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Theorem dihjatcclem4 35728
 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 35586 . . 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 34323 . . . . . . . . . . 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 34885 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐶𝐴 ∧ ¬ 𝐶 𝑊) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → 𝐺𝑇)
161, 11, 12, 15syl3anc 1318 . . . . . . . . 9 (𝜑𝐺𝑇)
17 dihjatcclem.q . . . . . . . . . . 11 (𝜑 → (𝑄𝐴 ∧ ¬ 𝑄 𝑊))
18 dihjatcc.dd . . . . . . . . . . . 12 𝐷 = (𝑑𝑇 (𝑑𝐶) = 𝑄)
197, 8, 2, 13, 18ltrniotacl 34885 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐶𝐴 ∧ ¬ 𝐶 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → 𝐷𝑇)
201, 11, 17, 19syl3anc 1318 . . . . . . . . . 10 (𝜑𝐷𝑇)
212, 13ltrncnv 34450 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐷𝑇) → 𝐷𝑇)
221, 20, 21syl2anc 691 . . . . . . . . 9 (𝜑𝐷𝑇)
232, 13ltrnco 35025 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐺𝑇𝐷𝑇) → (𝐺𝐷) ∈ 𝑇)
241, 16, 22, 23syl3anc 1318 . . . . . . . 8 (𝜑 → (𝐺𝐷) ∈ 𝑇)
2524adantr 480 . . . . . . 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 35727 . . . . . . . . 9 (𝜑 → (𝑅‘(𝐺𝐷)) = 𝑉)
3736adantr 480 . . . . . . . 8 ((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) → (𝑅‘(𝐺𝐷)) = 𝑉)
3827, 37breqtrrd 4611 . . . . . . 7 ((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) → (𝑅𝑓) (𝑅‘(𝐺𝐷)))
397, 2, 13, 34, 35tendoex 35281 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝐺𝐷) ∈ 𝑇𝑓𝑇) ∧ (𝑅𝑓) (𝑅‘(𝐺𝐷))) → ∃𝑡𝐸 (𝑡‘(𝐺𝐷)) = 𝑓)
406, 25, 26, 38, 39syl121anc 1323 . . . . . 6 ((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) → ∃𝑡𝐸 (𝑡‘(𝐺𝐷)) = 𝑓)
41 df-rex 2902 . . . . . 6 (∃𝑡𝐸 (𝑡‘(𝐺𝐷)) = 𝑓 ↔ ∃𝑡(𝑡𝐸 ∧ (𝑡‘(𝐺𝐷)) = 𝑓))
4240, 41sylib 207 . . . . 5 ((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) → ∃𝑡(𝑡𝐸 ∧ (𝑡‘(𝐺𝐷)) = 𝑓))
43 eqidd 2611 . . . . . . . . . 10 (((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡𝐸 ∧ (𝑡‘(𝐺𝐷)) = 𝑓)) → (𝑡𝐺) = (𝑡𝐺))
44 simprl 790 . . . . . . . . . 10 (((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡𝐸 ∧ (𝑡‘(𝐺𝐷)) = 𝑓)) → 𝑡𝐸)
451ad2antrr 758 . . . . . . . . . . 11 (((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡𝐸 ∧ (𝑡‘(𝐺𝐷)) = 𝑓)) → (𝐾 ∈ HL ∧ 𝑊𝐻))
4612ad2antrr 758 . . . . . . . . . . 11 (((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡𝐸 ∧ (𝑡‘(𝐺𝐷)) = 𝑓)) → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
47 fvex 6113 . . . . . . . . . . . 12 (𝑡𝐺) ∈ V
48 vex 3176 . . . . . . . . . . . 12 𝑡 ∈ V
497, 8, 2, 9, 13, 35, 3, 14, 47, 48dihopelvalcqat 35553 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (⟨(𝑡𝐺), 𝑡⟩ ∈ (𝐼𝑃) ↔ ((𝑡𝐺) = (𝑡𝐺) ∧ 𝑡𝐸)))
5045, 46, 49syl2anc 691 . . . . . . . . . 10 (((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡𝐸 ∧ (𝑡‘(𝐺𝐷)) = 𝑓)) → (⟨(𝑡𝐺), 𝑡⟩ ∈ (𝐼𝑃) ↔ ((𝑡𝐺) = (𝑡𝐺) ∧ 𝑡𝐸)))
5143, 44, 50mpbir2and 959 . . . . . . . . 9 (((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡𝐸 ∧ (𝑡‘(𝐺𝐷)) = 𝑓)) → ⟨(𝑡𝐺), 𝑡⟩ ∈ (𝐼𝑃))
52 eqidd 2611 . . . . . . . . . 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 35102 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑡𝐸) → (𝑁𝑡) ∈ 𝐸)
5545, 44, 54syl2anc 691 . . . . . . . . . 10 (((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡𝐸 ∧ (𝑡‘(𝐺𝐷)) = 𝑓)) → (𝑁𝑡) ∈ 𝐸)
5617ad2antrr 758 . . . . . . . . . . 11 (((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡𝐸 ∧ (𝑡‘(𝐺𝐷)) = 𝑓)) → (𝑄𝐴 ∧ ¬ 𝑄 𝑊))
57 fvex 6113 . . . . . . . . . . . 12 ((𝑁𝑡)‘𝐷) ∈ V
58 fvex 6113 . . . . . . . . . . . 12 (𝑁𝑡) ∈ V
597, 8, 2, 9, 13, 35, 3, 18, 57, 58dihopelvalcqat 35553 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (⟨((𝑁𝑡)‘𝐷), (𝑁𝑡)⟩ ∈ (𝐼𝑄) ↔ (((𝑁𝑡)‘𝐷) = ((𝑁𝑡)‘𝐷) ∧ (𝑁𝑡) ∈ 𝐸)))
6045, 56, 59syl2anc 691 . . . . . . . . . 10 (((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡𝐸 ∧ (𝑡‘(𝐺𝐷)) = 𝑓)) → (⟨((𝑁𝑡)‘𝐷), (𝑁𝑡)⟩ ∈ (𝐼𝑄) ↔ (((𝑁𝑡)‘𝐷) = ((𝑁𝑡)‘𝐷) ∧ (𝑁𝑡) ∈ 𝐸)))
6152, 55, 60mpbir2and 959 . . . . . . . . 9 (((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡𝐸 ∧ (𝑡‘(𝐺𝐷)) = 𝑓)) → ⟨((𝑁𝑡)‘𝐷), (𝑁𝑡)⟩ ∈ (𝐼𝑄))
6216ad2antrr 758 . . . . . . . . . . 11 (((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡𝐸 ∧ (𝑡‘(𝐺𝐷)) = 𝑓)) → 𝐺𝑇)
6322ad2antrr 758 . . . . . . . . . . 11 (((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡𝐸 ∧ (𝑡‘(𝐺𝐷)) = 𝑓)) → 𝐷𝑇)
642, 13, 35tendospdi1 35327 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑡𝐸𝐺𝑇𝐷𝑇)) → (𝑡‘(𝐺𝐷)) = ((𝑡𝐺) ∘ (𝑡𝐷)))
6545, 44, 62, 63, 64syl13anc 1320 . . . . . . . . . 10 (((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡𝐸 ∧ (𝑡‘(𝐺𝐷)) = 𝑓)) → (𝑡‘(𝐺𝐷)) = ((𝑡𝐺) ∘ (𝑡𝐷)))
66 simprr 792 . . . . . . . . . 10 (((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡𝐸 ∧ (𝑡‘(𝐺𝐷)) = 𝑓)) → (𝑡‘(𝐺𝐷)) = 𝑓)
6720ad2antrr 758 . . . . . . . . . . . . 13 (((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡𝐸 ∧ (𝑡‘(𝐺𝐷)) = 𝑓)) → 𝐷𝑇)
6853, 13tendoi2 35101 . . . . . . . . . . . . 13 ((𝑡𝐸𝐷𝑇) → ((𝑁𝑡)‘𝐷) = (𝑡𝐷))
6944, 67, 68syl2anc 691 . . . . . . . . . . . 12 (((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡𝐸 ∧ (𝑡‘(𝐺𝐷)) = 𝑓)) → ((𝑁𝑡)‘𝐷) = (𝑡𝐷))
702, 13, 35tendocnv 35328 . . . . . . . . . . . . 13 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑡𝐸𝐷𝑇) → (𝑡𝐷) = (𝑡𝐷))
7145, 44, 67, 70syl3anc 1318 . . . . . . . . . . . 12 (((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡𝐸 ∧ (𝑡‘(𝐺𝐷)) = 𝑓)) → (𝑡𝐷) = (𝑡𝐷))
7269, 71eqtr2d 2645 . . . . . . . . . . 11 (((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡𝐸 ∧ (𝑡‘(𝐺𝐷)) = 𝑓)) → (𝑡𝐷) = ((𝑁𝑡)‘𝐷))
7372coeq2d 5206 . . . . . . . . . 10 (((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡𝐸 ∧ (𝑡‘(𝐺𝐷)) = 𝑓)) → ((𝑡𝐺) ∘ (𝑡𝐷)) = ((𝑡𝐺) ∘ ((𝑁𝑡)‘𝐷)))
7465, 66, 733eqtr3d 2652 . . . . . . . . 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 35104 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑡𝐸) → (𝑡𝐽(𝑁𝑡)) = 0 )
7945, 44, 78syl2anc 691 . . . . . . . . . 10 (((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡𝐸 ∧ (𝑡‘(𝐺𝐷)) = 𝑓)) → (𝑡𝐽(𝑁𝑡)) = 0 )
8075, 79eqtr4d 2647 . . . . . . . . 9 (((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡𝐸 ∧ (𝑡‘(𝐺𝐷)) = 𝑓)) → 𝑠 = (𝑡𝐽(𝑁𝑡)))
81 opeq1 4340 . . . . . . . . . . . . . . 15 (𝑔 = (𝑡𝐺) → ⟨𝑔, 𝑡⟩ = ⟨(𝑡𝐺), 𝑡⟩)
8281eleq1d 2672 . . . . . . . . . . . . . 14 (𝑔 = (𝑡𝐺) → (⟨𝑔, 𝑡⟩ ∈ (𝐼𝑃) ↔ ⟨(𝑡𝐺), 𝑡⟩ ∈ (𝐼𝑃)))
8382anbi1d 737 . . . . . . . . . . . . 13 (𝑔 = (𝑡𝐺) → ((⟨𝑔, 𝑡⟩ ∈ (𝐼𝑃) ∧ ⟨, 𝑢⟩ ∈ (𝐼𝑄)) ↔ (⟨(𝑡𝐺), 𝑡⟩ ∈ (𝐼𝑃) ∧ ⟨, 𝑢⟩ ∈ (𝐼𝑄))))
84 coeq1 5201 . . . . . . . . . . . . . . 15 (𝑔 = (𝑡𝐺) → (𝑔) = ((𝑡𝐺) ∘ ))
8584eqeq2d 2620 . . . . . . . . . . . . . 14 (𝑔 = (𝑡𝐺) → (𝑓 = (𝑔) ↔ 𝑓 = ((𝑡𝐺) ∘ )))
8685anbi1d 737 . . . . . . . . . . . . 13 (𝑔 = (𝑡𝐺) → ((𝑓 = (𝑔) ∧ 𝑠 = (𝑡𝐽𝑢)) ↔ (𝑓 = ((𝑡𝐺) ∘ ) ∧ 𝑠 = (𝑡𝐽𝑢))))
8783, 86anbi12d 743 . . . . . . . . . . . 12 (𝑔 = (𝑡𝐺) → (((⟨𝑔, 𝑡⟩ ∈ (𝐼𝑃) ∧ ⟨, 𝑢⟩ ∈ (𝐼𝑄)) ∧ (𝑓 = (𝑔) ∧ 𝑠 = (𝑡𝐽𝑢))) ↔ ((⟨(𝑡𝐺), 𝑡⟩ ∈ (𝐼𝑃) ∧ ⟨, 𝑢⟩ ∈ (𝐼𝑄)) ∧ (𝑓 = ((𝑡𝐺) ∘ ) ∧ 𝑠 = (𝑡𝐽𝑢)))))
88 opeq1 4340 . . . . . . . . . . . . . . 15 ( = ((𝑁𝑡)‘𝐷) → ⟨, 𝑢⟩ = ⟨((𝑁𝑡)‘𝐷), 𝑢⟩)
8988eleq1d 2672 . . . . . . . . . . . . . 14 ( = ((𝑁𝑡)‘𝐷) → (⟨, 𝑢⟩ ∈ (𝐼𝑄) ↔ ⟨((𝑁𝑡)‘𝐷), 𝑢⟩ ∈ (𝐼𝑄)))
9089anbi2d 736 . . . . . . . . . . . . 13 ( = ((𝑁𝑡)‘𝐷) → ((⟨(𝑡𝐺), 𝑡⟩ ∈ (𝐼𝑃) ∧ ⟨, 𝑢⟩ ∈ (𝐼𝑄)) ↔ (⟨(𝑡𝐺), 𝑡⟩ ∈ (𝐼𝑃) ∧ ⟨((𝑁𝑡)‘𝐷), 𝑢⟩ ∈ (𝐼𝑄))))
91 coeq2 5202 . . . . . . . . . . . . . . 15 ( = ((𝑁𝑡)‘𝐷) → ((𝑡𝐺) ∘ ) = ((𝑡𝐺) ∘ ((𝑁𝑡)‘𝐷)))
9291eqeq2d 2620 . . . . . . . . . . . . . 14 ( = ((𝑁𝑡)‘𝐷) → (𝑓 = ((𝑡𝐺) ∘ ) ↔ 𝑓 = ((𝑡𝐺) ∘ ((𝑁𝑡)‘𝐷))))
9392anbi1d 737 . . . . . . . . . . . . 13 ( = ((𝑁𝑡)‘𝐷) → ((𝑓 = ((𝑡𝐺) ∘ ) ∧ 𝑠 = (𝑡𝐽𝑢)) ↔ (𝑓 = ((𝑡𝐺) ∘ ((𝑁𝑡)‘𝐷)) ∧ 𝑠 = (𝑡𝐽𝑢))))
9490, 93anbi12d 743 . . . . . . . . . . . 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 4341 . . . . . . . . . . . . . . 15 (𝑢 = (𝑁𝑡) → ⟨((𝑁𝑡)‘𝐷), 𝑢⟩ = ⟨((𝑁𝑡)‘𝐷), (𝑁𝑡)⟩)
9695eleq1d 2672 . . . . . . . . . . . . . 14 (𝑢 = (𝑁𝑡) → (⟨((𝑁𝑡)‘𝐷), 𝑢⟩ ∈ (𝐼𝑄) ↔ ⟨((𝑁𝑡)‘𝐷), (𝑁𝑡)⟩ ∈ (𝐼𝑄)))
9796anbi2d 736 . . . . . . . . . . . . 13 (𝑢 = (𝑁𝑡) → ((⟨(𝑡𝐺), 𝑡⟩ ∈ (𝐼𝑃) ∧ ⟨((𝑁𝑡)‘𝐷), 𝑢⟩ ∈ (𝐼𝑄)) ↔ (⟨(𝑡𝐺), 𝑡⟩ ∈ (𝐼𝑃) ∧ ⟨((𝑁𝑡)‘𝐷), (𝑁𝑡)⟩ ∈ (𝐼𝑄))))
98 oveq2 6557 . . . . . . . . . . . . . . 15 (𝑢 = (𝑁𝑡) → (𝑡𝐽𝑢) = (𝑡𝐽(𝑁𝑡)))
9998eqeq2d 2620 . . . . . . . . . . . . . 14 (𝑢 = (𝑁𝑡) → (𝑠 = (𝑡𝐽𝑢) ↔ 𝑠 = (𝑡𝐽(𝑁𝑡))))
10099anbi2d 736 . . . . . . . . . . . . 13 (𝑢 = (𝑁𝑡) → ((𝑓 = ((𝑡𝐺) ∘ ((𝑁𝑡)‘𝐷)) ∧ 𝑠 = (𝑡𝐽𝑢)) ↔ (𝑓 = ((𝑡𝐺) ∘ ((𝑁𝑡)‘𝐷)) ∧ 𝑠 = (𝑡𝐽(𝑁𝑡)))))
10197, 100anbi12d 743 . . . . . . . . . . . 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 1389 . . . . . . . . . . 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 3270 . . . . . . . . . 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 1416 . . . . . . . . 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 1319 . . . . . . . 8 (((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡𝐸 ∧ (𝑡‘(𝐺𝐷)) = 𝑓)) → ∃𝑔𝑢((⟨𝑔, 𝑡⟩ ∈ (𝐼𝑃) ∧ ⟨, 𝑢⟩ ∈ (𝐼𝑄)) ∧ (𝑓 = (𝑔) ∧ 𝑠 = (𝑡𝐽𝑢))))
106105ex 449 . . . . . . 7 ((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) → ((𝑡𝐸 ∧ (𝑡‘(𝐺𝐷)) = 𝑓) → ∃𝑔𝑢((⟨𝑔, 𝑡⟩ ∈ (𝐼𝑃) ∧ ⟨, 𝑢⟩ ∈ (𝐼𝑄)) ∧ (𝑓 = (𝑔) ∧ 𝑠 = (𝑡𝐽𝑢)))))
107106eximdv 1833 . . . . . 6 ((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) → (∃𝑡(𝑡𝐸 ∧ (𝑡‘(𝐺𝐷)) = 𝑓) → ∃𝑡𝑔𝑢((⟨𝑔, 𝑡⟩ ∈ (𝐼𝑃) ∧ ⟨, 𝑢⟩ ∈ (𝐼𝑄)) ∧ (𝑓 = (𝑔) ∧ 𝑠 = (𝑡𝐽𝑢)))))
108 excom 2029 . . . . . 6 (∃𝑡𝑔𝑢((⟨𝑔, 𝑡⟩ ∈ (𝐼𝑃) ∧ ⟨, 𝑢⟩ ∈ (𝐼𝑄)) ∧ (𝑓 = (𝑔) ∧ 𝑠 = (𝑡𝐽𝑢))) ↔ ∃𝑔𝑡𝑢((⟨𝑔, 𝑡⟩ ∈ (𝐼𝑃) ∧ ⟨, 𝑢⟩ ∈ (𝐼𝑄)) ∧ (𝑓 = (𝑔) ∧ 𝑠 = (𝑡𝐽𝑢))))
109107, 108syl6ib 240 . . . . 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 449 . . 3 (𝜑 → (((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 ) → ∃𝑔𝑡𝑢((⟨𝑔, 𝑡⟩ ∈ (𝐼𝑃) ∧ ⟨, 𝑢⟩ ∈ (𝐼𝑄)) ∧ (𝑓 = (𝑔) ∧ 𝑠 = (𝑡𝐽𝑢)))))
1121simpld 474 . . . . . . . . 9 (𝜑𝐾 ∈ HL)
113 hllat 33668 . . . . . . . . 9 (𝐾 ∈ HL → 𝐾 ∈ Lat)
114112, 113syl 17 . . . . . . . 8 (𝜑𝐾 ∈ Lat)
11512simpld 474 . . . . . . . . 9 (𝜑𝑃𝐴)
11617simpld 474 . . . . . . . . 9 (𝜑𝑄𝐴)
11728, 29, 8hlatjcl 33671 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → (𝑃 𝑄) ∈ 𝐵)
118112, 115, 116, 117syl3anc 1318 . . . . . . . 8 (𝜑 → (𝑃 𝑄) ∈ 𝐵)
1191simprd 478 . . . . . . . . 9 (𝜑𝑊𝐻)
12028, 2lhpbase 34302 . . . . . . . . 9 (𝑊𝐻𝑊𝐵)
121119, 120syl 17 . . . . . . . 8 (𝜑𝑊𝐵)
12228, 30latmcl 16875 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ 𝐵𝑊𝐵) → ((𝑃 𝑄) 𝑊) ∈ 𝐵)
123114, 118, 121, 122syl3anc 1318 . . . . . . 7 (𝜑 → ((𝑃 𝑄) 𝑊) ∈ 𝐵)
12433, 123syl5eqel 2692 . . . . . 6 (𝜑𝑉𝐵)
12528, 7, 30latmle2 16900 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ 𝐵𝑊𝐵) → ((𝑃 𝑄) 𝑊) 𝑊)
126114, 118, 121, 125syl3anc 1318 . . . . . . 7 (𝜑 → ((𝑃 𝑄) 𝑊) 𝑊)
12733, 126syl5eqbr 4618 . . . . . 6 (𝜑𝑉 𝑊)
128 eqid 2610 . . . . . . 7 ((DIsoB‘𝐾)‘𝑊) = ((DIsoB‘𝐾)‘𝑊)
12928, 7, 2, 3, 128dihvalb 35544 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑉𝐵𝑉 𝑊)) → (𝐼𝑉) = (((DIsoB‘𝐾)‘𝑊)‘𝑉))
1301, 124, 127, 129syl12anc 1316 . . . . 5 (𝜑 → (𝐼𝑉) = (((DIsoB‘𝐾)‘𝑊)‘𝑉))
131130eleq2d 2673 . . . 4 (𝜑 → (⟨𝑓, 𝑠⟩ ∈ (𝐼𝑉) ↔ ⟨𝑓, 𝑠⟩ ∈ (((DIsoB‘𝐾)‘𝑊)‘𝑉)))
13228, 7, 2, 13, 34, 77, 128dibopelval3 35455 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑉𝐵𝑉 𝑊)) → (⟨𝑓, 𝑠⟩ ∈ (((DIsoB‘𝐾)‘𝑊)‘𝑉) ↔ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )))
1331, 124, 127, 132syl12anc 1316 . . . 4 (𝜑 → (⟨𝑓, 𝑠⟩ ∈ (((DIsoB‘𝐾)‘𝑊)‘𝑉) ↔ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )))
134131, 133bitrd 267 . . 3 (𝜑 → (⟨𝑓, 𝑠⟩ ∈ (𝐼𝑉) ↔ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )))
135 eqid 2610 . . . 4 (LSubSp‘𝑈) = (LSubSp‘𝑈)
13628, 8atbase 33594 . . . . 5 (𝑃𝐴𝑃𝐵)
137115, 136syl 17 . . . 4 (𝜑𝑃𝐵)
13828, 8atbase 33594 . . . . 5 (𝑄𝐴𝑄𝐵)
139116, 138syl 17 . . . 4 (𝜑𝑄𝐵)
14028, 2, 13, 35, 76, 31, 135, 32, 3, 1, 137, 139dihopellsm 35562 . . 3 (𝜑 → (⟨𝑓, 𝑠⟩ ∈ ((𝐼𝑃) (𝐼𝑄)) ↔ ∃𝑔𝑡𝑢((⟨𝑔, 𝑡⟩ ∈ (𝐼𝑃) ∧ ⟨, 𝑢⟩ ∈ (𝐼𝑄)) ∧ (𝑓 = (𝑔) ∧ 𝑠 = (𝑡𝐽𝑢)))))
141111, 134, 1403imtr4d 282 . 2 (𝜑 → (⟨𝑓, 𝑠⟩ ∈ (𝐼𝑉) → ⟨𝑓, 𝑠⟩ ∈ ((𝐼𝑃) (𝐼𝑄))))
1425, 141relssdv 5135 1 (𝜑 → (𝐼𝑉) ⊆ ((𝐼𝑃) (𝐼𝑄)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977  ∃wrex 2897  Vcvv 3173   ⊆ wss 3540  ⟨cop 4131   class class class wbr 4583   ↦ cmpt 4643   I cid 4948  ◡ccnv 5037   ↾ cres 5040   ∘ ccom 5042  Rel wrel 5043  ‘cfv 5804  ℩crio 6510  (class class class)co 6549   ↦ cmpt2 6551  Basecbs 15695  lecple 15775  occoc 15776  joincjn 16767  meetcmee 16768  Latclat 16868  LSSumclsm 17872  LSubSpclss 18753  Atomscatm 33568  HLchlt 33655  LHypclh 34288  LTrncltrn 34405  trLctrl 34463  TEndoctendo 35058  DVecHcdvh 35385  DIsoBcdib 35445  DIsoHcdih 35535 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-riotaBAD 33257 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-tpos 7239  df-undef 7286  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-mulr 15782  df-sca 15784  df-vsca 15785  df-0g 15925  df-preset 16751  df-poset 16769  df-plt 16781  df-lub 16797  df-glb 16798  df-join 16799  df-meet 16800  df-p0 16862  df-p1 16863  df-lat 16869  df-clat 16931  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-submnd 17159  df-grp 17248  df-minusg 17249  df-sbg 17250  df-subg 17414  df-cntz 17573  df-lsm 17874  df-cmn 18018  df-abl 18019  df-mgp 18313  df-ur 18325  df-ring 18372  df-oppr 18446  df-dvdsr 18464  df-unit 18465  df-invr 18495  df-dvr 18506  df-drng 18572  df-lmod 18688  df-lss 18754  df-lsp 18793  df-lvec 18924  df-oposet 33481  df-ol 33483  df-oml 33484  df-covers 33571  df-ats 33572  df-atl 33603  df-cvlat 33627  df-hlat 33656  df-llines 33802  df-lplanes 33803  df-lvols 33804  df-lines 33805  df-psubsp 33807  df-pmap 33808  df-padd 34100  df-lhyp 34292  df-laut 34293  df-ldil 34408  df-ltrn 34409  df-trl 34464  df-tendo 35061  df-edring 35063  df-disoa 35336  df-dvech 35386  df-dib 35446  df-dic 35480  df-dih 35536 This theorem is referenced by:  dihjatcc  35729
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