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Theorem dihmeetlem4preN 36912
Description: Lemma for isomorphism H of a lattice meet. (Contributed by NM, 30-Mar-2014.) (New usage is discouraged.)
Hypotheses
Ref Expression
dihmeetlem4.b 𝐵 = (Base‘𝐾)
dihmeetlem4.l = (le‘𝐾)
dihmeetlem4.m = (meet‘𝐾)
dihmeetlem4.a 𝐴 = (Atoms‘𝐾)
dihmeetlem4.h 𝐻 = (LHyp‘𝐾)
dihmeetlem4.i 𝐼 = ((DIsoH‘𝐾)‘𝑊)
dihmeetlem4.u 𝑈 = ((DVecH‘𝐾)‘𝑊)
dihmeetlem4.z 0 = (0g𝑈)
dihmeetlem4.g 𝐺 = (𝑔𝑇 (𝑔𝑃) = 𝑄)
dihmeetlem4.p 𝑃 = ((oc‘𝐾)‘𝑊)
dihmeetlem4.t 𝑇 = ((LTrn‘𝐾)‘𝑊)
dihmeetlem4.r 𝑅 = ((trL‘𝐾)‘𝑊)
dihmeetlem4.e 𝐸 = ((TEndo‘𝐾)‘𝑊)
dihmeetlem4.o 𝑂 = (𝑇 ↦ ( I ↾ 𝐵))
Assertion
Ref Expression
dihmeetlem4preN (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → ((𝐼𝑄) ∩ (𝐼‘(𝑋 𝑊))) = { 0 })
Distinct variable groups:   ,𝑔   𝐴,𝑔   𝑔,,𝐻   𝐵,   𝑔,𝐾,   𝑄,𝑔   𝑇,𝑔,   𝑔,𝑊,   𝑃,𝑔
Allowed substitution hints:   𝐴()   𝐵(𝑔)   𝑃()   𝑄()   𝑅(𝑔,)   𝑈(𝑔,)   𝐸(𝑔,)   𝐺(𝑔,)   𝐼(𝑔,)   ()   (𝑔,)   𝑂(𝑔,)   𝑋(𝑔,)   0 (𝑔,)

Proof of Theorem dihmeetlem4preN
Dummy variables 𝑓 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dihmeetlem4.h . . . . 5 𝐻 = (LHyp‘𝐾)
2 dihmeetlem4.i . . . . 5 𝐼 = ((DIsoH‘𝐾)‘𝑊)
31, 2dihvalrel 36885 . . . 4 ((𝐾 ∈ HL ∧ 𝑊𝐻) → Rel (𝐼𝑄))
4 relin1 5269 . . . 4 (Rel (𝐼𝑄) → Rel ((𝐼𝑄) ∩ (𝐼‘(𝑋 𝑊))))
53, 4syl 17 . . 3 ((𝐾 ∈ HL ∧ 𝑊𝐻) → Rel ((𝐼𝑄) ∩ (𝐼‘(𝑋 𝑊))))
653ad2ant1 1102 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → Rel ((𝐼𝑄) ∩ (𝐼‘(𝑋 𝑊))))
71, 2dihvalrel 36885 . . . 4 ((𝐾 ∈ HL ∧ 𝑊𝐻) → Rel (𝐼‘(0.‘𝐾)))
8 eqid 2651 . . . . . 6 (0.‘𝐾) = (0.‘𝐾)
9 dihmeetlem4.u . . . . . 6 𝑈 = ((DVecH‘𝐾)‘𝑊)
10 dihmeetlem4.z . . . . . 6 0 = (0g𝑈)
118, 1, 2, 9, 10dih0 36886 . . . . 5 ((𝐾 ∈ HL ∧ 𝑊𝐻) → (𝐼‘(0.‘𝐾)) = { 0 })
1211releqd 5237 . . . 4 ((𝐾 ∈ HL ∧ 𝑊𝐻) → (Rel (𝐼‘(0.‘𝐾)) ↔ Rel { 0 }))
137, 12mpbid 222 . . 3 ((𝐾 ∈ HL ∧ 𝑊𝐻) → Rel { 0 })
14133ad2ant1 1102 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → Rel { 0 })
15 id 22 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → ((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)))
16 elin 3829 . . . 4 (⟨𝑓, 𝑠⟩ ∈ ((𝐼𝑄) ∩ (𝐼‘(𝑋 𝑊))) ↔ (⟨𝑓, 𝑠⟩ ∈ (𝐼𝑄) ∧ ⟨𝑓, 𝑠⟩ ∈ (𝐼‘(𝑋 𝑊))))
17 dihmeetlem4.l . . . . . . . . . 10 = (le‘𝐾)
18 dihmeetlem4.a . . . . . . . . . 10 𝐴 = (Atoms‘𝐾)
19 dihmeetlem4.p . . . . . . . . . 10 𝑃 = ((oc‘𝐾)‘𝑊)
20 dihmeetlem4.t . . . . . . . . . 10 𝑇 = ((LTrn‘𝐾)‘𝑊)
21 dihmeetlem4.e . . . . . . . . . 10 𝐸 = ((TEndo‘𝐾)‘𝑊)
22 dihmeetlem4.g . . . . . . . . . 10 𝐺 = (𝑔𝑇 (𝑔𝑃) = 𝑄)
23 vex 3234 . . . . . . . . . 10 𝑓 ∈ V
24 vex 3234 . . . . . . . . . 10 𝑠 ∈ V
2517, 18, 1, 19, 20, 21, 2, 22, 23, 24dihopelvalcqat 36852 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (⟨𝑓, 𝑠⟩ ∈ (𝐼𝑄) ↔ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸)))
26253adant2 1100 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (⟨𝑓, 𝑠⟩ ∈ (𝐼𝑄) ↔ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸)))
27 simp1 1081 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝐾 ∈ HL ∧ 𝑊𝐻))
28 simp1l 1105 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → 𝐾 ∈ HL)
29 hllat 34968 . . . . . . . . . . 11 (𝐾 ∈ HL → 𝐾 ∈ Lat)
3028, 29syl 17 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → 𝐾 ∈ Lat)
31 simp2l 1107 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → 𝑋𝐵)
32 simp1r 1106 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → 𝑊𝐻)
33 dihmeetlem4.b . . . . . . . . . . . 12 𝐵 = (Base‘𝐾)
3433, 1lhpbase 35602 . . . . . . . . . . 11 (𝑊𝐻𝑊𝐵)
3532, 34syl 17 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → 𝑊𝐵)
36 dihmeetlem4.m . . . . . . . . . . 11 = (meet‘𝐾)
3733, 36latmcl 17099 . . . . . . . . . 10 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑊𝐵) → (𝑋 𝑊) ∈ 𝐵)
3830, 31, 35, 37syl3anc 1366 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝑋 𝑊) ∈ 𝐵)
3933, 17, 36latmle2 17124 . . . . . . . . . 10 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑊𝐵) → (𝑋 𝑊) 𝑊)
4030, 31, 35, 39syl3anc 1366 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝑋 𝑊) 𝑊)
41 dihmeetlem4.r . . . . . . . . . 10 𝑅 = ((trL‘𝐾)‘𝑊)
42 dihmeetlem4.o . . . . . . . . . 10 𝑂 = (𝑇 ↦ ( I ↾ 𝐵))
4333, 17, 1, 20, 41, 42, 2dihopelvalbN 36844 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑋 𝑊) ∈ 𝐵 ∧ (𝑋 𝑊) 𝑊)) → (⟨𝑓, 𝑠⟩ ∈ (𝐼‘(𝑋 𝑊)) ↔ ((𝑓𝑇 ∧ (𝑅𝑓) (𝑋 𝑊)) ∧ 𝑠 = 𝑂)))
4427, 38, 40, 43syl12anc 1364 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (⟨𝑓, 𝑠⟩ ∈ (𝐼‘(𝑋 𝑊)) ↔ ((𝑓𝑇 ∧ (𝑅𝑓) (𝑋 𝑊)) ∧ 𝑠 = 𝑂)))
4526, 44anbi12d 747 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → ((⟨𝑓, 𝑠⟩ ∈ (𝐼𝑄) ∧ ⟨𝑓, 𝑠⟩ ∈ (𝐼‘(𝑋 𝑊))) ↔ ((𝑓 = (𝑠𝐺) ∧ 𝑠𝐸) ∧ ((𝑓𝑇 ∧ (𝑅𝑓) (𝑋 𝑊)) ∧ 𝑠 = 𝑂))))
46 simprll 819 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑓 = (𝑠𝐺) ∧ 𝑠𝐸) ∧ ((𝑓𝑇 ∧ (𝑅𝑓) (𝑋 𝑊)) ∧ 𝑠 = 𝑂))) → 𝑓 = (𝑠𝐺))
47 simprrr 822 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑓 = (𝑠𝐺) ∧ 𝑠𝐸) ∧ ((𝑓𝑇 ∧ (𝑅𝑓) (𝑋 𝑊)) ∧ 𝑠 = 𝑂))) → 𝑠 = 𝑂)
4847fveq1d 6231 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑓 = (𝑠𝐺) ∧ 𝑠𝐸) ∧ ((𝑓𝑇 ∧ (𝑅𝑓) (𝑋 𝑊)) ∧ 𝑠 = 𝑂))) → (𝑠𝐺) = (𝑂𝐺))
49 simpl1 1084 . . . . . . . . . . . 12 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑓 = (𝑠𝐺) ∧ 𝑠𝐸) ∧ ((𝑓𝑇 ∧ (𝑅𝑓) (𝑋 𝑊)) ∧ 𝑠 = 𝑂))) → (𝐾 ∈ HL ∧ 𝑊𝐻))
5017, 18, 1, 19lhpocnel2 35623 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ 𝑊𝐻) → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
5149, 50syl 17 . . . . . . . . . . . 12 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑓 = (𝑠𝐺) ∧ 𝑠𝐸) ∧ ((𝑓𝑇 ∧ (𝑅𝑓) (𝑋 𝑊)) ∧ 𝑠 = 𝑂))) → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
52 simpl3 1086 . . . . . . . . . . . 12 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑓 = (𝑠𝐺) ∧ 𝑠𝐸) ∧ ((𝑓𝑇 ∧ (𝑅𝑓) (𝑋 𝑊)) ∧ 𝑠 = 𝑂))) → (𝑄𝐴 ∧ ¬ 𝑄 𝑊))
5317, 18, 1, 20, 22ltrniotacl 36184 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → 𝐺𝑇)
5449, 51, 52, 53syl3anc 1366 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑓 = (𝑠𝐺) ∧ 𝑠𝐸) ∧ ((𝑓𝑇 ∧ (𝑅𝑓) (𝑋 𝑊)) ∧ 𝑠 = 𝑂))) → 𝐺𝑇)
5542, 33tendo02 36392 . . . . . . . . . . 11 (𝐺𝑇 → (𝑂𝐺) = ( I ↾ 𝐵))
5654, 55syl 17 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑓 = (𝑠𝐺) ∧ 𝑠𝐸) ∧ ((𝑓𝑇 ∧ (𝑅𝑓) (𝑋 𝑊)) ∧ 𝑠 = 𝑂))) → (𝑂𝐺) = ( I ↾ 𝐵))
5746, 48, 563eqtrd 2689 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑓 = (𝑠𝐺) ∧ 𝑠𝐸) ∧ ((𝑓𝑇 ∧ (𝑅𝑓) (𝑋 𝑊)) ∧ 𝑠 = 𝑂))) → 𝑓 = ( I ↾ 𝐵))
5857, 47jca 553 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑓 = (𝑠𝐺) ∧ 𝑠𝐸) ∧ ((𝑓𝑇 ∧ (𝑅𝑓) (𝑋 𝑊)) ∧ 𝑠 = 𝑂))) → (𝑓 = ( I ↾ 𝐵) ∧ 𝑠 = 𝑂))
59 simpl1 1084 . . . . . . . . . . . 12 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑓 = ( I ↾ 𝐵) ∧ 𝑠 = 𝑂)) → (𝐾 ∈ HL ∧ 𝑊𝐻))
6059, 50syl 17 . . . . . . . . . . . 12 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑓 = ( I ↾ 𝐵) ∧ 𝑠 = 𝑂)) → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
61 simpl3 1086 . . . . . . . . . . . 12 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑓 = ( I ↾ 𝐵) ∧ 𝑠 = 𝑂)) → (𝑄𝐴 ∧ ¬ 𝑄 𝑊))
6259, 60, 61, 53syl3anc 1366 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑓 = ( I ↾ 𝐵) ∧ 𝑠 = 𝑂)) → 𝐺𝑇)
6362, 55syl 17 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑓 = ( I ↾ 𝐵) ∧ 𝑠 = 𝑂)) → (𝑂𝐺) = ( I ↾ 𝐵))
64 simprr 811 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑓 = ( I ↾ 𝐵) ∧ 𝑠 = 𝑂)) → 𝑠 = 𝑂)
6564fveq1d 6231 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑓 = ( I ↾ 𝐵) ∧ 𝑠 = 𝑂)) → (𝑠𝐺) = (𝑂𝐺))
66 simprl 809 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑓 = ( I ↾ 𝐵) ∧ 𝑠 = 𝑂)) → 𝑓 = ( I ↾ 𝐵))
6763, 65, 663eqtr4rd 2696 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑓 = ( I ↾ 𝐵) ∧ 𝑠 = 𝑂)) → 𝑓 = (𝑠𝐺))
6833, 1, 20, 21, 42tendo0cl 36395 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ 𝑊𝐻) → 𝑂𝐸)
6959, 68syl 17 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑓 = ( I ↾ 𝐵) ∧ 𝑠 = 𝑂)) → 𝑂𝐸)
7064, 69eqeltrd 2730 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑓 = ( I ↾ 𝐵) ∧ 𝑠 = 𝑂)) → 𝑠𝐸)
7133, 1, 20idltrn 35754 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ 𝑊𝐻) → ( I ↾ 𝐵) ∈ 𝑇)
7259, 71syl 17 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑓 = ( I ↾ 𝐵) ∧ 𝑠 = 𝑂)) → ( I ↾ 𝐵) ∈ 𝑇)
7366, 72eqeltrd 2730 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑓 = ( I ↾ 𝐵) ∧ 𝑠 = 𝑂)) → 𝑓𝑇)
7466fveq2d 6233 . . . . . . . . . . . 12 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑓 = ( I ↾ 𝐵) ∧ 𝑠 = 𝑂)) → (𝑅𝑓) = (𝑅‘( I ↾ 𝐵)))
7533, 8, 1, 41trlid0 35781 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ 𝑊𝐻) → (𝑅‘( I ↾ 𝐵)) = (0.‘𝐾))
7659, 75syl 17 . . . . . . . . . . . 12 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑓 = ( I ↾ 𝐵) ∧ 𝑠 = 𝑂)) → (𝑅‘( I ↾ 𝐵)) = (0.‘𝐾))
7774, 76eqtrd 2685 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑓 = ( I ↾ 𝐵) ∧ 𝑠 = 𝑂)) → (𝑅𝑓) = (0.‘𝐾))
78 simpl1l 1132 . . . . . . . . . . . . 13 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑓 = ( I ↾ 𝐵) ∧ 𝑠 = 𝑂)) → 𝐾 ∈ HL)
79 hlatl 34965 . . . . . . . . . . . . 13 (𝐾 ∈ HL → 𝐾 ∈ AtLat)
8078, 79syl 17 . . . . . . . . . . . 12 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑓 = ( I ↾ 𝐵) ∧ 𝑠 = 𝑂)) → 𝐾 ∈ AtLat)
8138adantr 480 . . . . . . . . . . . 12 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑓 = ( I ↾ 𝐵) ∧ 𝑠 = 𝑂)) → (𝑋 𝑊) ∈ 𝐵)
8233, 17, 8atl0le 34909 . . . . . . . . . . . 12 ((𝐾 ∈ AtLat ∧ (𝑋 𝑊) ∈ 𝐵) → (0.‘𝐾) (𝑋 𝑊))
8380, 81, 82syl2anc 694 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑓 = ( I ↾ 𝐵) ∧ 𝑠 = 𝑂)) → (0.‘𝐾) (𝑋 𝑊))
8477, 83eqbrtrd 4707 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑓 = ( I ↾ 𝐵) ∧ 𝑠 = 𝑂)) → (𝑅𝑓) (𝑋 𝑊))
8573, 84, 64jca31 556 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑓 = ( I ↾ 𝐵) ∧ 𝑠 = 𝑂)) → ((𝑓𝑇 ∧ (𝑅𝑓) (𝑋 𝑊)) ∧ 𝑠 = 𝑂))
8667, 70, 85jca31 556 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑓 = ( I ↾ 𝐵) ∧ 𝑠 = 𝑂)) → ((𝑓 = (𝑠𝐺) ∧ 𝑠𝐸) ∧ ((𝑓𝑇 ∧ (𝑅𝑓) (𝑋 𝑊)) ∧ 𝑠 = 𝑂)))
8758, 86impbida 895 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (((𝑓 = (𝑠𝐺) ∧ 𝑠𝐸) ∧ ((𝑓𝑇 ∧ (𝑅𝑓) (𝑋 𝑊)) ∧ 𝑠 = 𝑂)) ↔ (𝑓 = ( I ↾ 𝐵) ∧ 𝑠 = 𝑂)))
8845, 87bitrd 268 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → ((⟨𝑓, 𝑠⟩ ∈ (𝐼𝑄) ∧ ⟨𝑓, 𝑠⟩ ∈ (𝐼‘(𝑋 𝑊))) ↔ (𝑓 = ( I ↾ 𝐵) ∧ 𝑠 = 𝑂)))
89 opex 4962 . . . . . . . 8 𝑓, 𝑠⟩ ∈ V
9089elsn 4225 . . . . . . 7 (⟨𝑓, 𝑠⟩ ∈ {⟨( I ↾ 𝐵), 𝑂⟩} ↔ ⟨𝑓, 𝑠⟩ = ⟨( I ↾ 𝐵), 𝑂⟩)
9123, 24opth 4974 . . . . . . 7 (⟨𝑓, 𝑠⟩ = ⟨( I ↾ 𝐵), 𝑂⟩ ↔ (𝑓 = ( I ↾ 𝐵) ∧ 𝑠 = 𝑂))
9290, 91bitr2i 265 . . . . . 6 ((𝑓 = ( I ↾ 𝐵) ∧ 𝑠 = 𝑂) ↔ ⟨𝑓, 𝑠⟩ ∈ {⟨( I ↾ 𝐵), 𝑂⟩})
9388, 92syl6bb 276 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → ((⟨𝑓, 𝑠⟩ ∈ (𝐼𝑄) ∧ ⟨𝑓, 𝑠⟩ ∈ (𝐼‘(𝑋 𝑊))) ↔ ⟨𝑓, 𝑠⟩ ∈ {⟨( I ↾ 𝐵), 𝑂⟩}))
9433, 1, 20, 9, 10, 42dvh0g 36717 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑊𝐻) → 0 = ⟨( I ↾ 𝐵), 𝑂⟩)
95943ad2ant1 1102 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → 0 = ⟨( I ↾ 𝐵), 𝑂⟩)
9695sneqd 4222 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → { 0 } = {⟨( I ↾ 𝐵), 𝑂⟩})
9796eleq2d 2716 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (⟨𝑓, 𝑠⟩ ∈ { 0 } ↔ ⟨𝑓, 𝑠⟩ ∈ {⟨( I ↾ 𝐵), 𝑂⟩}))
9893, 97bitr4d 271 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → ((⟨𝑓, 𝑠⟩ ∈ (𝐼𝑄) ∧ ⟨𝑓, 𝑠⟩ ∈ (𝐼‘(𝑋 𝑊))) ↔ ⟨𝑓, 𝑠⟩ ∈ { 0 }))
9916, 98syl5bb 272 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (⟨𝑓, 𝑠⟩ ∈ ((𝐼𝑄) ∩ (𝐼‘(𝑋 𝑊))) ↔ ⟨𝑓, 𝑠⟩ ∈ { 0 }))
10099eqrelrdv2 5253 . 2 (((Rel ((𝐼𝑄) ∩ (𝐼‘(𝑋 𝑊))) ∧ Rel { 0 }) ∧ ((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊))) → ((𝐼𝑄) ∩ (𝐼‘(𝑋 𝑊))) = { 0 })
1016, 14, 15, 100syl21anc 1365 1 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → ((𝐼𝑄) ∩ (𝐼‘(𝑋 𝑊))) = { 0 })
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  cin 3606  {csn 4210  cop 4216   class class class wbr 4685  cmpt 4762   I cid 5052  cres 5145  Rel wrel 5148  cfv 5926  crio 6650  (class class class)co 6690  Basecbs 15904  lecple 15995  occoc 15996  0gc0g 16147  meetcmee 16992  0.cp0 17084  Latclat 17092  Atomscatm 34868  AtLatcal 34869  HLchlt 34955  LHypclh 35588  LTrncltrn 35705  trLctrl 35763  TEndoctendo 36357  DVecHcdvh 36684  DIsoHcdih 36834
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-riotaBAD 34557
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-iin 4555  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-tpos 7397  df-undef 7444  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-map 7901  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-2 11117  df-3 11118  df-4 11119  df-5 11120  df-6 11121  df-n0 11331  df-z 11416  df-uz 11726  df-fz 12365  df-struct 15906  df-ndx 15907  df-slot 15908  df-base 15910  df-sets 15911  df-ress 15912  df-plusg 16001  df-mulr 16002  df-sca 16004  df-vsca 16005  df-0g 16149  df-preset 16975  df-poset 16993  df-plt 17005  df-lub 17021  df-glb 17022  df-join 17023  df-meet 17024  df-p0 17086  df-p1 17087  df-lat 17093  df-clat 17155  df-mgm 17289  df-sgrp 17331  df-mnd 17342  df-submnd 17383  df-grp 17472  df-minusg 17473  df-sbg 17474  df-subg 17638  df-cntz 17796  df-lsm 18097  df-cmn 18241  df-abl 18242  df-mgp 18536  df-ur 18548  df-ring 18595  df-oppr 18669  df-dvdsr 18687  df-unit 18688  df-invr 18718  df-dvr 18729  df-drng 18797  df-lmod 18913  df-lss 18981  df-lsp 19020  df-lvec 19151  df-oposet 34781  df-ol 34783  df-oml 34784  df-covers 34871  df-ats 34872  df-atl 34903  df-cvlat 34927  df-hlat 34956  df-llines 35102  df-lplanes 35103  df-lvols 35104  df-lines 35105  df-psubsp 35107  df-pmap 35108  df-padd 35400  df-lhyp 35592  df-laut 35593  df-ldil 35708  df-ltrn 35709  df-trl 35764  df-tendo 36360  df-edring 36362  df-disoa 36635  df-dvech 36685  df-dib 36745  df-dic 36779  df-dih 36835
This theorem is referenced by:  dihmeetlem4N  36913
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