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 35586 |
. . 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 34323 |
. . . . . . . . . . 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 34885 |
. . . . . . . . . 10
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐶 ∈ 𝐴 ∧ ¬ 𝐶 ≤ 𝑊) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝐺 ∈ 𝑇) |
16 | 1, 11, 12, 15 | syl3anc 1318 |
. . . . . . . . 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 34885 |
. . . . . . . . . . 11
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐶 ∈ 𝐴 ∧ ¬ 𝐶 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → 𝐷 ∈ 𝑇) |
20 | 1, 11, 17, 19 | syl3anc 1318 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐷 ∈ 𝑇) |
21 | 2, 13 | ltrncnv 34450 |
. . . . . . . . . 10
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐷 ∈ 𝑇) → ◡𝐷 ∈ 𝑇) |
22 | 1, 20, 21 | syl2anc 691 |
. . . . . . . . 9
⊢ (𝜑 → ◡𝐷 ∈ 𝑇) |
23 | 2, 13 | ltrnco 35025 |
. . . . . . . . 9
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇 ∧ ◡𝐷 ∈ 𝑇) → (𝐺 ∘ ◡𝐷) ∈ 𝑇) |
24 | 1, 16, 22, 23 | syl3anc 1318 |
. . . . . . . 8
⊢ (𝜑 → (𝐺 ∘ ◡𝐷) ∈ 𝑇) |
25 | 24 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑉) ∧ 𝑠 = 0 )) → (𝐺 ∘ ◡𝐷) ∈ 𝑇) |
26 | | simprll 798 |
. . . . . . 7
⊢ ((𝜑 ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑉) ∧ 𝑠 = 0 )) → 𝑓 ∈ 𝑇) |
27 | | simprlr 799 |
. . . . . . . 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 35727 |
. . . . . . . . 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 4611 |
. . . . . . 7
⊢ ((𝜑 ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑉) ∧ 𝑠 = 0 )) → (𝑅‘𝑓) ≤ (𝑅‘(𝐺 ∘ ◡𝐷))) |
39 | 7, 2, 13, 34, 35 | tendoex 35281 |
. . . . . . 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 1323 |
. . . . . 6
⊢ ((𝜑 ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑉) ∧ 𝑠 = 0 )) → ∃𝑡 ∈ 𝐸 (𝑡‘(𝐺 ∘ ◡𝐷)) = 𝑓) |
41 | | df-rex 2902 |
. . . . . 6
⊢
(∃𝑡 ∈
𝐸 (𝑡‘(𝐺 ∘ ◡𝐷)) = 𝑓 ↔ ∃𝑡(𝑡 ∈ 𝐸 ∧ (𝑡‘(𝐺 ∘ ◡𝐷)) = 𝑓)) |
42 | 40, 41 | sylib 207 |
. . . . 5
⊢ ((𝜑 ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑉) ∧ 𝑠 = 0 )) → ∃𝑡(𝑡 ∈ 𝐸 ∧ (𝑡‘(𝐺 ∘ ◡𝐷)) = 𝑓)) |
43 | | eqidd 2611 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡 ∈ 𝐸 ∧ (𝑡‘(𝐺 ∘ ◡𝐷)) = 𝑓)) → (𝑡‘𝐺) = (𝑡‘𝐺)) |
44 | | simprl 790 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡 ∈ 𝐸 ∧ (𝑡‘(𝐺 ∘ ◡𝐷)) = 𝑓)) → 𝑡 ∈ 𝐸) |
45 | 1 | ad2antrr 758 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡 ∈ 𝐸 ∧ (𝑡‘(𝐺 ∘ ◡𝐷)) = 𝑓)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
46 | 12 | ad2antrr 758 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡 ∈ 𝐸 ∧ (𝑡‘(𝐺 ∘ ◡𝐷)) = 𝑓)) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) |
47 | | fvex 6113 |
. . . . . . . . . . . 12
⊢ (𝑡‘𝐺) ∈ V |
48 | | vex 3176 |
. . . . . . . . . . . 12
⊢ 𝑡 ∈ V |
49 | 7, 8, 2, 9, 13, 35, 3, 14, 47, 48 | dihopelvalcqat 35553 |
. . . . . . . . . . 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 691 |
. . . . . . . . . 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 959 |
. . . . . . . . 9
⊢ (((𝜑 ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡 ∈ 𝐸 ∧ (𝑡‘(𝐺 ∘ ◡𝐷)) = 𝑓)) → 〈(𝑡‘𝐺), 𝑡〉 ∈ (𝐼‘𝑃)) |
52 | | eqidd 2611 |
. . . . . . . . . 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 35102 |
. . . . . . . . . . 11
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑡 ∈ 𝐸) → (𝑁‘𝑡) ∈ 𝐸) |
55 | 45, 44, 54 | syl2anc 691 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡 ∈ 𝐸 ∧ (𝑡‘(𝐺 ∘ ◡𝐷)) = 𝑓)) → (𝑁‘𝑡) ∈ 𝐸) |
56 | 17 | ad2antrr 758 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡 ∈ 𝐸 ∧ (𝑡‘(𝐺 ∘ ◡𝐷)) = 𝑓)) → (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) |
57 | | fvex 6113 |
. . . . . . . . . . . 12
⊢ ((𝑁‘𝑡)‘𝐷) ∈ V |
58 | | fvex 6113 |
. . . . . . . . . . . 12
⊢ (𝑁‘𝑡) ∈ V |
59 | 7, 8, 2, 9, 13, 35, 3, 18, 57, 58 | dihopelvalcqat 35553 |
. . . . . . . . . . 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 691 |
. . . . . . . . . 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 959 |
. . . . . . . . 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 758 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡 ∈ 𝐸 ∧ (𝑡‘(𝐺 ∘ ◡𝐷)) = 𝑓)) → 𝐺 ∈ 𝑇) |
63 | 22 | ad2antrr 758 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡 ∈ 𝐸 ∧ (𝑡‘(𝐺 ∘ ◡𝐷)) = 𝑓)) → ◡𝐷 ∈ 𝑇) |
64 | 2, 13, 35 | tendospdi1 35327 |
. . . . . . . . . . 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 1320 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡 ∈ 𝐸 ∧ (𝑡‘(𝐺 ∘ ◡𝐷)) = 𝑓)) → (𝑡‘(𝐺 ∘ ◡𝐷)) = ((𝑡‘𝐺) ∘ (𝑡‘◡𝐷))) |
66 | | simprr 792 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡 ∈ 𝐸 ∧ (𝑡‘(𝐺 ∘ ◡𝐷)) = 𝑓)) → (𝑡‘(𝐺 ∘ ◡𝐷)) = 𝑓) |
67 | 20 | ad2antrr 758 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡 ∈ 𝐸 ∧ (𝑡‘(𝐺 ∘ ◡𝐷)) = 𝑓)) → 𝐷 ∈ 𝑇) |
68 | 53, 13 | tendoi2 35101 |
. . . . . . . . . . . . 13
⊢ ((𝑡 ∈ 𝐸 ∧ 𝐷 ∈ 𝑇) → ((𝑁‘𝑡)‘𝐷) = ◡(𝑡‘𝐷)) |
69 | 44, 67, 68 | syl2anc 691 |
. . . . . . . . . . . 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 35328 |
. . . . . . . . . . . . 13
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑡 ∈ 𝐸 ∧ 𝐷 ∈ 𝑇) → ◡(𝑡‘𝐷) = (𝑡‘◡𝐷)) |
71 | 45, 44, 67, 70 | syl3anc 1318 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡 ∈ 𝐸 ∧ (𝑡‘(𝐺 ∘ ◡𝐷)) = 𝑓)) → ◡(𝑡‘𝐷) = (𝑡‘◡𝐷)) |
72 | 69, 71 | eqtr2d 2645 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡 ∈ 𝐸 ∧ (𝑡‘(𝐺 ∘ ◡𝐷)) = 𝑓)) → (𝑡‘◡𝐷) = ((𝑁‘𝑡)‘𝐷)) |
73 | 72 | coeq2d 5206 |
. . . . . . . . . 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 2652 |
. . . . . . . . 9
⊢ (((𝜑 ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡 ∈ 𝐸 ∧ (𝑡‘(𝐺 ∘ ◡𝐷)) = 𝑓)) → 𝑓 = ((𝑡‘𝐺) ∘ ((𝑁‘𝑡)‘𝐷))) |
75 | | simplrr 797 |
. . . . . . . . . 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 35104 |
. . . . . . . . . . 11
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑡 ∈ 𝐸) → (𝑡𝐽(𝑁‘𝑡)) = 0 ) |
79 | 45, 44, 78 | syl2anc 691 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡 ∈ 𝐸 ∧ (𝑡‘(𝐺 ∘ ◡𝐷)) = 𝑓)) → (𝑡𝐽(𝑁‘𝑡)) = 0 ) |
80 | 75, 79 | eqtr4d 2647 |
. . . . . . . . 9
⊢ (((𝜑 ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑉) ∧ 𝑠 = 0 )) ∧ (𝑡 ∈ 𝐸 ∧ (𝑡‘(𝐺 ∘ ◡𝐷)) = 𝑓)) → 𝑠 = (𝑡𝐽(𝑁‘𝑡))) |
81 | | opeq1 4340 |
. . . . . . . . . . . . . . 15
⊢ (𝑔 = (𝑡‘𝐺) → 〈𝑔, 𝑡〉 = 〈(𝑡‘𝐺), 𝑡〉) |
82 | 81 | eleq1d 2672 |
. . . . . . . . . . . . . 14
⊢ (𝑔 = (𝑡‘𝐺) → (〈𝑔, 𝑡〉 ∈ (𝐼‘𝑃) ↔ 〈(𝑡‘𝐺), 𝑡〉 ∈ (𝐼‘𝑃))) |
83 | 82 | anbi1d 737 |
. . . . . . . . . . . . 13
⊢ (𝑔 = (𝑡‘𝐺) → ((〈𝑔, 𝑡〉 ∈ (𝐼‘𝑃) ∧ 〈ℎ, 𝑢〉 ∈ (𝐼‘𝑄)) ↔ (〈(𝑡‘𝐺), 𝑡〉 ∈ (𝐼‘𝑃) ∧ 〈ℎ, 𝑢〉 ∈ (𝐼‘𝑄)))) |
84 | | coeq1 5201 |
. . . . . . . . . . . . . . 15
⊢ (𝑔 = (𝑡‘𝐺) → (𝑔 ∘ ℎ) = ((𝑡‘𝐺) ∘ ℎ)) |
85 | 84 | eqeq2d 2620 |
. . . . . . . . . . . . . 14
⊢ (𝑔 = (𝑡‘𝐺) → (𝑓 = (𝑔 ∘ ℎ) ↔ 𝑓 = ((𝑡‘𝐺) ∘ ℎ))) |
86 | 85 | anbi1d 737 |
. . . . . . . . . . . . 13
⊢ (𝑔 = (𝑡‘𝐺) → ((𝑓 = (𝑔 ∘ ℎ) ∧ 𝑠 = (𝑡𝐽𝑢)) ↔ (𝑓 = ((𝑡‘𝐺) ∘ ℎ) ∧ 𝑠 = (𝑡𝐽𝑢)))) |
87 | 83, 86 | anbi12d 743 |
. . . . . . . . . . . 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 4340 |
. . . . . . . . . . . . . . 15
⊢ (ℎ = ((𝑁‘𝑡)‘𝐷) → 〈ℎ, 𝑢〉 = 〈((𝑁‘𝑡)‘𝐷), 𝑢〉) |
89 | 88 | eleq1d 2672 |
. . . . . . . . . . . . . 14
⊢ (ℎ = ((𝑁‘𝑡)‘𝐷) → (〈ℎ, 𝑢〉 ∈ (𝐼‘𝑄) ↔ 〈((𝑁‘𝑡)‘𝐷), 𝑢〉 ∈ (𝐼‘𝑄))) |
90 | 89 | anbi2d 736 |
. . . . . . . . . . . . 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 5202 |
. . . . . . . . . . . . . . 15
⊢ (ℎ = ((𝑁‘𝑡)‘𝐷) → ((𝑡‘𝐺) ∘ ℎ) = ((𝑡‘𝐺) ∘ ((𝑁‘𝑡)‘𝐷))) |
92 | 91 | eqeq2d 2620 |
. . . . . . . . . . . . . 14
⊢ (ℎ = ((𝑁‘𝑡)‘𝐷) → (𝑓 = ((𝑡‘𝐺) ∘ ℎ) ↔ 𝑓 = ((𝑡‘𝐺) ∘ ((𝑁‘𝑡)‘𝐷)))) |
93 | 92 | anbi1d 737 |
. . . . . . . . . . . . 13
⊢ (ℎ = ((𝑁‘𝑡)‘𝐷) → ((𝑓 = ((𝑡‘𝐺) ∘ ℎ) ∧ 𝑠 = (𝑡𝐽𝑢)) ↔ (𝑓 = ((𝑡‘𝐺) ∘ ((𝑁‘𝑡)‘𝐷)) ∧ 𝑠 = (𝑡𝐽𝑢)))) |
94 | 90, 93 | anbi12d 743 |
. . . . . . . . . . . 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 4341 |
. . . . . . . . . . . . . . 15
⊢ (𝑢 = (𝑁‘𝑡) → 〈((𝑁‘𝑡)‘𝐷), 𝑢〉 = 〈((𝑁‘𝑡)‘𝐷), (𝑁‘𝑡)〉) |
96 | 95 | eleq1d 2672 |
. . . . . . . . . . . . . 14
⊢ (𝑢 = (𝑁‘𝑡) → (〈((𝑁‘𝑡)‘𝐷), 𝑢〉 ∈ (𝐼‘𝑄) ↔ 〈((𝑁‘𝑡)‘𝐷), (𝑁‘𝑡)〉 ∈ (𝐼‘𝑄))) |
97 | 96 | anbi2d 736 |
. . . . . . . . . . . . 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 6557 |
. . . . . . . . . . . . . . 15
⊢ (𝑢 = (𝑁‘𝑡) → (𝑡𝐽𝑢) = (𝑡𝐽(𝑁‘𝑡))) |
99 | 98 | eqeq2d 2620 |
. . . . . . . . . . . . . 14
⊢ (𝑢 = (𝑁‘𝑡) → (𝑠 = (𝑡𝐽𝑢) ↔ 𝑠 = (𝑡𝐽(𝑁‘𝑡)))) |
100 | 99 | anbi2d 736 |
. . . . . . . . . . . . 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 743 |
. . . . . . . . . . . 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 1389 |
. . . . . . . . . . 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 3270 |
. . . . . . . . . 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 1416 |
. . . . . . . . 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 1319 |
. . . . . . . 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 449 |
. . . . . . 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 1833 |
. . . . . 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 2029 |
. . . . . 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 | syl6ib 240 |
. . . . 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 449 |
. . 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 474 |
. . . . . . . . 9
⊢ (𝜑 → 𝐾 ∈ HL) |
113 | | hllat 33668 |
. . . . . . . . 9
⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat) |
114 | 112, 113 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → 𝐾 ∈ Lat) |
115 | 12 | simpld 474 |
. . . . . . . . 9
⊢ (𝜑 → 𝑃 ∈ 𝐴) |
116 | 17 | simpld 474 |
. . . . . . . . 9
⊢ (𝜑 → 𝑄 ∈ 𝐴) |
117 | 28, 29, 8 | hlatjcl 33671 |
. . . . . . . . 9
⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 ∨ 𝑄) ∈ 𝐵) |
118 | 112, 115,
116, 117 | syl3anc 1318 |
. . . . . . . 8
⊢ (𝜑 → (𝑃 ∨ 𝑄) ∈ 𝐵) |
119 | 1 | simprd 478 |
. . . . . . . . 9
⊢ (𝜑 → 𝑊 ∈ 𝐻) |
120 | 28, 2 | lhpbase 34302 |
. . . . . . . . 9
⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ 𝐵) |
121 | 119, 120 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → 𝑊 ∈ 𝐵) |
122 | 28, 30 | latmcl 16875 |
. . . . . . . 8
⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑄) ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) → ((𝑃 ∨ 𝑄) ∧ 𝑊) ∈ 𝐵) |
123 | 114, 118,
121, 122 | syl3anc 1318 |
. . . . . . 7
⊢ (𝜑 → ((𝑃 ∨ 𝑄) ∧ 𝑊) ∈ 𝐵) |
124 | 33, 123 | syl5eqel 2692 |
. . . . . 6
⊢ (𝜑 → 𝑉 ∈ 𝐵) |
125 | 28, 7, 30 | latmle2 16900 |
. . . . . . . 8
⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑄) ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) → ((𝑃 ∨ 𝑄) ∧ 𝑊) ≤ 𝑊) |
126 | 114, 118,
121, 125 | syl3anc 1318 |
. . . . . . 7
⊢ (𝜑 → ((𝑃 ∨ 𝑄) ∧ 𝑊) ≤ 𝑊) |
127 | 33, 126 | syl5eqbr 4618 |
. . . . . 6
⊢ (𝜑 → 𝑉 ≤ 𝑊) |
128 | | eqid 2610 |
. . . . . . 7
⊢
((DIsoB‘𝐾)‘𝑊) = ((DIsoB‘𝐾)‘𝑊) |
129 | 28, 7, 2, 3, 128 | dihvalb 35544 |
. . . . . 6
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑉 ∈ 𝐵 ∧ 𝑉 ≤ 𝑊)) → (𝐼‘𝑉) = (((DIsoB‘𝐾)‘𝑊)‘𝑉)) |
130 | 1, 124, 127, 129 | syl12anc 1316 |
. . . . 5
⊢ (𝜑 → (𝐼‘𝑉) = (((DIsoB‘𝐾)‘𝑊)‘𝑉)) |
131 | 130 | eleq2d 2673 |
. . . 4
⊢ (𝜑 → (〈𝑓, 𝑠〉 ∈ (𝐼‘𝑉) ↔ 〈𝑓, 𝑠〉 ∈ (((DIsoB‘𝐾)‘𝑊)‘𝑉))) |
132 | 28, 7, 2, 13, 34, 77, 128 | dibopelval3 35455 |
. . . . 5
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑉 ∈ 𝐵 ∧ 𝑉 ≤ 𝑊)) → (〈𝑓, 𝑠〉 ∈ (((DIsoB‘𝐾)‘𝑊)‘𝑉) ↔ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑉) ∧ 𝑠 = 0 ))) |
133 | 1, 124, 127, 132 | syl12anc 1316 |
. . . 4
⊢ (𝜑 → (〈𝑓, 𝑠〉 ∈ (((DIsoB‘𝐾)‘𝑊)‘𝑉) ↔ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑉) ∧ 𝑠 = 0 ))) |
134 | 131, 133 | bitrd 267 |
. . 3
⊢ (𝜑 → (〈𝑓, 𝑠〉 ∈ (𝐼‘𝑉) ↔ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑉) ∧ 𝑠 = 0 ))) |
135 | | eqid 2610 |
. . . 4
⊢
(LSubSp‘𝑈) =
(LSubSp‘𝑈) |
136 | 28, 8 | atbase 33594 |
. . . . 5
⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ 𝐵) |
137 | 115, 136 | syl 17 |
. . . 4
⊢ (𝜑 → 𝑃 ∈ 𝐵) |
138 | 28, 8 | atbase 33594 |
. . . . 5
⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ 𝐵) |
139 | 116, 138 | syl 17 |
. . . 4
⊢ (𝜑 → 𝑄 ∈ 𝐵) |
140 | 28, 2, 13, 35, 76, 31, 135, 32, 3, 1, 137, 139 | dihopellsm 35562 |
. . 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))))) |
141 | 111, 134,
140 | 3imtr4d 282 |
. 2
⊢ (𝜑 → (〈𝑓, 𝑠〉 ∈ (𝐼‘𝑉) → 〈𝑓, 𝑠〉 ∈ ((𝐼‘𝑃) ⊕ (𝐼‘𝑄)))) |
142 | 5, 141 | relssdv 5135 |
1
⊢ (𝜑 → (𝐼‘𝑉) ⊆ ((𝐼‘𝑃) ⊕ (𝐼‘𝑄))) |