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Theorem dihjatcclem4 41122
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 40980 . . 3 ((𝐾 ∈ HL ∧ 𝑊𝐻) → Rel (𝐼𝑉))
51, 4syl 17 . 2 (𝜑 → Rel (𝐼𝑉))
61adantr 479 . . . . . . 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 39720 . . . . . . . . . . 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 40280 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐶𝐴 ∧ ¬ 𝐶 𝑊) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → 𝐺𝑇)
161, 11, 12, 15syl3anc 1368 . . . . . . . . 9 (𝜑𝐺𝑇)
17 dihjatcclem.q . . . . . . . . . . 11 (𝜑 → (𝑄𝐴 ∧ ¬ 𝑄 𝑊))
18 dihjatcc.dd . . . . . . . . . . . 12 𝐷 = (𝑑𝑇 (𝑑𝐶) = 𝑄)
197, 8, 2, 13, 18ltrniotacl 40280 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐶𝐴 ∧ ¬ 𝐶 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → 𝐷𝑇)
201, 11, 17, 19syl3anc 1368 . . . . . . . . . 10 (𝜑𝐷𝑇)
212, 13ltrncnv 39847 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐷𝑇) → 𝐷𝑇)
221, 20, 21syl2anc 582 . . . . . . . . 9 (𝜑𝐷𝑇)
232, 13ltrnco 40420 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐺𝑇𝐷𝑇) → (𝐺𝐷) ∈ 𝑇)
241, 16, 22, 23syl3anc 1368 . . . . . . . 8 (𝜑 → (𝐺𝐷) ∈ 𝑇)
2524adantr 479 . . . . . . 7 ((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) → (𝐺𝐷) ∈ 𝑇)
26 simprll 777 . . . . . . 7 ((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) → 𝑓𝑇)
27 simprlr 778 . . . . . . . 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 41121 . . . . . . . . 9 (𝜑 → (𝑅‘(𝐺𝐷)) = 𝑉)
3736adantr 479 . . . . . . . 8 ((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) → (𝑅‘(𝐺𝐷)) = 𝑉)
3827, 37breqtrrd 5183 . . . . . . 7 ((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) → (𝑅𝑓) (𝑅‘(𝐺𝐷)))
397, 2, 13, 34, 35tendoex 40676 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝐺𝐷) ∈ 𝑇𝑓𝑇) ∧ (𝑅𝑓) (𝑅‘(𝐺𝐷))) → ∃𝑡𝐸 (𝑡‘(𝐺𝐷)) = 𝑓)
406, 25, 26, 38, 39syl121anc 1372 . . . . . 6 ((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) → ∃𝑡𝐸 (𝑡‘(𝐺𝐷)) = 𝑓)
41 df-rex 3061 . . . . . 6 (∃𝑡𝐸 (𝑡‘(𝐺𝐷)) = 𝑓 ↔ ∃𝑡(𝑡𝐸 ∧ (𝑡‘(𝐺𝐷)) = 𝑓))
4240, 41sylib 217 . . . . 5 ((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) → ∃𝑡(𝑡𝐸 ∧ (𝑡‘(𝐺𝐷)) = 𝑓))
43 eqidd 2727 . . . . . . . . . 10 (((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡𝐸 ∧ (𝑡‘(𝐺𝐷)) = 𝑓)) → (𝑡𝐺) = (𝑡𝐺))
44 simprl 769 . . . . . . . . . 10 (((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡𝐸 ∧ (𝑡‘(𝐺𝐷)) = 𝑓)) → 𝑡𝐸)
451ad2antrr 724 . . . . . . . . . . 11 (((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡𝐸 ∧ (𝑡‘(𝐺𝐷)) = 𝑓)) → (𝐾 ∈ HL ∧ 𝑊𝐻))
4612ad2antrr 724 . . . . . . . . . . 11 (((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡𝐸 ∧ (𝑡‘(𝐺𝐷)) = 𝑓)) → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
47 fvex 6916 . . . . . . . . . . . 12 (𝑡𝐺) ∈ V
48 vex 3466 . . . . . . . . . . . 12 𝑡 ∈ V
497, 8, 2, 9, 13, 35, 3, 14, 47, 48dihopelvalcqat 40947 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (⟨(𝑡𝐺), 𝑡⟩ ∈ (𝐼𝑃) ↔ ((𝑡𝐺) = (𝑡𝐺) ∧ 𝑡𝐸)))
5045, 46, 49syl2anc 582 . . . . . . . . . 10 (((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡𝐸 ∧ (𝑡‘(𝐺𝐷)) = 𝑓)) → (⟨(𝑡𝐺), 𝑡⟩ ∈ (𝐼𝑃) ↔ ((𝑡𝐺) = (𝑡𝐺) ∧ 𝑡𝐸)))
5143, 44, 50mpbir2and 711 . . . . . . . . 9 (((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡𝐸 ∧ (𝑡‘(𝐺𝐷)) = 𝑓)) → ⟨(𝑡𝐺), 𝑡⟩ ∈ (𝐼𝑃))
52 eqidd 2727 . . . . . . . . . 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 40497 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑡𝐸) → (𝑁𝑡) ∈ 𝐸)
5545, 44, 54syl2anc 582 . . . . . . . . . 10 (((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡𝐸 ∧ (𝑡‘(𝐺𝐷)) = 𝑓)) → (𝑁𝑡) ∈ 𝐸)
5617ad2antrr 724 . . . . . . . . . . 11 (((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡𝐸 ∧ (𝑡‘(𝐺𝐷)) = 𝑓)) → (𝑄𝐴 ∧ ¬ 𝑄 𝑊))
57 fvex 6916 . . . . . . . . . . . 12 ((𝑁𝑡)‘𝐷) ∈ V
58 fvex 6916 . . . . . . . . . . . 12 (𝑁𝑡) ∈ V
597, 8, 2, 9, 13, 35, 3, 18, 57, 58dihopelvalcqat 40947 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (⟨((𝑁𝑡)‘𝐷), (𝑁𝑡)⟩ ∈ (𝐼𝑄) ↔ (((𝑁𝑡)‘𝐷) = ((𝑁𝑡)‘𝐷) ∧ (𝑁𝑡) ∈ 𝐸)))
6045, 56, 59syl2anc 582 . . . . . . . . . 10 (((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡𝐸 ∧ (𝑡‘(𝐺𝐷)) = 𝑓)) → (⟨((𝑁𝑡)‘𝐷), (𝑁𝑡)⟩ ∈ (𝐼𝑄) ↔ (((𝑁𝑡)‘𝐷) = ((𝑁𝑡)‘𝐷) ∧ (𝑁𝑡) ∈ 𝐸)))
6152, 55, 60mpbir2and 711 . . . . . . . . 9 (((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡𝐸 ∧ (𝑡‘(𝐺𝐷)) = 𝑓)) → ⟨((𝑁𝑡)‘𝐷), (𝑁𝑡)⟩ ∈ (𝐼𝑄))
6216ad2antrr 724 . . . . . . . . . . 11 (((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡𝐸 ∧ (𝑡‘(𝐺𝐷)) = 𝑓)) → 𝐺𝑇)
6322ad2antrr 724 . . . . . . . . . . 11 (((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡𝐸 ∧ (𝑡‘(𝐺𝐷)) = 𝑓)) → 𝐷𝑇)
642, 13, 35tendospdi1 40721 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑡𝐸𝐺𝑇𝐷𝑇)) → (𝑡‘(𝐺𝐷)) = ((𝑡𝐺) ∘ (𝑡𝐷)))
6545, 44, 62, 63, 64syl13anc 1369 . . . . . . . . . 10 (((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡𝐸 ∧ (𝑡‘(𝐺𝐷)) = 𝑓)) → (𝑡‘(𝐺𝐷)) = ((𝑡𝐺) ∘ (𝑡𝐷)))
66 simprr 771 . . . . . . . . . 10 (((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡𝐸 ∧ (𝑡‘(𝐺𝐷)) = 𝑓)) → (𝑡‘(𝐺𝐷)) = 𝑓)
6720ad2antrr 724 . . . . . . . . . . . . 13 (((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡𝐸 ∧ (𝑡‘(𝐺𝐷)) = 𝑓)) → 𝐷𝑇)
6853, 13tendoi2 40496 . . . . . . . . . . . . 13 ((𝑡𝐸𝐷𝑇) → ((𝑁𝑡)‘𝐷) = (𝑡𝐷))
6944, 67, 68syl2anc 582 . . . . . . . . . . . 12 (((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡𝐸 ∧ (𝑡‘(𝐺𝐷)) = 𝑓)) → ((𝑁𝑡)‘𝐷) = (𝑡𝐷))
702, 13, 35tendocnv 40722 . . . . . . . . . . . . 13 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑡𝐸𝐷𝑇) → (𝑡𝐷) = (𝑡𝐷))
7145, 44, 67, 70syl3anc 1368 . . . . . . . . . . . 12 (((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡𝐸 ∧ (𝑡‘(𝐺𝐷)) = 𝑓)) → (𝑡𝐷) = (𝑡𝐷))
7269, 71eqtr2d 2767 . . . . . . . . . . 11 (((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡𝐸 ∧ (𝑡‘(𝐺𝐷)) = 𝑓)) → (𝑡𝐷) = ((𝑁𝑡)‘𝐷))
7372coeq2d 5871 . . . . . . . . . 10 (((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡𝐸 ∧ (𝑡‘(𝐺𝐷)) = 𝑓)) → ((𝑡𝐺) ∘ (𝑡𝐷)) = ((𝑡𝐺) ∘ ((𝑁𝑡)‘𝐷)))
7465, 66, 733eqtr3d 2774 . . . . . . . . 9 (((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡𝐸 ∧ (𝑡‘(𝐺𝐷)) = 𝑓)) → 𝑓 = ((𝑡𝐺) ∘ ((𝑁𝑡)‘𝐷)))
75 simplrr 776 . . . . . . . . . 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 40499 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑡𝐸) → (𝑡𝐽(𝑁𝑡)) = 0 )
7945, 44, 78syl2anc 582 . . . . . . . . . 10 (((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡𝐸 ∧ (𝑡‘(𝐺𝐷)) = 𝑓)) → (𝑡𝐽(𝑁𝑡)) = 0 )
8075, 79eqtr4d 2769 . . . . . . . . 9 (((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡𝐸 ∧ (𝑡‘(𝐺𝐷)) = 𝑓)) → 𝑠 = (𝑡𝐽(𝑁𝑡)))
81 opeq1 4881 . . . . . . . . . . . . . . 15 (𝑔 = (𝑡𝐺) → ⟨𝑔, 𝑡⟩ = ⟨(𝑡𝐺), 𝑡⟩)
8281eleq1d 2811 . . . . . . . . . . . . . 14 (𝑔 = (𝑡𝐺) → (⟨𝑔, 𝑡⟩ ∈ (𝐼𝑃) ↔ ⟨(𝑡𝐺), 𝑡⟩ ∈ (𝐼𝑃)))
8382anbi1d 629 . . . . . . . . . . . . 13 (𝑔 = (𝑡𝐺) → ((⟨𝑔, 𝑡⟩ ∈ (𝐼𝑃) ∧ ⟨, 𝑢⟩ ∈ (𝐼𝑄)) ↔ (⟨(𝑡𝐺), 𝑡⟩ ∈ (𝐼𝑃) ∧ ⟨, 𝑢⟩ ∈ (𝐼𝑄))))
84 coeq1 5866 . . . . . . . . . . . . . . 15 (𝑔 = (𝑡𝐺) → (𝑔) = ((𝑡𝐺) ∘ ))
8584eqeq2d 2737 . . . . . . . . . . . . . 14 (𝑔 = (𝑡𝐺) → (𝑓 = (𝑔) ↔ 𝑓 = ((𝑡𝐺) ∘ )))
8685anbi1d 629 . . . . . . . . . . . . 13 (𝑔 = (𝑡𝐺) → ((𝑓 = (𝑔) ∧ 𝑠 = (𝑡𝐽𝑢)) ↔ (𝑓 = ((𝑡𝐺) ∘ ) ∧ 𝑠 = (𝑡𝐽𝑢))))
8783, 86anbi12d 630 . . . . . . . . . . . 12 (𝑔 = (𝑡𝐺) → (((⟨𝑔, 𝑡⟩ ∈ (𝐼𝑃) ∧ ⟨, 𝑢⟩ ∈ (𝐼𝑄)) ∧ (𝑓 = (𝑔) ∧ 𝑠 = (𝑡𝐽𝑢))) ↔ ((⟨(𝑡𝐺), 𝑡⟩ ∈ (𝐼𝑃) ∧ ⟨, 𝑢⟩ ∈ (𝐼𝑄)) ∧ (𝑓 = ((𝑡𝐺) ∘ ) ∧ 𝑠 = (𝑡𝐽𝑢)))))
88 opeq1 4881 . . . . . . . . . . . . . . 15 ( = ((𝑁𝑡)‘𝐷) → ⟨, 𝑢⟩ = ⟨((𝑁𝑡)‘𝐷), 𝑢⟩)
8988eleq1d 2811 . . . . . . . . . . . . . 14 ( = ((𝑁𝑡)‘𝐷) → (⟨, 𝑢⟩ ∈ (𝐼𝑄) ↔ ⟨((𝑁𝑡)‘𝐷), 𝑢⟩ ∈ (𝐼𝑄)))
9089anbi2d 628 . . . . . . . . . . . . 13 ( = ((𝑁𝑡)‘𝐷) → ((⟨(𝑡𝐺), 𝑡⟩ ∈ (𝐼𝑃) ∧ ⟨, 𝑢⟩ ∈ (𝐼𝑄)) ↔ (⟨(𝑡𝐺), 𝑡⟩ ∈ (𝐼𝑃) ∧ ⟨((𝑁𝑡)‘𝐷), 𝑢⟩ ∈ (𝐼𝑄))))
91 coeq2 5867 . . . . . . . . . . . . . . 15 ( = ((𝑁𝑡)‘𝐷) → ((𝑡𝐺) ∘ ) = ((𝑡𝐺) ∘ ((𝑁𝑡)‘𝐷)))
9291eqeq2d 2737 . . . . . . . . . . . . . 14 ( = ((𝑁𝑡)‘𝐷) → (𝑓 = ((𝑡𝐺) ∘ ) ↔ 𝑓 = ((𝑡𝐺) ∘ ((𝑁𝑡)‘𝐷))))
9392anbi1d 629 . . . . . . . . . . . . 13 ( = ((𝑁𝑡)‘𝐷) → ((𝑓 = ((𝑡𝐺) ∘ ) ∧ 𝑠 = (𝑡𝐽𝑢)) ↔ (𝑓 = ((𝑡𝐺) ∘ ((𝑁𝑡)‘𝐷)) ∧ 𝑠 = (𝑡𝐽𝑢))))
9490, 93anbi12d 630 . . . . . . . . . . . 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 4882 . . . . . . . . . . . . . . 15 (𝑢 = (𝑁𝑡) → ⟨((𝑁𝑡)‘𝐷), 𝑢⟩ = ⟨((𝑁𝑡)‘𝐷), (𝑁𝑡)⟩)
9695eleq1d 2811 . . . . . . . . . . . . . 14 (𝑢 = (𝑁𝑡) → (⟨((𝑁𝑡)‘𝐷), 𝑢⟩ ∈ (𝐼𝑄) ↔ ⟨((𝑁𝑡)‘𝐷), (𝑁𝑡)⟩ ∈ (𝐼𝑄)))
9796anbi2d 628 . . . . . . . . . . . . 13 (𝑢 = (𝑁𝑡) → ((⟨(𝑡𝐺), 𝑡⟩ ∈ (𝐼𝑃) ∧ ⟨((𝑁𝑡)‘𝐷), 𝑢⟩ ∈ (𝐼𝑄)) ↔ (⟨(𝑡𝐺), 𝑡⟩ ∈ (𝐼𝑃) ∧ ⟨((𝑁𝑡)‘𝐷), (𝑁𝑡)⟩ ∈ (𝐼𝑄))))
98 oveq2 7434 . . . . . . . . . . . . . . 15 (𝑢 = (𝑁𝑡) → (𝑡𝐽𝑢) = (𝑡𝐽(𝑁𝑡)))
9998eqeq2d 2737 . . . . . . . . . . . . . 14 (𝑢 = (𝑁𝑡) → (𝑠 = (𝑡𝐽𝑢) ↔ 𝑠 = (𝑡𝐽(𝑁𝑡))))
10099anbi2d 628 . . . . . . . . . . . . 13 (𝑢 = (𝑁𝑡) → ((𝑓 = ((𝑡𝐺) ∘ ((𝑁𝑡)‘𝐷)) ∧ 𝑠 = (𝑡𝐽𝑢)) ↔ (𝑓 = ((𝑡𝐺) ∘ ((𝑁𝑡)‘𝐷)) ∧ 𝑠 = (𝑡𝐽(𝑁𝑡)))))
10197, 100anbi12d 630 . . . . . . . . . . . 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 1431 . . . . . . . . . . 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 3589 . . . . . . . . . 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 1458 . . . . . . . . 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 837 . . . . . . . 8 (((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡𝐸 ∧ (𝑡‘(𝐺𝐷)) = 𝑓)) → ∃𝑔𝑢((⟨𝑔, 𝑡⟩ ∈ (𝐼𝑃) ∧ ⟨, 𝑢⟩ ∈ (𝐼𝑄)) ∧ (𝑓 = (𝑔) ∧ 𝑠 = (𝑡𝐽𝑢))))
106105ex 411 . . . . . . 7 ((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) → ((𝑡𝐸 ∧ (𝑡‘(𝐺𝐷)) = 𝑓) → ∃𝑔𝑢((⟨𝑔, 𝑡⟩ ∈ (𝐼𝑃) ∧ ⟨, 𝑢⟩ ∈ (𝐼𝑄)) ∧ (𝑓 = (𝑔) ∧ 𝑠 = (𝑡𝐽𝑢)))))
107106eximdv 1913 . . . . . 6 ((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) → (∃𝑡(𝑡𝐸 ∧ (𝑡‘(𝐺𝐷)) = 𝑓) → ∃𝑡𝑔𝑢((⟨𝑔, 𝑡⟩ ∈ (𝐼𝑃) ∧ ⟨, 𝑢⟩ ∈ (𝐼𝑄)) ∧ (𝑓 = (𝑔) ∧ 𝑠 = (𝑡𝐽𝑢)))))
108 excom 2152 . . . . . 6 (∃𝑡𝑔𝑢((⟨𝑔, 𝑡⟩ ∈ (𝐼𝑃) ∧ ⟨, 𝑢⟩ ∈ (𝐼𝑄)) ∧ (𝑓 = (𝑔) ∧ 𝑠 = (𝑡𝐽𝑢))) ↔ ∃𝑔𝑡𝑢((⟨𝑔, 𝑡⟩ ∈ (𝐼𝑃) ∧ ⟨, 𝑢⟩ ∈ (𝐼𝑄)) ∧ (𝑓 = (𝑔) ∧ 𝑠 = (𝑡𝐽𝑢))))
109107, 108imbitrdi 250 . . . . 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 411 . . 3 (𝜑 → (((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 ) → ∃𝑔𝑡𝑢((⟨𝑔, 𝑡⟩ ∈ (𝐼𝑃) ∧ ⟨, 𝑢⟩ ∈ (𝐼𝑄)) ∧ (𝑓 = (𝑔) ∧ 𝑠 = (𝑡𝐽𝑢)))))
1121simpld 493 . . . . . . . . 9 (𝜑𝐾 ∈ HL)
113112hllatd 39064 . . . . . . . 8 (𝜑𝐾 ∈ Lat)
11412simpld 493 . . . . . . . . 9 (𝜑𝑃𝐴)
11517simpld 493 . . . . . . . . 9 (𝜑𝑄𝐴)
11628, 29, 8hlatjcl 39067 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → (𝑃 𝑄) ∈ 𝐵)
117112, 114, 115, 116syl3anc 1368 . . . . . . . 8 (𝜑 → (𝑃 𝑄) ∈ 𝐵)
1181simprd 494 . . . . . . . . 9 (𝜑𝑊𝐻)
11928, 2lhpbase 39699 . . . . . . . . 9 (𝑊𝐻𝑊𝐵)
120118, 119syl 17 . . . . . . . 8 (𝜑𝑊𝐵)
12128, 30latmcl 18467 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ 𝐵𝑊𝐵) → ((𝑃 𝑄) 𝑊) ∈ 𝐵)
122113, 117, 120, 121syl3anc 1368 . . . . . . 7 (𝜑 → ((𝑃 𝑄) 𝑊) ∈ 𝐵)
12333, 122eqeltrid 2830 . . . . . 6 (𝜑𝑉𝐵)
12428, 7, 30latmle2 18492 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ 𝐵𝑊𝐵) → ((𝑃 𝑄) 𝑊) 𝑊)
125113, 117, 120, 124syl3anc 1368 . . . . . . 7 (𝜑 → ((𝑃 𝑄) 𝑊) 𝑊)
12633, 125eqbrtrid 5190 . . . . . 6 (𝜑𝑉 𝑊)
127 eqid 2726 . . . . . . 7 ((DIsoB‘𝐾)‘𝑊) = ((DIsoB‘𝐾)‘𝑊)
12828, 7, 2, 3, 127dihvalb 40938 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑉𝐵𝑉 𝑊)) → (𝐼𝑉) = (((DIsoB‘𝐾)‘𝑊)‘𝑉))
1291, 123, 126, 128syl12anc 835 . . . . 5 (𝜑 → (𝐼𝑉) = (((DIsoB‘𝐾)‘𝑊)‘𝑉))
130129eleq2d 2812 . . . 4 (𝜑 → (⟨𝑓, 𝑠⟩ ∈ (𝐼𝑉) ↔ ⟨𝑓, 𝑠⟩ ∈ (((DIsoB‘𝐾)‘𝑊)‘𝑉)))
13128, 7, 2, 13, 34, 77, 127dibopelval3 40849 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑉𝐵𝑉 𝑊)) → (⟨𝑓, 𝑠⟩ ∈ (((DIsoB‘𝐾)‘𝑊)‘𝑉) ↔ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )))
1321, 123, 126, 131syl12anc 835 . . . 4 (𝜑 → (⟨𝑓, 𝑠⟩ ∈ (((DIsoB‘𝐾)‘𝑊)‘𝑉) ↔ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )))
133130, 132bitrd 278 . . 3 (𝜑 → (⟨𝑓, 𝑠⟩ ∈ (𝐼𝑉) ↔ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )))
134 eqid 2726 . . . 4 (LSubSp‘𝑈) = (LSubSp‘𝑈)
13528, 8atbase 38989 . . . . 5 (𝑃𝐴𝑃𝐵)
136114, 135syl 17 . . . 4 (𝜑𝑃𝐵)
13728, 8atbase 38989 . . . . 5 (𝑄𝐴𝑄𝐵)
138115, 137syl 17 . . . 4 (𝜑𝑄𝐵)
13928, 2, 13, 35, 76, 31, 134, 32, 3, 1, 136, 138dihopellsm 40956 . . 3 (𝜑 → (⟨𝑓, 𝑠⟩ ∈ ((𝐼𝑃) (𝐼𝑄)) ↔ ∃𝑔𝑡𝑢((⟨𝑔, 𝑡⟩ ∈ (𝐼𝑃) ∧ ⟨, 𝑢⟩ ∈ (𝐼𝑄)) ∧ (𝑓 = (𝑔) ∧ 𝑠 = (𝑡𝐽𝑢)))))
140111, 133, 1393imtr4d 293 . 2 (𝜑 → (⟨𝑓, 𝑠⟩ ∈ (𝐼𝑉) → ⟨𝑓, 𝑠⟩ ∈ ((𝐼𝑃) (𝐼𝑄))))
1415, 140relssdv 5796 1 (𝜑 → (𝐼𝑉) ⊆ ((𝐼𝑃) (𝐼𝑄)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 394   = wceq 1534  wex 1774  wcel 2099  wrex 3060  Vcvv 3462  wss 3947  cop 4639   class class class wbr 5155  cmpt 5238   I cid 5581  ccnv 5683  cres 5686  ccom 5688  Rel wrel 5689  cfv 6556  crio 7381  (class class class)co 7426  cmpo 7428  Basecbs 17215  lecple 17275  occoc 17276  joincjn 18338  meetcmee 18339  Latclat 18458  LSSumclsm 19634  LSubSpclss 20910  Atomscatm 38963  HLchlt 39050  LHypclh 39685  LTrncltrn 39802  trLctrl 39859  TEndoctendo 40453  DVecHcdvh 40779  DIsoBcdib 40839  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:  dihjatcc  41123
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