 16482  df-plusg 16569  df-mulr 16570  df-sca 16572  df-vsca 16573  df-0g 16706  df-proset 17529  df-poset 17547  df-plt 17559  df-lub 17575  df-glb 17576  df-join 17577  df-meet 17578  df-p0 17640  df-p1 17641  df-lat 17647  df-clat 17709  df-mgm 17843  df-sgrp 17892  df-mnd 17903  df-submnd 17948  df-grp 18097  df-minusg 18098  df-sbg 18099  df-subg 18267  df-cntz 18438  df-lsm 18752  df-cmn 18899  df-abl 18900  df-mgp 19231  df-ur 19243  df-ring 19290  df-oppr 19367  df-dvdsr 19385  df-unit 19386  df-invr 19416  df-dvr 19427  df-drng 19495  df-lmod 19627  df-lss 19695  df-lsp 19735  df-lvec 19866  df-oposet 36430  df-ol 36432  df-oml 36433  df-covers 36520  df-ats 36521  df-atl 36552  df-cvlat 36576  df-hlat 36605  df-llines 36752  df-lplanes 36753  df-lvols 36754  df-lines 36755  df-psubsp 36757  df-pmap 36758  df-padd 37050  df-lhyp 37242  df-laut 37243  df-ldil 37358  df-ltrn 37359  df-trl 37413  df-tendo 38009  df-edring 38011  df-disoa 38283  df-dvech 38333  df-dib 38393  df-dic 38427  df-dih 38483
This theorem is referenced by:  dihmeetlem15N  38575
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