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Theorem dihjatcclem4 36217
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 36075 . . 3 ((𝐾 ∈ HL ∧ 𝑊𝐻) → Rel (𝐼𝑉))
51, 4syl 17 . 2 (𝜑 → Rel (𝐼𝑉))
61adantr 481 . . . . . . 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 34812 . . . . . . . . . . 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 35374 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐶𝐴 ∧ ¬ 𝐶 𝑊) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → 𝐺𝑇)
161, 11, 12, 15syl3anc 1323 . . . . . . . . 9 (𝜑𝐺𝑇)
17 dihjatcclem.q . . . . . . . . . . 11 (𝜑 → (𝑄𝐴 ∧ ¬ 𝑄 𝑊))
18 dihjatcc.dd . . . . . . . . . . . 12 𝐷 = (𝑑𝑇 (𝑑𝐶) = 𝑄)
197, 8, 2, 13, 18ltrniotacl 35374 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐶𝐴 ∧ ¬ 𝐶 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → 𝐷𝑇)
201, 11, 17, 19syl3anc 1323 . . . . . . . . . 10 (𝜑𝐷𝑇)
212, 13ltrncnv 34939 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐷𝑇) → 𝐷𝑇)
221, 20, 21syl2anc 692 . . . . . . . . 9 (𝜑𝐷𝑇)
232, 13ltrnco 35514 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐺𝑇𝐷𝑇) → (𝐺𝐷) ∈ 𝑇)
241, 16, 22, 23syl3anc 1323 . . . . . . . 8 (𝜑 → (𝐺𝐷) ∈ 𝑇)
2524adantr 481 . . . . . . 7 ((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) → (𝐺𝐷) ∈ 𝑇)
26 simprll 801 . . . . . . 7 ((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) → 𝑓𝑇)
27 simprlr 802 . . . . . . . 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 36216 . . . . . . . . 9 (𝜑 → (𝑅‘(𝐺𝐷)) = 𝑉)
3736adantr 481 . . . . . . . 8 ((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) → (𝑅‘(𝐺𝐷)) = 𝑉)
3827, 37breqtrrd 4646 . . . . . . 7 ((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) → (𝑅𝑓) (𝑅‘(𝐺𝐷)))
397, 2, 13, 34, 35tendoex 35770 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝐺𝐷) ∈ 𝑇𝑓𝑇) ∧ (𝑅𝑓) (𝑅‘(𝐺𝐷))) → ∃𝑡𝐸 (𝑡‘(𝐺𝐷)) = 𝑓)
406, 25, 26, 38, 39syl121anc 1328 . . . . . 6 ((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) → ∃𝑡𝐸 (𝑡‘(𝐺𝐷)) = 𝑓)
41 df-rex 2913 . . . . . 6 (∃𝑡𝐸 (𝑡‘(𝐺𝐷)) = 𝑓 ↔ ∃𝑡(𝑡𝐸 ∧ (𝑡‘(𝐺𝐷)) = 𝑓))
4240, 41sylib 208 . . . . 5 ((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) → ∃𝑡(𝑡𝐸 ∧ (𝑡‘(𝐺𝐷)) = 𝑓))
43 eqidd 2622 . . . . . . . . . 10 (((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡𝐸 ∧ (𝑡‘(𝐺𝐷)) = 𝑓)) → (𝑡𝐺) = (𝑡𝐺))
44 simprl 793 . . . . . . . . . 10 (((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡𝐸 ∧ (𝑡‘(𝐺𝐷)) = 𝑓)) → 𝑡𝐸)
451ad2antrr 761 . . . . . . . . . . 11 (((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡𝐸 ∧ (𝑡‘(𝐺𝐷)) = 𝑓)) → (𝐾 ∈ HL ∧ 𝑊𝐻))
4612ad2antrr 761 . . . . . . . . . . 11 (((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡𝐸 ∧ (𝑡‘(𝐺𝐷)) = 𝑓)) → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
47 fvex 6163 . . . . . . . . . . . 12 (𝑡𝐺) ∈ V
48 vex 3192 . . . . . . . . . . . 12 𝑡 ∈ V
497, 8, 2, 9, 13, 35, 3, 14, 47, 48dihopelvalcqat 36042 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (⟨(𝑡𝐺), 𝑡⟩ ∈ (𝐼𝑃) ↔ ((𝑡𝐺) = (𝑡𝐺) ∧ 𝑡𝐸)))
5045, 46, 49syl2anc 692 . . . . . . . . . 10 (((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡𝐸 ∧ (𝑡‘(𝐺𝐷)) = 𝑓)) → (⟨(𝑡𝐺), 𝑡⟩ ∈ (𝐼𝑃) ↔ ((𝑡𝐺) = (𝑡𝐺) ∧ 𝑡𝐸)))
5143, 44, 50mpbir2and 956 . . . . . . . . 9 (((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡𝐸 ∧ (𝑡‘(𝐺𝐷)) = 𝑓)) → ⟨(𝑡𝐺), 𝑡⟩ ∈ (𝐼𝑃))
52 eqidd 2622 . . . . . . . . . 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 35591 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑡𝐸) → (𝑁𝑡) ∈ 𝐸)
5545, 44, 54syl2anc 692 . . . . . . . . . 10 (((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡𝐸 ∧ (𝑡‘(𝐺𝐷)) = 𝑓)) → (𝑁𝑡) ∈ 𝐸)
5617ad2antrr 761 . . . . . . . . . . 11 (((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡𝐸 ∧ (𝑡‘(𝐺𝐷)) = 𝑓)) → (𝑄𝐴 ∧ ¬ 𝑄 𝑊))
57 fvex 6163 . . . . . . . . . . . 12 ((𝑁𝑡)‘𝐷) ∈ V
58 fvex 6163 . . . . . . . . . . . 12 (𝑁𝑡) ∈ V
597, 8, 2, 9, 13, 35, 3, 18, 57, 58dihopelvalcqat 36042 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (⟨((𝑁𝑡)‘𝐷), (𝑁𝑡)⟩ ∈ (𝐼𝑄) ↔ (((𝑁𝑡)‘𝐷) = ((𝑁𝑡)‘𝐷) ∧ (𝑁𝑡) ∈ 𝐸)))
6045, 56, 59syl2anc 692 . . . . . . . . . 10 (((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡𝐸 ∧ (𝑡‘(𝐺𝐷)) = 𝑓)) → (⟨((𝑁𝑡)‘𝐷), (𝑁𝑡)⟩ ∈ (𝐼𝑄) ↔ (((𝑁𝑡)‘𝐷) = ((𝑁𝑡)‘𝐷) ∧ (𝑁𝑡) ∈ 𝐸)))
6152, 55, 60mpbir2and 956 . . . . . . . . 9 (((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡𝐸 ∧ (𝑡‘(𝐺𝐷)) = 𝑓)) → ⟨((𝑁𝑡)‘𝐷), (𝑁𝑡)⟩ ∈ (𝐼𝑄))
6216ad2antrr 761 . . . . . . . . . . 11 (((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡𝐸 ∧ (𝑡‘(𝐺𝐷)) = 𝑓)) → 𝐺𝑇)
6322ad2antrr 761 . . . . . . . . . . 11 (((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡𝐸 ∧ (𝑡‘(𝐺𝐷)) = 𝑓)) → 𝐷𝑇)
642, 13, 35tendospdi1 35816 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑡𝐸𝐺𝑇𝐷𝑇)) → (𝑡‘(𝐺𝐷)) = ((𝑡𝐺) ∘ (𝑡𝐷)))
6545, 44, 62, 63, 64syl13anc 1325 . . . . . . . . . 10 (((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡𝐸 ∧ (𝑡‘(𝐺𝐷)) = 𝑓)) → (𝑡‘(𝐺𝐷)) = ((𝑡𝐺) ∘ (𝑡𝐷)))
66 simprr 795 . . . . . . . . . 10 (((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡𝐸 ∧ (𝑡‘(𝐺𝐷)) = 𝑓)) → (𝑡‘(𝐺𝐷)) = 𝑓)
6720ad2antrr 761 . . . . . . . . . . . . 13 (((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡𝐸 ∧ (𝑡‘(𝐺𝐷)) = 𝑓)) → 𝐷𝑇)
6853, 13tendoi2 35590 . . . . . . . . . . . . 13 ((𝑡𝐸𝐷𝑇) → ((𝑁𝑡)‘𝐷) = (𝑡𝐷))
6944, 67, 68syl2anc 692 . . . . . . . . . . . 12 (((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡𝐸 ∧ (𝑡‘(𝐺𝐷)) = 𝑓)) → ((𝑁𝑡)‘𝐷) = (𝑡𝐷))
702, 13, 35tendocnv 35817 . . . . . . . . . . . . 13 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑡𝐸𝐷𝑇) → (𝑡𝐷) = (𝑡𝐷))
7145, 44, 67, 70syl3anc 1323 . . . . . . . . . . . 12 (((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡𝐸 ∧ (𝑡‘(𝐺𝐷)) = 𝑓)) → (𝑡𝐷) = (𝑡𝐷))
7269, 71eqtr2d 2656 . . . . . . . . . . 11 (((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡𝐸 ∧ (𝑡‘(𝐺𝐷)) = 𝑓)) → (𝑡𝐷) = ((𝑁𝑡)‘𝐷))
7372coeq2d 5249 . . . . . . . . . 10 (((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡𝐸 ∧ (𝑡‘(𝐺𝐷)) = 𝑓)) → ((𝑡𝐺) ∘ (𝑡𝐷)) = ((𝑡𝐺) ∘ ((𝑁𝑡)‘𝐷)))
7465, 66, 733eqtr3d 2663 . . . . . . . . 9 (((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡𝐸 ∧ (𝑡‘(𝐺𝐷)) = 𝑓)) → 𝑓 = ((𝑡𝐺) ∘ ((𝑁𝑡)‘𝐷)))
75 simplrr 800 . . . . . . . . . 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 35593 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑡𝐸) → (𝑡𝐽(𝑁𝑡)) = 0 )
7945, 44, 78syl2anc 692 . . . . . . . . . 10 (((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡𝐸 ∧ (𝑡‘(𝐺𝐷)) = 𝑓)) → (𝑡𝐽(𝑁𝑡)) = 0 )
8075, 79eqtr4d 2658 . . . . . . . . 9 (((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡𝐸 ∧ (𝑡‘(𝐺𝐷)) = 𝑓)) → 𝑠 = (𝑡𝐽(𝑁𝑡)))
81 opeq1 4375 . . . . . . . . . . . . . . 15 (𝑔 = (𝑡𝐺) → ⟨𝑔, 𝑡⟩ = ⟨(𝑡𝐺), 𝑡⟩)
8281eleq1d 2683 . . . . . . . . . . . . . 14 (𝑔 = (𝑡𝐺) → (⟨𝑔, 𝑡⟩ ∈ (𝐼𝑃) ↔ ⟨(𝑡𝐺), 𝑡⟩ ∈ (𝐼𝑃)))
8382anbi1d 740 . . . . . . . . . . . . 13 (𝑔 = (𝑡𝐺) → ((⟨𝑔, 𝑡⟩ ∈ (𝐼𝑃) ∧ ⟨, 𝑢⟩ ∈ (𝐼𝑄)) ↔ (⟨(𝑡𝐺), 𝑡⟩ ∈ (𝐼𝑃) ∧ ⟨, 𝑢⟩ ∈ (𝐼𝑄))))
84 coeq1 5244 . . . . . . . . . . . . . . 15 (𝑔 = (𝑡𝐺) → (𝑔) = ((𝑡𝐺) ∘ ))
8584eqeq2d 2631 . . . . . . . . . . . . . 14 (𝑔 = (𝑡𝐺) → (𝑓 = (𝑔) ↔ 𝑓 = ((𝑡𝐺) ∘ )))
8685anbi1d 740 . . . . . . . . . . . . 13 (𝑔 = (𝑡𝐺) → ((𝑓 = (𝑔) ∧ 𝑠 = (𝑡𝐽𝑢)) ↔ (𝑓 = ((𝑡𝐺) ∘ ) ∧ 𝑠 = (𝑡𝐽𝑢))))
8783, 86anbi12d 746 . . . . . . . . . . . 12 (𝑔 = (𝑡𝐺) → (((⟨𝑔, 𝑡⟩ ∈ (𝐼𝑃) ∧ ⟨, 𝑢⟩ ∈ (𝐼𝑄)) ∧ (𝑓 = (𝑔) ∧ 𝑠 = (𝑡𝐽𝑢))) ↔ ((⟨(𝑡𝐺), 𝑡⟩ ∈ (𝐼𝑃) ∧ ⟨, 𝑢⟩ ∈ (𝐼𝑄)) ∧ (𝑓 = ((𝑡𝐺) ∘ ) ∧ 𝑠 = (𝑡𝐽𝑢)))))
88 opeq1 4375 . . . . . . . . . . . . . . 15 ( = ((𝑁𝑡)‘𝐷) → ⟨, 𝑢⟩ = ⟨((𝑁𝑡)‘𝐷), 𝑢⟩)
8988eleq1d 2683 . . . . . . . . . . . . . 14 ( = ((𝑁𝑡)‘𝐷) → (⟨, 𝑢⟩ ∈ (𝐼𝑄) ↔ ⟨((𝑁𝑡)‘𝐷), 𝑢⟩ ∈ (𝐼𝑄)))
9089anbi2d 739 . . . . . . . . . . . . 13 ( = ((𝑁𝑡)‘𝐷) → ((⟨(𝑡𝐺), 𝑡⟩ ∈ (𝐼𝑃) ∧ ⟨, 𝑢⟩ ∈ (𝐼𝑄)) ↔ (⟨(𝑡𝐺), 𝑡⟩ ∈ (𝐼𝑃) ∧ ⟨((𝑁𝑡)‘𝐷), 𝑢⟩ ∈ (𝐼𝑄))))
91 coeq2 5245 . . . . . . . . . . . . . . 15 ( = ((𝑁𝑡)‘𝐷) → ((𝑡𝐺) ∘ ) = ((𝑡𝐺) ∘ ((𝑁𝑡)‘𝐷)))
9291eqeq2d 2631 . . . . . . . . . . . . . 14 ( = ((𝑁𝑡)‘𝐷) → (𝑓 = ((𝑡𝐺) ∘ ) ↔ 𝑓 = ((𝑡𝐺) ∘ ((𝑁𝑡)‘𝐷))))
9392anbi1d 740 . . . . . . . . . . . . 13 ( = ((𝑁𝑡)‘𝐷) → ((𝑓 = ((𝑡𝐺) ∘ ) ∧ 𝑠 = (𝑡𝐽𝑢)) ↔ (𝑓 = ((𝑡𝐺) ∘ ((𝑁𝑡)‘𝐷)) ∧ 𝑠 = (𝑡𝐽𝑢))))
9490, 93anbi12d 746 . . . . . . . . . . . 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 4376 . . . . . . . . . . . . . . 15 (𝑢 = (𝑁𝑡) → ⟨((𝑁𝑡)‘𝐷), 𝑢⟩ = ⟨((𝑁𝑡)‘𝐷), (𝑁𝑡)⟩)
9695eleq1d 2683 . . . . . . . . . . . . . 14 (𝑢 = (𝑁𝑡) → (⟨((𝑁𝑡)‘𝐷), 𝑢⟩ ∈ (𝐼𝑄) ↔ ⟨((𝑁𝑡)‘𝐷), (𝑁𝑡)⟩ ∈ (𝐼𝑄)))
9796anbi2d 739 . . . . . . . . . . . . 13 (𝑢 = (𝑁𝑡) → ((⟨(𝑡𝐺), 𝑡⟩ ∈ (𝐼𝑃) ∧ ⟨((𝑁𝑡)‘𝐷), 𝑢⟩ ∈ (𝐼𝑄)) ↔ (⟨(𝑡𝐺), 𝑡⟩ ∈ (𝐼𝑃) ∧ ⟨((𝑁𝑡)‘𝐷), (𝑁𝑡)⟩ ∈ (𝐼𝑄))))
98 oveq2 6618 . . . . . . . . . . . . . . 15 (𝑢 = (𝑁𝑡) → (𝑡𝐽𝑢) = (𝑡𝐽(𝑁𝑡)))
9998eqeq2d 2631 . . . . . . . . . . . . . 14 (𝑢 = (𝑁𝑡) → (𝑠 = (𝑡𝐽𝑢) ↔ 𝑠 = (𝑡𝐽(𝑁𝑡))))
10099anbi2d 739 . . . . . . . . . . . . 13 (𝑢 = (𝑁𝑡) → ((𝑓 = ((𝑡𝐺) ∘ ((𝑁𝑡)‘𝐷)) ∧ 𝑠 = (𝑡𝐽𝑢)) ↔ (𝑓 = ((𝑡𝐺) ∘ ((𝑁𝑡)‘𝐷)) ∧ 𝑠 = (𝑡𝐽(𝑁𝑡)))))
10197, 100anbi12d 746 . . . . . . . . . . . 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 1394 . . . . . . . . . . 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 3286 . . . . . . . . . 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 1421 . . . . . . . . 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 1324 . . . . . . . 8 (((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡𝐸 ∧ (𝑡‘(𝐺𝐷)) = 𝑓)) → ∃𝑔𝑢((⟨𝑔, 𝑡⟩ ∈ (𝐼𝑃) ∧ ⟨, 𝑢⟩ ∈ (𝐼𝑄)) ∧ (𝑓 = (𝑔) ∧ 𝑠 = (𝑡𝐽𝑢))))
106105ex 450 . . . . . . 7 ((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) → ((𝑡𝐸 ∧ (𝑡‘(𝐺𝐷)) = 𝑓) → ∃𝑔𝑢((⟨𝑔, 𝑡⟩ ∈ (𝐼𝑃) ∧ ⟨, 𝑢⟩ ∈ (𝐼𝑄)) ∧ (𝑓 = (𝑔) ∧ 𝑠 = (𝑡𝐽𝑢)))))
107106eximdv 1843 . . . . . 6 ((𝜑 ∧ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )) → (∃𝑡(𝑡𝐸 ∧ (𝑡‘(𝐺𝐷)) = 𝑓) → ∃𝑡𝑔𝑢((⟨𝑔, 𝑡⟩ ∈ (𝐼𝑃) ∧ ⟨, 𝑢⟩ ∈ (𝐼𝑄)) ∧ (𝑓 = (𝑔) ∧ 𝑠 = (𝑡𝐽𝑢)))))
108 excom 2039 . . . . . 6 (∃𝑡𝑔𝑢((⟨𝑔, 𝑡⟩ ∈ (𝐼𝑃) ∧ ⟨, 𝑢⟩ ∈ (𝐼𝑄)) ∧ (𝑓 = (𝑔) ∧ 𝑠 = (𝑡𝐽𝑢))) ↔ ∃𝑔𝑡𝑢((⟨𝑔, 𝑡⟩ ∈ (𝐼𝑃) ∧ ⟨, 𝑢⟩ ∈ (𝐼𝑄)) ∧ (𝑓 = (𝑔) ∧ 𝑠 = (𝑡𝐽𝑢))))
109107, 108syl6ib 241 . . . . 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 450 . . 3 (𝜑 → (((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 ) → ∃𝑔𝑡𝑢((⟨𝑔, 𝑡⟩ ∈ (𝐼𝑃) ∧ ⟨, 𝑢⟩ ∈ (𝐼𝑄)) ∧ (𝑓 = (𝑔) ∧ 𝑠 = (𝑡𝐽𝑢)))))
1121simpld 475 . . . . . . . . 9 (𝜑𝐾 ∈ HL)
113 hllat 34157 . . . . . . . . 9 (𝐾 ∈ HL → 𝐾 ∈ Lat)
114112, 113syl 17 . . . . . . . 8 (𝜑𝐾 ∈ Lat)
11512simpld 475 . . . . . . . . 9 (𝜑𝑃𝐴)
11617simpld 475 . . . . . . . . 9 (𝜑𝑄𝐴)
11728, 29, 8hlatjcl 34160 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → (𝑃 𝑄) ∈ 𝐵)
118112, 115, 116, 117syl3anc 1323 . . . . . . . 8 (𝜑 → (𝑃 𝑄) ∈ 𝐵)
1191simprd 479 . . . . . . . . 9 (𝜑𝑊𝐻)
12028, 2lhpbase 34791 . . . . . . . . 9 (𝑊𝐻𝑊𝐵)
121119, 120syl 17 . . . . . . . 8 (𝜑𝑊𝐵)
12228, 30latmcl 16980 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ 𝐵𝑊𝐵) → ((𝑃 𝑄) 𝑊) ∈ 𝐵)
123114, 118, 121, 122syl3anc 1323 . . . . . . 7 (𝜑 → ((𝑃 𝑄) 𝑊) ∈ 𝐵)
12433, 123syl5eqel 2702 . . . . . 6 (𝜑𝑉𝐵)
12528, 7, 30latmle2 17005 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ 𝐵𝑊𝐵) → ((𝑃 𝑄) 𝑊) 𝑊)
126114, 118, 121, 125syl3anc 1323 . . . . . . 7 (𝜑 → ((𝑃 𝑄) 𝑊) 𝑊)
12733, 126syl5eqbr 4653 . . . . . 6 (𝜑𝑉 𝑊)
128 eqid 2621 . . . . . . 7 ((DIsoB‘𝐾)‘𝑊) = ((DIsoB‘𝐾)‘𝑊)
12928, 7, 2, 3, 128dihvalb 36033 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑉𝐵𝑉 𝑊)) → (𝐼𝑉) = (((DIsoB‘𝐾)‘𝑊)‘𝑉))
1301, 124, 127, 129syl12anc 1321 . . . . 5 (𝜑 → (𝐼𝑉) = (((DIsoB‘𝐾)‘𝑊)‘𝑉))
131130eleq2d 2684 . . . 4 (𝜑 → (⟨𝑓, 𝑠⟩ ∈ (𝐼𝑉) ↔ ⟨𝑓, 𝑠⟩ ∈ (((DIsoB‘𝐾)‘𝑊)‘𝑉)))
13228, 7, 2, 13, 34, 77, 128dibopelval3 35944 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑉𝐵𝑉 𝑊)) → (⟨𝑓, 𝑠⟩ ∈ (((DIsoB‘𝐾)‘𝑊)‘𝑉) ↔ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )))
1331, 124, 127, 132syl12anc 1321 . . . 4 (𝜑 → (⟨𝑓, 𝑠⟩ ∈ (((DIsoB‘𝐾)‘𝑊)‘𝑉) ↔ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )))
134131, 133bitrd 268 . . 3 (𝜑 → (⟨𝑓, 𝑠⟩ ∈ (𝐼𝑉) ↔ ((𝑓𝑇 ∧ (𝑅𝑓) 𝑉) ∧ 𝑠 = 0 )))
135 eqid 2621 . . . 4 (LSubSp‘𝑈) = (LSubSp‘𝑈)
13628, 8atbase 34083 . . . . 5 (𝑃𝐴𝑃𝐵)
137115, 136syl 17 . . . 4 (𝜑𝑃𝐵)
13828, 8atbase 34083 . . . . 5 (𝑄𝐴𝑄𝐵)
139116, 138syl 17 . . . 4 (𝜑𝑄𝐵)
14028, 2, 13, 35, 76, 31, 135, 32, 3, 1, 137, 139dihopellsm 36051 . . 3 (𝜑 → (⟨𝑓, 𝑠⟩ ∈ ((𝐼𝑃) (𝐼𝑄)) ↔ ∃𝑔𝑡𝑢((⟨𝑔, 𝑡⟩ ∈ (𝐼𝑃) ∧ ⟨, 𝑢⟩ ∈ (𝐼𝑄)) ∧ (𝑓 = (𝑔) ∧ 𝑠 = (𝑡𝐽𝑢)))))
141111, 134, 1403imtr4d 283 . 2 (𝜑 → (⟨𝑓, 𝑠⟩ ∈ (𝐼𝑉) → ⟨𝑓, 𝑠⟩ ∈ ((𝐼𝑃) (𝐼𝑄))))
1425, 141relssdv 5178 1 (𝜑 → (𝐼𝑉) ⊆ ((𝐼𝑃) (𝐼𝑄)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384   = wceq 1480  wex 1701  wcel 1987  wrex 2908  Vcvv 3189  wss 3559  cop 4159   class class class wbr 4618  cmpt 4678   I cid 4989  ccnv 5078  cres 5081  ccom 5083  Rel wrel 5084  cfv 5852  crio 6570  (class class class)co 6610  cmpt2 6612  Basecbs 15788  lecple 15876  occoc 15877  joincjn 16872  meetcmee 16873  Latclat 16973  LSSumclsm 17977  LSubSpclss 18860  Atomscatm 34057  HLchlt 34144  LHypclh 34777  LTrncltrn 34894  trLctrl 34952  TEndoctendo 35547  DVecHcdvh 35874  DIsoBcdib 35934  DIsoHcdih 36024
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-cnex 9943  ax-resscn 9944  ax-1cn 9945  ax-icn 9946  ax-addcl 9947  ax-addrcl 9948  ax-mulcl 9949  ax-mulrcl 9950  ax-mulcom 9951  ax-addass 9952  ax-mulass 9953  ax-distr 9954  ax-i2m1 9955  ax-1ne0 9956  ax-1rid 9957  ax-rnegex 9958  ax-rrecex 9959  ax-cnre 9960  ax-pre-lttri 9961  ax-pre-lttrn 9962  ax-pre-ltadd 9963  ax-pre-mulgt0 9964  ax-riotaBAD 33746
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-iin 4493  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-om 7020  df-1st 7120  df-2nd 7121  df-tpos 7304  df-undef 7351  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-1o 7512  df-oadd 7516  df-er 7694  df-map 7811  df-en 7907  df-dom 7908  df-sdom 7909  df-fin 7910  df-pnf 10027  df-mnf 10028  df-xr 10029  df-ltxr 10030  df-le 10031  df-sub 10219  df-neg 10220  df-nn 10972  df-2 11030  df-3 11031  df-4 11032  df-5 11033  df-6 11034  df-n0 11244  df-z 11329  df-uz 11639  df-fz 12276  df-struct 15790  df-ndx 15791  df-slot 15792  df-base 15793  df-sets 15794  df-ress 15795  df-plusg 15882  df-mulr 15883  df-sca 15885  df-vsca 15886  df-0g 16030  df-preset 16856  df-poset 16874  df-plt 16886  df-lub 16902  df-glb 16903  df-join 16904  df-meet 16905  df-p0 16967  df-p1 16968  df-lat 16974  df-clat 17036  df-mgm 17170  df-sgrp 17212  df-mnd 17223  df-submnd 17264  df-grp 17353  df-minusg 17354  df-sbg 17355  df-subg 17519  df-cntz 17678  df-lsm 17979  df-cmn 18123  df-abl 18124  df-mgp 18418  df-ur 18430  df-ring 18477  df-oppr 18551  df-dvdsr 18569  df-unit 18570  df-invr 18600  df-dvr 18611  df-drng 18677  df-lmod 18793  df-lss 18861  df-lsp 18900  df-lvec 19031  df-oposet 33970  df-ol 33972  df-oml 33973  df-covers 34060  df-ats 34061  df-atl 34092  df-cvlat 34116  df-hlat 34145  df-llines 34291  df-lplanes 34292  df-lvols 34293  df-lines 34294  df-psubsp 34296  df-pmap 34297  df-padd 34589  df-lhyp 34781  df-laut 34782  df-ldil 34897  df-ltrn 34898  df-trl 34953  df-tendo 35550  df-edring 35552  df-disoa 35825  df-dvech 35875  df-dib 35935  df-dic 35969  df-dih 36025
This theorem is referenced by:  dihjatcc  36218
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