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Theorem dihmeetlem13N 41020
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 40980 . . . . 5 ((𝐾 ∈ HL ∧ 𝑊𝐻) → Rel (𝐼𝑄))
433ad2ant1 1130 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) → Rel (𝐼𝑄))
5 relin1 5820 . . . 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 3963 . . . . . 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 3466 . . . . . . . . 9 𝑓 ∈ V
17 vex 3466 . . . . . . . . 9 𝑠 ∈ V
1810, 11, 1, 12, 13, 14, 2, 15, 16, 17dihopelvalcqat 40947 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (⟨𝑓, 𝑠⟩ ∈ (𝐼𝑄) ↔ (𝑓 = (𝑠𝐹) ∧ 𝑠𝐸)))
198, 9, 18syl2anc 582 . . . . . . 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 40947 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) → (⟨𝑓, 𝑠⟩ ∈ (𝐼𝑅) ↔ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸)))
238, 20, 22syl2anc 582 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) → (⟨𝑓, 𝑠⟩ ∈ (𝐼𝑅) ↔ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸)))
2419, 23anbi12d 630 . . . . . 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 1190 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) ∧ ((𝑓 = (𝑠𝐹) ∧ 𝑠𝐸) ∧ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸))) → 𝑄𝑅)
28 fveq1 6902 . . . . . . . . . . . . 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 39720 . . . . . . . . . . . . . . . 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 40282 . . . . . . . . . . . . . . 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 40282 . . . . . . . . . . . . . . 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 2742 . . . . . . . . . . . . 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 2951 . . . . . . . . . . 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 2768 . . . . . . . . . . . . 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 40280 . . . . . . . . . . . . . . 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 40280 . . . . . . . . . . . . . . 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 40724 . . . . . . . . . . . . . 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 231 . . . . . . . . . . . 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 2952 . . . . . . . . . 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 6905 . . . . . . . 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 40488 . . . . . . . . 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 2770 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) ∧ ((𝑓 = (𝑠𝐹) ∧ 𝑠𝐸) ∧ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸))) → 𝑓 = ( I ↾ 𝐵))
6867, 62jca 510 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) ∧ ((𝑓 = (𝑠𝐹) ∧ 𝑠𝐸) ∧ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸))) → (𝑓 = ( I ↾ 𝐵) ∧ 𝑠 = 𝑂))
6968ex 411 . . . . 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 5472 . . . . . . 7 𝑓, 𝑠⟩ ∈ V
7271elsn 4648 . . . . . 6 (⟨𝑓, 𝑠⟩ ∈ {⟨( I ↾ 𝐵), 𝑂⟩} ↔ ⟨𝑓, 𝑠⟩ = ⟨( I ↾ 𝐵), 𝑂⟩)
7316, 17opth 5484 . . . . . 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 40812 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑊𝐻) → 0 = ⟨( I ↾ 𝐵), 𝑂⟩)
78773ad2ant1 1130 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) → 0 = ⟨( I ↾ 𝐵), 𝑂⟩)
7978sneqd 4645 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) → { 0 } = {⟨( I ↾ 𝐵), 𝑂⟩})
8079eleq2d 2812 . . . . 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 5796 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) → ((𝐼𝑄) ∩ (𝐼𝑅)) ⊆ { 0 })
841, 75, 8dvhlmod 40811 . . 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 38989 . . . . . 6 (𝑄𝐴𝑄𝐵)
8785, 86syl 17 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) → 𝑄𝐵)
88 eqid 2726 . . . . . 6 (LSubSp‘𝑈) = (LSubSp‘𝑈)
8955, 1, 2, 75, 88dihlss 40951 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑄𝐵) → (𝐼𝑄) ∈ (LSubSp‘𝑈))
908, 87, 89syl2anc 582 . . . 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 38989 . . . . . 6 (𝑅𝐴𝑅𝐵)
9391, 92syl 17 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) → 𝑅𝐵)
9455, 1, 2, 75, 88dihlss 40951 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑅𝐵) → (𝐼𝑅) ∈ (LSubSp‘𝑈))
958, 93, 94syl2anc 582 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) → (𝐼𝑅) ∈ (LSubSp‘𝑈))
9688lssincl 20944 . . . 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))
9876, 88lss0ss 20928 . . 3 ((𝑈 ∈ LMod ∧ ((𝐼𝑄) ∩ (𝐼𝑅)) ∈ (LSubSp‘𝑈)) → { 0 } ⊆ ((𝐼𝑄) ∩ (𝐼𝑅)))
9984, 97, 98syl2anc 582 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) → { 0 } ⊆ ((𝐼𝑄) ∩ (𝐼𝑅)))
10083, 99eqssd 3997 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 394  w3a 1084   = wceq 1534  wcel 2099  wne 2930  cin 3946  wss 3947  {csn 4633  cop 4639   class class class wbr 5155  cmpt 5238   I cid 5581  cres 5686  Rel wrel 5689  cfv 6556  crio 7381  Basecbs 17215  lecple 17275  occoc 17276  0gc0g 17456  joincjn 18338  LModclmod 20838  LSubSpclss 20910  Atomscatm 38963  HLchlt 39050  LHypclh 39685  LTrncltrn 39802  TEndoctendo 40453  DVecHcdvh 40779  DIsoHcdih 40929
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5292  ax-sep 5306  ax-nul 5313  ax-pow 5371  ax-pr 5435  ax-un 7748  ax-cnex 11216  ax-resscn 11217  ax-1cn 11218  ax-icn 11219  ax-addcl 11220  ax-addrcl 11221  ax-mulcl 11222  ax-mulrcl 11223  ax-mulcom 11224  ax-addass 11225  ax-mulass 11226  ax-distr 11227  ax-i2m1 11228  ax-1ne0 11229  ax-1rid 11230  ax-rnegex 11231  ax-rrecex 11232  ax-cnre 11233  ax-pre-lttri 11234  ax-pre-lttrn 11235  ax-pre-ltadd 11236  ax-pre-mulgt0 11237  ax-riotaBAD 38653
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3967  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-tp 4638  df-op 4640  df-uni 4916  df-int 4957  df-iun 5005  df-iin 5006  df-br 5156  df-opab 5218  df-mpt 5239  df-tr 5273  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5639  df-we 5641  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6314  df-ord 6381  df-on 6382  df-lim 6383  df-suc 6384  df-iota 6508  df-fun 6558  df-fn 6559  df-f 6560  df-f1 6561  df-fo 6562  df-f1o 6563  df-fv 6564  df-riota 7382  df-ov 7429  df-oprab 7430  df-mpo 7431  df-om 7879  df-1st 8005  df-2nd 8006  df-tpos 8243  df-undef 8290  df-frecs 8298  df-wrecs 8329  df-recs 8403  df-rdg 8442  df-1o 8498  df-er 8736  df-map 8859  df-en 8977  df-dom 8978  df-sdom 8979  df-fin 8980  df-pnf 11302  df-mnf 11303  df-xr 11304  df-ltxr 11305  df-le 11306  df-sub 11498  df-neg 11499  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-5 12332  df-6 12333  df-n0 12527  df-z 12613  df-uz 12877  df-fz 13541  df-struct 17151  df-sets 17168  df-slot 17186  df-ndx 17198  df-base 17216  df-ress 17245  df-plusg 17281  df-mulr 17282  df-sca 17284  df-vsca 17285  df-0g 17458  df-proset 18322  df-poset 18340  df-plt 18357  df-lub 18373  df-glb 18374  df-join 18375  df-meet 18376  df-p0 18452  df-p1 18453  df-lat 18459  df-clat 18526  df-mgm 18635  df-sgrp 18714  df-mnd 18730  df-submnd 18776  df-grp 18933  df-minusg 18934  df-sbg 18935  df-subg 19119  df-cntz 19313  df-lsm 19636  df-cmn 19782  df-abl 19783  df-mgp 20120  df-rng 20138  df-ur 20167  df-ring 20220  df-oppr 20318  df-dvdsr 20341  df-unit 20342  df-invr 20372  df-dvr 20385  df-drng 20711  df-lmod 20840  df-lss 20911  df-lsp 20951  df-lvec 21083  df-oposet 38876  df-ol 38878  df-oml 38879  df-covers 38966  df-ats 38967  df-atl 38998  df-cvlat 39022  df-hlat 39051  df-llines 39199  df-lplanes 39200  df-lvols 39201  df-lines 39202  df-psubsp 39204  df-pmap 39205  df-padd 39497  df-lhyp 39689  df-laut 39690  df-ldil 39805  df-ltrn 39806  df-trl 39860  df-tendo 40456  df-edring 40458  df-disoa 40730  df-dvech 40780  df-dib 40840  df-dic 40874  df-dih 40930
This theorem is referenced by:  dihmeetlem15N  41022
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