Theorem dihjatcclem4 41378

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 41236	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → Rel (𝐼‘𝑉))
5	1, 4	syl 17	. 2 ⊢ (𝜑 → Rel (𝐼‘𝑉))
6	1	adantr 480	. . . . . . 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 39976	. . . . . . . . . . 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 40536	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐶 ∈ 𝐴 ∧ ¬ 𝐶 ≤ 𝑊) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝐺 ∈ 𝑇)
16	1, 11, 12, 15	syl3anc 1371	. . . . . . . . 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 40536	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐶 ∈ 𝐴 ∧ ¬ 𝐶 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → 𝐷 ∈ 𝑇)
20	1, 11, 17, 19	syl3anc 1371	. . . . . . . . . 10 ⊢ (𝜑 → 𝐷 ∈ 𝑇)
21	2, 13	ltrncnv 40103	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐷 ∈ 𝑇) → ◡𝐷 ∈ 𝑇)
22	1, 20, 21	syl2anc 583	. . . . . . . . 9 ⊢ (𝜑 → ◡𝐷 ∈ 𝑇)
23	2, 13	ltrnco 40676	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇 ∧ ◡𝐷 ∈ 𝑇) → (𝐺 ∘ ◡𝐷) ∈ 𝑇)
24	1, 16, 22, 23	syl3anc 1371	. . . . . . . 8 ⊢ (𝜑 → (𝐺 ∘ ◡𝐷) ∈ 𝑇)
25	24	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑉) ∧ 𝑠 = 0 )) → (𝐺 ∘ ◡𝐷) ∈ 𝑇)
26		simprll 778	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑉) ∧ 𝑠 = 0 )) → 𝑓 ∈ 𝑇)
27		simprlr 779	. . . . . . . 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 41377	. . . . . . . . 9 ⊢ (𝜑 → (𝑅‘(𝐺 ∘ ◡𝐷)) = 𝑉)
37	36	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑉) ∧ 𝑠 = 0 )) → (𝑅‘(𝐺 ∘ ◡𝐷)) = 𝑉)
38	27, 37	breqtrrd 5194	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑉) ∧ 𝑠 = 0 )) → (𝑅‘𝑓) ≤ (𝑅‘(𝐺 ∘ ◡𝐷)))
39	7, 2, 13, 34, 35	tendoex 40932	. . . . . . 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 1375	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑉) ∧ 𝑠 = 0 )) → ∃𝑡 ∈ 𝐸 (𝑡‘(𝐺 ∘ ◡𝐷)) = 𝑓)
41		df-rex 3077	. . . . . 6 ⊢ (∃𝑡 ∈ 𝐸 (𝑡‘(𝐺 ∘ ◡𝐷)) = 𝑓 ↔ ∃𝑡(𝑡 ∈ 𝐸 ∧ (𝑡‘(𝐺 ∘ ◡𝐷)) = 𝑓))
42	40, 41	sylib 218	. . . . 5 ⊢ ((𝜑 ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑉) ∧ 𝑠 = 0 )) → ∃𝑡(𝑡 ∈ 𝐸 ∧ (𝑡‘(𝐺 ∘ ◡𝐷)) = 𝑓))
43		eqidd 2741	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡 ∈ 𝐸 ∧ (𝑡‘(𝐺 ∘ ◡𝐷)) = 𝑓)) → (𝑡‘𝐺) = (𝑡‘𝐺))
44		simprl 770	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡 ∈ 𝐸 ∧ (𝑡‘(𝐺 ∘ ◡𝐷)) = 𝑓)) → 𝑡 ∈ 𝐸)
45	1	ad2antrr 725	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡 ∈ 𝐸 ∧ (𝑡‘(𝐺 ∘ ◡𝐷)) = 𝑓)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
46	12	ad2antrr 725	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡 ∈ 𝐸 ∧ (𝑡‘(𝐺 ∘ ◡𝐷)) = 𝑓)) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
47		fvex 6933	. . . . . . . . . . . 12 ⊢ (𝑡‘𝐺) ∈ V
48		vex 3492	. . . . . . . . . . . 12 ⊢ 𝑡 ∈ V
49	7, 8, 2, 9, 13, 35, 3, 14, 47, 48	dihopelvalcqat 41203	. . . . . . . . . . 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 583	. . . . . . . . . 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 712	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡 ∈ 𝐸 ∧ (𝑡‘(𝐺 ∘ ◡𝐷)) = 𝑓)) → ⟨(𝑡‘𝐺), 𝑡⟩ ∈ (𝐼‘𝑃))
52		eqidd 2741	. . . . . . . . . 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 40753	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑡 ∈ 𝐸) → (𝑁‘𝑡) ∈ 𝐸)
55	45, 44, 54	syl2anc 583	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡 ∈ 𝐸 ∧ (𝑡‘(𝐺 ∘ ◡𝐷)) = 𝑓)) → (𝑁‘𝑡) ∈ 𝐸)
56	17	ad2antrr 725	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡 ∈ 𝐸 ∧ (𝑡‘(𝐺 ∘ ◡𝐷)) = 𝑓)) → (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))
57		fvex 6933	. . . . . . . . . . . 12 ⊢ ((𝑁‘𝑡)‘𝐷) ∈ V
58		fvex 6933	. . . . . . . . . . . 12 ⊢ (𝑁‘𝑡) ∈ V
59	7, 8, 2, 9, 13, 35, 3, 18, 57, 58	dihopelvalcqat 41203	. . . . . . . . . . 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 583	. . . . . . . . . 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 712	. . . . . . . . 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 725	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡 ∈ 𝐸 ∧ (𝑡‘(𝐺 ∘ ◡𝐷)) = 𝑓)) → 𝐺 ∈ 𝑇)
63	22	ad2antrr 725	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡 ∈ 𝐸 ∧ (𝑡‘(𝐺 ∘ ◡𝐷)) = 𝑓)) → ◡𝐷 ∈ 𝑇)
64	2, 13, 35	tendospdi1 40977	. . . . . . . . . . 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 1372	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡 ∈ 𝐸 ∧ (𝑡‘(𝐺 ∘ ◡𝐷)) = 𝑓)) → (𝑡‘(𝐺 ∘ ◡𝐷)) = ((𝑡‘𝐺) ∘ (𝑡‘◡𝐷)))
66		simprr 772	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡 ∈ 𝐸 ∧ (𝑡‘(𝐺 ∘ ◡𝐷)) = 𝑓)) → (𝑡‘(𝐺 ∘ ◡𝐷)) = 𝑓)
67	20	ad2antrr 725	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡 ∈ 𝐸 ∧ (𝑡‘(𝐺 ∘ ◡𝐷)) = 𝑓)) → 𝐷 ∈ 𝑇)
68	53, 13	tendoi2 40752	. . . . . . . . . . . . 13 ⊢ ((𝑡 ∈ 𝐸 ∧ 𝐷 ∈ 𝑇) → ((𝑁‘𝑡)‘𝐷) = ◡(𝑡‘𝐷))
69	44, 67, 68	syl2anc 583	. . . . . . . . . . . 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 40978	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑡 ∈ 𝐸 ∧ 𝐷 ∈ 𝑇) → ◡(𝑡‘𝐷) = (𝑡‘◡𝐷))
71	45, 44, 67, 70	syl3anc 1371	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡 ∈ 𝐸 ∧ (𝑡‘(𝐺 ∘ ◡𝐷)) = 𝑓)) → ◡(𝑡‘𝐷) = (𝑡‘◡𝐷))
72	69, 71	eqtr2d 2781	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡 ∈ 𝐸 ∧ (𝑡‘(𝐺 ∘ ◡𝐷)) = 𝑓)) → (𝑡‘◡𝐷) = ((𝑁‘𝑡)‘𝐷))
73	72	coeq2d 5887	. . . . . . . . . 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 2788	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡 ∈ 𝐸 ∧ (𝑡‘(𝐺 ∘ ◡𝐷)) = 𝑓)) → 𝑓 = ((𝑡‘𝐺) ∘ ((𝑁‘𝑡)‘𝐷)))
75		simplrr 777	. . . . . . . . . 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 40755	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑡 ∈ 𝐸) → (𝑡𝐽(𝑁‘𝑡)) = 0 )
79	45, 44, 78	syl2anc 583	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡 ∈ 𝐸 ∧ (𝑡‘(𝐺 ∘ ◡𝐷)) = 𝑓)) → (𝑡𝐽(𝑁‘𝑡)) = 0 )
80	75, 79	eqtr4d 2783	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡 ∈ 𝐸 ∧ (𝑡‘(𝐺 ∘ ◡𝐷)) = 𝑓)) → 𝑠 = (𝑡𝐽(𝑁‘𝑡)))
81		opeq1 4897	. . . . . . . . . . . . . . 15 ⊢ (𝑔 = (𝑡‘𝐺) → ⟨𝑔, 𝑡⟩ = ⟨(𝑡‘𝐺), 𝑡⟩)
82	81	eleq1d 2829	. . . . . . . . . . . . . 14 ⊢ (𝑔 = (𝑡‘𝐺) → (⟨𝑔, 𝑡⟩ ∈ (𝐼‘𝑃) ↔ ⟨(𝑡‘𝐺), 𝑡⟩ ∈ (𝐼‘𝑃)))
83	82	anbi1d 630	. . . . . . . . . . . . 13 ⊢ (𝑔 = (𝑡‘𝐺) → ((⟨𝑔, 𝑡⟩ ∈ (𝐼‘𝑃) ∧ ⟨ℎ, 𝑢⟩ ∈ (𝐼‘𝑄)) ↔ (⟨(𝑡‘𝐺), 𝑡⟩ ∈ (𝐼‘𝑃) ∧ ⟨ℎ, 𝑢⟩ ∈ (𝐼‘𝑄))))
84		coeq1 5882	. . . . . . . . . . . . . . 15 ⊢ (𝑔 = (𝑡‘𝐺) → (𝑔 ∘ ℎ) = ((𝑡‘𝐺) ∘ ℎ))
85	84	eqeq2d 2751	. . . . . . . . . . . . . 14 ⊢ (𝑔 = (𝑡‘𝐺) → (𝑓 = (𝑔 ∘ ℎ) ↔ 𝑓 = ((𝑡‘𝐺) ∘ ℎ)))
86	85	anbi1d 630	. . . . . . . . . . . . 13 ⊢ (𝑔 = (𝑡‘𝐺) → ((𝑓 = (𝑔 ∘ ℎ) ∧ 𝑠 = (𝑡𝐽𝑢)) ↔ (𝑓 = ((𝑡‘𝐺) ∘ ℎ) ∧ 𝑠 = (𝑡𝐽𝑢))))
87	83, 86	anbi12d 631	. . . . . . . . . . . 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 4897	. . . . . . . . . . . . . . 15 ⊢ (ℎ = ((𝑁‘𝑡)‘𝐷) → ⟨ℎ, 𝑢⟩ = ⟨((𝑁‘𝑡)‘𝐷), 𝑢⟩)
89	88	eleq1d 2829	. . . . . . . . . . . . . 14 ⊢ (ℎ = ((𝑁‘𝑡)‘𝐷) → (⟨ℎ, 𝑢⟩ ∈ (𝐼‘𝑄) ↔ ⟨((𝑁‘𝑡)‘𝐷), 𝑢⟩ ∈ (𝐼‘𝑄)))
90	89	anbi2d 629	. . . . . . . . . . . . 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 5883	. . . . . . . . . . . . . . 15 ⊢ (ℎ = ((𝑁‘𝑡)‘𝐷) → ((𝑡‘𝐺) ∘ ℎ) = ((𝑡‘𝐺) ∘ ((𝑁‘𝑡)‘𝐷)))
92	91	eqeq2d 2751	. . . . . . . . . . . . . 14 ⊢ (ℎ = ((𝑁‘𝑡)‘𝐷) → (𝑓 = ((𝑡‘𝐺) ∘ ℎ) ↔ 𝑓 = ((𝑡‘𝐺) ∘ ((𝑁‘𝑡)‘𝐷))))
93	92	anbi1d 630	. . . . . . . . . . . . 13 ⊢ (ℎ = ((𝑁‘𝑡)‘𝐷) → ((𝑓 = ((𝑡‘𝐺) ∘ ℎ) ∧ 𝑠 = (𝑡𝐽𝑢)) ↔ (𝑓 = ((𝑡‘𝐺) ∘ ((𝑁‘𝑡)‘𝐷)) ∧ 𝑠 = (𝑡𝐽𝑢))))
94	90, 93	anbi12d 631	. . . . . . . . . . . 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 4898	. . . . . . . . . . . . . . 15 ⊢ (𝑢 = (𝑁‘𝑡) → ⟨((𝑁‘𝑡)‘𝐷), 𝑢⟩ = ⟨((𝑁‘𝑡)‘𝐷), (𝑁‘𝑡)⟩)
96	95	eleq1d 2829	. . . . . . . . . . . . . 14 ⊢ (𝑢 = (𝑁‘𝑡) → (⟨((𝑁‘𝑡)‘𝐷), 𝑢⟩ ∈ (𝐼‘𝑄) ↔ ⟨((𝑁‘𝑡)‘𝐷), (𝑁‘𝑡)⟩ ∈ (𝐼‘𝑄)))
97	96	anbi2d 629	. . . . . . . . . . . . 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 7456	. . . . . . . . . . . . . . 15 ⊢ (𝑢 = (𝑁‘𝑡) → (𝑡𝐽𝑢) = (𝑡𝐽(𝑁‘𝑡)))
99	98	eqeq2d 2751	. . . . . . . . . . . . . 14 ⊢ (𝑢 = (𝑁‘𝑡) → (𝑠 = (𝑡𝐽𝑢) ↔ 𝑠 = (𝑡𝐽(𝑁‘𝑡))))
100	99	anbi2d 629	. . . . . . . . . . . . 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 631	. . . . . . . . . . . 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 1434	. . . . . . . . . . 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 3616	. . . . . . . . . 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 1461	. . . . . . . . 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 838	. . . . . . . 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 412	. . . . . . 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 1916	. . . . . 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 2163	. . . . . 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 251	. . . . 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 412	. . 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 494	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ∈ HL)
113	112	hllatd 39320	. . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ Lat)
114	12	simpld 494	. . . . . . . . 9 ⊢ (𝜑 → 𝑃 ∈ 𝐴)
115	17	simpld 494	. . . . . . . . 9 ⊢ (𝜑 → 𝑄 ∈ 𝐴)
116	28, 29, 8	hlatjcl 39323	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 ∨ 𝑄) ∈ 𝐵)
117	112, 114, 115, 116	syl3anc 1371	. . . . . . . 8 ⊢ (𝜑 → (𝑃 ∨ 𝑄) ∈ 𝐵)
118	1	simprd 495	. . . . . . . . 9 ⊢ (𝜑 → 𝑊 ∈ 𝐻)
119	28, 2	lhpbase 39955	. . . . . . . . 9 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ 𝐵)
120	118, 119	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑊 ∈ 𝐵)
121	28, 30	latmcl 18510	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑄) ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) → ((𝑃 ∨ 𝑄) ∧ 𝑊) ∈ 𝐵)
122	113, 117, 120, 121	syl3anc 1371	. . . . . . 7 ⊢ (𝜑 → ((𝑃 ∨ 𝑄) ∧ 𝑊) ∈ 𝐵)
123	33, 122	eqeltrid 2848	. . . . . 6 ⊢ (𝜑 → 𝑉 ∈ 𝐵)
124	28, 7, 30	latmle2 18535	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑄) ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) → ((𝑃 ∨ 𝑄) ∧ 𝑊) ≤ 𝑊)
125	113, 117, 120, 124	syl3anc 1371	. . . . . . 7 ⊢ (𝜑 → ((𝑃 ∨ 𝑄) ∧ 𝑊) ≤ 𝑊)
126	33, 125	eqbrtrid 5201	. . . . . 6 ⊢ (𝜑 → 𝑉 ≤ 𝑊)
127		eqid 2740	. . . . . . 7 ⊢ ((DIsoB‘𝐾)‘𝑊) = ((DIsoB‘𝐾)‘𝑊)
128	28, 7, 2, 3, 127	dihvalb 41194	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑉 ∈ 𝐵 ∧ 𝑉 ≤ 𝑊)) → (𝐼‘𝑉) = (((DIsoB‘𝐾)‘𝑊)‘𝑉))
129	1, 123, 126, 128	syl12anc 836	. . . . 5 ⊢ (𝜑 → (𝐼‘𝑉) = (((DIsoB‘𝐾)‘𝑊)‘𝑉))
130	129	eleq2d 2830	. . . 4 ⊢ (𝜑 → (⟨𝑓, 𝑠⟩ ∈ (𝐼‘𝑉) ↔ ⟨𝑓, 𝑠⟩ ∈ (((DIsoB‘𝐾)‘𝑊)‘𝑉)))
131	28, 7, 2, 13, 34, 77, 127	dibopelval3 41105	. . . . 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 836	. . . 4 ⊢ (𝜑 → (⟨𝑓, 𝑠⟩ ∈ (((DIsoB‘𝐾)‘𝑊)‘𝑉) ↔ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑉) ∧ 𝑠 = 0 )))
133	130, 132	bitrd 279	. . 3 ⊢ (𝜑 → (⟨𝑓, 𝑠⟩ ∈ (𝐼‘𝑉) ↔ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑉) ∧ 𝑠 = 0 )))
134		eqid 2740	. . . 4 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
135	28, 8	atbase 39245	. . . . 5 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ 𝐵)
136	114, 135	syl 17	. . . 4 ⊢ (𝜑 → 𝑃 ∈ 𝐵)
137	28, 8	atbase 39245	. . . . 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 41212	. . 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 294	. 2 ⊢ (𝜑 → (⟨𝑓, 𝑠⟩ ∈ (𝐼‘𝑉) → ⟨𝑓, 𝑠⟩ ∈ ((𝐼‘𝑃) ⊕ (𝐼‘𝑄))))
141	5, 140	relssdv 5812	1 ⊢ (𝜑 → (𝐼‘𝑉) ⊆ ((𝐼‘𝑃) ⊕ (𝐼‘𝑄)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537  ∃wex 1777   ∈ wcel 2108  ∃wrex 3076  Vcvv 3488   ⊆ wss 3976  ⟨cop 4654   class class class wbr 5166   ↦ cmpt 5249   I cid 5592  ^◡ccnv 5699   ↾ cres 5702   ∘ ccom 5704  Rel wrel 5705  ‘cfv 6573  ℩crio 7403  (class class class)co 7448   ∈ cmpo 7450  Basecbs 17258  lecple 17318  occoc 17319  joincjn 18381  meetcmee 18382  Latclat 18501  LSSumclsm 19676  LSubSpclss 20952  Atomscatm 39219  HLchlt 39306  LHypclh 39941  LTrncltrn 40058  trLctrl 40115  TEndoctendo 40709  DVecHcdvh 41035  DIsoBcdib 41095  DIsoHcdih 41185

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261  ax-riotaBAD 38909

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-tp 4653  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-iin 5018  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-tpos 8267  df-undef 8314  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-er 8763  df-map 8886  df-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-nn 12294  df-2 12356  df-3 12357  df-4 12358  df-5 12359  df-6 12360  df-n0 12554  df-z 12640  df-uz 12904  df-fz 13568  df-struct 17194  df-sets 17211  df-slot 17229  df-ndx 17241  df-base 17259  df-ress 17288  df-plusg 17324  df-mulr 17325  df-sca 17327  df-vsca 17328  df-0g 17501  df-proset 18365  df-poset 18383  df-plt 18400  df-lub 18416  df-glb 18417  df-join 18418  df-meet 18419  df-p0 18495  df-p1 18496  df-lat 18502  df-clat 18569  df-mgm 18678  df-sgrp 18757  df-mnd 18773  df-submnd 18819  df-grp 18976  df-minusg 18977  df-sbg 18978  df-subg 19163  df-cntz 19357  df-lsm 19678  df-cmn 19824  df-abl 19825  df-mgp 20162  df-rng 20180  df-ur 20209  df-ring 20262  df-oppr 20360  df-dvdsr 20383  df-unit 20384  df-invr 20414  df-dvr 20427  df-drng 20753  df-lmod 20882  df-lss 20953  df-lsp 20993  df-lvec 21125  df-oposet 39132  df-ol 39134  df-oml 39135  df-covers 39222  df-ats 39223  df-atl 39254  df-cvlat 39278  df-hlat 39307  df-llines 39455  df-lplanes 39456  df-lvols 39457  df-lines 39458  df-psubsp 39460  df-pmap 39461  df-padd 39753  df-lhyp 39945  df-laut 39946  df-ldil 40061  df-ltrn 40062  df-trl 40116  df-tendo 40712  df-edring 40714  df-disoa 40986  df-dvech 41036  df-dib 41096  df-dic 41130  df-dih 41186

This theorem is referenced by:  dihjatcc  41379