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Theorem dihjatcclem4 41423
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 41281 . . 3 ((𝐾 ∈ HL ∧ 𝑊𝐻) → Rel (𝐼𝑉))
51, 4syl 17 . 2 (𝜑 → Rel (𝐼𝑉))
61adantr 480 . . . . . . 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 40021 . . . . . . . . . . 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 40581 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐶𝐴 ∧ ¬ 𝐶 𝑊) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → 𝐺𝑇)
161, 11, 12, 15syl3anc 1373 . . . . . . . . 9 (𝜑𝐺𝑇)
17 dihjatcclem.q . . . . . . . . . . 11 (𝜑 → (𝑄𝐴 ∧ ¬ 𝑄 𝑊))
18 dihjatcc.dd . . . . . . . . . . . 12 𝐷 = (𝑑𝑇 (𝑑𝐶) = 𝑄)
197, 8, 2, 13, 18ltrniotacl 40581 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐶𝐴 ∧ ¬ 𝐶 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → 𝐷𝑇)
201, 11, 17, 19syl3anc 1373 . . . . . . . . . 10 (𝜑𝐷𝑇)
212, 13ltrncnv 40148 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐷𝑇) → 𝐷𝑇)
221, 20, 21syl2anc 584 . . . . . . . . 9 (𝜑𝐷𝑇)
232, 13ltrnco 40721 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐺𝑇𝐷𝑇) → (𝐺𝐷) ∈ 𝑇)
241, 16, 22, 23syl3anc 1373 . . . . . . . 8 (𝜑 → (𝐺𝐷) ∈ 𝑇)
2524adantr 480 . . . . . . 7 ((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) → (𝐺𝐷) ∈ 𝑇)
26 simprll 779 . . . . . . 7 ((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) → 𝑓𝑇)
27 simprlr 780 . . . . . . . 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 41422 . . . . . . . . 9 (𝜑 → (𝑅‘(𝐺𝐷)) = 𝑉)
3736adantr 480 . . . . . . . 8 ((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) → (𝑅‘(𝐺𝐷)) = 𝑉)
3827, 37breqtrrd 5171 . . . . . . 7 ((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) → (𝑅𝑓) (𝑅‘(𝐺𝐷)))
397, 2, 13, 34, 35tendoex 40977 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝐺𝐷) ∈ 𝑇𝑓𝑇) ∧ (𝑅𝑓) (𝑅‘(𝐺𝐷))) → ∃𝑡𝐸 (𝑡‘(𝐺𝐷)) = 𝑓)
406, 25, 26, 38, 39syl121anc 1377 . . . . . 6 ((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) → ∃𝑡𝐸 (𝑡‘(𝐺𝐷)) = 𝑓)
41 df-rex 3071 . . . . . 6 (∃𝑡𝐸 (𝑡‘(𝐺𝐷)) = 𝑓 ↔ ∃𝑡(𝑡𝐸 ∧ (𝑡‘(𝐺𝐷)) = 𝑓))
4240, 41sylib 218 . . . . 5 ((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) → ∃𝑡(𝑡𝐸 ∧ (𝑡‘(𝐺𝐷)) = 𝑓))
43 eqidd 2738 . . . . . . . . . 10 (((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡𝐸 ∧ (𝑡‘(𝐺𝐷)) = 𝑓)) → (𝑡𝐺) = (𝑡𝐺))
44 simprl 771 . . . . . . . . . 10 (((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡𝐸 ∧ (𝑡‘(𝐺𝐷)) = 𝑓)) → 𝑡𝐸)
451ad2antrr 726 . . . . . . . . . . 11 (((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡𝐸 ∧ (𝑡‘(𝐺𝐷)) = 𝑓)) → (𝐾 ∈ HL ∧ 𝑊𝐻))
4612ad2antrr 726 . . . . . . . . . . 11 (((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡𝐸 ∧ (𝑡‘(𝐺𝐷)) = 𝑓)) → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
47 fvex 6919 . . . . . . . . . . . 12 (𝑡𝐺) ∈ V
48 vex 3484 . . . . . . . . . . . 12 𝑡 ∈ V
497, 8, 2, 9, 13, 35, 3, 14, 47, 48dihopelvalcqat 41248 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (⟨(𝑡𝐺), 𝑡⟩ ∈ (𝐼𝑃) ↔ ((𝑡𝐺) = (𝑡𝐺) ∧ 𝑡𝐸)))
5045, 46, 49syl2anc 584 . . . . . . . . . 10 (((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡𝐸 ∧ (𝑡‘(𝐺𝐷)) = 𝑓)) → (⟨(𝑡𝐺), 𝑡⟩ ∈ (𝐼𝑃) ↔ ((𝑡𝐺) = (𝑡𝐺) ∧ 𝑡𝐸)))
5143, 44, 50mpbir2and 713 . . . . . . . . 9 (((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡𝐸 ∧ (𝑡‘(𝐺𝐷)) = 𝑓)) → ⟨(𝑡𝐺), 𝑡⟩ ∈ (𝐼𝑃))
52 eqidd 2738 . . . . . . . . . 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 40798 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑡𝐸) → (𝑁𝑡) ∈ 𝐸)
5545, 44, 54syl2anc 584 . . . . . . . . . 10 (((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡𝐸 ∧ (𝑡‘(𝐺𝐷)) = 𝑓)) → (𝑁𝑡) ∈ 𝐸)
5617ad2antrr 726 . . . . . . . . . . 11 (((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡𝐸 ∧ (𝑡‘(𝐺𝐷)) = 𝑓)) → (𝑄𝐴 ∧ ¬ 𝑄 𝑊))
57 fvex 6919 . . . . . . . . . . . 12 ((𝑁𝑡)‘𝐷) ∈ V
58 fvex 6919 . . . . . . . . . . . 12 (𝑁𝑡) ∈ V
597, 8, 2, 9, 13, 35, 3, 18, 57, 58dihopelvalcqat 41248 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (⟨((𝑁𝑡)‘𝐷), (𝑁𝑡)⟩ ∈ (𝐼𝑄) ↔ (((𝑁𝑡)‘𝐷) = ((𝑁𝑡)‘𝐷) ∧ (𝑁𝑡) ∈ 𝐸)))
6045, 56, 59syl2anc 584 . . . . . . . . . 10 (((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡𝐸 ∧ (𝑡‘(𝐺𝐷)) = 𝑓)) → (⟨((𝑁𝑡)‘𝐷), (𝑁𝑡)⟩ ∈ (𝐼𝑄) ↔ (((𝑁𝑡)‘𝐷) = ((𝑁𝑡)‘𝐷) ∧ (𝑁𝑡) ∈ 𝐸)))
6152, 55, 60mpbir2and 713 . . . . . . . . 9 (((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡𝐸 ∧ (𝑡‘(𝐺𝐷)) = 𝑓)) → ⟨((𝑁𝑡)‘𝐷), (𝑁𝑡)⟩ ∈ (𝐼𝑄))
6216ad2antrr 726 . . . . . . . . . . 11 (((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡𝐸 ∧ (𝑡‘(𝐺𝐷)) = 𝑓)) → 𝐺𝑇)
6322ad2antrr 726 . . . . . . . . . . 11 (((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡𝐸 ∧ (𝑡‘(𝐺𝐷)) = 𝑓)) → 𝐷𝑇)
642, 13, 35tendospdi1 41022 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑡𝐸𝐺𝑇𝐷𝑇)) → (𝑡‘(𝐺𝐷)) = ((𝑡𝐺) ∘ (𝑡𝐷)))
6545, 44, 62, 63, 64syl13anc 1374 . . . . . . . . . 10 (((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡𝐸 ∧ (𝑡‘(𝐺𝐷)) = 𝑓)) → (𝑡‘(𝐺𝐷)) = ((𝑡𝐺) ∘ (𝑡𝐷)))
66 simprr 773 . . . . . . . . . 10 (((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡𝐸 ∧ (𝑡‘(𝐺𝐷)) = 𝑓)) → (𝑡‘(𝐺𝐷)) = 𝑓)
6720ad2antrr 726 . . . . . . . . . . . . 13 (((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡𝐸 ∧ (𝑡‘(𝐺𝐷)) = 𝑓)) → 𝐷𝑇)
6853, 13tendoi2 40797 . . . . . . . . . . . . 13 ((𝑡𝐸𝐷𝑇) → ((𝑁𝑡)‘𝐷) = (𝑡𝐷))
6944, 67, 68syl2anc 584 . . . . . . . . . . . 12 (((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡𝐸 ∧ (𝑡‘(𝐺𝐷)) = 𝑓)) → ((𝑁𝑡)‘𝐷) = (𝑡𝐷))
702, 13, 35tendocnv 41023 . . . . . . . . . . . . 13 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑡𝐸𝐷𝑇) → (𝑡𝐷) = (𝑡𝐷))
7145, 44, 67, 70syl3anc 1373 . . . . . . . . . . . 12 (((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡𝐸 ∧ (𝑡‘(𝐺𝐷)) = 𝑓)) → (𝑡𝐷) = (𝑡𝐷))
7269, 71eqtr2d 2778 . . . . . . . . . . 11 (((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡𝐸 ∧ (𝑡‘(𝐺𝐷)) = 𝑓)) → (𝑡𝐷) = ((𝑁𝑡)‘𝐷))
7372coeq2d 5873 . . . . . . . . . 10 (((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡𝐸 ∧ (𝑡‘(𝐺𝐷)) = 𝑓)) → ((𝑡𝐺) ∘ (𝑡𝐷)) = ((𝑡𝐺) ∘ ((𝑁𝑡)‘𝐷)))
7465, 66, 733eqtr3d 2785 . . . . . . . . 9 (((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡𝐸 ∧ (𝑡‘(𝐺𝐷)) = 𝑓)) → 𝑓 = ((𝑡𝐺) ∘ ((𝑁𝑡)‘𝐷)))
75 simplrr 778 . . . . . . . . . 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 40800 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑡𝐸) → (𝑡𝐽(𝑁𝑡)) = 0 )
7945, 44, 78syl2anc 584 . . . . . . . . . 10 (((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡𝐸 ∧ (𝑡‘(𝐺𝐷)) = 𝑓)) → (𝑡𝐽(𝑁𝑡)) = 0 )
8075, 79eqtr4d 2780 . . . . . . . . 9 (((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡𝐸 ∧ (𝑡‘(𝐺𝐷)) = 𝑓)) → 𝑠 = (𝑡𝐽(𝑁𝑡)))
81 opeq1 4873 . . . . . . . . . . . . . . 15 (𝑔 = (𝑡𝐺) → ⟨𝑔, 𝑡⟩ = ⟨(𝑡𝐺), 𝑡⟩)
8281eleq1d 2826 . . . . . . . . . . . . . 14 (𝑔 = (𝑡𝐺) → (⟨𝑔, 𝑡⟩ ∈ (𝐼𝑃) ↔ ⟨(𝑡𝐺), 𝑡⟩ ∈ (𝐼𝑃)))
8382anbi1d 631 . . . . . . . . . . . . 13 (𝑔 = (𝑡𝐺) → ((⟨𝑔, 𝑡⟩ ∈ (𝐼𝑃) ∧ ⟨, 𝑢⟩ ∈ (𝐼𝑄)) ↔ (⟨(𝑡𝐺), 𝑡⟩ ∈ (𝐼𝑃) ∧ ⟨, 𝑢⟩ ∈ (𝐼𝑄))))
84 coeq1 5868 . . . . . . . . . . . . . . 15 (𝑔 = (𝑡𝐺) → (𝑔) = ((𝑡𝐺) ∘ ))
8584eqeq2d 2748 . . . . . . . . . . . . . 14 (𝑔 = (𝑡𝐺) → (𝑓 = (𝑔) ↔ 𝑓 = ((𝑡𝐺) ∘ )))
8685anbi1d 631 . . . . . . . . . . . . 13 (𝑔 = (𝑡𝐺) → ((𝑓 = (𝑔) ∧ 𝑠 = (𝑡𝐽𝑢)) ↔ (𝑓 = ((𝑡𝐺) ∘ ) ∧ 𝑠 = (𝑡𝐽𝑢))))
8783, 86anbi12d 632 . . . . . . . . . . . 12 (𝑔 = (𝑡𝐺) → (((⟨𝑔, 𝑡⟩ ∈ (𝐼𝑃) ∧ ⟨, 𝑢⟩ ∈ (𝐼𝑄)) ∧ (𝑓 = (𝑔) ∧ 𝑠 = (𝑡𝐽𝑢))) ↔ ((⟨(𝑡𝐺), 𝑡⟩ ∈ (𝐼𝑃) ∧ ⟨, 𝑢⟩ ∈ (𝐼𝑄)) ∧ (𝑓 = ((𝑡𝐺) ∘ ) ∧ 𝑠 = (𝑡𝐽𝑢)))))
88 opeq1 4873 . . . . . . . . . . . . . . 15 ( = ((𝑁𝑡)‘𝐷) → ⟨, 𝑢⟩ = ⟨((𝑁𝑡)‘𝐷), 𝑢⟩)
8988eleq1d 2826 . . . . . . . . . . . . . 14 ( = ((𝑁𝑡)‘𝐷) → (⟨, 𝑢⟩ ∈ (𝐼𝑄) ↔ ⟨((𝑁𝑡)‘𝐷), 𝑢⟩ ∈ (𝐼𝑄)))
9089anbi2d 630 . . . . . . . . . . . . 13 ( = ((𝑁𝑡)‘𝐷) → ((⟨(𝑡𝐺), 𝑡⟩ ∈ (𝐼𝑃) ∧ ⟨, 𝑢⟩ ∈ (𝐼𝑄)) ↔ (⟨(𝑡𝐺), 𝑡⟩ ∈ (𝐼𝑃) ∧ ⟨((𝑁𝑡)‘𝐷), 𝑢⟩ ∈ (𝐼𝑄))))
91 coeq2 5869 . . . . . . . . . . . . . . 15 ( = ((𝑁𝑡)‘𝐷) → ((𝑡𝐺) ∘ ) = ((𝑡𝐺) ∘ ((𝑁𝑡)‘𝐷)))
9291eqeq2d 2748 . . . . . . . . . . . . . 14 ( = ((𝑁𝑡)‘𝐷) → (𝑓 = ((𝑡𝐺) ∘ ) ↔ 𝑓 = ((𝑡𝐺) ∘ ((𝑁𝑡)‘𝐷))))
9392anbi1d 631 . . . . . . . . . . . . 13 ( = ((𝑁𝑡)‘𝐷) → ((𝑓 = ((𝑡𝐺) ∘ ) ∧ 𝑠 = (𝑡𝐽𝑢)) ↔ (𝑓 = ((𝑡𝐺) ∘ ((𝑁𝑡)‘𝐷)) ∧ 𝑠 = (𝑡𝐽𝑢))))
9490, 93anbi12d 632 . . . . . . . . . . . 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 4874 . . . . . . . . . . . . . . 15 (𝑢 = (𝑁𝑡) → ⟨((𝑁𝑡)‘𝐷), 𝑢⟩ = ⟨((𝑁𝑡)‘𝐷), (𝑁𝑡)⟩)
9695eleq1d 2826 . . . . . . . . . . . . . 14 (𝑢 = (𝑁𝑡) → (⟨((𝑁𝑡)‘𝐷), 𝑢⟩ ∈ (𝐼𝑄) ↔ ⟨((𝑁𝑡)‘𝐷), (𝑁𝑡)⟩ ∈ (𝐼𝑄)))
9796anbi2d 630 . . . . . . . . . . . . 13 (𝑢 = (𝑁𝑡) → ((⟨(𝑡𝐺), 𝑡⟩ ∈ (𝐼𝑃) ∧ ⟨((𝑁𝑡)‘𝐷), 𝑢⟩ ∈ (𝐼𝑄)) ↔ (⟨(𝑡𝐺), 𝑡⟩ ∈ (𝐼𝑃) ∧ ⟨((𝑁𝑡)‘𝐷), (𝑁𝑡)⟩ ∈ (𝐼𝑄))))
98 oveq2 7439 . . . . . . . . . . . . . . 15 (𝑢 = (𝑁𝑡) → (𝑡𝐽𝑢) = (𝑡𝐽(𝑁𝑡)))
9998eqeq2d 2748 . . . . . . . . . . . . . 14 (𝑢 = (𝑁𝑡) → (𝑠 = (𝑡𝐽𝑢) ↔ 𝑠 = (𝑡𝐽(𝑁𝑡))))
10099anbi2d 630 . . . . . . . . . . . . 13 (𝑢 = (𝑁𝑡) → ((𝑓 = ((𝑡𝐺) ∘ ((𝑁𝑡)‘𝐷)) ∧ 𝑠 = (𝑡𝐽𝑢)) ↔ (𝑓 = ((𝑡𝐺) ∘ ((𝑁𝑡)‘𝐷)) ∧ 𝑠 = (𝑡𝐽(𝑁𝑡)))))
10197, 100anbi12d 632 . . . . . . . . . . . 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 1436 . . . . . . . . . . 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 3603 . . . . . . . . . 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 1463 . . . . . . . . 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 839 . . . . . . . 8 (((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡𝐸 ∧ (𝑡‘(𝐺𝐷)) = 𝑓)) → ∃𝑔𝑢((⟨𝑔, 𝑡⟩ ∈ (𝐼𝑃) ∧ ⟨, 𝑢⟩ ∈ (𝐼𝑄)) ∧ (𝑓 = (𝑔) ∧ 𝑠 = (𝑡𝐽𝑢))))
106105ex 412 . . . . . . 7 ((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) → ((𝑡𝐸 ∧ (𝑡‘(𝐺𝐷)) = 𝑓) → ∃𝑔𝑢((⟨𝑔, 𝑡⟩ ∈ (𝐼𝑃) ∧ ⟨, 𝑢⟩ ∈ (𝐼𝑄)) ∧ (𝑓 = (𝑔) ∧ 𝑠 = (𝑡𝐽𝑢)))))
107106eximdv 1917 . . . . . 6 ((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) → (∃𝑡(𝑡𝐸 ∧ (𝑡‘(𝐺𝐷)) = 𝑓) → ∃𝑡𝑔𝑢((⟨𝑔, 𝑡⟩ ∈ (𝐼𝑃) ∧ ⟨, 𝑢⟩ ∈ (𝐼𝑄)) ∧ (𝑓 = (𝑔) ∧ 𝑠 = (𝑡𝐽𝑢)))))
108 excom 2162 . . . . . 6 (∃𝑡𝑔𝑢((⟨𝑔, 𝑡⟩ ∈ (𝐼𝑃) ∧ ⟨, 𝑢⟩ ∈ (𝐼𝑄)) ∧ (𝑓 = (𝑔) ∧ 𝑠 = (𝑡𝐽𝑢))) ↔ ∃𝑔𝑡𝑢((⟨𝑔, 𝑡⟩ ∈ (𝐼𝑃) ∧ ⟨, 𝑢⟩ ∈ (𝐼𝑄)) ∧ (𝑓 = (𝑔) ∧ 𝑠 = (𝑡𝐽𝑢))))
109107, 108imbitrdi 251 . . . . 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 412 . . 3 (𝜑 → (((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 ) → ∃𝑔𝑡𝑢((⟨𝑔, 𝑡⟩ ∈ (𝐼𝑃) ∧ ⟨, 𝑢⟩ ∈ (𝐼𝑄)) ∧ (𝑓 = (𝑔) ∧ 𝑠 = (𝑡𝐽𝑢)))))
1121simpld 494 . . . . . . . . 9 (𝜑𝐾 ∈ HL)
113112hllatd 39365 . . . . . . . 8 (𝜑𝐾 ∈ Lat)
11412simpld 494 . . . . . . . . 9 (𝜑𝑃𝐴)
11517simpld 494 . . . . . . . . 9 (𝜑𝑄𝐴)
11628, 29, 8hlatjcl 39368 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → (𝑃 𝑄) ∈ 𝐵)
117112, 114, 115, 116syl3anc 1373 . . . . . . . 8 (𝜑 → (𝑃 𝑄) ∈ 𝐵)
1181simprd 495 . . . . . . . . 9 (𝜑𝑊𝐻)
11928, 2lhpbase 40000 . . . . . . . . 9 (𝑊𝐻𝑊𝐵)
120118, 119syl 17 . . . . . . . 8 (𝜑𝑊𝐵)
12128, 30latmcl 18485 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ 𝐵𝑊𝐵) → ((𝑃 𝑄) 𝑊) ∈ 𝐵)
122113, 117, 120, 121syl3anc 1373 . . . . . . 7 (𝜑 → ((𝑃 𝑄) 𝑊) ∈ 𝐵)
12333, 122eqeltrid 2845 . . . . . 6 (𝜑𝑉𝐵)
12428, 7, 30latmle2 18510 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ 𝐵𝑊𝐵) → ((𝑃 𝑄) 𝑊) 𝑊)
125113, 117, 120, 124syl3anc 1373 . . . . . . 7 (𝜑 → ((𝑃 𝑄) 𝑊) 𝑊)
12633, 125eqbrtrid 5178 . . . . . 6 (𝜑𝑉 𝑊)
127 eqid 2737 . . . . . . 7 ((DIsoB‘𝐾)‘𝑊) = ((DIsoB‘𝐾)‘𝑊)
12828, 7, 2, 3, 127dihvalb 41239 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑉𝐵𝑉 𝑊)) → (𝐼𝑉) = (((DIsoB‘𝐾)‘𝑊)‘𝑉))
1291, 123, 126, 128syl12anc 837 . . . . 5 (𝜑 → (𝐼𝑉) = (((DIsoB‘𝐾)‘𝑊)‘𝑉))
130129eleq2d 2827 . . . 4 (𝜑 → (⟨𝑓, 𝑠⟩ ∈ (𝐼𝑉) ↔ ⟨𝑓, 𝑠⟩ ∈ (((DIsoB‘𝐾)‘𝑊)‘𝑉)))
13128, 7, 2, 13, 34, 77, 127dibopelval3 41150 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑉𝐵𝑉 𝑊)) → (⟨𝑓, 𝑠⟩ ∈ (((DIsoB‘𝐾)‘𝑊)‘𝑉) ↔ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )))
1321, 123, 126, 131syl12anc 837 . . . 4 (𝜑 → (⟨𝑓, 𝑠⟩ ∈ (((DIsoB‘𝐾)‘𝑊)‘𝑉) ↔ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )))
133130, 132bitrd 279 . . 3 (𝜑 → (⟨𝑓, 𝑠⟩ ∈ (𝐼𝑉) ↔ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )))
134 eqid 2737 . . . 4 (LSubSp‘𝑈) = (LSubSp‘𝑈)
13528, 8atbase 39290 . . . . 5 (𝑃𝐴𝑃𝐵)
136114, 135syl 17 . . . 4 (𝜑𝑃𝐵)
13728, 8atbase 39290 . . . . 5 (𝑄𝐴𝑄𝐵)
138115, 137syl 17 . . . 4 (𝜑𝑄𝐵)
13928, 2, 13, 35, 76, 31, 134, 32, 3, 1, 136, 138dihopellsm 41257 . . 3 (𝜑 → (⟨𝑓, 𝑠⟩ ∈ ((𝐼𝑃) (𝐼𝑄)) ↔ ∃𝑔𝑡𝑢((⟨𝑔, 𝑡⟩ ∈ (𝐼𝑃) ∧ ⟨, 𝑢⟩ ∈ (𝐼𝑄)) ∧ (𝑓 = (𝑔) ∧ 𝑠 = (𝑡𝐽𝑢)))))
140111, 133, 1393imtr4d 294 . 2 (𝜑 → (⟨𝑓, 𝑠⟩ ∈ (𝐼𝑉) → ⟨𝑓, 𝑠⟩ ∈ ((𝐼𝑃) (𝐼𝑄))))
1415, 140relssdv 5798 1 (𝜑 → (𝐼𝑉) ⊆ ((𝐼𝑃) (𝐼𝑄)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395   = wceq 1540  wex 1779  wcel 2108  wrex 3070  Vcvv 3480  wss 3951  cop 4632   class class class wbr 5143  cmpt 5225   I cid 5577  ccnv 5684  cres 5687  ccom 5689  Rel wrel 5690  cfv 6561  crio 7387  (class class class)co 7431  cmpo 7433  Basecbs 17247  lecple 17304  occoc 17305  joincjn 18357  meetcmee 18358  Latclat 18476  LSSumclsm 19652  LSubSpclss 20929  Atomscatm 39264  HLchlt 39351  LHypclh 39986  LTrncltrn 40103  trLctrl 40160  TEndoctendo 40754  DVecHcdvh 41080  DIsoBcdib 41140  DIsoHcdih 41230
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-riotaBAD 38954
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-tpos 8251  df-undef 8298  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-er 8745  df-map 8868  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-5 12332  df-6 12333  df-n0 12527  df-z 12614  df-uz 12879  df-fz 13548  df-struct 17184  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  df-plusg 17310  df-mulr 17311  df-sca 17313  df-vsca 17314  df-0g 17486  df-proset 18340  df-poset 18359  df-plt 18375  df-lub 18391  df-glb 18392  df-join 18393  df-meet 18394  df-p0 18470  df-p1 18471  df-lat 18477  df-clat 18544  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-submnd 18797  df-grp 18954  df-minusg 18955  df-sbg 18956  df-subg 19141  df-cntz 19335  df-lsm 19654  df-cmn 19800  df-abl 19801  df-mgp 20138  df-rng 20150  df-ur 20179  df-ring 20232  df-oppr 20334  df-dvdsr 20357  df-unit 20358  df-invr 20388  df-dvr 20401  df-drng 20731  df-lmod 20860  df-lss 20930  df-lsp 20970  df-lvec 21102  df-oposet 39177  df-ol 39179  df-oml 39180  df-covers 39267  df-ats 39268  df-atl 39299  df-cvlat 39323  df-hlat 39352  df-llines 39500  df-lplanes 39501  df-lvols 39502  df-lines 39503  df-psubsp 39505  df-pmap 39506  df-padd 39798  df-lhyp 39990  df-laut 39991  df-ldil 40106  df-ltrn 40107  df-trl 40161  df-tendo 40757  df-edring 40759  df-disoa 41031  df-dvech 41081  df-dib 41141  df-dic 41175  df-dih 41231
This theorem is referenced by:  dihjatcc  41424
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