Theorem dihjatcclem4 41422

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 41280	. . 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 40020	. . . . . . . . . . 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 40580	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐶 ∈ 𝐴 ∧ ¬ 𝐶 ≤ 𝑊) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝐺 ∈ 𝑇)
16	1, 11, 12, 15	syl3anc 1373	. . . . . . . . 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 40580	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐶 ∈ 𝐴 ∧ ¬ 𝐶 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → 𝐷 ∈ 𝑇)
20	1, 11, 17, 19	syl3anc 1373	. . . . . . . . . 10 ⊢ (𝜑 → 𝐷 ∈ 𝑇)
21	2, 13	ltrncnv 40147	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐷 ∈ 𝑇) → ◡𝐷 ∈ 𝑇)
22	1, 20, 21	syl2anc 584	. . . . . . . . 9 ⊢ (𝜑 → ◡𝐷 ∈ 𝑇)
23	2, 13	ltrnco 40720	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇 ∧ ◡𝐷 ∈ 𝑇) → (𝐺 ∘ ◡𝐷) ∈ 𝑇)
24	1, 16, 22, 23	syl3anc 1373	. . . . . . . 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 41421	. . . . . . . . 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 5138	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑉) ∧ 𝑠 = 0 )) → (𝑅‘𝑓) ≤ (𝑅‘(𝐺 ∘ ◡𝐷)))
39	7, 2, 13, 34, 35	tendoex 40976	. . . . . . 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 1377	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑉) ∧ 𝑠 = 0 )) → ∃𝑡 ∈ 𝐸 (𝑡‘(𝐺 ∘ ◡𝐷)) = 𝑓)
41		df-rex 3055	. . . . . 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 2731	. . . . . . . . . 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 726	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡 ∈ 𝐸 ∧ (𝑡‘(𝐺 ∘ ◡𝐷)) = 𝑓)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
46	12	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡 ∈ 𝐸 ∧ (𝑡‘(𝐺 ∘ ◡𝐷)) = 𝑓)) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
47		fvex 6874	. . . . . . . . . . . 12 ⊢ (𝑡‘𝐺) ∈ V
48		vex 3454	. . . . . . . . . . . 12 ⊢ 𝑡 ∈ V
49	7, 8, 2, 9, 13, 35, 3, 14, 47, 48	dihopelvalcqat 41247	. . . . . . . . . . 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 584	. . . . . . . . . 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 713	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡 ∈ 𝐸 ∧ (𝑡‘(𝐺 ∘ ◡𝐷)) = 𝑓)) → ⟨(𝑡‘𝐺), 𝑡⟩ ∈ (𝐼‘𝑃))
52		eqidd 2731	. . . . . . . . . 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 40797	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑡 ∈ 𝐸) → (𝑁‘𝑡) ∈ 𝐸)
55	45, 44, 54	syl2anc 584	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡 ∈ 𝐸 ∧ (𝑡‘(𝐺 ∘ ◡𝐷)) = 𝑓)) → (𝑁‘𝑡) ∈ 𝐸)
56	17	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡 ∈ 𝐸 ∧ (𝑡‘(𝐺 ∘ ◡𝐷)) = 𝑓)) → (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))
57		fvex 6874	. . . . . . . . . . . 12 ⊢ ((𝑁‘𝑡)‘𝐷) ∈ V
58		fvex 6874	. . . . . . . . . . . 12 ⊢ (𝑁‘𝑡) ∈ V
59	7, 8, 2, 9, 13, 35, 3, 18, 57, 58	dihopelvalcqat 41247	. . . . . . . . . . 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 584	. . . . . . . . . 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 713	. . . . . . . . 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 726	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡 ∈ 𝐸 ∧ (𝑡‘(𝐺 ∘ ◡𝐷)) = 𝑓)) → 𝐺 ∈ 𝑇)
63	22	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡 ∈ 𝐸 ∧ (𝑡‘(𝐺 ∘ ◡𝐷)) = 𝑓)) → ◡𝐷 ∈ 𝑇)
64	2, 13, 35	tendospdi1 41021	. . . . . . . . . . 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 1374	. . . . . . . . . 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 726	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡 ∈ 𝐸 ∧ (𝑡‘(𝐺 ∘ ◡𝐷)) = 𝑓)) → 𝐷 ∈ 𝑇)
68	53, 13	tendoi2 40796	. . . . . . . . . . . . 13 ⊢ ((𝑡 ∈ 𝐸 ∧ 𝐷 ∈ 𝑇) → ((𝑁‘𝑡)‘𝐷) = ◡(𝑡‘𝐷))
69	44, 67, 68	syl2anc 584	. . . . . . . . . . . 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 41022	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑡 ∈ 𝐸 ∧ 𝐷 ∈ 𝑇) → ◡(𝑡‘𝐷) = (𝑡‘◡𝐷))
71	45, 44, 67, 70	syl3anc 1373	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡 ∈ 𝐸 ∧ (𝑡‘(𝐺 ∘ ◡𝐷)) = 𝑓)) → ◡(𝑡‘𝐷) = (𝑡‘◡𝐷))
72	69, 71	eqtr2d 2766	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡 ∈ 𝐸 ∧ (𝑡‘(𝐺 ∘ ◡𝐷)) = 𝑓)) → (𝑡‘◡𝐷) = ((𝑁‘𝑡)‘𝐷))
73	72	coeq2d 5829	. . . . . . . . . 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 2773	. . . . . . . . 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 40799	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑡 ∈ 𝐸) → (𝑡𝐽(𝑁‘𝑡)) = 0 )
79	45, 44, 78	syl2anc 584	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡 ∈ 𝐸 ∧ (𝑡‘(𝐺 ∘ ◡𝐷)) = 𝑓)) → (𝑡𝐽(𝑁‘𝑡)) = 0 )
80	75, 79	eqtr4d 2768	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡 ∈ 𝐸 ∧ (𝑡‘(𝐺 ∘ ◡𝐷)) = 𝑓)) → 𝑠 = (𝑡𝐽(𝑁‘𝑡)))
81		opeq1 4840	. . . . . . . . . . . . . . 15 ⊢ (𝑔 = (𝑡‘𝐺) → ⟨𝑔, 𝑡⟩ = ⟨(𝑡‘𝐺), 𝑡⟩)
82	81	eleq1d 2814	. . . . . . . . . . . . . 14 ⊢ (𝑔 = (𝑡‘𝐺) → (⟨𝑔, 𝑡⟩ ∈ (𝐼‘𝑃) ↔ ⟨(𝑡‘𝐺), 𝑡⟩ ∈ (𝐼‘𝑃)))
83	82	anbi1d 631	. . . . . . . . . . . . 13 ⊢ (𝑔 = (𝑡‘𝐺) → ((⟨𝑔, 𝑡⟩ ∈ (𝐼‘𝑃) ∧ ⟨ℎ, 𝑢⟩ ∈ (𝐼‘𝑄)) ↔ (⟨(𝑡‘𝐺), 𝑡⟩ ∈ (𝐼‘𝑃) ∧ ⟨ℎ, 𝑢⟩ ∈ (𝐼‘𝑄))))
84		coeq1 5824	. . . . . . . . . . . . . . 15 ⊢ (𝑔 = (𝑡‘𝐺) → (𝑔 ∘ ℎ) = ((𝑡‘𝐺) ∘ ℎ))
85	84	eqeq2d 2741	. . . . . . . . . . . . . 14 ⊢ (𝑔 = (𝑡‘𝐺) → (𝑓 = (𝑔 ∘ ℎ) ↔ 𝑓 = ((𝑡‘𝐺) ∘ ℎ)))
86	85	anbi1d 631	. . . . . . . . . . . . 13 ⊢ (𝑔 = (𝑡‘𝐺) → ((𝑓 = (𝑔 ∘ ℎ) ∧ 𝑠 = (𝑡𝐽𝑢)) ↔ (𝑓 = ((𝑡‘𝐺) ∘ ℎ) ∧ 𝑠 = (𝑡𝐽𝑢))))
87	83, 86	anbi12d 632	. . . . . . . . . . . 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 4840	. . . . . . . . . . . . . . 15 ⊢ (ℎ = ((𝑁‘𝑡)‘𝐷) → ⟨ℎ, 𝑢⟩ = ⟨((𝑁‘𝑡)‘𝐷), 𝑢⟩)
89	88	eleq1d 2814	. . . . . . . . . . . . . 14 ⊢ (ℎ = ((𝑁‘𝑡)‘𝐷) → (⟨ℎ, 𝑢⟩ ∈ (𝐼‘𝑄) ↔ ⟨((𝑁‘𝑡)‘𝐷), 𝑢⟩ ∈ (𝐼‘𝑄)))
90	89	anbi2d 630	. . . . . . . . . . . . 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 5825	. . . . . . . . . . . . . . 15 ⊢ (ℎ = ((𝑁‘𝑡)‘𝐷) → ((𝑡‘𝐺) ∘ ℎ) = ((𝑡‘𝐺) ∘ ((𝑁‘𝑡)‘𝐷)))
92	91	eqeq2d 2741	. . . . . . . . . . . . . 14 ⊢ (ℎ = ((𝑁‘𝑡)‘𝐷) → (𝑓 = ((𝑡‘𝐺) ∘ ℎ) ↔ 𝑓 = ((𝑡‘𝐺) ∘ ((𝑁‘𝑡)‘𝐷))))
93	92	anbi1d 631	. . . . . . . . . . . . 13 ⊢ (ℎ = ((𝑁‘𝑡)‘𝐷) → ((𝑓 = ((𝑡‘𝐺) ∘ ℎ) ∧ 𝑠 = (𝑡𝐽𝑢)) ↔ (𝑓 = ((𝑡‘𝐺) ∘ ((𝑁‘𝑡)‘𝐷)) ∧ 𝑠 = (𝑡𝐽𝑢))))
94	90, 93	anbi12d 632	. . . . . . . . . . . 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 4841	. . . . . . . . . . . . . . 15 ⊢ (𝑢 = (𝑁‘𝑡) → ⟨((𝑁‘𝑡)‘𝐷), 𝑢⟩ = ⟨((𝑁‘𝑡)‘𝐷), (𝑁‘𝑡)⟩)
96	95	eleq1d 2814	. . . . . . . . . . . . . 14 ⊢ (𝑢 = (𝑁‘𝑡) → (⟨((𝑁‘𝑡)‘𝐷), 𝑢⟩ ∈ (𝐼‘𝑄) ↔ ⟨((𝑁‘𝑡)‘𝐷), (𝑁‘𝑡)⟩ ∈ (𝐼‘𝑄)))
97	96	anbi2d 630	. . . . . . . . . . . . 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 7398	. . . . . . . . . . . . . . 15 ⊢ (𝑢 = (𝑁‘𝑡) → (𝑡𝐽𝑢) = (𝑡𝐽(𝑁‘𝑡)))
99	98	eqeq2d 2741	. . . . . . . . . . . . . 14 ⊢ (𝑢 = (𝑁‘𝑡) → (𝑠 = (𝑡𝐽𝑢) ↔ 𝑠 = (𝑡𝐽(𝑁‘𝑡))))
100	99	anbi2d 630	. . . . . . . . . . . . 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 632	. . . . . . . . . . . 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 1436	. . . . . . . . . . 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 3572	. . . . . . . . . 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 1463	. . . . . . . . 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 1917	. . . . . 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 39364	. . . . . . . 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 39367	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 ∨ 𝑄) ∈ 𝐵)
117	112, 114, 115, 116	syl3anc 1373	. . . . . . . 8 ⊢ (𝜑 → (𝑃 ∨ 𝑄) ∈ 𝐵)
118	1	simprd 495	. . . . . . . . 9 ⊢ (𝜑 → 𝑊 ∈ 𝐻)
119	28, 2	lhpbase 39999	. . . . . . . . 9 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ 𝐵)
120	118, 119	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑊 ∈ 𝐵)
121	28, 30	latmcl 18406	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑄) ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) → ((𝑃 ∨ 𝑄) ∧ 𝑊) ∈ 𝐵)
122	113, 117, 120, 121	syl3anc 1373	. . . . . . 7 ⊢ (𝜑 → ((𝑃 ∨ 𝑄) ∧ 𝑊) ∈ 𝐵)
123	33, 122	eqeltrid 2833	. . . . . 6 ⊢ (𝜑 → 𝑉 ∈ 𝐵)
124	28, 7, 30	latmle2 18431	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑄) ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) → ((𝑃 ∨ 𝑄) ∧ 𝑊) ≤ 𝑊)
125	113, 117, 120, 124	syl3anc 1373	. . . . . . 7 ⊢ (𝜑 → ((𝑃 ∨ 𝑄) ∧ 𝑊) ≤ 𝑊)
126	33, 125	eqbrtrid 5145	. . . . . 6 ⊢ (𝜑 → 𝑉 ≤ 𝑊)
127		eqid 2730	. . . . . . 7 ⊢ ((DIsoB‘𝐾)‘𝑊) = ((DIsoB‘𝐾)‘𝑊)
128	28, 7, 2, 3, 127	dihvalb 41238	. . . . . 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 2815	. . . 4 ⊢ (𝜑 → (⟨𝑓, 𝑠⟩ ∈ (𝐼‘𝑉) ↔ ⟨𝑓, 𝑠⟩ ∈ (((DIsoB‘𝐾)‘𝑊)‘𝑉)))
131	28, 7, 2, 13, 34, 77, 127	dibopelval3 41149	. . . . 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 2730	. . . 4 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
135	28, 8	atbase 39289	. . . . 5 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ 𝐵)
136	114, 135	syl 17	. . . 4 ⊢ (𝜑 → 𝑃 ∈ 𝐵)
137	28, 8	atbase 39289	. . . . 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 41256	. . 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 5754	1 ⊢ (𝜑 → (𝐼‘𝑉) ⊆ ((𝐼‘𝑃) ⊕ (𝐼‘𝑄)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2109  ∃wrex 3054  Vcvv 3450   ⊆ wss 3917  ⟨cop 4598   class class class wbr 5110   ↦ cmpt 5191   I cid 5535  ^◡ccnv 5640   ↾ cres 5643   ∘ ccom 5645  Rel wrel 5646  ‘cfv 6514  ℩crio 7346  (class class class)co 7390   ∈ cmpo 7392  Basecbs 17186  lecple 17234  occoc 17235  joincjn 18279  meetcmee 18280  Latclat 18397  LSSumclsm 19571  LSubSpclss 20844  Atomscatm 39263  HLchlt 39350  LHypclh 39985  LTrncltrn 40102  trLctrl 40159  TEndoctendo 40753  DVecHcdvh 41079  DIsoBcdib 41139  DIsoHcdih 41229

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-cnex 11131  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151  ax-pre-mulgt0 11152  ax-riotaBAD 38953

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-tp 4597  df-op 4599  df-uni 4875  df-int 4914  df-iun 4960  df-iin 4961  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7846  df-1st 7971  df-2nd 7972  df-tpos 8208  df-undef 8255  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-1o 8437  df-er 8674  df-map 8804  df-en 8922  df-dom 8923  df-sdom 8924  df-fin 8925  df-pnf 11217  df-mnf 11218  df-xr 11219  df-ltxr 11220  df-le 11221  df-sub 11414  df-neg 11415  df-nn 12194  df-2 12256  df-3 12257  df-4 12258  df-5 12259  df-6 12260  df-n0 12450  df-z 12537  df-uz 12801  df-fz 13476  df-struct 17124  df-sets 17141  df-slot 17159  df-ndx 17171  df-base 17187  df-ress 17208  df-plusg 17240  df-mulr 17241  df-sca 17243  df-vsca 17244  df-0g 17411  df-proset 18262  df-poset 18281  df-plt 18296  df-lub 18312  df-glb 18313  df-join 18314  df-meet 18315  df-p0 18391  df-p1 18392  df-lat 18398  df-clat 18465  df-mgm 18574  df-sgrp 18653  df-mnd 18669  df-submnd 18718  df-grp 18875  df-minusg 18876  df-sbg 18877  df-subg 19062  df-cntz 19256  df-lsm 19573  df-cmn 19719  df-abl 19720  df-mgp 20057  df-rng 20069  df-ur 20098  df-ring 20151  df-oppr 20253  df-dvdsr 20273  df-unit 20274  df-invr 20304  df-dvr 20317  df-drng 20647  df-lmod 20775  df-lss 20845  df-lsp 20885  df-lvec 21017  df-oposet 39176  df-ol 39178  df-oml 39179  df-covers 39266  df-ats 39267  df-atl 39298  df-cvlat 39322  df-hlat 39351  df-llines 39499  df-lplanes 39500  df-lvols 39501  df-lines 39502  df-psubsp 39504  df-pmap 39505  df-padd 39797  df-lhyp 39989  df-laut 39990  df-ldil 40105  df-ltrn 40106  df-trl 40160  df-tendo 40756  df-edring 40758  df-disoa 41030  df-dvech 41080  df-dib 41140  df-dic 41174  df-dih 41230

This theorem is referenced by:  dihjatcc  41423