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 ∧ 𝑊 ∈ 𝐻))
dihjatcclem.p	⊢ (𝜑 → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
dihjatcclem.q	⊢ (𝜑 → (𝑄 ∈ 𝐴 ∧ ¬ 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 = (℩𝑑 ∈ 𝑇 (𝑑‘𝐶) = 𝑃)
dihjatcc.dd	⊢ 𝐷 = (℩𝑑 ∈ 𝑇 (𝑑‘𝐶) = 𝑄)
dihjatcc.n	⊢ 𝑁 = (𝑎 ∈ 𝐸 ↦ (𝑑 ∈ 𝑇 ↦ ◡(𝑎‘𝑑)))
dihjatcc.o	⊢ 0 = (𝑑 ∈ 𝑇 ↦ ( I ↾ 𝐵))
dihjatcc.d	⊢ 𝐽 = (𝑎 ∈ 𝐸, 𝑏 ∈ 𝐸 ↦ (𝑑 ∈ 𝑇 ↦ ((𝑎‘𝑑) ∘ (𝑏‘𝑑))))

Assertion

Ref	Expression
dihjatcclem4	⊢ (𝜑 → (𝐼‘𝑉) ⊆ ((𝐼‘𝑃) ⊕ (𝐼‘𝑄)))

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 ℎ 𝑢 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dihjatcclem.k	. . 3 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
2		dihjatcclem.h	. . . 4 ⊢ 𝐻 = (LHyp‘𝐾)
3		dihjatcclem.i	. . . 4 ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)
4	2, 3	dihvalrel 40980	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → Rel (𝐼‘𝑉))
5	1, 4	syl 17	. 2 ⊢ (𝜑 → Rel (𝐼‘𝑉))
6	1	adantr 479	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑉) ∧ 𝑠 = 0 )) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
7		dihjatcclem.l	. . . . . . . . . . . 12 ⊢ ≤ = (le‘𝐾)
8		dihjatcclem.a	. . . . . . . . . . . 12 ⊢ 𝐴 = (Atoms‘𝐾)
9		dihjatcc.w	. . . . . . . . . . . 12 ⊢ 𝐶 = ((oc‘𝐾)‘𝑊)
10	7, 8, 2, 9	lhpocnel2 39720	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝐶 ∈ 𝐴 ∧ ¬ 𝐶 ≤ 𝑊))
11	1, 10	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (𝐶 ∈ 𝐴 ∧ ¬ 𝐶 ≤ 𝑊))
12		dihjatcclem.p	. . . . . . . . . 10 ⊢ (𝜑 → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
13		dihjatcc.t	. . . . . . . . . . 11 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
14		dihjatcc.g	. . . . . . . . . . 11 ⊢ 𝐺 = (℩𝑑 ∈ 𝑇 (𝑑‘𝐶) = 𝑃)
15	7, 8, 2, 13, 14	ltrniotacl 40280	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐶 ∈ 𝐴 ∧ ¬ 𝐶 ≤ 𝑊) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝐺 ∈ 𝑇)
16	1, 11, 12, 15	syl3anc 1368	. . . . . . . . 9 ⊢ (𝜑 → 𝐺 ∈ 𝑇)
17		dihjatcclem.q	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))
18		dihjatcc.dd	. . . . . . . . . . . 12 ⊢ 𝐷 = (℩𝑑 ∈ 𝑇 (𝑑‘𝐶) = 𝑄)
19	7, 8, 2, 13, 18	ltrniotacl 40280	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐶 ∈ 𝐴 ∧ ¬ 𝐶 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → 𝐷 ∈ 𝑇)
20	1, 11, 17, 19	syl3anc 1368	. . . . . . . . . 10 ⊢ (𝜑 → 𝐷 ∈ 𝑇)
21	2, 13	ltrncnv 39847	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐷 ∈ 𝑇) → ◡𝐷 ∈ 𝑇)
22	1, 20, 21	syl2anc 582	. . . . . . . . 9 ⊢ (𝜑 → ◡𝐷 ∈ 𝑇)
23	2, 13	ltrnco 40420	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇 ∧ ◡𝐷 ∈ 𝑇) → (𝐺 ∘ ◡𝐷) ∈ 𝑇)
24	1, 16, 22, 23	syl3anc 1368	. . . . . . . 8 ⊢ (𝜑 → (𝐺 ∘ ◡𝐷) ∈ 𝑇)
25	24	adantr 479	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑉) ∧ 𝑠 = 0 )) → (𝐺 ∘ ◡𝐷) ∈ 𝑇)
26		simprll 777	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑉) ∧ 𝑠 = 0 )) → 𝑓 ∈ 𝑇)
27		simprlr 778	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑉) ∧ 𝑠 = 0 )) → (𝑅‘𝑓) ≤ 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)
36	28, 7, 2, 29, 30, 8, 31, 32, 3, 33, 1, 12, 17, 9, 13, 34, 35, 14, 18	dihjatcclem3 41121	. . . . . . . . 9 ⊢ (𝜑 → (𝑅‘(𝐺 ∘ ◡𝐷)) = 𝑉)
37	36	adantr 479	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑉) ∧ 𝑠 = 0 )) → (𝑅‘(𝐺 ∘ ◡𝐷)) = 𝑉)
38	27, 37	breqtrrd 5183	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑉) ∧ 𝑠 = 0 )) → (𝑅‘𝑓) ≤ (𝑅‘(𝐺 ∘ ◡𝐷)))
39	7, 2, 13, 34, 35	tendoex 40676	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝐺 ∘ ◡𝐷) ∈ 𝑇 ∧ 𝑓 ∈ 𝑇) ∧ (𝑅‘𝑓) ≤ (𝑅‘(𝐺 ∘ ◡𝐷))) → ∃𝑡 ∈ 𝐸 (𝑡‘(𝐺 ∘ ◡𝐷)) = 𝑓)
40	6, 25, 26, 38, 39	syl121anc 1372	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑉) ∧ 𝑠 = 0 )) → ∃𝑡 ∈ 𝐸 (𝑡‘(𝐺 ∘ ◡𝐷)) = 𝑓)
41		df-rex 3061	. . . . . 6 ⊢ (∃𝑡 ∈ 𝐸 (𝑡‘(𝐺 ∘ ◡𝐷)) = 𝑓 ↔ ∃𝑡(𝑡 ∈ 𝐸 ∧ (𝑡‘(𝐺 ∘ ◡𝐷)) = 𝑓))
42	40, 41	sylib 217	. . . . 5 ⊢ ((𝜑 ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑉) ∧ 𝑠 = 0 )) → ∃𝑡(𝑡 ∈ 𝐸 ∧ (𝑡‘(𝐺 ∘ ◡𝐷)) = 𝑓))
43		eqidd 2727	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡 ∈ 𝐸 ∧ (𝑡‘(𝐺 ∘ ◡𝐷)) = 𝑓)) → (𝑡‘𝐺) = (𝑡‘𝐺))
44		simprl 769	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡 ∈ 𝐸 ∧ (𝑡‘(𝐺 ∘ ◡𝐷)) = 𝑓)) → 𝑡 ∈ 𝐸)
45	1	ad2antrr 724	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡 ∈ 𝐸 ∧ (𝑡‘(𝐺 ∘ ◡𝐷)) = 𝑓)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
46	12	ad2antrr 724	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡 ∈ 𝐸 ∧ (𝑡‘(𝐺 ∘ ◡𝐷)) = 𝑓)) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
47		fvex 6916	. . . . . . . . . . . 12 ⊢ (𝑡‘𝐺) ∈ V
48		vex 3466	. . . . . . . . . . . 12 ⊢ 𝑡 ∈ V
49	7, 8, 2, 9, 13, 35, 3, 14, 47, 48	dihopelvalcqat 40947	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (⟨(𝑡‘𝐺), 𝑡⟩ ∈ (𝐼‘𝑃) ↔ ((𝑡‘𝐺) = (𝑡‘𝐺) ∧ 𝑡 ∈ 𝐸)))
50	45, 46, 49	syl2anc 582	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡 ∈ 𝐸 ∧ (𝑡‘(𝐺 ∘ ◡𝐷)) = 𝑓)) → (⟨(𝑡‘𝐺), 𝑡⟩ ∈ (𝐼‘𝑃) ↔ ((𝑡‘𝐺) = (𝑡‘𝐺) ∧ 𝑡 ∈ 𝐸)))
51	43, 44, 50	mpbir2and 711	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡 ∈ 𝐸 ∧ (𝑡‘(𝐺 ∘ ◡𝐷)) = 𝑓)) → ⟨(𝑡‘𝐺), 𝑡⟩ ∈ (𝐼‘𝑃))
52		eqidd 2727	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡 ∈ 𝐸 ∧ (𝑡‘(𝐺 ∘ ◡𝐷)) = 𝑓)) → ((𝑁‘𝑡)‘𝐷) = ((𝑁‘𝑡)‘𝐷))
53		dihjatcc.n	. . . . . . . . . . . 12 ⊢ 𝑁 = (𝑎 ∈ 𝐸 ↦ (𝑑 ∈ 𝑇 ↦ ◡(𝑎‘𝑑)))
54	2, 13, 35, 53	tendoicl 40497	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑡 ∈ 𝐸) → (𝑁‘𝑡) ∈ 𝐸)
55	45, 44, 54	syl2anc 582	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡 ∈ 𝐸 ∧ (𝑡‘(𝐺 ∘ ◡𝐷)) = 𝑓)) → (𝑁‘𝑡) ∈ 𝐸)
56	17	ad2antrr 724	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡 ∈ 𝐸 ∧ (𝑡‘(𝐺 ∘ ◡𝐷)) = 𝑓)) → (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))
57		fvex 6916	. . . . . . . . . . . 12 ⊢ ((𝑁‘𝑡)‘𝐷) ∈ V
58		fvex 6916	. . . . . . . . . . . 12 ⊢ (𝑁‘𝑡) ∈ V
59	7, 8, 2, 9, 13, 35, 3, 18, 57, 58	dihopelvalcqat 40947	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (⟨((𝑁‘𝑡)‘𝐷), (𝑁‘𝑡)⟩ ∈ (𝐼‘𝑄) ↔ (((𝑁‘𝑡)‘𝐷) = ((𝑁‘𝑡)‘𝐷) ∧ (𝑁‘𝑡) ∈ 𝐸)))
60	45, 56, 59	syl2anc 582	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡 ∈ 𝐸 ∧ (𝑡‘(𝐺 ∘ ◡𝐷)) = 𝑓)) → (⟨((𝑁‘𝑡)‘𝐷), (𝑁‘𝑡)⟩ ∈ (𝐼‘𝑄) ↔ (((𝑁‘𝑡)‘𝐷) = ((𝑁‘𝑡)‘𝐷) ∧ (𝑁‘𝑡) ∈ 𝐸)))
61	52, 55, 60	mpbir2and 711	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡 ∈ 𝐸 ∧ (𝑡‘(𝐺 ∘ ◡𝐷)) = 𝑓)) → ⟨((𝑁‘𝑡)‘𝐷), (𝑁‘𝑡)⟩ ∈ (𝐼‘𝑄))
62	16	ad2antrr 724	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡 ∈ 𝐸 ∧ (𝑡‘(𝐺 ∘ ◡𝐷)) = 𝑓)) → 𝐺 ∈ 𝑇)
63	22	ad2antrr 724	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡 ∈ 𝐸 ∧ (𝑡‘(𝐺 ∘ ◡𝐷)) = 𝑓)) → ◡𝐷 ∈ 𝑇)
64	2, 13, 35	tendospdi1 40721	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑡 ∈ 𝐸 ∧ 𝐺 ∈ 𝑇 ∧ ◡𝐷 ∈ 𝑇)) → (𝑡‘(𝐺 ∘ ◡𝐷)) = ((𝑡‘𝐺) ∘ (𝑡‘◡𝐷)))
65	45, 44, 62, 63, 64	syl13anc 1369	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡 ∈ 𝐸 ∧ (𝑡‘(𝐺 ∘ ◡𝐷)) = 𝑓)) → (𝑡‘(𝐺 ∘ ◡𝐷)) = ((𝑡‘𝐺) ∘ (𝑡‘◡𝐷)))
66		simprr 771	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡 ∈ 𝐸 ∧ (𝑡‘(𝐺 ∘ ◡𝐷)) = 𝑓)) → (𝑡‘(𝐺 ∘ ◡𝐷)) = 𝑓)
67	20	ad2antrr 724	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡 ∈ 𝐸 ∧ (𝑡‘(𝐺 ∘ ◡𝐷)) = 𝑓)) → 𝐷 ∈ 𝑇)
68	53, 13	tendoi2 40496	. . . . . . . . . . . . 13 ⊢ ((𝑡 ∈ 𝐸 ∧ 𝐷 ∈ 𝑇) → ((𝑁‘𝑡)‘𝐷) = ◡(𝑡‘𝐷))
69	44, 67, 68	syl2anc 582	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡 ∈ 𝐸 ∧ (𝑡‘(𝐺 ∘ ◡𝐷)) = 𝑓)) → ((𝑁‘𝑡)‘𝐷) = ◡(𝑡‘𝐷))
70	2, 13, 35	tendocnv 40722	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑡 ∈ 𝐸 ∧ 𝐷 ∈ 𝑇) → ◡(𝑡‘𝐷) = (𝑡‘◡𝐷))
71	45, 44, 67, 70	syl3anc 1368	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡 ∈ 𝐸 ∧ (𝑡‘(𝐺 ∘ ◡𝐷)) = 𝑓)) → ◡(𝑡‘𝐷) = (𝑡‘◡𝐷))
72	69, 71	eqtr2d 2767	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡 ∈ 𝐸 ∧ (𝑡‘(𝐺 ∘ ◡𝐷)) = 𝑓)) → (𝑡‘◡𝐷) = ((𝑁‘𝑡)‘𝐷))
73	72	coeq2d 5871	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡 ∈ 𝐸 ∧ (𝑡‘(𝐺 ∘ ◡𝐷)) = 𝑓)) → ((𝑡‘𝐺) ∘ (𝑡‘◡𝐷)) = ((𝑡‘𝐺) ∘ ((𝑁‘𝑡)‘𝐷)))
74	65, 66, 73	3eqtr3d 2774	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡 ∈ 𝐸 ∧ (𝑡‘(𝐺 ∘ ◡𝐷)) = 𝑓)) → 𝑓 = ((𝑡‘𝐺) ∘ ((𝑁‘𝑡)‘𝐷)))
75		simplrr 776	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡 ∈ 𝐸 ∧ (𝑡‘(𝐺 ∘ ◡𝐷)) = 𝑓)) → 𝑠 = 0 )
76		dihjatcc.d	. . . . . . . . . . . 12 ⊢ 𝐽 = (𝑎 ∈ 𝐸, 𝑏 ∈ 𝐸 ↦ (𝑑 ∈ 𝑇 ↦ ((𝑎‘𝑑) ∘ (𝑏‘𝑑))))
77		dihjatcc.o	. . . . . . . . . . . 12 ⊢ 0 = (𝑑 ∈ 𝑇 ↦ ( I ↾ 𝐵))
78	2, 13, 35, 53, 28, 76, 77	tendoipl2 40499	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑡 ∈ 𝐸) → (𝑡𝐽(𝑁‘𝑡)) = 0 )
79	45, 44, 78	syl2anc 582	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡 ∈ 𝐸 ∧ (𝑡‘(𝐺 ∘ ◡𝐷)) = 𝑓)) → (𝑡𝐽(𝑁‘𝑡)) = 0 )
80	75, 79	eqtr4d 2769	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡 ∈ 𝐸 ∧ (𝑡‘(𝐺 ∘ ◡𝐷)) = 𝑓)) → 𝑠 = (𝑡𝐽(𝑁‘𝑡)))
81		opeq1 4881	. . . . . . . . . . . . . . 15 ⊢ (𝑔 = (𝑡‘𝐺) → ⟨𝑔, 𝑡⟩ = ⟨(𝑡‘𝐺), 𝑡⟩)
82	81	eleq1d 2811	. . . . . . . . . . . . . 14 ⊢ (𝑔 = (𝑡‘𝐺) → (⟨𝑔, 𝑡⟩ ∈ (𝐼‘𝑃) ↔ ⟨(𝑡‘𝐺), 𝑡⟩ ∈ (𝐼‘𝑃)))
83	82	anbi1d 629	. . . . . . . . . . . . 13 ⊢ (𝑔 = (𝑡‘𝐺) → ((⟨𝑔, 𝑡⟩ ∈ (𝐼‘𝑃) ∧ ⟨ℎ, 𝑢⟩ ∈ (𝐼‘𝑄)) ↔ (⟨(𝑡‘𝐺), 𝑡⟩ ∈ (𝐼‘𝑃) ∧ ⟨ℎ, 𝑢⟩ ∈ (𝐼‘𝑄))))
84		coeq1 5866	. . . . . . . . . . . . . . 15 ⊢ (𝑔 = (𝑡‘𝐺) → (𝑔 ∘ ℎ) = ((𝑡‘𝐺) ∘ ℎ))
85	84	eqeq2d 2737	. . . . . . . . . . . . . 14 ⊢ (𝑔 = (𝑡‘𝐺) → (𝑓 = (𝑔 ∘ ℎ) ↔ 𝑓 = ((𝑡‘𝐺) ∘ ℎ)))
86	85	anbi1d 629	. . . . . . . . . . . . 13 ⊢ (𝑔 = (𝑡‘𝐺) → ((𝑓 = (𝑔 ∘ ℎ) ∧ 𝑠 = (𝑡𝐽𝑢)) ↔ (𝑓 = ((𝑡‘𝐺) ∘ ℎ) ∧ 𝑠 = (𝑡𝐽𝑢))))
87	83, 86	anbi12d 630	. . . . . . . . . . . 12 ⊢ (𝑔 = (𝑡‘𝐺) → (((⟨𝑔, 𝑡⟩ ∈ (𝐼‘𝑃) ∧ ⟨ℎ, 𝑢⟩ ∈ (𝐼‘𝑄)) ∧ (𝑓 = (𝑔 ∘ ℎ) ∧ 𝑠 = (𝑡𝐽𝑢))) ↔ ((⟨(𝑡‘𝐺), 𝑡⟩ ∈ (𝐼‘𝑃) ∧ ⟨ℎ, 𝑢⟩ ∈ (𝐼‘𝑄)) ∧ (𝑓 = ((𝑡‘𝐺) ∘ ℎ) ∧ 𝑠 = (𝑡𝐽𝑢)))))
88		opeq1 4881	. . . . . . . . . . . . . . 15 ⊢ (ℎ = ((𝑁‘𝑡)‘𝐷) → ⟨ℎ, 𝑢⟩ = ⟨((𝑁‘𝑡)‘𝐷), 𝑢⟩)
89	88	eleq1d 2811	. . . . . . . . . . . . . 14 ⊢ (ℎ = ((𝑁‘𝑡)‘𝐷) → (⟨ℎ, 𝑢⟩ ∈ (𝐼‘𝑄) ↔ ⟨((𝑁‘𝑡)‘𝐷), 𝑢⟩ ∈ (𝐼‘𝑄)))
90	89	anbi2d 628	. . . . . . . . . . . . 13 ⊢ (ℎ = ((𝑁‘𝑡)‘𝐷) → ((⟨(𝑡‘𝐺), 𝑡⟩ ∈ (𝐼‘𝑃) ∧ ⟨ℎ, 𝑢⟩ ∈ (𝐼‘𝑄)) ↔ (⟨(𝑡‘𝐺), 𝑡⟩ ∈ (𝐼‘𝑃) ∧ ⟨((𝑁‘𝑡)‘𝐷), 𝑢⟩ ∈ (𝐼‘𝑄))))
91		coeq2 5867	. . . . . . . . . . . . . . 15 ⊢ (ℎ = ((𝑁‘𝑡)‘𝐷) → ((𝑡‘𝐺) ∘ ℎ) = ((𝑡‘𝐺) ∘ ((𝑁‘𝑡)‘𝐷)))
92	91	eqeq2d 2737	. . . . . . . . . . . . . 14 ⊢ (ℎ = ((𝑁‘𝑡)‘𝐷) → (𝑓 = ((𝑡‘𝐺) ∘ ℎ) ↔ 𝑓 = ((𝑡‘𝐺) ∘ ((𝑁‘𝑡)‘𝐷))))
93	92	anbi1d 629	. . . . . . . . . . . . 13 ⊢ (ℎ = ((𝑁‘𝑡)‘𝐷) → ((𝑓 = ((𝑡‘𝐺) ∘ ℎ) ∧ 𝑠 = (𝑡𝐽𝑢)) ↔ (𝑓 = ((𝑡‘𝐺) ∘ ((𝑁‘𝑡)‘𝐷)) ∧ 𝑠 = (𝑡𝐽𝑢))))
94	90, 93	anbi12d 630	. . . . . . . . . . . 12 ⊢ (ℎ = ((𝑁‘𝑡)‘𝐷) → (((⟨(𝑡‘𝐺), 𝑡⟩ ∈ (𝐼‘𝑃) ∧ ⟨ℎ, 𝑢⟩ ∈ (𝐼‘𝑄)) ∧ (𝑓 = ((𝑡‘𝐺) ∘ ℎ) ∧ 𝑠 = (𝑡𝐽𝑢))) ↔ ((⟨(𝑡‘𝐺), 𝑡⟩ ∈ (𝐼‘𝑃) ∧ ⟨((𝑁‘𝑡)‘𝐷), 𝑢⟩ ∈ (𝐼‘𝑄)) ∧ (𝑓 = ((𝑡‘𝐺) ∘ ((𝑁‘𝑡)‘𝐷)) ∧ 𝑠 = (𝑡𝐽𝑢)))))
95		opeq2 4882	. . . . . . . . . . . . . . 15 ⊢ (𝑢 = (𝑁‘𝑡) → ⟨((𝑁‘𝑡)‘𝐷), 𝑢⟩ = ⟨((𝑁‘𝑡)‘𝐷), (𝑁‘𝑡)⟩)
96	95	eleq1d 2811	. . . . . . . . . . . . . 14 ⊢ (𝑢 = (𝑁‘𝑡) → (⟨((𝑁‘𝑡)‘𝐷), 𝑢⟩ ∈ (𝐼‘𝑄) ↔ ⟨((𝑁‘𝑡)‘𝐷), (𝑁‘𝑡)⟩ ∈ (𝐼‘𝑄)))
97	96	anbi2d 628	. . . . . . . . . . . . 13 ⊢ (𝑢 = (𝑁‘𝑡) → ((⟨(𝑡‘𝐺), 𝑡⟩ ∈ (𝐼‘𝑃) ∧ ⟨((𝑁‘𝑡)‘𝐷), 𝑢⟩ ∈ (𝐼‘𝑄)) ↔ (⟨(𝑡‘𝐺), 𝑡⟩ ∈ (𝐼‘𝑃) ∧ ⟨((𝑁‘𝑡)‘𝐷), (𝑁‘𝑡)⟩ ∈ (𝐼‘𝑄))))
98		oveq2 7434	. . . . . . . . . . . . . . 15 ⊢ (𝑢 = (𝑁‘𝑡) → (𝑡𝐽𝑢) = (𝑡𝐽(𝑁‘𝑡)))
99	98	eqeq2d 2737	. . . . . . . . . . . . . 14 ⊢ (𝑢 = (𝑁‘𝑡) → (𝑠 = (𝑡𝐽𝑢) ↔ 𝑠 = (𝑡𝐽(𝑁‘𝑡))))
100	99	anbi2d 628	. . . . . . . . . . . . 13 ⊢ (𝑢 = (𝑁‘𝑡) → ((𝑓 = ((𝑡‘𝐺) ∘ ((𝑁‘𝑡)‘𝐷)) ∧ 𝑠 = (𝑡𝐽𝑢)) ↔ (𝑓 = ((𝑡‘𝐺) ∘ ((𝑁‘𝑡)‘𝐷)) ∧ 𝑠 = (𝑡𝐽(𝑁‘𝑡)))))
101	97, 100	anbi12d 630	. . . . . . . . . . . 12 ⊢ (𝑢 = (𝑁‘𝑡) → (((⟨(𝑡‘𝐺), 𝑡⟩ ∈ (𝐼‘𝑃) ∧ ⟨((𝑁‘𝑡)‘𝐷), 𝑢⟩ ∈ (𝐼‘𝑄)) ∧ (𝑓 = ((𝑡‘𝐺) ∘ ((𝑁‘𝑡)‘𝐷)) ∧ 𝑠 = (𝑡𝐽𝑢))) ↔ ((⟨(𝑡‘𝐺), 𝑡⟩ ∈ (𝐼‘𝑃) ∧ ⟨((𝑁‘𝑡)‘𝐷), (𝑁‘𝑡)⟩ ∈ (𝐼‘𝑄)) ∧ (𝑓 = ((𝑡‘𝐺) ∘ ((𝑁‘𝑡)‘𝐷)) ∧ 𝑠 = (𝑡𝐽(𝑁‘𝑡))))))
102	87, 94, 101	syl3an9b 1431	. . . . . . . . . . 11 ⊢ ((𝑔 = (𝑡‘𝐺) ∧ ℎ = ((𝑁‘𝑡)‘𝐷) ∧ 𝑢 = (𝑁‘𝑡)) → (((⟨𝑔, 𝑡⟩ ∈ (𝐼‘𝑃) ∧ ⟨ℎ, 𝑢⟩ ∈ (𝐼‘𝑄)) ∧ (𝑓 = (𝑔 ∘ ℎ) ∧ 𝑠 = (𝑡𝐽𝑢))) ↔ ((⟨(𝑡‘𝐺), 𝑡⟩ ∈ (𝐼‘𝑃) ∧ ⟨((𝑁‘𝑡)‘𝐷), (𝑁‘𝑡)⟩ ∈ (𝐼‘𝑄)) ∧ (𝑓 = ((𝑡‘𝐺) ∘ ((𝑁‘𝑡)‘𝐷)) ∧ 𝑠 = (𝑡𝐽(𝑁‘𝑡))))))
103	102	spc3egv 3589	. . . . . . . . . 10 ⊢ (((𝑡‘𝐺) ∈ V ∧ ((𝑁‘𝑡)‘𝐷) ∈ V ∧ (𝑁‘𝑡) ∈ V) → (((⟨(𝑡‘𝐺), 𝑡⟩ ∈ (𝐼‘𝑃) ∧ ⟨((𝑁‘𝑡)‘𝐷), (𝑁‘𝑡)⟩ ∈ (𝐼‘𝑄)) ∧ (𝑓 = ((𝑡‘𝐺) ∘ ((𝑁‘𝑡)‘𝐷)) ∧ 𝑠 = (𝑡𝐽(𝑁‘𝑡)))) → ∃𝑔∃ℎ∃𝑢((⟨𝑔, 𝑡⟩ ∈ (𝐼‘𝑃) ∧ ⟨ℎ, 𝑢⟩ ∈ (𝐼‘𝑄)) ∧ (𝑓 = (𝑔 ∘ ℎ) ∧ 𝑠 = (𝑡𝐽𝑢)))))
104	47, 57, 58, 103	mp3an 1458	. . . . . . . . 9 ⊢ (((⟨(𝑡‘𝐺), 𝑡⟩ ∈ (𝐼‘𝑃) ∧ ⟨((𝑁‘𝑡)‘𝐷), (𝑁‘𝑡)⟩ ∈ (𝐼‘𝑄)) ∧ (𝑓 = ((𝑡‘𝐺) ∘ ((𝑁‘𝑡)‘𝐷)) ∧ 𝑠 = (𝑡𝐽(𝑁‘𝑡)))) → ∃𝑔∃ℎ∃𝑢((⟨𝑔, 𝑡⟩ ∈ (𝐼‘𝑃) ∧ ⟨ℎ, 𝑢⟩ ∈ (𝐼‘𝑄)) ∧ (𝑓 = (𝑔 ∘ ℎ) ∧ 𝑠 = (𝑡𝐽𝑢))))
105	51, 61, 74, 80, 104	syl22anc 837	. . . . . . . 8 ⊢ (((𝜑 ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡 ∈ 𝐸 ∧ (𝑡‘(𝐺 ∘ ◡𝐷)) = 𝑓)) → ∃𝑔∃ℎ∃𝑢((⟨𝑔, 𝑡⟩ ∈ (𝐼‘𝑃) ∧ ⟨ℎ, 𝑢⟩ ∈ (𝐼‘𝑄)) ∧ (𝑓 = (𝑔 ∘ ℎ) ∧ 𝑠 = (𝑡𝐽𝑢))))
106	105	ex 411	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑉) ∧ 𝑠 = 0 )) → ((𝑡 ∈ 𝐸 ∧ (𝑡‘(𝐺 ∘ ◡𝐷)) = 𝑓) → ∃𝑔∃ℎ∃𝑢((⟨𝑔, 𝑡⟩ ∈ (𝐼‘𝑃) ∧ ⟨ℎ, 𝑢⟩ ∈ (𝐼‘𝑄)) ∧ (𝑓 = (𝑔 ∘ ℎ) ∧ 𝑠 = (𝑡𝐽𝑢)))))
107	106	eximdv 1913	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑉) ∧ 𝑠 = 0 )) → (∃𝑡(𝑡 ∈ 𝐸 ∧ (𝑡‘(𝐺 ∘ ◡𝐷)) = 𝑓) → ∃𝑡∃𝑔∃ℎ∃𝑢((⟨𝑔, 𝑡⟩ ∈ (𝐼‘𝑃) ∧ ⟨ℎ, 𝑢⟩ ∈ (𝐼‘𝑄)) ∧ (𝑓 = (𝑔 ∘ ℎ) ∧ 𝑠 = (𝑡𝐽𝑢)))))
108		excom 2152	. . . . . 6 ⊢ (∃𝑡∃𝑔∃ℎ∃𝑢((⟨𝑔, 𝑡⟩ ∈ (𝐼‘𝑃) ∧ ⟨ℎ, 𝑢⟩ ∈ (𝐼‘𝑄)) ∧ (𝑓 = (𝑔 ∘ ℎ) ∧ 𝑠 = (𝑡𝐽𝑢))) ↔ ∃𝑔∃𝑡∃ℎ∃𝑢((⟨𝑔, 𝑡⟩ ∈ (𝐼‘𝑃) ∧ ⟨ℎ, 𝑢⟩ ∈ (𝐼‘𝑄)) ∧ (𝑓 = (𝑔 ∘ ℎ) ∧ 𝑠 = (𝑡𝐽𝑢))))
109	107, 108	imbitrdi 250	. . . . 5 ⊢ ((𝜑 ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑉) ∧ 𝑠 = 0 )) → (∃𝑡(𝑡 ∈ 𝐸 ∧ (𝑡‘(𝐺 ∘ ◡𝐷)) = 𝑓) → ∃𝑔∃𝑡∃ℎ∃𝑢((⟨𝑔, 𝑡⟩ ∈ (𝐼‘𝑃) ∧ ⟨ℎ, 𝑢⟩ ∈ (𝐼‘𝑄)) ∧ (𝑓 = (𝑔 ∘ ℎ) ∧ 𝑠 = (𝑡𝐽𝑢)))))
110	42, 109	mpd 15	. . . 4 ⊢ ((𝜑 ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑉) ∧ 𝑠 = 0 )) → ∃𝑔∃𝑡∃ℎ∃𝑢((⟨𝑔, 𝑡⟩ ∈ (𝐼‘𝑃) ∧ ⟨ℎ, 𝑢⟩ ∈ (𝐼‘𝑄)) ∧ (𝑓 = (𝑔 ∘ ℎ) ∧ 𝑠 = (𝑡𝐽𝑢))))
111	110	ex 411	. . 3 ⊢ (𝜑 → (((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑉) ∧ 𝑠 = 0 ) → ∃𝑔∃𝑡∃ℎ∃𝑢((⟨𝑔, 𝑡⟩ ∈ (𝐼‘𝑃) ∧ ⟨ℎ, 𝑢⟩ ∈ (𝐼‘𝑄)) ∧ (𝑓 = (𝑔 ∘ ℎ) ∧ 𝑠 = (𝑡𝐽𝑢)))))
112	1	simpld 493	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ∈ HL)
113	112	hllatd 39064	. . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ Lat)
114	12	simpld 493	. . . . . . . . 9 ⊢ (𝜑 → 𝑃 ∈ 𝐴)
115	17	simpld 493	. . . . . . . . 9 ⊢ (𝜑 → 𝑄 ∈ 𝐴)
116	28, 29, 8	hlatjcl 39067	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 ∨ 𝑄) ∈ 𝐵)
117	112, 114, 115, 116	syl3anc 1368	. . . . . . . 8 ⊢ (𝜑 → (𝑃 ∨ 𝑄) ∈ 𝐵)
118	1	simprd 494	. . . . . . . . 9 ⊢ (𝜑 → 𝑊 ∈ 𝐻)
119	28, 2	lhpbase 39699	. . . . . . . . 9 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ 𝐵)
120	118, 119	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑊 ∈ 𝐵)
121	28, 30	latmcl 18467	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑄) ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) → ((𝑃 ∨ 𝑄) ∧ 𝑊) ∈ 𝐵)
122	113, 117, 120, 121	syl3anc 1368	. . . . . . 7 ⊢ (𝜑 → ((𝑃 ∨ 𝑄) ∧ 𝑊) ∈ 𝐵)
123	33, 122	eqeltrid 2830	. . . . . 6 ⊢ (𝜑 → 𝑉 ∈ 𝐵)
124	28, 7, 30	latmle2 18492	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑄) ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) → ((𝑃 ∨ 𝑄) ∧ 𝑊) ≤ 𝑊)
125	113, 117, 120, 124	syl3anc 1368	. . . . . . 7 ⊢ (𝜑 → ((𝑃 ∨ 𝑄) ∧ 𝑊) ≤ 𝑊)
126	33, 125	eqbrtrid 5190	. . . . . 6 ⊢ (𝜑 → 𝑉 ≤ 𝑊)
127		eqid 2726	. . . . . . 7 ⊢ ((DIsoB‘𝐾)‘𝑊) = ((DIsoB‘𝐾)‘𝑊)
128	28, 7, 2, 3, 127	dihvalb 40938	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑉 ∈ 𝐵 ∧ 𝑉 ≤ 𝑊)) → (𝐼‘𝑉) = (((DIsoB‘𝐾)‘𝑊)‘𝑉))
129	1, 123, 126, 128	syl12anc 835	. . . . 5 ⊢ (𝜑 → (𝐼‘𝑉) = (((DIsoB‘𝐾)‘𝑊)‘𝑉))
130	129	eleq2d 2812	. . . 4 ⊢ (𝜑 → (⟨𝑓, 𝑠⟩ ∈ (𝐼‘𝑉) ↔ ⟨𝑓, 𝑠⟩ ∈ (((DIsoB‘𝐾)‘𝑊)‘𝑉)))
131	28, 7, 2, 13, 34, 77, 127	dibopelval3 40849	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑉 ∈ 𝐵 ∧ 𝑉 ≤ 𝑊)) → (⟨𝑓, 𝑠⟩ ∈ (((DIsoB‘𝐾)‘𝑊)‘𝑉) ↔ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑉) ∧ 𝑠 = 0 )))
132	1, 123, 126, 131	syl12anc 835	. . . 4 ⊢ (𝜑 → (⟨𝑓, 𝑠⟩ ∈ (((DIsoB‘𝐾)‘𝑊)‘𝑉) ↔ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑉) ∧ 𝑠 = 0 )))
133	130, 132	bitrd 278	. . 3 ⊢ (𝜑 → (⟨𝑓, 𝑠⟩ ∈ (𝐼‘𝑉) ↔ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑉) ∧ 𝑠 = 0 )))
134		eqid 2726	. . . 4 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
135	28, 8	atbase 38989	. . . . 5 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ 𝐵)
136	114, 135	syl 17	. . . 4 ⊢ (𝜑 → 𝑃 ∈ 𝐵)
137	28, 8	atbase 38989	. . . . 5 ⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ 𝐵)
138	115, 137	syl 17	. . . 4 ⊢ (𝜑 → 𝑄 ∈ 𝐵)
139	28, 2, 13, 35, 76, 31, 134, 32, 3, 1, 136, 138	dihopellsm 40956	. . 3 ⊢ (𝜑 → (⟨𝑓, 𝑠⟩ ∈ ((𝐼‘𝑃) ⊕ (𝐼‘𝑄)) ↔ ∃𝑔∃𝑡∃ℎ∃𝑢((⟨𝑔, 𝑡⟩ ∈ (𝐼‘𝑃) ∧ ⟨ℎ, 𝑢⟩ ∈ (𝐼‘𝑄)) ∧ (𝑓 = (𝑔 ∘ ℎ) ∧ 𝑠 = (𝑡𝐽𝑢)))))
140	111, 133, 139	3imtr4d 293	. 2 ⊢ (𝜑 → (⟨𝑓, 𝑠⟩ ∈ (𝐼‘𝑉) → ⟨𝑓, 𝑠⟩ ∈ ((𝐼‘𝑃) ⊕ (𝐼‘𝑄))))
141	5, 140	relssdv 5796	1 ⊢ (𝜑 → (𝐼‘𝑉) ⊆ ((𝐼‘𝑃) ⊕ (𝐼‘𝑄)))

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