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Theorem dihmeetlem13N 36427
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 36387 . . . . 5 ((𝐾 ∈ HL ∧ 𝑊𝐻) → Rel (𝐼𝑄))
433ad2ant1 1080 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) → Rel (𝐼𝑄))
5 relin1 5226 . . . 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 3788 . . . . . 6 (⟨𝑓, 𝑠⟩ ∈ ((𝐼𝑄) ∩ (𝐼𝑅)) ↔ (⟨𝑓, 𝑠⟩ ∈ (𝐼𝑄) ∧ ⟨𝑓, 𝑠⟩ ∈ (𝐼𝑅)))
8 simp1 1059 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) → (𝐾 ∈ HL ∧ 𝑊𝐻))
9 simp2l 1085 . . . . . . . 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 3198 . . . . . . . . 9 𝑓 ∈ V
17 vex 3198 . . . . . . . . 9 𝑠 ∈ V
1810, 11, 1, 12, 13, 14, 2, 15, 16, 17dihopelvalcqat 36354 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (⟨𝑓, 𝑠⟩ ∈ (𝐼𝑄) ↔ (𝑓 = (𝑠𝐹) ∧ 𝑠𝐸)))
198, 9, 18syl2anc 692 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) → (⟨𝑓, 𝑠⟩ ∈ (𝐼𝑄) ↔ (𝑓 = (𝑠𝐹) ∧ 𝑠𝐸)))
20 simp2r 1086 . . . . . . . 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 36354 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) → (⟨𝑓, 𝑠⟩ ∈ (𝐼𝑅) ↔ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸)))
238, 20, 22syl2anc 692 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) → (⟨𝑓, 𝑠⟩ ∈ (𝐼𝑅) ↔ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸)))
2419, 23anbi12d 746 . . . . . 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 272 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) → (⟨𝑓, 𝑠⟩ ∈ ((𝐼𝑄) ∩ (𝐼𝑅)) ↔ ((𝑓 = (𝑠𝐹) ∧ 𝑠𝐸) ∧ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸))))
26 simprll 801 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) ∧ ((𝑓 = (𝑠𝐹) ∧ 𝑠𝐸) ∧ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸))) → 𝑓 = (𝑠𝐹))
27 simpl3 1064 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) ∧ ((𝑓 = (𝑠𝐹) ∧ 𝑠𝐸) ∧ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸))) → 𝑄𝑅)
28 fveq1 6177 . . . . . . . . . . . . 13 (𝐹 = 𝐺 → (𝐹𝑃) = (𝐺𝑃))
29 simpl1 1062 . . . . . . . . . . . . . . 15 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) ∧ ((𝑓 = (𝑠𝐹) ∧ 𝑠𝐸) ∧ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸))) → (𝐾 ∈ HL ∧ 𝑊𝐻))
3010, 11, 1, 12lhpocnel2 35124 . . . . . . . . . . . . . . . 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 1112 . . . . . . . . . . . . . . 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 35688 . . . . . . . . . . . . . . 15 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝐹𝑃) = 𝑄)
3429, 31, 32, 33syl3anc 1324 . . . . . . . . . . . . . 14 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) ∧ ((𝑓 = (𝑠𝐹) ∧ 𝑠𝐸) ∧ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸))) → (𝐹𝑃) = 𝑄)
35 simpl2r 1113 . . . . . . . . . . . . . . 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 35688 . . . . . . . . . . . . . . 15 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) → (𝐺𝑃) = 𝑅)
3729, 31, 35, 36syl3anc 1324 . . . . . . . . . . . . . 14 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) ∧ ((𝑓 = (𝑠𝐹) ∧ 𝑠𝐸) ∧ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸))) → (𝐺𝑃) = 𝑅)
3834, 37eqeq12d 2635 . . . . . . . . . . . . 13 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) ∧ ((𝑓 = (𝑠𝐹) ∧ 𝑠𝐸) ∧ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸))) → ((𝐹𝑃) = (𝐺𝑃) ↔ 𝑄 = 𝑅))
3928, 38syl5ib 234 . . . . . . . . . . . 12 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) ∧ ((𝑓 = (𝑠𝐹) ∧ 𝑠𝐸) ∧ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸))) → (𝐹 = 𝐺𝑄 = 𝑅))
4039necon3d 2812 . . . . . . . . . . 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 1126 . . . . . . . . . . . . . 14 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) ∧ ((𝑓 = (𝑠𝐹) ∧ 𝑠𝐸) ∧ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸)) ∧ 𝑠𝑂) → 𝑓 = (𝑠𝐹))
43 simp2rl 1128 . . . . . . . . . . . . . 14 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) ∧ ((𝑓 = (𝑠𝐹) ∧ 𝑠𝐸) ∧ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸)) ∧ 𝑠𝑂) → 𝑓 = (𝑠𝐺))
4442, 43eqtr3d 2656 . . . . . . . . . . . . 13 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) ∧ ((𝑓 = (𝑠𝐹) ∧ 𝑠𝐸) ∧ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸)) ∧ 𝑠𝑂) → (𝑠𝐹) = (𝑠𝐺))
45 simp11 1089 . . . . . . . . . . . . . 14 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) ∧ ((𝑓 = (𝑠𝐹) ∧ 𝑠𝐸) ∧ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸)) ∧ 𝑠𝑂) → (𝐾 ∈ HL ∧ 𝑊𝐻))
46 simp2rr 1129 . . . . . . . . . . . . . 14 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) ∧ ((𝑓 = (𝑠𝐹) ∧ 𝑠𝐸) ∧ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸)) ∧ 𝑠𝑂) → 𝑠𝐸)
47 simp3 1061 . . . . . . . . . . . . . 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 1172 . . . . . . . . . . . . . . 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 35686 . . . . . . . . . . . . . . 15 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → 𝐹𝑇)
5145, 48, 49, 50syl3anc 1324 . . . . . . . . . . . . . 14 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) ∧ ((𝑓 = (𝑠𝐹) ∧ 𝑠𝐸) ∧ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸)) ∧ 𝑠𝑂) → 𝐹𝑇)
52 simp12r 1173 . . . . . . . . . . . . . . 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 35686 . . . . . . . . . . . . . . 15 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) → 𝐺𝑇)
5445, 48, 52, 53syl3anc 1324 . . . . . . . . . . . . . 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 36131 . . . . . . . . . . . . . 14 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝑠𝑂) ∧ (𝐹𝑇𝐺𝑇)) → ((𝑠𝐹) = (𝑠𝐺) ↔ 𝐹 = 𝐺))
5845, 46, 47, 51, 54, 57syl122anc 1333 . . . . . . . . . . . . 13 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) ∧ ((𝑓 = (𝑠𝐹) ∧ 𝑠𝐸) ∧ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸)) ∧ 𝑠𝑂) → ((𝑠𝐹) = (𝑠𝐺) ↔ 𝐹 = 𝐺))
5944, 58mpbid 222 . . . . . . . . . . . 12 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) ∧ ((𝑓 = (𝑠𝐹) ∧ 𝑠𝐸) ∧ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸)) ∧ 𝑠𝑂) → 𝐹 = 𝐺)
60593expia 1265 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) ∧ ((𝑓 = (𝑠𝐹) ∧ 𝑠𝐸) ∧ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸))) → (𝑠𝑂𝐹 = 𝐺))
6160necon1d 2813 . . . . . . . . . 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 6180 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) ∧ ((𝑓 = (𝑠𝐹) ∧ 𝑠𝐸) ∧ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸))) → (𝑠𝐹) = (𝑂𝐹))
6429, 31, 32, 50syl3anc 1324 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) ∧ ((𝑓 = (𝑠𝐹) ∧ 𝑠𝐸) ∧ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸))) → 𝐹𝑇)
6556, 55tendo02 35894 . . . . . . . . 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 2658 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) ∧ ((𝑓 = (𝑠𝐹) ∧ 𝑠𝐸) ∧ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸))) → 𝑓 = ( I ↾ 𝐵))
6867, 62jca 554 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) ∧ ((𝑓 = (𝑠𝐹) ∧ 𝑠𝐸) ∧ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸))) → (𝑓 = ( I ↾ 𝐵) ∧ 𝑠 = 𝑂))
6968ex 450 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) → (((𝑓 = (𝑠𝐹) ∧ 𝑠𝐸) ∧ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸)) → (𝑓 = ( I ↾ 𝐵) ∧ 𝑠 = 𝑂)))
7025, 69sylbid 230 . . . 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 36219 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑊𝐻) → 0 = ⟨( I ↾ 𝐵), 𝑂⟩)
74733ad2ant1 1080 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) → 0 = ⟨( I ↾ 𝐵), 𝑂⟩)
7574sneqd 4180 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) → { 0 } = {⟨( I ↾ 𝐵), 𝑂⟩})
7675eleq2d 2685 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) → (⟨𝑓, 𝑠⟩ ∈ { 0 } ↔ ⟨𝑓, 𝑠⟩ ∈ {⟨( I ↾ 𝐵), 𝑂⟩}))
77 opex 4923 . . . . . . 7 𝑓, 𝑠⟩ ∈ V
7877elsn 4183 . . . . . 6 (⟨𝑓, 𝑠⟩ ∈ {⟨( I ↾ 𝐵), 𝑂⟩} ↔ ⟨𝑓, 𝑠⟩ = ⟨( I ↾ 𝐵), 𝑂⟩)
7916, 17opth 4935 . . . . . 6 (⟨𝑓, 𝑠⟩ = ⟨( I ↾ 𝐵), 𝑂⟩ ↔ (𝑓 = ( I ↾ 𝐵) ∧ 𝑠 = 𝑂))
8078, 79bitr2i 265 . . . . 5 ((𝑓 = ( I ↾ 𝐵) ∧ 𝑠 = 𝑂) ↔ ⟨𝑓, 𝑠⟩ ∈ {⟨( I ↾ 𝐵), 𝑂⟩})
8176, 80syl6rbbr 279 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) → ((𝑓 = ( I ↾ 𝐵) ∧ 𝑠 = 𝑂) ↔ ⟨𝑓, 𝑠⟩ ∈ { 0 }))
8270, 81sylibd 229 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) → (⟨𝑓, 𝑠⟩ ∈ ((𝐼𝑄) ∩ (𝐼𝑅)) → ⟨𝑓, 𝑠⟩ ∈ { 0 }))
836, 82relssdv 5202 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) → ((𝐼𝑄) ∩ (𝐼𝑅)) ⊆ { 0 })
841, 71, 8dvhlmod 36218 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) → 𝑈 ∈ LMod)
85 simp2ll 1126 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) → 𝑄𝐴)
8655, 11atbase 34395 . . . . . 6 (𝑄𝐴𝑄𝐵)
8785, 86syl 17 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) → 𝑄𝐵)
88 eqid 2620 . . . . . 6 (LSubSp‘𝑈) = (LSubSp‘𝑈)
8955, 1, 2, 71, 88dihlss 36358 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑄𝐵) → (𝐼𝑄) ∈ (LSubSp‘𝑈))
908, 87, 89syl2anc 692 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) → (𝐼𝑄) ∈ (LSubSp‘𝑈))
91 simp2rl 1128 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) → 𝑅𝐴)
9255, 11atbase 34395 . . . . . 6 (𝑅𝐴𝑅𝐵)
9391, 92syl 17 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) → 𝑅𝐵)
9455, 1, 2, 71, 88dihlss 36358 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑅𝐵) → (𝐼𝑅) ∈ (LSubSp‘𝑈))
958, 93, 94syl2anc 692 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) → (𝐼𝑅) ∈ (LSubSp‘𝑈))
9688lssincl 18946 . . . 4 ((𝑈 ∈ LMod ∧ (𝐼𝑄) ∈ (LSubSp‘𝑈) ∧ (𝐼𝑅) ∈ (LSubSp‘𝑈)) → ((𝐼𝑄) ∩ (𝐼𝑅)) ∈ (LSubSp‘𝑈))
9784, 90, 95, 96syl3anc 1324 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) → ((𝐼𝑄) ∩ (𝐼𝑅)) ∈ (LSubSp‘𝑈))
9872, 88lss0ss 18930 . . 3 ((𝑈 ∈ LMod ∧ ((𝐼𝑄) ∩ (𝐼𝑅)) ∈ (LSubSp‘𝑈)) → { 0 } ⊆ ((𝐼𝑄) ∩ (𝐼𝑅)))
9984, 97, 98syl2anc 692 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑄𝑅) → { 0 } ⊆ ((𝐼𝑄) ∩ (𝐼𝑅)))
10083, 99eqssd 3612 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 196  wa 384  w3a 1036   = wceq 1481  wcel 1988  wne 2791  cin 3566  wss 3567  {csn 4168  cop 4174   class class class wbr 4644  cmpt 4720   I cid 5013  cres 5106  Rel wrel 5109  cfv 5876  crio 6595  Basecbs 15838  lecple 15929  occoc 15930  0gc0g 16081  joincjn 16925  LModclmod 18844  LSubSpclss 18913  Atomscatm 34369  HLchlt 34456  LHypclh 35089  LTrncltrn 35206  TEndoctendo 35859  DVecHcdvh 36186  DIsoHcdih 36336
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  df-dih 36337
This theorem is referenced by:  dihmeetlem15N  36429
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