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Theorem dihjatcclem4 37442
Description: Lemma for isomorphism H of lattice join of two atoms not under the fiducial hyperplane. (Contributed by NM, 29-Sep-2014.)
Hypotheses
Ref Expression
dihjatcclem.b 𝐵 = (Base‘𝐾)
dihjatcclem.l = (le‘𝐾)
dihjatcclem.h 𝐻 = (LHyp‘𝐾)
dihjatcclem.j = (join‘𝐾)
dihjatcclem.m = (meet‘𝐾)
dihjatcclem.a 𝐴 = (Atoms‘𝐾)
dihjatcclem.u 𝑈 = ((DVecH‘𝐾)‘𝑊)
dihjatcclem.s = (LSSum‘𝑈)
dihjatcclem.i 𝐼 = ((DIsoH‘𝐾)‘𝑊)
dihjatcclem.v 𝑉 = ((𝑃 𝑄) 𝑊)
dihjatcclem.k (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))
dihjatcclem.p (𝜑 → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
dihjatcclem.q (𝜑 → (𝑄𝐴 ∧ ¬ 𝑄 𝑊))
dihjatcc.w 𝐶 = ((oc‘𝐾)‘𝑊)
dihjatcc.t 𝑇 = ((LTrn‘𝐾)‘𝑊)
dihjatcc.r 𝑅 = ((trL‘𝐾)‘𝑊)
dihjatcc.e 𝐸 = ((TEndo‘𝐾)‘𝑊)
dihjatcc.g 𝐺 = (𝑑𝑇 (𝑑𝐶) = 𝑃)
dihjatcc.dd 𝐷 = (𝑑𝑇 (𝑑𝐶) = 𝑄)
dihjatcc.n 𝑁 = (𝑎𝐸 ↦ (𝑑𝑇(𝑎𝑑)))
dihjatcc.o 0 = (𝑑𝑇 ↦ ( I ↾ 𝐵))
dihjatcc.d 𝐽 = (𝑎𝐸, 𝑏𝐸 ↦ (𝑑𝑇 ↦ ((𝑎𝑑) ∘ (𝑏𝑑))))
Assertion
Ref Expression
dihjatcclem4 (𝜑 → (𝐼𝑉) ⊆ ((𝐼𝑃) (𝐼𝑄)))
Distinct variable groups:   ,𝑑   𝐴,𝑑   𝐵,𝑑   𝐶,𝑑   𝑎,𝑏,𝐸   𝐻,𝑑   𝑃,𝑑   𝑎,𝑑,𝐾,𝑏   𝑄,𝑑   𝑇,𝑎,𝑏,𝑑   𝑊,𝑎,𝑏,𝑑
Allowed substitution hints:   𝜑(𝑎,𝑏,𝑑)   𝐴(𝑎,𝑏)   𝐵(𝑎,𝑏)   𝐶(𝑎,𝑏)   𝐷(𝑎,𝑏,𝑑)   𝑃(𝑎,𝑏)   (𝑎,𝑏,𝑑)   𝑄(𝑎,𝑏)   𝑅(𝑎,𝑏,𝑑)   𝑈(𝑎,𝑏,𝑑)   𝐸(𝑑)   𝐺(𝑎,𝑏,𝑑)   𝐻(𝑎,𝑏)   𝐼(𝑎,𝑏,𝑑)   𝐽(𝑎,𝑏,𝑑)   (𝑎,𝑏,𝑑)   (𝑎,𝑏)   (𝑎,𝑏,𝑑)   𝑁(𝑎,𝑏,𝑑)   𝑉(𝑎,𝑏,𝑑)   0 (𝑎,𝑏,𝑑)

Proof of Theorem dihjatcclem4
Dummy variables 𝑡 𝑓 𝑠 𝑔 𝑢 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dihjatcclem.k . . 3 (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))
2 dihjatcclem.h . . . 4 𝐻 = (LHyp‘𝐾)
3 dihjatcclem.i . . . 4 𝐼 = ((DIsoH‘𝐾)‘𝑊)
42, 3dihvalrel 37300 . . 3 ((𝐾 ∈ HL ∧ 𝑊𝐻) → Rel (𝐼𝑉))
51, 4syl 17 . 2 (𝜑 → Rel (𝐼𝑉))
61adantr 473 . . . . . . 7 ((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) → (𝐾 ∈ HL ∧ 𝑊𝐻))
7 dihjatcclem.l . . . . . . . . . . . 12 = (le‘𝐾)
8 dihjatcclem.a . . . . . . . . . . . 12 𝐴 = (Atoms‘𝐾)
9 dihjatcc.w . . . . . . . . . . . 12 𝐶 = ((oc‘𝐾)‘𝑊)
107, 8, 2, 9lhpocnel2 36040 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ 𝑊𝐻) → (𝐶𝐴 ∧ ¬ 𝐶 𝑊))
111, 10syl 17 . . . . . . . . . 10 (𝜑 → (𝐶𝐴 ∧ ¬ 𝐶 𝑊))
12 dihjatcclem.p . . . . . . . . . 10 (𝜑 → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
13 dihjatcc.t . . . . . . . . . . 11 𝑇 = ((LTrn‘𝐾)‘𝑊)
14 dihjatcc.g . . . . . . . . . . 11 𝐺 = (𝑑𝑇 (𝑑𝐶) = 𝑃)
157, 8, 2, 13, 14ltrniotacl 36600 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐶𝐴 ∧ ¬ 𝐶 𝑊) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → 𝐺𝑇)
161, 11, 12, 15syl3anc 1491 . . . . . . . . 9 (𝜑𝐺𝑇)
17 dihjatcclem.q . . . . . . . . . . 11 (𝜑 → (𝑄𝐴 ∧ ¬ 𝑄 𝑊))
18 dihjatcc.dd . . . . . . . . . . . 12 𝐷 = (𝑑𝑇 (𝑑𝐶) = 𝑄)
197, 8, 2, 13, 18ltrniotacl 36600 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐶𝐴 ∧ ¬ 𝐶 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → 𝐷𝑇)
201, 11, 17, 19syl3anc 1491 . . . . . . . . . 10 (𝜑𝐷𝑇)
212, 13ltrncnv 36167 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐷𝑇) → 𝐷𝑇)
221, 20, 21syl2anc 580 . . . . . . . . 9 (𝜑𝐷𝑇)
232, 13ltrnco 36740 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐺𝑇𝐷𝑇) → (𝐺𝐷) ∈ 𝑇)
241, 16, 22, 23syl3anc 1491 . . . . . . . 8 (𝜑 → (𝐺𝐷) ∈ 𝑇)
2524adantr 473 . . . . . . 7 ((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) → (𝐺𝐷) ∈ 𝑇)
26 simprll 798 . . . . . . 7 ((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) → 𝑓𝑇)
27 simprlr 799 . . . . . . . 8 ((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) → (𝑅𝑓) 𝑉)
28 dihjatcclem.b . . . . . . . . . 10 𝐵 = (Base‘𝐾)
29 dihjatcclem.j . . . . . . . . . 10 = (join‘𝐾)
30 dihjatcclem.m . . . . . . . . . 10 = (meet‘𝐾)
31 dihjatcclem.u . . . . . . . . . 10 𝑈 = ((DVecH‘𝐾)‘𝑊)
32 dihjatcclem.s . . . . . . . . . 10 = (LSSum‘𝑈)
33 dihjatcclem.v . . . . . . . . . 10 𝑉 = ((𝑃 𝑄) 𝑊)
34 dihjatcc.r . . . . . . . . . 10 𝑅 = ((trL‘𝐾)‘𝑊)
35 dihjatcc.e . . . . . . . . . 10 𝐸 = ((TEndo‘𝐾)‘𝑊)
3628, 7, 2, 29, 30, 8, 31, 32, 3, 33, 1, 12, 17, 9, 13, 34, 35, 14, 18dihjatcclem3 37441 . . . . . . . . 9 (𝜑 → (𝑅‘(𝐺𝐷)) = 𝑉)
3736adantr 473 . . . . . . . 8 ((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) → (𝑅‘(𝐺𝐷)) = 𝑉)
3827, 37breqtrrd 4871 . . . . . . 7 ((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) → (𝑅𝑓) (𝑅‘(𝐺𝐷)))
397, 2, 13, 34, 35tendoex 36996 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝐺𝐷) ∈ 𝑇𝑓𝑇) ∧ (𝑅𝑓) (𝑅‘(𝐺𝐷))) → ∃𝑡𝐸 (𝑡‘(𝐺𝐷)) = 𝑓)
406, 25, 26, 38, 39syl121anc 1495 . . . . . 6 ((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) → ∃𝑡𝐸 (𝑡‘(𝐺𝐷)) = 𝑓)
41 df-rex 3095 . . . . . 6 (∃𝑡𝐸 (𝑡‘(𝐺𝐷)) = 𝑓 ↔ ∃𝑡(𝑡𝐸 ∧ (𝑡‘(𝐺𝐷)) = 𝑓))
4240, 41sylib 210 . . . . 5 ((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) → ∃𝑡(𝑡𝐸 ∧ (𝑡‘(𝐺𝐷)) = 𝑓))
43 eqidd 2800 . . . . . . . . . 10 (((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡𝐸 ∧ (𝑡‘(𝐺𝐷)) = 𝑓)) → (𝑡𝐺) = (𝑡𝐺))
44 simprl 788 . . . . . . . . . 10 (((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡𝐸 ∧ (𝑡‘(𝐺𝐷)) = 𝑓)) → 𝑡𝐸)
451ad2antrr 718 . . . . . . . . . . 11 (((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡𝐸 ∧ (𝑡‘(𝐺𝐷)) = 𝑓)) → (𝐾 ∈ HL ∧ 𝑊𝐻))
4612ad2antrr 718 . . . . . . . . . . 11 (((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡𝐸 ∧ (𝑡‘(𝐺𝐷)) = 𝑓)) → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
47 fvex 6424 . . . . . . . . . . . 12 (𝑡𝐺) ∈ V
48 vex 3388 . . . . . . . . . . . 12 𝑡 ∈ V
497, 8, 2, 9, 13, 35, 3, 14, 47, 48dihopelvalcqat 37267 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (⟨(𝑡𝐺), 𝑡⟩ ∈ (𝐼𝑃) ↔ ((𝑡𝐺) = (𝑡𝐺) ∧ 𝑡𝐸)))
5045, 46, 49syl2anc 580 . . . . . . . . . 10 (((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡𝐸 ∧ (𝑡‘(𝐺𝐷)) = 𝑓)) → (⟨(𝑡𝐺), 𝑡⟩ ∈ (𝐼𝑃) ↔ ((𝑡𝐺) = (𝑡𝐺) ∧ 𝑡𝐸)))
5143, 44, 50mpbir2and 705 . . . . . . . . 9 (((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡𝐸 ∧ (𝑡‘(𝐺𝐷)) = 𝑓)) → ⟨(𝑡𝐺), 𝑡⟩ ∈ (𝐼𝑃))
52 eqidd 2800 . . . . . . . . . 10 (((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡𝐸 ∧ (𝑡‘(𝐺𝐷)) = 𝑓)) → ((𝑁𝑡)‘𝐷) = ((𝑁𝑡)‘𝐷))
53 dihjatcc.n . . . . . . . . . . . 12 𝑁 = (𝑎𝐸 ↦ (𝑑𝑇(𝑎𝑑)))
542, 13, 35, 53tendoicl 36817 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑡𝐸) → (𝑁𝑡) ∈ 𝐸)
5545, 44, 54syl2anc 580 . . . . . . . . . 10 (((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡𝐸 ∧ (𝑡‘(𝐺𝐷)) = 𝑓)) → (𝑁𝑡) ∈ 𝐸)
5617ad2antrr 718 . . . . . . . . . . 11 (((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡𝐸 ∧ (𝑡‘(𝐺𝐷)) = 𝑓)) → (𝑄𝐴 ∧ ¬ 𝑄 𝑊))
57 fvex 6424 . . . . . . . . . . . 12 ((𝑁𝑡)‘𝐷) ∈ V
58 fvex 6424 . . . . . . . . . . . 12 (𝑁𝑡) ∈ V
597, 8, 2, 9, 13, 35, 3, 18, 57, 58dihopelvalcqat 37267 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (⟨((𝑁𝑡)‘𝐷), (𝑁𝑡)⟩ ∈ (𝐼𝑄) ↔ (((𝑁𝑡)‘𝐷) = ((𝑁𝑡)‘𝐷) ∧ (𝑁𝑡) ∈ 𝐸)))
6045, 56, 59syl2anc 580 . . . . . . . . . 10 (((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡𝐸 ∧ (𝑡‘(𝐺𝐷)) = 𝑓)) → (⟨((𝑁𝑡)‘𝐷), (𝑁𝑡)⟩ ∈ (𝐼𝑄) ↔ (((𝑁𝑡)‘𝐷) = ((𝑁𝑡)‘𝐷) ∧ (𝑁𝑡) ∈ 𝐸)))
6152, 55, 60mpbir2and 705 . . . . . . . . 9 (((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡𝐸 ∧ (𝑡‘(𝐺𝐷)) = 𝑓)) → ⟨((𝑁𝑡)‘𝐷), (𝑁𝑡)⟩ ∈ (𝐼𝑄))
6216ad2antrr 718 . . . . . . . . . . 11 (((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡𝐸 ∧ (𝑡‘(𝐺𝐷)) = 𝑓)) → 𝐺𝑇)
6322ad2antrr 718 . . . . . . . . . . 11 (((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡𝐸 ∧ (𝑡‘(𝐺𝐷)) = 𝑓)) → 𝐷𝑇)
642, 13, 35tendospdi1 37041 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑡𝐸𝐺𝑇𝐷𝑇)) → (𝑡‘(𝐺𝐷)) = ((𝑡𝐺) ∘ (𝑡𝐷)))
6545, 44, 62, 63, 64syl13anc 1492 . . . . . . . . . 10 (((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡𝐸 ∧ (𝑡‘(𝐺𝐷)) = 𝑓)) → (𝑡‘(𝐺𝐷)) = ((𝑡𝐺) ∘ (𝑡𝐷)))
66 simprr 790 . . . . . . . . . 10 (((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡𝐸 ∧ (𝑡‘(𝐺𝐷)) = 𝑓)) → (𝑡‘(𝐺𝐷)) = 𝑓)
6720ad2antrr 718 . . . . . . . . . . . . 13 (((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡𝐸 ∧ (𝑡‘(𝐺𝐷)) = 𝑓)) → 𝐷𝑇)
6853, 13tendoi2 36816 . . . . . . . . . . . . 13 ((𝑡𝐸𝐷𝑇) → ((𝑁𝑡)‘𝐷) = (𝑡𝐷))
6944, 67, 68syl2anc 580 . . . . . . . . . . . 12 (((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡𝐸 ∧ (𝑡‘(𝐺𝐷)) = 𝑓)) → ((𝑁𝑡)‘𝐷) = (𝑡𝐷))
702, 13, 35tendocnv 37042 . . . . . . . . . . . . 13 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑡𝐸𝐷𝑇) → (𝑡𝐷) = (𝑡𝐷))
7145, 44, 67, 70syl3anc 1491 . . . . . . . . . . . 12 (((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡𝐸 ∧ (𝑡‘(𝐺𝐷)) = 𝑓)) → (𝑡𝐷) = (𝑡𝐷))
7269, 71eqtr2d 2834 . . . . . . . . . . 11 (((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡𝐸 ∧ (𝑡‘(𝐺𝐷)) = 𝑓)) → (𝑡𝐷) = ((𝑁𝑡)‘𝐷))
7372coeq2d 5488 . . . . . . . . . 10 (((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡𝐸 ∧ (𝑡‘(𝐺𝐷)) = 𝑓)) → ((𝑡𝐺) ∘ (𝑡𝐷)) = ((𝑡𝐺) ∘ ((𝑁𝑡)‘𝐷)))
7465, 66, 733eqtr3d 2841 . . . . . . . . 9 (((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡𝐸 ∧ (𝑡‘(𝐺𝐷)) = 𝑓)) → 𝑓 = ((𝑡𝐺) ∘ ((𝑁𝑡)‘𝐷)))
75 simplrr 797 . . . . . . . . . 10 (((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡𝐸 ∧ (𝑡‘(𝐺𝐷)) = 𝑓)) → 𝑠 = 0 )
76 dihjatcc.d . . . . . . . . . . . 12 𝐽 = (𝑎𝐸, 𝑏𝐸 ↦ (𝑑𝑇 ↦ ((𝑎𝑑) ∘ (𝑏𝑑))))
77 dihjatcc.o . . . . . . . . . . . 12 0 = (𝑑𝑇 ↦ ( I ↾ 𝐵))
782, 13, 35, 53, 28, 76, 77tendoipl2 36819 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑡𝐸) → (𝑡𝐽(𝑁𝑡)) = 0 )
7945, 44, 78syl2anc 580 . . . . . . . . . 10 (((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡𝐸 ∧ (𝑡‘(𝐺𝐷)) = 𝑓)) → (𝑡𝐽(𝑁𝑡)) = 0 )
8075, 79eqtr4d 2836 . . . . . . . . 9 (((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡𝐸 ∧ (𝑡‘(𝐺𝐷)) = 𝑓)) → 𝑠 = (𝑡𝐽(𝑁𝑡)))
81 opeq1 4593 . . . . . . . . . . . . . . 15 (𝑔 = (𝑡𝐺) → ⟨𝑔, 𝑡⟩ = ⟨(𝑡𝐺), 𝑡⟩)
8281eleq1d 2863 . . . . . . . . . . . . . 14 (𝑔 = (𝑡𝐺) → (⟨𝑔, 𝑡⟩ ∈ (𝐼𝑃) ↔ ⟨(𝑡𝐺), 𝑡⟩ ∈ (𝐼𝑃)))
8382anbi1d 624 . . . . . . . . . . . . 13 (𝑔 = (𝑡𝐺) → ((⟨𝑔, 𝑡⟩ ∈ (𝐼𝑃) ∧ ⟨, 𝑢⟩ ∈ (𝐼𝑄)) ↔ (⟨(𝑡𝐺), 𝑡⟩ ∈ (𝐼𝑃) ∧ ⟨, 𝑢⟩ ∈ (𝐼𝑄))))
84 coeq1 5483 . . . . . . . . . . . . . . 15 (𝑔 = (𝑡𝐺) → (𝑔) = ((𝑡𝐺) ∘ ))
8584eqeq2d 2809 . . . . . . . . . . . . . 14 (𝑔 = (𝑡𝐺) → (𝑓 = (𝑔) ↔ 𝑓 = ((𝑡𝐺) ∘ )))
8685anbi1d 624 . . . . . . . . . . . . 13 (𝑔 = (𝑡𝐺) → ((𝑓 = (𝑔) ∧ 𝑠 = (𝑡𝐽𝑢)) ↔ (𝑓 = ((𝑡𝐺) ∘ ) ∧ 𝑠 = (𝑡𝐽𝑢))))
8783, 86anbi12d 625 . . . . . . . . . . . 12 (𝑔 = (𝑡𝐺) → (((⟨𝑔, 𝑡⟩ ∈ (𝐼𝑃) ∧ ⟨, 𝑢⟩ ∈ (𝐼𝑄)) ∧ (𝑓 = (𝑔) ∧ 𝑠 = (𝑡𝐽𝑢))) ↔ ((⟨(𝑡𝐺), 𝑡⟩ ∈ (𝐼𝑃) ∧ ⟨, 𝑢⟩ ∈ (𝐼𝑄)) ∧ (𝑓 = ((𝑡𝐺) ∘ ) ∧ 𝑠 = (𝑡𝐽𝑢)))))
88 opeq1 4593 . . . . . . . . . . . . . . 15 ( = ((𝑁𝑡)‘𝐷) → ⟨, 𝑢⟩ = ⟨((𝑁𝑡)‘𝐷), 𝑢⟩)
8988eleq1d 2863 . . . . . . . . . . . . . 14 ( = ((𝑁𝑡)‘𝐷) → (⟨, 𝑢⟩ ∈ (𝐼𝑄) ↔ ⟨((𝑁𝑡)‘𝐷), 𝑢⟩ ∈ (𝐼𝑄)))
9089anbi2d 623 . . . . . . . . . . . . 13 ( = ((𝑁𝑡)‘𝐷) → ((⟨(𝑡𝐺), 𝑡⟩ ∈ (𝐼𝑃) ∧ ⟨, 𝑢⟩ ∈ (𝐼𝑄)) ↔ (⟨(𝑡𝐺), 𝑡⟩ ∈ (𝐼𝑃) ∧ ⟨((𝑁𝑡)‘𝐷), 𝑢⟩ ∈ (𝐼𝑄))))
91 coeq2 5484 . . . . . . . . . . . . . . 15 ( = ((𝑁𝑡)‘𝐷) → ((𝑡𝐺) ∘ ) = ((𝑡𝐺) ∘ ((𝑁𝑡)‘𝐷)))
9291eqeq2d 2809 . . . . . . . . . . . . . 14 ( = ((𝑁𝑡)‘𝐷) → (𝑓 = ((𝑡𝐺) ∘ ) ↔ 𝑓 = ((𝑡𝐺) ∘ ((𝑁𝑡)‘𝐷))))
9392anbi1d 624 . . . . . . . . . . . . 13 ( = ((𝑁𝑡)‘𝐷) → ((𝑓 = ((𝑡𝐺) ∘ ) ∧ 𝑠 = (𝑡𝐽𝑢)) ↔ (𝑓 = ((𝑡𝐺) ∘ ((𝑁𝑡)‘𝐷)) ∧ 𝑠 = (𝑡𝐽𝑢))))
9490, 93anbi12d 625 . . . . . . . . . . . 12 ( = ((𝑁𝑡)‘𝐷) → (((⟨(𝑡𝐺), 𝑡⟩ ∈ (𝐼𝑃) ∧ ⟨, 𝑢⟩ ∈ (𝐼𝑄)) ∧ (𝑓 = ((𝑡𝐺) ∘ ) ∧ 𝑠 = (𝑡𝐽𝑢))) ↔ ((⟨(𝑡𝐺), 𝑡⟩ ∈ (𝐼𝑃) ∧ ⟨((𝑁𝑡)‘𝐷), 𝑢⟩ ∈ (𝐼𝑄)) ∧ (𝑓 = ((𝑡𝐺) ∘ ((𝑁𝑡)‘𝐷)) ∧ 𝑠 = (𝑡𝐽𝑢)))))
95 opeq2 4594 . . . . . . . . . . . . . . 15 (𝑢 = (𝑁𝑡) → ⟨((𝑁𝑡)‘𝐷), 𝑢⟩ = ⟨((𝑁𝑡)‘𝐷), (𝑁𝑡)⟩)
9695eleq1d 2863 . . . . . . . . . . . . . 14 (𝑢 = (𝑁𝑡) → (⟨((𝑁𝑡)‘𝐷), 𝑢⟩ ∈ (𝐼𝑄) ↔ ⟨((𝑁𝑡)‘𝐷), (𝑁𝑡)⟩ ∈ (𝐼𝑄)))
9796anbi2d 623 . . . . . . . . . . . . 13 (𝑢 = (𝑁𝑡) → ((⟨(𝑡𝐺), 𝑡⟩ ∈ (𝐼𝑃) ∧ ⟨((𝑁𝑡)‘𝐷), 𝑢⟩ ∈ (𝐼𝑄)) ↔ (⟨(𝑡𝐺), 𝑡⟩ ∈ (𝐼𝑃) ∧ ⟨((𝑁𝑡)‘𝐷), (𝑁𝑡)⟩ ∈ (𝐼𝑄))))
98 oveq2 6886 . . . . . . . . . . . . . . 15 (𝑢 = (𝑁𝑡) → (𝑡𝐽𝑢) = (𝑡𝐽(𝑁𝑡)))
9998eqeq2d 2809 . . . . . . . . . . . . . 14 (𝑢 = (𝑁𝑡) → (𝑠 = (𝑡𝐽𝑢) ↔ 𝑠 = (𝑡𝐽(𝑁𝑡))))
10099anbi2d 623 . . . . . . . . . . . . 13 (𝑢 = (𝑁𝑡) → ((𝑓 = ((𝑡𝐺) ∘ ((𝑁𝑡)‘𝐷)) ∧ 𝑠 = (𝑡𝐽𝑢)) ↔ (𝑓 = ((𝑡𝐺) ∘ ((𝑁𝑡)‘𝐷)) ∧ 𝑠 = (𝑡𝐽(𝑁𝑡)))))
10197, 100anbi12d 625 . . . . . . . . . . . 12 (𝑢 = (𝑁𝑡) → (((⟨(𝑡𝐺), 𝑡⟩ ∈ (𝐼𝑃) ∧ ⟨((𝑁𝑡)‘𝐷), 𝑢⟩ ∈ (𝐼𝑄)) ∧ (𝑓 = ((𝑡𝐺) ∘ ((𝑁𝑡)‘𝐷)) ∧ 𝑠 = (𝑡𝐽𝑢))) ↔ ((⟨(𝑡𝐺), 𝑡⟩ ∈ (𝐼𝑃) ∧ ⟨((𝑁𝑡)‘𝐷), (𝑁𝑡)⟩ ∈ (𝐼𝑄)) ∧ (𝑓 = ((𝑡𝐺) ∘ ((𝑁𝑡)‘𝐷)) ∧ 𝑠 = (𝑡𝐽(𝑁𝑡))))))
10287, 94, 101syl3an9b 1559 . . . . . . . . . . 11 ((𝑔 = (𝑡𝐺) ∧ = ((𝑁𝑡)‘𝐷) ∧ 𝑢 = (𝑁𝑡)) → (((⟨𝑔, 𝑡⟩ ∈ (𝐼𝑃) ∧ ⟨, 𝑢⟩ ∈ (𝐼𝑄)) ∧ (𝑓 = (𝑔) ∧ 𝑠 = (𝑡𝐽𝑢))) ↔ ((⟨(𝑡𝐺), 𝑡⟩ ∈ (𝐼𝑃) ∧ ⟨((𝑁𝑡)‘𝐷), (𝑁𝑡)⟩ ∈ (𝐼𝑄)) ∧ (𝑓 = ((𝑡𝐺) ∘ ((𝑁𝑡)‘𝐷)) ∧ 𝑠 = (𝑡𝐽(𝑁𝑡))))))
103102spc3egv 3485 . . . . . . . . . 10 (((𝑡𝐺) ∈ V ∧ ((𝑁𝑡)‘𝐷) ∈ V ∧ (𝑁𝑡) ∈ V) → (((⟨(𝑡𝐺), 𝑡⟩ ∈ (𝐼𝑃) ∧ ⟨((𝑁𝑡)‘𝐷), (𝑁𝑡)⟩ ∈ (𝐼𝑄)) ∧ (𝑓 = ((𝑡𝐺) ∘ ((𝑁𝑡)‘𝐷)) ∧ 𝑠 = (𝑡𝐽(𝑁𝑡)))) → ∃𝑔𝑢((⟨𝑔, 𝑡⟩ ∈ (𝐼𝑃) ∧ ⟨, 𝑢⟩ ∈ (𝐼𝑄)) ∧ (𝑓 = (𝑔) ∧ 𝑠 = (𝑡𝐽𝑢)))))
10447, 57, 58, 103mp3an 1586 . . . . . . . . 9 (((⟨(𝑡𝐺), 𝑡⟩ ∈ (𝐼𝑃) ∧ ⟨((𝑁𝑡)‘𝐷), (𝑁𝑡)⟩ ∈ (𝐼𝑄)) ∧ (𝑓 = ((𝑡𝐺) ∘ ((𝑁𝑡)‘𝐷)) ∧ 𝑠 = (𝑡𝐽(𝑁𝑡)))) → ∃𝑔𝑢((⟨𝑔, 𝑡⟩ ∈ (𝐼𝑃) ∧ ⟨, 𝑢⟩ ∈ (𝐼𝑄)) ∧ (𝑓 = (𝑔) ∧ 𝑠 = (𝑡𝐽𝑢))))
10551, 61, 74, 80, 104syl22anc 868 . . . . . . . 8 (((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡𝐸 ∧ (𝑡‘(𝐺𝐷)) = 𝑓)) → ∃𝑔𝑢((⟨𝑔, 𝑡⟩ ∈ (𝐼𝑃) ∧ ⟨, 𝑢⟩ ∈ (𝐼𝑄)) ∧ (𝑓 = (𝑔) ∧ 𝑠 = (𝑡𝐽𝑢))))
106105ex 402 . . . . . . 7 ((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) → ((𝑡𝐸 ∧ (𝑡‘(𝐺𝐷)) = 𝑓) → ∃𝑔𝑢((⟨𝑔, 𝑡⟩ ∈ (𝐼𝑃) ∧ ⟨, 𝑢⟩ ∈ (𝐼𝑄)) ∧ (𝑓 = (𝑔) ∧ 𝑠 = (𝑡𝐽𝑢)))))
107106eximdv 2013 . . . . . 6 ((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) → (∃𝑡(𝑡𝐸 ∧ (𝑡‘(𝐺𝐷)) = 𝑓) → ∃𝑡𝑔𝑢((⟨𝑔, 𝑡⟩ ∈ (𝐼𝑃) ∧ ⟨, 𝑢⟩ ∈ (𝐼𝑄)) ∧ (𝑓 = (𝑔) ∧ 𝑠 = (𝑡𝐽𝑢)))))
108 excom 2205 . . . . . 6 (∃𝑡𝑔𝑢((⟨𝑔, 𝑡⟩ ∈ (𝐼𝑃) ∧ ⟨, 𝑢⟩ ∈ (𝐼𝑄)) ∧ (𝑓 = (𝑔) ∧ 𝑠 = (𝑡𝐽𝑢))) ↔ ∃𝑔𝑡𝑢((⟨𝑔, 𝑡⟩ ∈ (𝐼𝑃) ∧ ⟨, 𝑢⟩ ∈ (𝐼𝑄)) ∧ (𝑓 = (𝑔) ∧ 𝑠 = (𝑡𝐽𝑢))))
109107, 108syl6ib 243 . . . . 5 ((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) → (∃𝑡(𝑡𝐸 ∧ (𝑡‘(𝐺𝐷)) = 𝑓) → ∃𝑔𝑡𝑢((⟨𝑔, 𝑡⟩ ∈ (𝐼𝑃) ∧ ⟨, 𝑢⟩ ∈ (𝐼𝑄)) ∧ (𝑓 = (𝑔) ∧ 𝑠 = (𝑡𝐽𝑢)))))
11042, 109mpd 15 . . . 4 ((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) → ∃𝑔𝑡𝑢((⟨𝑔, 𝑡⟩ ∈ (𝐼𝑃) ∧ ⟨, 𝑢⟩ ∈ (𝐼𝑄)) ∧ (𝑓 = (𝑔) ∧ 𝑠 = (𝑡𝐽𝑢))))
111110ex 402 . . 3 (𝜑 → (((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 ) → ∃𝑔𝑡𝑢((⟨𝑔, 𝑡⟩ ∈ (𝐼𝑃) ∧ ⟨, 𝑢⟩ ∈ (𝐼𝑄)) ∧ (𝑓 = (𝑔) ∧ 𝑠 = (𝑡𝐽𝑢)))))
1121simpld 489 . . . . . . . . 9 (𝜑𝐾 ∈ HL)
113112hllatd 35385 . . . . . . . 8 (𝜑𝐾 ∈ Lat)
11412simpld 489 . . . . . . . . 9 (𝜑𝑃𝐴)
11517simpld 489 . . . . . . . . 9 (𝜑𝑄𝐴)
11628, 29, 8hlatjcl 35388 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → (𝑃 𝑄) ∈ 𝐵)
117112, 114, 115, 116syl3anc 1491 . . . . . . . 8 (𝜑 → (𝑃 𝑄) ∈ 𝐵)
1181simprd 490 . . . . . . . . 9 (𝜑𝑊𝐻)
11928, 2lhpbase 36019 . . . . . . . . 9 (𝑊𝐻𝑊𝐵)
120118, 119syl 17 . . . . . . . 8 (𝜑𝑊𝐵)
12128, 30latmcl 17367 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ 𝐵𝑊𝐵) → ((𝑃 𝑄) 𝑊) ∈ 𝐵)
122113, 117, 120, 121syl3anc 1491 . . . . . . 7 (𝜑 → ((𝑃 𝑄) 𝑊) ∈ 𝐵)
12333, 122syl5eqel 2882 . . . . . 6 (𝜑𝑉𝐵)
12428, 7, 30latmle2 17392 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ 𝐵𝑊𝐵) → ((𝑃 𝑄) 𝑊) 𝑊)
125113, 117, 120, 124syl3anc 1491 . . . . . . 7 (𝜑 → ((𝑃 𝑄) 𝑊) 𝑊)
12633, 125syl5eqbr 4878 . . . . . 6 (𝜑𝑉 𝑊)
127 eqid 2799 . . . . . . 7 ((DIsoB‘𝐾)‘𝑊) = ((DIsoB‘𝐾)‘𝑊)
12828, 7, 2, 3, 127dihvalb 37258 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑉𝐵𝑉 𝑊)) → (𝐼𝑉) = (((DIsoB‘𝐾)‘𝑊)‘𝑉))
1291, 123, 126, 128syl12anc 866 . . . . 5 (𝜑 → (𝐼𝑉) = (((DIsoB‘𝐾)‘𝑊)‘𝑉))
130129eleq2d 2864 . . . 4 (𝜑 → (⟨𝑓, 𝑠⟩ ∈ (𝐼𝑉) ↔ ⟨𝑓, 𝑠⟩ ∈ (((DIsoB‘𝐾)‘𝑊)‘𝑉)))
13128, 7, 2, 13, 34, 77, 127dibopelval3 37169 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑉𝐵𝑉 𝑊)) → (⟨𝑓, 𝑠⟩ ∈ (((DIsoB‘𝐾)‘𝑊)‘𝑉) ↔ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )))
1321, 123, 126, 131syl12anc 866 . . . 4 (𝜑 → (⟨𝑓, 𝑠⟩ ∈ (((DIsoB‘𝐾)‘𝑊)‘𝑉) ↔ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )))
133130, 132bitrd 271 . . 3 (𝜑 → (⟨𝑓, 𝑠⟩ ∈ (𝐼𝑉) ↔ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )))
134 eqid 2799 . . . 4 (LSubSp‘𝑈) = (LSubSp‘𝑈)
13528, 8atbase 35310 . . . . 5 (𝑃𝐴𝑃𝐵)
136114, 135syl 17 . . . 4 (𝜑𝑃𝐵)
13728, 8atbase 35310 . . . . 5 (𝑄𝐴𝑄𝐵)
138115, 137syl 17 . . . 4 (𝜑𝑄𝐵)
13928, 2, 13, 35, 76, 31, 134, 32, 3, 1, 136, 138dihopellsm 37276 . . 3 (𝜑 → (⟨𝑓, 𝑠⟩ ∈ ((𝐼𝑃) (𝐼𝑄)) ↔ ∃𝑔𝑡𝑢((⟨𝑔, 𝑡⟩ ∈ (𝐼𝑃) ∧ ⟨, 𝑢⟩ ∈ (𝐼𝑄)) ∧ (𝑓 = (𝑔) ∧ 𝑠 = (𝑡𝐽𝑢)))))
140111, 133, 1393imtr4d 286 . 2 (𝜑 → (⟨𝑓, 𝑠⟩ ∈ (𝐼𝑉) → ⟨𝑓, 𝑠⟩ ∈ ((𝐼𝑃) (𝐼𝑄))))
1415, 140relssdv 5416 1 (𝜑 → (𝐼𝑉) ⊆ ((𝐼𝑃) (𝐼𝑄)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wa 385   = wceq 1653  wex 1875  wcel 2157  wrex 3090  Vcvv 3385  wss 3769  cop 4374   class class class wbr 4843  cmpt 4922   I cid 5219  ccnv 5311  cres 5314  ccom 5316  Rel wrel 5317  cfv 6101  crio 6838  (class class class)co 6878  cmpt2 6880  Basecbs 16184  lecple 16274  occoc 16275  joincjn 17259  meetcmee 17260  Latclat 17360  LSSumclsm 18362  LSubSpclss 19250  Atomscatm 35284  HLchlt 35371  LHypclh 36005  LTrncltrn 36122  trLctrl 36179  TEndoctendo 36773  DVecHcdvh 37099  DIsoBcdib 37159  DIsoHcdih 37249
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183  ax-cnex 10280  ax-resscn 10281  ax-1cn 10282  ax-icn 10283  ax-addcl 10284  ax-addrcl 10285  ax-mulcl 10286  ax-mulrcl 10287  ax-mulcom 10288  ax-addass 10289  ax-mulass 10290  ax-distr 10291  ax-i2m1 10292  ax-1ne0 10293  ax-1rid 10294  ax-rnegex 10295  ax-rrecex 10296  ax-cnre 10297  ax-pre-lttri 10298  ax-pre-lttrn 10299  ax-pre-ltadd 10300  ax-pre-mulgt0 10301  ax-riotaBAD 34974
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-fal 1667  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-nel 3075  df-ral 3094  df-rex 3095  df-reu 3096  df-rmo 3097  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-tp 4373  df-op 4375  df-uni 4629  df-int 4668  df-iun 4712  df-iin 4713  df-br 4844  df-opab 4906  df-mpt 4923  df-tr 4946  df-id 5220  df-eprel 5225  df-po 5233  df-so 5234  df-fr 5271  df-we 5273  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-pred 5898  df-ord 5944  df-on 5945  df-lim 5946  df-suc 5947  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-riota 6839  df-ov 6881  df-oprab 6882  df-mpt2 6883  df-om 7300  df-1st 7401  df-2nd 7402  df-tpos 7590  df-undef 7637  df-wrecs 7645  df-recs 7707  df-rdg 7745  df-1o 7799  df-oadd 7803  df-er 7982  df-map 8097  df-en 8196  df-dom 8197  df-sdom 8198  df-fin 8199  df-pnf 10365  df-mnf 10366  df-xr 10367  df-ltxr 10368  df-le 10369  df-sub 10558  df-neg 10559  df-nn 11313  df-2 11376  df-3 11377  df-4 11378  df-5 11379  df-6 11380  df-n0 11581  df-z 11667  df-uz 11931  df-fz 12581  df-struct 16186  df-ndx 16187  df-slot 16188  df-base 16190  df-sets 16191  df-ress 16192  df-plusg 16280  df-mulr 16281  df-sca 16283  df-vsca 16284  df-0g 16417  df-proset 17243  df-poset 17261  df-plt 17273  df-lub 17289  df-glb 17290  df-join 17291  df-meet 17292  df-p0 17354  df-p1 17355  df-lat 17361  df-clat 17423  df-mgm 17557  df-sgrp 17599  df-mnd 17610  df-submnd 17651  df-grp 17741  df-minusg 17742  df-sbg 17743  df-subg 17904  df-cntz 18062  df-lsm 18364  df-cmn 18510  df-abl 18511  df-mgp 18806  df-ur 18818  df-ring 18865  df-oppr 18939  df-dvdsr 18957  df-unit 18958  df-invr 18988  df-dvr 18999  df-drng 19067  df-lmod 19183  df-lss 19251  df-lsp 19293  df-lvec 19424  df-oposet 35197  df-ol 35199  df-oml 35200  df-covers 35287  df-ats 35288  df-atl 35319  df-cvlat 35343  df-hlat 35372  df-llines 35519  df-lplanes 35520  df-lvols 35521  df-lines 35522  df-psubsp 35524  df-pmap 35525  df-padd 35817  df-lhyp 36009  df-laut 36010  df-ldil 36125  df-ltrn 36126  df-trl 36180  df-tendo 36776  df-edring 36778  df-disoa 37050  df-dvech 37100  df-dib 37160  df-dic 37194  df-dih 37250
This theorem is referenced by:  dihjatcc  37443
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