Step | Hyp | Ref
| Expression |
1 | | frlmup.b |
. 2
⊢ 𝐵 = (Base‘𝐹) |
2 | | eqid 2610 |
. 2
⊢ (
·𝑠 ‘𝐹) = ( ·𝑠
‘𝐹) |
3 | | frlmup.v |
. 2
⊢ · = (
·𝑠 ‘𝑇) |
4 | | eqid 2610 |
. 2
⊢
(Scalar‘𝐹) =
(Scalar‘𝐹) |
5 | | eqid 2610 |
. 2
⊢
(Scalar‘𝑇) =
(Scalar‘𝑇) |
6 | | eqid 2610 |
. 2
⊢
(Base‘(Scalar‘𝐹)) = (Base‘(Scalar‘𝐹)) |
7 | | frlmup.r |
. . . 4
⊢ (𝜑 → 𝑅 = (Scalar‘𝑇)) |
8 | | frlmup.t |
. . . . 5
⊢ (𝜑 → 𝑇 ∈ LMod) |
9 | 5 | lmodring 18694 |
. . . . 5
⊢ (𝑇 ∈ LMod →
(Scalar‘𝑇) ∈
Ring) |
10 | 8, 9 | syl 17 |
. . . 4
⊢ (𝜑 → (Scalar‘𝑇) ∈ Ring) |
11 | 7, 10 | eqeltrd 2688 |
. . 3
⊢ (𝜑 → 𝑅 ∈ Ring) |
12 | | frlmup.i |
. . 3
⊢ (𝜑 → 𝐼 ∈ 𝑋) |
13 | | frlmup.f |
. . . 4
⊢ 𝐹 = (𝑅 freeLMod 𝐼) |
14 | 13 | frlmlmod 19912 |
. . 3
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑋) → 𝐹 ∈ LMod) |
15 | 11, 12, 14 | syl2anc 691 |
. 2
⊢ (𝜑 → 𝐹 ∈ LMod) |
16 | 13 | frlmsca 19916 |
. . . 4
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑋) → 𝑅 = (Scalar‘𝐹)) |
17 | 11, 12, 16 | syl2anc 691 |
. . 3
⊢ (𝜑 → 𝑅 = (Scalar‘𝐹)) |
18 | 7, 17 | eqtr3d 2646 |
. 2
⊢ (𝜑 → (Scalar‘𝑇) = (Scalar‘𝐹)) |
19 | | frlmup.c |
. . 3
⊢ 𝐶 = (Base‘𝑇) |
20 | | eqid 2610 |
. . 3
⊢
(+g‘𝐹) = (+g‘𝐹) |
21 | | eqid 2610 |
. . 3
⊢
(+g‘𝑇) = (+g‘𝑇) |
22 | | lmodgrp 18693 |
. . . 4
⊢ (𝐹 ∈ LMod → 𝐹 ∈ Grp) |
23 | 15, 22 | syl 17 |
. . 3
⊢ (𝜑 → 𝐹 ∈ Grp) |
24 | | lmodgrp 18693 |
. . . 4
⊢ (𝑇 ∈ LMod → 𝑇 ∈ Grp) |
25 | 8, 24 | syl 17 |
. . 3
⊢ (𝜑 → 𝑇 ∈ Grp) |
26 | | eleq1 2676 |
. . . . . . 7
⊢ (𝑧 = 𝑥 → (𝑧 ∈ 𝐵 ↔ 𝑥 ∈ 𝐵)) |
27 | 26 | anbi2d 736 |
. . . . . 6
⊢ (𝑧 = 𝑥 → ((𝜑 ∧ 𝑧 ∈ 𝐵) ↔ (𝜑 ∧ 𝑥 ∈ 𝐵))) |
28 | | oveq1 6556 |
. . . . . . . 8
⊢ (𝑧 = 𝑥 → (𝑧 ∘𝑓 · 𝐴) = (𝑥 ∘𝑓 · 𝐴)) |
29 | 28 | oveq2d 6565 |
. . . . . . 7
⊢ (𝑧 = 𝑥 → (𝑇 Σg (𝑧 ∘𝑓
·
𝐴)) = (𝑇 Σg (𝑥 ∘𝑓
·
𝐴))) |
30 | 29 | eleq1d 2672 |
. . . . . 6
⊢ (𝑧 = 𝑥 → ((𝑇 Σg (𝑧 ∘𝑓
·
𝐴)) ∈ 𝐶 ↔ (𝑇 Σg (𝑥 ∘𝑓
·
𝐴)) ∈ 𝐶)) |
31 | 27, 30 | imbi12d 333 |
. . . . 5
⊢ (𝑧 = 𝑥 → (((𝜑 ∧ 𝑧 ∈ 𝐵) → (𝑇 Σg (𝑧 ∘𝑓
·
𝐴)) ∈ 𝐶) ↔ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑇 Σg (𝑥 ∘𝑓
·
𝐴)) ∈ 𝐶))) |
32 | | eqid 2610 |
. . . . . 6
⊢
(0g‘𝑇) = (0g‘𝑇) |
33 | | lmodcmn 18734 |
. . . . . . . 8
⊢ (𝑇 ∈ LMod → 𝑇 ∈ CMnd) |
34 | 8, 33 | syl 17 |
. . . . . . 7
⊢ (𝜑 → 𝑇 ∈ CMnd) |
35 | 34 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → 𝑇 ∈ CMnd) |
36 | 12 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → 𝐼 ∈ 𝑋) |
37 | 8 | ad2antrr 758 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑧 ∈ 𝐵) ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ 𝐶)) → 𝑇 ∈ LMod) |
38 | | simprl 790 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑧 ∈ 𝐵) ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ 𝐶)) → 𝑥 ∈ (Base‘𝑅)) |
39 | 7 | fveq2d 6107 |
. . . . . . . . . 10
⊢ (𝜑 → (Base‘𝑅) =
(Base‘(Scalar‘𝑇))) |
40 | 39 | ad2antrr 758 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑧 ∈ 𝐵) ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ 𝐶)) → (Base‘𝑅) = (Base‘(Scalar‘𝑇))) |
41 | 38, 40 | eleqtrd 2690 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑧 ∈ 𝐵) ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ 𝐶)) → 𝑥 ∈ (Base‘(Scalar‘𝑇))) |
42 | | simprr 792 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑧 ∈ 𝐵) ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ 𝐶)) → 𝑦 ∈ 𝐶) |
43 | | eqid 2610 |
. . . . . . . . 9
⊢
(Base‘(Scalar‘𝑇)) = (Base‘(Scalar‘𝑇)) |
44 | 19, 5, 3, 43 | lmodvscl 18703 |
. . . . . . . 8
⊢ ((𝑇 ∈ LMod ∧ 𝑥 ∈
(Base‘(Scalar‘𝑇)) ∧ 𝑦 ∈ 𝐶) → (𝑥 · 𝑦) ∈ 𝐶) |
45 | 37, 41, 42, 44 | syl3anc 1318 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑧 ∈ 𝐵) ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ 𝐶)) → (𝑥 · 𝑦) ∈ 𝐶) |
46 | | eqid 2610 |
. . . . . . . . 9
⊢
(Base‘𝑅) =
(Base‘𝑅) |
47 | 13, 46, 1 | frlmbasf 19923 |
. . . . . . . 8
⊢ ((𝐼 ∈ 𝑋 ∧ 𝑧 ∈ 𝐵) → 𝑧:𝐼⟶(Base‘𝑅)) |
48 | 12, 47 | sylan 487 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → 𝑧:𝐼⟶(Base‘𝑅)) |
49 | | frlmup.a |
. . . . . . . 8
⊢ (𝜑 → 𝐴:𝐼⟶𝐶) |
50 | 49 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → 𝐴:𝐼⟶𝐶) |
51 | | inidm 3784 |
. . . . . . 7
⊢ (𝐼 ∩ 𝐼) = 𝐼 |
52 | 45, 48, 50, 36, 36, 51 | off 6810 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → (𝑧 ∘𝑓 · 𝐴):𝐼⟶𝐶) |
53 | | ovex 6577 |
. . . . . . . 8
⊢ (𝑧 ∘𝑓
·
𝐴) ∈
V |
54 | 53 | a1i 11 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → (𝑧 ∘𝑓 · 𝐴) ∈ V) |
55 | | ffun 5961 |
. . . . . . . 8
⊢ ((𝑧 ∘𝑓
·
𝐴):𝐼⟶𝐶 → Fun (𝑧 ∘𝑓 · 𝐴)) |
56 | 52, 55 | syl 17 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → Fun (𝑧 ∘𝑓 · 𝐴)) |
57 | | fvex 6113 |
. . . . . . . 8
⊢
(0g‘𝑇) ∈ V |
58 | 57 | a1i 11 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → (0g‘𝑇) ∈ V) |
59 | | eqid 2610 |
. . . . . . . . . . 11
⊢
(0g‘𝑅) = (0g‘𝑅) |
60 | 13, 59, 1 | frlmbasfsupp 19921 |
. . . . . . . . . 10
⊢ ((𝐼 ∈ 𝑋 ∧ 𝑧 ∈ 𝐵) → 𝑧 finSupp (0g‘𝑅)) |
61 | 12, 60 | sylan 487 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → 𝑧 finSupp (0g‘𝑅)) |
62 | 7 | fveq2d 6107 |
. . . . . . . . . . . 12
⊢ (𝜑 → (0g‘𝑅) =
(0g‘(Scalar‘𝑇))) |
63 | 62 | eqcomd 2616 |
. . . . . . . . . . 11
⊢ (𝜑 →
(0g‘(Scalar‘𝑇)) = (0g‘𝑅)) |
64 | 63 | breq2d 4595 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑧 finSupp
(0g‘(Scalar‘𝑇)) ↔ 𝑧 finSupp (0g‘𝑅))) |
65 | 64 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → (𝑧 finSupp
(0g‘(Scalar‘𝑇)) ↔ 𝑧 finSupp (0g‘𝑅))) |
66 | 61, 65 | mpbird 246 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → 𝑧 finSupp
(0g‘(Scalar‘𝑇))) |
67 | 66 | fsuppimpd 8165 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → (𝑧 supp
(0g‘(Scalar‘𝑇))) ∈ Fin) |
68 | | ssid 3587 |
. . . . . . . . 9
⊢ (𝑧 supp
(0g‘(Scalar‘𝑇))) ⊆ (𝑧 supp
(0g‘(Scalar‘𝑇))) |
69 | 68 | a1i 11 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → (𝑧 supp
(0g‘(Scalar‘𝑇))) ⊆ (𝑧 supp
(0g‘(Scalar‘𝑇)))) |
70 | 8 | ad2antrr 758 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑧 ∈ 𝐵) ∧ 𝑤 ∈ 𝐶) → 𝑇 ∈ LMod) |
71 | | eqid 2610 |
. . . . . . . . . 10
⊢
(0g‘(Scalar‘𝑇)) =
(0g‘(Scalar‘𝑇)) |
72 | 19, 5, 3, 71, 32 | lmod0vs 18719 |
. . . . . . . . 9
⊢ ((𝑇 ∈ LMod ∧ 𝑤 ∈ 𝐶) →
((0g‘(Scalar‘𝑇)) · 𝑤) = (0g‘𝑇)) |
73 | 70, 72 | sylancom 698 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑧 ∈ 𝐵) ∧ 𝑤 ∈ 𝐶) →
((0g‘(Scalar‘𝑇)) · 𝑤) = (0g‘𝑇)) |
74 | | fvex 6113 |
. . . . . . . . 9
⊢
(0g‘(Scalar‘𝑇)) ∈ V |
75 | 74 | a1i 11 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) →
(0g‘(Scalar‘𝑇)) ∈ V) |
76 | 69, 73, 48, 50, 36, 75 | suppssof1 7215 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → ((𝑧 ∘𝑓 · 𝐴) supp
(0g‘𝑇))
⊆ (𝑧 supp
(0g‘(Scalar‘𝑇)))) |
77 | | suppssfifsupp 8173 |
. . . . . . 7
⊢ ((((𝑧 ∘𝑓
·
𝐴) ∈ V ∧ Fun
(𝑧
∘𝑓 · 𝐴) ∧ (0g‘𝑇) ∈ V) ∧ ((𝑧 supp
(0g‘(Scalar‘𝑇))) ∈ Fin ∧ ((𝑧 ∘𝑓 · 𝐴) supp
(0g‘𝑇))
⊆ (𝑧 supp
(0g‘(Scalar‘𝑇))))) → (𝑧 ∘𝑓 · 𝐴) finSupp
(0g‘𝑇)) |
78 | 54, 56, 58, 67, 76, 77 | syl32anc 1326 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → (𝑧 ∘𝑓 · 𝐴) finSupp
(0g‘𝑇)) |
79 | 19, 32, 35, 36, 52, 78 | gsumcl 18139 |
. . . . 5
⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → (𝑇 Σg (𝑧 ∘𝑓
·
𝐴)) ∈ 𝐶) |
80 | 31, 79 | chvarv 2251 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑇 Σg (𝑥 ∘𝑓
·
𝐴)) ∈ 𝐶) |
81 | | frlmup.e |
. . . 4
⊢ 𝐸 = (𝑥 ∈ 𝐵 ↦ (𝑇 Σg (𝑥 ∘𝑓
·
𝐴))) |
82 | 80, 81 | fmptd 6292 |
. . 3
⊢ (𝜑 → 𝐸:𝐵⟶𝐶) |
83 | 34 | adantr 480 |
. . . . 5
⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → 𝑇 ∈ CMnd) |
84 | 12 | adantr 480 |
. . . . 5
⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → 𝐼 ∈ 𝑋) |
85 | | eleq1 2676 |
. . . . . . . . 9
⊢ (𝑧 = 𝑦 → (𝑧 ∈ 𝐵 ↔ 𝑦 ∈ 𝐵)) |
86 | 85 | anbi2d 736 |
. . . . . . . 8
⊢ (𝑧 = 𝑦 → ((𝜑 ∧ 𝑧 ∈ 𝐵) ↔ (𝜑 ∧ 𝑦 ∈ 𝐵))) |
87 | | oveq1 6556 |
. . . . . . . . 9
⊢ (𝑧 = 𝑦 → (𝑧 ∘𝑓 · 𝐴) = (𝑦 ∘𝑓 · 𝐴)) |
88 | 87 | feq1d 5943 |
. . . . . . . 8
⊢ (𝑧 = 𝑦 → ((𝑧 ∘𝑓 · 𝐴):𝐼⟶𝐶 ↔ (𝑦 ∘𝑓 · 𝐴):𝐼⟶𝐶)) |
89 | 86, 88 | imbi12d 333 |
. . . . . . 7
⊢ (𝑧 = 𝑦 → (((𝜑 ∧ 𝑧 ∈ 𝐵) → (𝑧 ∘𝑓 · 𝐴):𝐼⟶𝐶) ↔ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (𝑦 ∘𝑓 · 𝐴):𝐼⟶𝐶))) |
90 | 89, 52 | chvarv 2251 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (𝑦 ∘𝑓 · 𝐴):𝐼⟶𝐶) |
91 | 90 | adantrr 749 |
. . . . 5
⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → (𝑦 ∘𝑓 · 𝐴):𝐼⟶𝐶) |
92 | 52 | adantrl 748 |
. . . . 5
⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → (𝑧 ∘𝑓 · 𝐴):𝐼⟶𝐶) |
93 | 87 | breq1d 4593 |
. . . . . . . 8
⊢ (𝑧 = 𝑦 → ((𝑧 ∘𝑓 · 𝐴) finSupp
(0g‘𝑇)
↔ (𝑦
∘𝑓 · 𝐴) finSupp (0g‘𝑇))) |
94 | 86, 93 | imbi12d 333 |
. . . . . . 7
⊢ (𝑧 = 𝑦 → (((𝜑 ∧ 𝑧 ∈ 𝐵) → (𝑧 ∘𝑓 · 𝐴) finSupp
(0g‘𝑇))
↔ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (𝑦 ∘𝑓 · 𝐴) finSupp
(0g‘𝑇)))) |
95 | 94, 78 | chvarv 2251 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (𝑦 ∘𝑓 · 𝐴) finSupp
(0g‘𝑇)) |
96 | 95 | adantrr 749 |
. . . . 5
⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → (𝑦 ∘𝑓 · 𝐴) finSupp
(0g‘𝑇)) |
97 | 78 | adantrl 748 |
. . . . 5
⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → (𝑧 ∘𝑓 · 𝐴) finSupp
(0g‘𝑇)) |
98 | 19, 32, 21, 83, 84, 91, 92, 96, 97 | gsumadd 18146 |
. . . 4
⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → (𝑇 Σg ((𝑦 ∘𝑓
·
𝐴)
∘𝑓 (+g‘𝑇)(𝑧 ∘𝑓 · 𝐴))) = ((𝑇 Σg (𝑦 ∘𝑓
·
𝐴))(+g‘𝑇)(𝑇 Σg (𝑧 ∘𝑓
·
𝐴)))) |
99 | 1, 20 | lmodvacl 18700 |
. . . . . . . 8
⊢ ((𝐹 ∈ LMod ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → (𝑦(+g‘𝐹)𝑧) ∈ 𝐵) |
100 | 99 | 3expb 1258 |
. . . . . . 7
⊢ ((𝐹 ∈ LMod ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → (𝑦(+g‘𝐹)𝑧) ∈ 𝐵) |
101 | 15, 100 | sylan 487 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → (𝑦(+g‘𝐹)𝑧) ∈ 𝐵) |
102 | | oveq1 6556 |
. . . . . . . 8
⊢ (𝑥 = (𝑦(+g‘𝐹)𝑧) → (𝑥 ∘𝑓 · 𝐴) = ((𝑦(+g‘𝐹)𝑧) ∘𝑓 · 𝐴)) |
103 | 102 | oveq2d 6565 |
. . . . . . 7
⊢ (𝑥 = (𝑦(+g‘𝐹)𝑧) → (𝑇 Σg (𝑥 ∘𝑓
·
𝐴)) = (𝑇 Σg ((𝑦(+g‘𝐹)𝑧) ∘𝑓 · 𝐴))) |
104 | | ovex 6577 |
. . . . . . 7
⊢ (𝑇 Σg
((𝑦(+g‘𝐹)𝑧) ∘𝑓 · 𝐴)) ∈ V |
105 | 103, 81, 104 | fvmpt 6191 |
. . . . . 6
⊢ ((𝑦(+g‘𝐹)𝑧) ∈ 𝐵 → (𝐸‘(𝑦(+g‘𝐹)𝑧)) = (𝑇 Σg ((𝑦(+g‘𝐹)𝑧) ∘𝑓 · 𝐴))) |
106 | 101, 105 | syl 17 |
. . . . 5
⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → (𝐸‘(𝑦(+g‘𝐹)𝑧)) = (𝑇 Σg ((𝑦(+g‘𝐹)𝑧) ∘𝑓 · 𝐴))) |
107 | 11 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → 𝑅 ∈ Ring) |
108 | | simprl 790 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → 𝑦 ∈ 𝐵) |
109 | | simprr 792 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → 𝑧 ∈ 𝐵) |
110 | | eqid 2610 |
. . . . . . . . 9
⊢
(+g‘𝑅) = (+g‘𝑅) |
111 | 13, 1, 107, 84, 108, 109, 110, 20 | frlmplusgval 19926 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → (𝑦(+g‘𝐹)𝑧) = (𝑦 ∘𝑓
(+g‘𝑅)𝑧)) |
112 | 111 | oveq1d 6564 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((𝑦(+g‘𝐹)𝑧) ∘𝑓 · 𝐴) = ((𝑦 ∘𝑓
(+g‘𝑅)𝑧) ∘𝑓 · 𝐴)) |
113 | 13, 46, 1 | frlmbasf 19923 |
. . . . . . . . . . . . 13
⊢ ((𝐼 ∈ 𝑋 ∧ 𝑦 ∈ 𝐵) → 𝑦:𝐼⟶(Base‘𝑅)) |
114 | 12, 113 | sylan 487 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝑦:𝐼⟶(Base‘𝑅)) |
115 | 114 | adantrr 749 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → 𝑦:𝐼⟶(Base‘𝑅)) |
116 | | ffn 5958 |
. . . . . . . . . . 11
⊢ (𝑦:𝐼⟶(Base‘𝑅) → 𝑦 Fn 𝐼) |
117 | 115, 116 | syl 17 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → 𝑦 Fn 𝐼) |
118 | 48 | adantrl 748 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → 𝑧:𝐼⟶(Base‘𝑅)) |
119 | | ffn 5958 |
. . . . . . . . . . 11
⊢ (𝑧:𝐼⟶(Base‘𝑅) → 𝑧 Fn 𝐼) |
120 | 118, 119 | syl 17 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → 𝑧 Fn 𝐼) |
121 | 117, 120,
84, 84, 51 | offn 6806 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → (𝑦 ∘𝑓
(+g‘𝑅)𝑧) Fn 𝐼) |
122 | | ffn 5958 |
. . . . . . . . . . 11
⊢ (𝐴:𝐼⟶𝐶 → 𝐴 Fn 𝐼) |
123 | 49, 122 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐴 Fn 𝐼) |
124 | 123 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → 𝐴 Fn 𝐼) |
125 | 121, 124,
84, 84, 51 | offn 6806 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((𝑦 ∘𝑓
(+g‘𝑅)𝑧) ∘𝑓 · 𝐴) Fn 𝐼) |
126 | | ffn 5958 |
. . . . . . . . . . 11
⊢ ((𝑦 ∘𝑓
·
𝐴):𝐼⟶𝐶 → (𝑦 ∘𝑓 · 𝐴) Fn 𝐼) |
127 | 90, 126 | syl 17 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (𝑦 ∘𝑓 · 𝐴) Fn 𝐼) |
128 | 127 | adantrr 749 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → (𝑦 ∘𝑓 · 𝐴) Fn 𝐼) |
129 | | ffn 5958 |
. . . . . . . . . . 11
⊢ ((𝑧 ∘𝑓
·
𝐴):𝐼⟶𝐶 → (𝑧 ∘𝑓 · 𝐴) Fn 𝐼) |
130 | 52, 129 | syl 17 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → (𝑧 ∘𝑓 · 𝐴) Fn 𝐼) |
131 | 130 | adantrl 748 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → (𝑧 ∘𝑓 · 𝐴) Fn 𝐼) |
132 | 128, 131,
84, 84, 51 | offn 6806 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((𝑦 ∘𝑓 · 𝐴) ∘𝑓
(+g‘𝑇)(𝑧 ∘𝑓 · 𝐴)) Fn 𝐼) |
133 | 7 | fveq2d 6107 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (+g‘𝑅) =
(+g‘(Scalar‘𝑇))) |
134 | 133 | ad2antrr 758 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑥 ∈ 𝐼) → (+g‘𝑅) =
(+g‘(Scalar‘𝑇))) |
135 | 134 | oveqd 6566 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑥 ∈ 𝐼) → ((𝑦‘𝑥)(+g‘𝑅)(𝑧‘𝑥)) = ((𝑦‘𝑥)(+g‘(Scalar‘𝑇))(𝑧‘𝑥))) |
136 | 135 | oveq1d 6564 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑥 ∈ 𝐼) → (((𝑦‘𝑥)(+g‘𝑅)(𝑧‘𝑥)) · (𝐴‘𝑥)) = (((𝑦‘𝑥)(+g‘(Scalar‘𝑇))(𝑧‘𝑥)) · (𝐴‘𝑥))) |
137 | 8 | ad2antrr 758 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑥 ∈ 𝐼) → 𝑇 ∈ LMod) |
138 | 115 | ffvelrnda 6267 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑥 ∈ 𝐼) → (𝑦‘𝑥) ∈ (Base‘𝑅)) |
139 | 39 | ad2antrr 758 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑥 ∈ 𝐼) → (Base‘𝑅) = (Base‘(Scalar‘𝑇))) |
140 | 138, 139 | eleqtrd 2690 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑥 ∈ 𝐼) → (𝑦‘𝑥) ∈ (Base‘(Scalar‘𝑇))) |
141 | 118 | ffvelrnda 6267 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑥 ∈ 𝐼) → (𝑧‘𝑥) ∈ (Base‘𝑅)) |
142 | 141, 139 | eleqtrd 2690 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑥 ∈ 𝐼) → (𝑧‘𝑥) ∈ (Base‘(Scalar‘𝑇))) |
143 | 49 | adantr 480 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → 𝐴:𝐼⟶𝐶) |
144 | 143 | ffvelrnda 6267 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑥 ∈ 𝐼) → (𝐴‘𝑥) ∈ 𝐶) |
145 | | eqid 2610 |
. . . . . . . . . . . . 13
⊢
(+g‘(Scalar‘𝑇)) =
(+g‘(Scalar‘𝑇)) |
146 | 19, 21, 5, 3, 43, 145 | lmodvsdir 18710 |
. . . . . . . . . . . 12
⊢ ((𝑇 ∈ LMod ∧ ((𝑦‘𝑥) ∈ (Base‘(Scalar‘𝑇)) ∧ (𝑧‘𝑥) ∈ (Base‘(Scalar‘𝑇)) ∧ (𝐴‘𝑥) ∈ 𝐶)) → (((𝑦‘𝑥)(+g‘(Scalar‘𝑇))(𝑧‘𝑥)) · (𝐴‘𝑥)) = (((𝑦‘𝑥) · (𝐴‘𝑥))(+g‘𝑇)((𝑧‘𝑥) · (𝐴‘𝑥)))) |
147 | 137, 140,
142, 144, 146 | syl13anc 1320 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑥 ∈ 𝐼) → (((𝑦‘𝑥)(+g‘(Scalar‘𝑇))(𝑧‘𝑥)) · (𝐴‘𝑥)) = (((𝑦‘𝑥) · (𝐴‘𝑥))(+g‘𝑇)((𝑧‘𝑥) · (𝐴‘𝑥)))) |
148 | 136, 147 | eqtrd 2644 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑥 ∈ 𝐼) → (((𝑦‘𝑥)(+g‘𝑅)(𝑧‘𝑥)) · (𝐴‘𝑥)) = (((𝑦‘𝑥) · (𝐴‘𝑥))(+g‘𝑇)((𝑧‘𝑥) · (𝐴‘𝑥)))) |
149 | 117 | adantr 480 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑥 ∈ 𝐼) → 𝑦 Fn 𝐼) |
150 | 120 | adantr 480 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑥 ∈ 𝐼) → 𝑧 Fn 𝐼) |
151 | 12 | ad2antrr 758 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑥 ∈ 𝐼) → 𝐼 ∈ 𝑋) |
152 | | simpr 476 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑥 ∈ 𝐼) → 𝑥 ∈ 𝐼) |
153 | | fnfvof 6809 |
. . . . . . . . . . . 12
⊢ (((𝑦 Fn 𝐼 ∧ 𝑧 Fn 𝐼) ∧ (𝐼 ∈ 𝑋 ∧ 𝑥 ∈ 𝐼)) → ((𝑦 ∘𝑓
(+g‘𝑅)𝑧)‘𝑥) = ((𝑦‘𝑥)(+g‘𝑅)(𝑧‘𝑥))) |
154 | 149, 150,
151, 152, 153 | syl22anc 1319 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑥 ∈ 𝐼) → ((𝑦 ∘𝑓
(+g‘𝑅)𝑧)‘𝑥) = ((𝑦‘𝑥)(+g‘𝑅)(𝑧‘𝑥))) |
155 | 154 | oveq1d 6564 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑥 ∈ 𝐼) → (((𝑦 ∘𝑓
(+g‘𝑅)𝑧)‘𝑥) · (𝐴‘𝑥)) = (((𝑦‘𝑥)(+g‘𝑅)(𝑧‘𝑥)) · (𝐴‘𝑥))) |
156 | 123 | ad2antrr 758 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑥 ∈ 𝐼) → 𝐴 Fn 𝐼) |
157 | | fnfvof 6809 |
. . . . . . . . . . . 12
⊢ (((𝑦 Fn 𝐼 ∧ 𝐴 Fn 𝐼) ∧ (𝐼 ∈ 𝑋 ∧ 𝑥 ∈ 𝐼)) → ((𝑦 ∘𝑓 · 𝐴)‘𝑥) = ((𝑦‘𝑥) · (𝐴‘𝑥))) |
158 | 149, 156,
151, 152, 157 | syl22anc 1319 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑥 ∈ 𝐼) → ((𝑦 ∘𝑓 · 𝐴)‘𝑥) = ((𝑦‘𝑥) · (𝐴‘𝑥))) |
159 | | fnfvof 6809 |
. . . . . . . . . . . 12
⊢ (((𝑧 Fn 𝐼 ∧ 𝐴 Fn 𝐼) ∧ (𝐼 ∈ 𝑋 ∧ 𝑥 ∈ 𝐼)) → ((𝑧 ∘𝑓 · 𝐴)‘𝑥) = ((𝑧‘𝑥) · (𝐴‘𝑥))) |
160 | 150, 156,
151, 152, 159 | syl22anc 1319 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑥 ∈ 𝐼) → ((𝑧 ∘𝑓 · 𝐴)‘𝑥) = ((𝑧‘𝑥) · (𝐴‘𝑥))) |
161 | 158, 160 | oveq12d 6567 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑥 ∈ 𝐼) → (((𝑦 ∘𝑓 · 𝐴)‘𝑥)(+g‘𝑇)((𝑧 ∘𝑓 · 𝐴)‘𝑥)) = (((𝑦‘𝑥) · (𝐴‘𝑥))(+g‘𝑇)((𝑧‘𝑥) · (𝐴‘𝑥)))) |
162 | 148, 155,
161 | 3eqtr4d 2654 |
. . . . . . . . 9
⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑥 ∈ 𝐼) → (((𝑦 ∘𝑓
(+g‘𝑅)𝑧)‘𝑥) · (𝐴‘𝑥)) = (((𝑦 ∘𝑓 · 𝐴)‘𝑥)(+g‘𝑇)((𝑧 ∘𝑓 · 𝐴)‘𝑥))) |
163 | 121 | adantr 480 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑥 ∈ 𝐼) → (𝑦 ∘𝑓
(+g‘𝑅)𝑧) Fn 𝐼) |
164 | | fnfvof 6809 |
. . . . . . . . . 10
⊢ ((((𝑦 ∘𝑓
(+g‘𝑅)𝑧) Fn 𝐼 ∧ 𝐴 Fn 𝐼) ∧ (𝐼 ∈ 𝑋 ∧ 𝑥 ∈ 𝐼)) → (((𝑦 ∘𝑓
(+g‘𝑅)𝑧) ∘𝑓 · 𝐴)‘𝑥) = (((𝑦 ∘𝑓
(+g‘𝑅)𝑧)‘𝑥) · (𝐴‘𝑥))) |
165 | 163, 156,
151, 152, 164 | syl22anc 1319 |
. . . . . . . . 9
⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑥 ∈ 𝐼) → (((𝑦 ∘𝑓
(+g‘𝑅)𝑧) ∘𝑓 · 𝐴)‘𝑥) = (((𝑦 ∘𝑓
(+g‘𝑅)𝑧)‘𝑥) · (𝐴‘𝑥))) |
166 | 128 | adantr 480 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑥 ∈ 𝐼) → (𝑦 ∘𝑓 · 𝐴) Fn 𝐼) |
167 | 131 | adantr 480 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑥 ∈ 𝐼) → (𝑧 ∘𝑓 · 𝐴) Fn 𝐼) |
168 | | fnfvof 6809 |
. . . . . . . . . 10
⊢ ((((𝑦 ∘𝑓
·
𝐴) Fn 𝐼 ∧ (𝑧 ∘𝑓 · 𝐴) Fn 𝐼) ∧ (𝐼 ∈ 𝑋 ∧ 𝑥 ∈ 𝐼)) → (((𝑦 ∘𝑓 · 𝐴) ∘𝑓
(+g‘𝑇)(𝑧 ∘𝑓 · 𝐴))‘𝑥) = (((𝑦 ∘𝑓 · 𝐴)‘𝑥)(+g‘𝑇)((𝑧 ∘𝑓 · 𝐴)‘𝑥))) |
169 | 166, 167,
151, 152, 168 | syl22anc 1319 |
. . . . . . . . 9
⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑥 ∈ 𝐼) → (((𝑦 ∘𝑓 · 𝐴) ∘𝑓
(+g‘𝑇)(𝑧 ∘𝑓 · 𝐴))‘𝑥) = (((𝑦 ∘𝑓 · 𝐴)‘𝑥)(+g‘𝑇)((𝑧 ∘𝑓 · 𝐴)‘𝑥))) |
170 | 162, 165,
169 | 3eqtr4d 2654 |
. . . . . . . 8
⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑥 ∈ 𝐼) → (((𝑦 ∘𝑓
(+g‘𝑅)𝑧) ∘𝑓 · 𝐴)‘𝑥) = (((𝑦 ∘𝑓 · 𝐴) ∘𝑓
(+g‘𝑇)(𝑧 ∘𝑓 · 𝐴))‘𝑥)) |
171 | 125, 132,
170 | eqfnfvd 6222 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((𝑦 ∘𝑓
(+g‘𝑅)𝑧) ∘𝑓 · 𝐴) = ((𝑦 ∘𝑓 · 𝐴) ∘𝑓
(+g‘𝑇)(𝑧 ∘𝑓 · 𝐴))) |
172 | 112, 171 | eqtrd 2644 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((𝑦(+g‘𝐹)𝑧) ∘𝑓 · 𝐴) = ((𝑦 ∘𝑓 · 𝐴) ∘𝑓
(+g‘𝑇)(𝑧 ∘𝑓 · 𝐴))) |
173 | 172 | oveq2d 6565 |
. . . . 5
⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → (𝑇 Σg ((𝑦(+g‘𝐹)𝑧) ∘𝑓 · 𝐴)) = (𝑇 Σg ((𝑦 ∘𝑓
·
𝐴)
∘𝑓 (+g‘𝑇)(𝑧 ∘𝑓 · 𝐴)))) |
174 | 106, 173 | eqtrd 2644 |
. . . 4
⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → (𝐸‘(𝑦(+g‘𝐹)𝑧)) = (𝑇 Σg ((𝑦 ∘𝑓
·
𝐴)
∘𝑓 (+g‘𝑇)(𝑧 ∘𝑓 · 𝐴)))) |
175 | | oveq1 6556 |
. . . . . . . 8
⊢ (𝑥 = 𝑦 → (𝑥 ∘𝑓 · 𝐴) = (𝑦 ∘𝑓 · 𝐴)) |
176 | 175 | oveq2d 6565 |
. . . . . . 7
⊢ (𝑥 = 𝑦 → (𝑇 Σg (𝑥 ∘𝑓
·
𝐴)) = (𝑇 Σg (𝑦 ∘𝑓
·
𝐴))) |
177 | | ovex 6577 |
. . . . . . 7
⊢ (𝑇 Σg
(𝑦
∘𝑓 · 𝐴)) ∈ V |
178 | 176, 81, 177 | fvmpt 6191 |
. . . . . 6
⊢ (𝑦 ∈ 𝐵 → (𝐸‘𝑦) = (𝑇 Σg (𝑦 ∘𝑓
·
𝐴))) |
179 | 178 | ad2antrl 760 |
. . . . 5
⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → (𝐸‘𝑦) = (𝑇 Σg (𝑦 ∘𝑓
·
𝐴))) |
180 | | oveq1 6556 |
. . . . . . . 8
⊢ (𝑥 = 𝑧 → (𝑥 ∘𝑓 · 𝐴) = (𝑧 ∘𝑓 · 𝐴)) |
181 | 180 | oveq2d 6565 |
. . . . . . 7
⊢ (𝑥 = 𝑧 → (𝑇 Σg (𝑥 ∘𝑓
·
𝐴)) = (𝑇 Σg (𝑧 ∘𝑓
·
𝐴))) |
182 | | ovex 6577 |
. . . . . . 7
⊢ (𝑇 Σg
(𝑧
∘𝑓 · 𝐴)) ∈ V |
183 | 181, 81, 182 | fvmpt 6191 |
. . . . . 6
⊢ (𝑧 ∈ 𝐵 → (𝐸‘𝑧) = (𝑇 Σg (𝑧 ∘𝑓
·
𝐴))) |
184 | 183 | ad2antll 761 |
. . . . 5
⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → (𝐸‘𝑧) = (𝑇 Σg (𝑧 ∘𝑓
·
𝐴))) |
185 | 179, 184 | oveq12d 6567 |
. . . 4
⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((𝐸‘𝑦)(+g‘𝑇)(𝐸‘𝑧)) = ((𝑇 Σg (𝑦 ∘𝑓
·
𝐴))(+g‘𝑇)(𝑇 Σg (𝑧 ∘𝑓
·
𝐴)))) |
186 | 98, 174, 185 | 3eqtr4d 2654 |
. . 3
⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → (𝐸‘(𝑦(+g‘𝐹)𝑧)) = ((𝐸‘𝑦)(+g‘𝑇)(𝐸‘𝑧))) |
187 | 1, 19, 20, 21, 23, 25, 82, 186 | isghmd 17492 |
. 2
⊢ (𝜑 → 𝐸 ∈ (𝐹 GrpHom 𝑇)) |
188 | 8 | adantr 480 |
. . . . 5
⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘(Scalar‘𝐹)) ∧ 𝑧 ∈ 𝐵)) → 𝑇 ∈ LMod) |
189 | 12 | adantr 480 |
. . . . 5
⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘(Scalar‘𝐹)) ∧ 𝑧 ∈ 𝐵)) → 𝐼 ∈ 𝑋) |
190 | 18 | fveq2d 6107 |
. . . . . . . 8
⊢ (𝜑 →
(Base‘(Scalar‘𝑇)) = (Base‘(Scalar‘𝐹))) |
191 | 190 | eleq2d 2673 |
. . . . . . 7
⊢ (𝜑 → (𝑦 ∈ (Base‘(Scalar‘𝑇)) ↔ 𝑦 ∈ (Base‘(Scalar‘𝐹)))) |
192 | 191 | biimpar 501 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘(Scalar‘𝐹))) → 𝑦 ∈ (Base‘(Scalar‘𝑇))) |
193 | 192 | adantrr 749 |
. . . . 5
⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘(Scalar‘𝐹)) ∧ 𝑧 ∈ 𝐵)) → 𝑦 ∈ (Base‘(Scalar‘𝑇))) |
194 | 52 | adantrl 748 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘(Scalar‘𝐹)) ∧ 𝑧 ∈ 𝐵)) → (𝑧 ∘𝑓 · 𝐴):𝐼⟶𝐶) |
195 | 194 | ffvelrnda 6267 |
. . . . 5
⊢ (((𝜑 ∧ (𝑦 ∈ (Base‘(Scalar‘𝐹)) ∧ 𝑧 ∈ 𝐵)) ∧ 𝑥 ∈ 𝐼) → ((𝑧 ∘𝑓 · 𝐴)‘𝑥) ∈ 𝐶) |
196 | 52 | feqmptd 6159 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → (𝑧 ∘𝑓 · 𝐴) = (𝑥 ∈ 𝐼 ↦ ((𝑧 ∘𝑓 · 𝐴)‘𝑥))) |
197 | 196, 78 | eqbrtrrd 4607 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → (𝑥 ∈ 𝐼 ↦ ((𝑧 ∘𝑓 · 𝐴)‘𝑥)) finSupp (0g‘𝑇)) |
198 | 197 | adantrl 748 |
. . . . 5
⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘(Scalar‘𝐹)) ∧ 𝑧 ∈ 𝐵)) → (𝑥 ∈ 𝐼 ↦ ((𝑧 ∘𝑓 · 𝐴)‘𝑥)) finSupp (0g‘𝑇)) |
199 | 19, 5, 43, 32, 21, 3, 188, 189, 193, 195, 198 | gsumvsmul 18750 |
. . . 4
⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘(Scalar‘𝐹)) ∧ 𝑧 ∈ 𝐵)) → (𝑇 Σg (𝑥 ∈ 𝐼 ↦ (𝑦 · ((𝑧 ∘𝑓
·
𝐴)‘𝑥)))) = (𝑦 · (𝑇 Σg (𝑥 ∈ 𝐼 ↦ ((𝑧 ∘𝑓 · 𝐴)‘𝑥))))) |
200 | 15 | adantr 480 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘(Scalar‘𝐹)) ∧ 𝑧 ∈ 𝐵)) → 𝐹 ∈ LMod) |
201 | | simprl 790 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘(Scalar‘𝐹)) ∧ 𝑧 ∈ 𝐵)) → 𝑦 ∈ (Base‘(Scalar‘𝐹))) |
202 | | simprr 792 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘(Scalar‘𝐹)) ∧ 𝑧 ∈ 𝐵)) → 𝑧 ∈ 𝐵) |
203 | 1, 4, 2, 6 | lmodvscl 18703 |
. . . . . . . . . . . 12
⊢ ((𝐹 ∈ LMod ∧ 𝑦 ∈
(Base‘(Scalar‘𝐹)) ∧ 𝑧 ∈ 𝐵) → (𝑦( ·𝑠
‘𝐹)𝑧) ∈ 𝐵) |
204 | 200, 201,
202, 203 | syl3anc 1318 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘(Scalar‘𝐹)) ∧ 𝑧 ∈ 𝐵)) → (𝑦( ·𝑠
‘𝐹)𝑧) ∈ 𝐵) |
205 | 13, 46, 1 | frlmbasf 19923 |
. . . . . . . . . . 11
⊢ ((𝐼 ∈ 𝑋 ∧ (𝑦( ·𝑠
‘𝐹)𝑧) ∈ 𝐵) → (𝑦( ·𝑠
‘𝐹)𝑧):𝐼⟶(Base‘𝑅)) |
206 | 189, 204,
205 | syl2anc 691 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘(Scalar‘𝐹)) ∧ 𝑧 ∈ 𝐵)) → (𝑦( ·𝑠
‘𝐹)𝑧):𝐼⟶(Base‘𝑅)) |
207 | | ffn 5958 |
. . . . . . . . . 10
⊢ ((𝑦(
·𝑠 ‘𝐹)𝑧):𝐼⟶(Base‘𝑅) → (𝑦( ·𝑠
‘𝐹)𝑧) Fn 𝐼) |
208 | 206, 207 | syl 17 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘(Scalar‘𝐹)) ∧ 𝑧 ∈ 𝐵)) → (𝑦( ·𝑠
‘𝐹)𝑧) Fn 𝐼) |
209 | 123 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘(Scalar‘𝐹)) ∧ 𝑧 ∈ 𝐵)) → 𝐴 Fn 𝐼) |
210 | 208, 209,
189, 189, 51 | offn 6806 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘(Scalar‘𝐹)) ∧ 𝑧 ∈ 𝐵)) → ((𝑦( ·𝑠
‘𝐹)𝑧) ∘𝑓 · 𝐴) Fn 𝐼) |
211 | | dffn2 5960 |
. . . . . . . 8
⊢ (((𝑦(
·𝑠 ‘𝐹)𝑧) ∘𝑓 · 𝐴) Fn 𝐼 ↔ ((𝑦( ·𝑠
‘𝐹)𝑧) ∘𝑓 · 𝐴):𝐼⟶V) |
212 | 210, 211 | sylib 207 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘(Scalar‘𝐹)) ∧ 𝑧 ∈ 𝐵)) → ((𝑦( ·𝑠
‘𝐹)𝑧) ∘𝑓 · 𝐴):𝐼⟶V) |
213 | 212 | feqmptd 6159 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘(Scalar‘𝐹)) ∧ 𝑧 ∈ 𝐵)) → ((𝑦( ·𝑠
‘𝐹)𝑧) ∘𝑓 · 𝐴) = (𝑥 ∈ 𝐼 ↦ (((𝑦( ·𝑠
‘𝐹)𝑧) ∘𝑓 · 𝐴)‘𝑥))) |
214 | 7 | fveq2d 6107 |
. . . . . . . . . . . 12
⊢ (𝜑 → (.r‘𝑅) =
(.r‘(Scalar‘𝑇))) |
215 | 214 | ad2antrr 758 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑦 ∈ (Base‘(Scalar‘𝐹)) ∧ 𝑧 ∈ 𝐵)) ∧ 𝑥 ∈ 𝐼) → (.r‘𝑅) =
(.r‘(Scalar‘𝑇))) |
216 | 215 | oveqd 6566 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑦 ∈ (Base‘(Scalar‘𝐹)) ∧ 𝑧 ∈ 𝐵)) ∧ 𝑥 ∈ 𝐼) → (𝑦(.r‘𝑅)(𝑧‘𝑥)) = (𝑦(.r‘(Scalar‘𝑇))(𝑧‘𝑥))) |
217 | 216 | oveq1d 6564 |
. . . . . . . . 9
⊢ (((𝜑 ∧ (𝑦 ∈ (Base‘(Scalar‘𝐹)) ∧ 𝑧 ∈ 𝐵)) ∧ 𝑥 ∈ 𝐼) → ((𝑦(.r‘𝑅)(𝑧‘𝑥)) · (𝐴‘𝑥)) = ((𝑦(.r‘(Scalar‘𝑇))(𝑧‘𝑥)) · (𝐴‘𝑥))) |
218 | 8 | ad2antrr 758 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑦 ∈ (Base‘(Scalar‘𝐹)) ∧ 𝑧 ∈ 𝐵)) ∧ 𝑥 ∈ 𝐼) → 𝑇 ∈ LMod) |
219 | | simplrl 796 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑦 ∈ (Base‘(Scalar‘𝐹)) ∧ 𝑧 ∈ 𝐵)) ∧ 𝑥 ∈ 𝐼) → 𝑦 ∈ (Base‘(Scalar‘𝐹))) |
220 | 190 | ad2antrr 758 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑦 ∈ (Base‘(Scalar‘𝐹)) ∧ 𝑧 ∈ 𝐵)) ∧ 𝑥 ∈ 𝐼) → (Base‘(Scalar‘𝑇)) =
(Base‘(Scalar‘𝐹))) |
221 | 219, 220 | eleqtrrd 2691 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑦 ∈ (Base‘(Scalar‘𝐹)) ∧ 𝑧 ∈ 𝐵)) ∧ 𝑥 ∈ 𝐼) → 𝑦 ∈ (Base‘(Scalar‘𝑇))) |
222 | 48 | ffvelrnda 6267 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑧 ∈ 𝐵) ∧ 𝑥 ∈ 𝐼) → (𝑧‘𝑥) ∈ (Base‘𝑅)) |
223 | 39 | ad2antrr 758 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑧 ∈ 𝐵) ∧ 𝑥 ∈ 𝐼) → (Base‘𝑅) = (Base‘(Scalar‘𝑇))) |
224 | 222, 223 | eleqtrd 2690 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑧 ∈ 𝐵) ∧ 𝑥 ∈ 𝐼) → (𝑧‘𝑥) ∈ (Base‘(Scalar‘𝑇))) |
225 | 224 | adantlrl 752 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑦 ∈ (Base‘(Scalar‘𝐹)) ∧ 𝑧 ∈ 𝐵)) ∧ 𝑥 ∈ 𝐼) → (𝑧‘𝑥) ∈ (Base‘(Scalar‘𝑇))) |
226 | 49 | ffvelrnda 6267 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝐴‘𝑥) ∈ 𝐶) |
227 | 226 | adantlr 747 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑦 ∈ (Base‘(Scalar‘𝐹)) ∧ 𝑧 ∈ 𝐵)) ∧ 𝑥 ∈ 𝐼) → (𝐴‘𝑥) ∈ 𝐶) |
228 | | eqid 2610 |
. . . . . . . . . . 11
⊢
(.r‘(Scalar‘𝑇)) =
(.r‘(Scalar‘𝑇)) |
229 | 19, 5, 3, 43, 228 | lmodvsass 18711 |
. . . . . . . . . 10
⊢ ((𝑇 ∈ LMod ∧ (𝑦 ∈
(Base‘(Scalar‘𝑇)) ∧ (𝑧‘𝑥) ∈ (Base‘(Scalar‘𝑇)) ∧ (𝐴‘𝑥) ∈ 𝐶)) → ((𝑦(.r‘(Scalar‘𝑇))(𝑧‘𝑥)) · (𝐴‘𝑥)) = (𝑦 · ((𝑧‘𝑥) · (𝐴‘𝑥)))) |
230 | 218, 221,
225, 227, 229 | syl13anc 1320 |
. . . . . . . . 9
⊢ (((𝜑 ∧ (𝑦 ∈ (Base‘(Scalar‘𝐹)) ∧ 𝑧 ∈ 𝐵)) ∧ 𝑥 ∈ 𝐼) → ((𝑦(.r‘(Scalar‘𝑇))(𝑧‘𝑥)) · (𝐴‘𝑥)) = (𝑦 · ((𝑧‘𝑥) · (𝐴‘𝑥)))) |
231 | 217, 230 | eqtrd 2644 |
. . . . . . . 8
⊢ (((𝜑 ∧ (𝑦 ∈ (Base‘(Scalar‘𝐹)) ∧ 𝑧 ∈ 𝐵)) ∧ 𝑥 ∈ 𝐼) → ((𝑦(.r‘𝑅)(𝑧‘𝑥)) · (𝐴‘𝑥)) = (𝑦 · ((𝑧‘𝑥) · (𝐴‘𝑥)))) |
232 | 208 | adantr 480 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑦 ∈ (Base‘(Scalar‘𝐹)) ∧ 𝑧 ∈ 𝐵)) ∧ 𝑥 ∈ 𝐼) → (𝑦( ·𝑠
‘𝐹)𝑧) Fn 𝐼) |
233 | 123 | ad2antrr 758 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑦 ∈ (Base‘(Scalar‘𝐹)) ∧ 𝑧 ∈ 𝐵)) ∧ 𝑥 ∈ 𝐼) → 𝐴 Fn 𝐼) |
234 | 12 | ad2antrr 758 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑦 ∈ (Base‘(Scalar‘𝐹)) ∧ 𝑧 ∈ 𝐵)) ∧ 𝑥 ∈ 𝐼) → 𝐼 ∈ 𝑋) |
235 | | simpr 476 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑦 ∈ (Base‘(Scalar‘𝐹)) ∧ 𝑧 ∈ 𝐵)) ∧ 𝑥 ∈ 𝐼) → 𝑥 ∈ 𝐼) |
236 | | fnfvof 6809 |
. . . . . . . . . 10
⊢ ((((𝑦(
·𝑠 ‘𝐹)𝑧) Fn 𝐼 ∧ 𝐴 Fn 𝐼) ∧ (𝐼 ∈ 𝑋 ∧ 𝑥 ∈ 𝐼)) → (((𝑦( ·𝑠
‘𝐹)𝑧) ∘𝑓 · 𝐴)‘𝑥) = (((𝑦( ·𝑠
‘𝐹)𝑧)‘𝑥) · (𝐴‘𝑥))) |
237 | 232, 233,
234, 235, 236 | syl22anc 1319 |
. . . . . . . . 9
⊢ (((𝜑 ∧ (𝑦 ∈ (Base‘(Scalar‘𝐹)) ∧ 𝑧 ∈ 𝐵)) ∧ 𝑥 ∈ 𝐼) → (((𝑦( ·𝑠
‘𝐹)𝑧) ∘𝑓 · 𝐴)‘𝑥) = (((𝑦( ·𝑠
‘𝐹)𝑧)‘𝑥) · (𝐴‘𝑥))) |
238 | 17 | fveq2d 6107 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (Base‘𝑅) =
(Base‘(Scalar‘𝐹))) |
239 | 238 | ad2antrr 758 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑦 ∈ (Base‘(Scalar‘𝐹)) ∧ 𝑧 ∈ 𝐵)) ∧ 𝑥 ∈ 𝐼) → (Base‘𝑅) = (Base‘(Scalar‘𝐹))) |
240 | 219, 239 | eleqtrrd 2691 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑦 ∈ (Base‘(Scalar‘𝐹)) ∧ 𝑧 ∈ 𝐵)) ∧ 𝑥 ∈ 𝐼) → 𝑦 ∈ (Base‘𝑅)) |
241 | | simplrr 797 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑦 ∈ (Base‘(Scalar‘𝐹)) ∧ 𝑧 ∈ 𝐵)) ∧ 𝑥 ∈ 𝐼) → 𝑧 ∈ 𝐵) |
242 | | eqid 2610 |
. . . . . . . . . . 11
⊢
(.r‘𝑅) = (.r‘𝑅) |
243 | 13, 1, 46, 234, 240, 241, 235, 2, 242 | frlmvscaval 19929 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑦 ∈ (Base‘(Scalar‘𝐹)) ∧ 𝑧 ∈ 𝐵)) ∧ 𝑥 ∈ 𝐼) → ((𝑦( ·𝑠
‘𝐹)𝑧)‘𝑥) = (𝑦(.r‘𝑅)(𝑧‘𝑥))) |
244 | 243 | oveq1d 6564 |
. . . . . . . . 9
⊢ (((𝜑 ∧ (𝑦 ∈ (Base‘(Scalar‘𝐹)) ∧ 𝑧 ∈ 𝐵)) ∧ 𝑥 ∈ 𝐼) → (((𝑦( ·𝑠
‘𝐹)𝑧)‘𝑥) · (𝐴‘𝑥)) = ((𝑦(.r‘𝑅)(𝑧‘𝑥)) · (𝐴‘𝑥))) |
245 | 237, 244 | eqtrd 2644 |
. . . . . . . 8
⊢ (((𝜑 ∧ (𝑦 ∈ (Base‘(Scalar‘𝐹)) ∧ 𝑧 ∈ 𝐵)) ∧ 𝑥 ∈ 𝐼) → (((𝑦( ·𝑠
‘𝐹)𝑧) ∘𝑓 · 𝐴)‘𝑥) = ((𝑦(.r‘𝑅)(𝑧‘𝑥)) · (𝐴‘𝑥))) |
246 | 48, 119 | syl 17 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → 𝑧 Fn 𝐼) |
247 | 246 | adantrl 748 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘(Scalar‘𝐹)) ∧ 𝑧 ∈ 𝐵)) → 𝑧 Fn 𝐼) |
248 | 247 | adantr 480 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑦 ∈ (Base‘(Scalar‘𝐹)) ∧ 𝑧 ∈ 𝐵)) ∧ 𝑥 ∈ 𝐼) → 𝑧 Fn 𝐼) |
249 | 248, 233,
234, 235, 159 | syl22anc 1319 |
. . . . . . . . 9
⊢ (((𝜑 ∧ (𝑦 ∈ (Base‘(Scalar‘𝐹)) ∧ 𝑧 ∈ 𝐵)) ∧ 𝑥 ∈ 𝐼) → ((𝑧 ∘𝑓 · 𝐴)‘𝑥) = ((𝑧‘𝑥) · (𝐴‘𝑥))) |
250 | 249 | oveq2d 6565 |
. . . . . . . 8
⊢ (((𝜑 ∧ (𝑦 ∈ (Base‘(Scalar‘𝐹)) ∧ 𝑧 ∈ 𝐵)) ∧ 𝑥 ∈ 𝐼) → (𝑦 · ((𝑧 ∘𝑓
·
𝐴)‘𝑥)) = (𝑦 · ((𝑧‘𝑥) · (𝐴‘𝑥)))) |
251 | 231, 245,
250 | 3eqtr4d 2654 |
. . . . . . 7
⊢ (((𝜑 ∧ (𝑦 ∈ (Base‘(Scalar‘𝐹)) ∧ 𝑧 ∈ 𝐵)) ∧ 𝑥 ∈ 𝐼) → (((𝑦( ·𝑠
‘𝐹)𝑧) ∘𝑓 · 𝐴)‘𝑥) = (𝑦 · ((𝑧 ∘𝑓
·
𝐴)‘𝑥))) |
252 | 251 | mpteq2dva 4672 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘(Scalar‘𝐹)) ∧ 𝑧 ∈ 𝐵)) → (𝑥 ∈ 𝐼 ↦ (((𝑦( ·𝑠
‘𝐹)𝑧) ∘𝑓 · 𝐴)‘𝑥)) = (𝑥 ∈ 𝐼 ↦ (𝑦 · ((𝑧 ∘𝑓
·
𝐴)‘𝑥)))) |
253 | 213, 252 | eqtrd 2644 |
. . . . 5
⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘(Scalar‘𝐹)) ∧ 𝑧 ∈ 𝐵)) → ((𝑦( ·𝑠
‘𝐹)𝑧) ∘𝑓 · 𝐴) = (𝑥 ∈ 𝐼 ↦ (𝑦 · ((𝑧 ∘𝑓
·
𝐴)‘𝑥)))) |
254 | 253 | oveq2d 6565 |
. . . 4
⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘(Scalar‘𝐹)) ∧ 𝑧 ∈ 𝐵)) → (𝑇 Σg ((𝑦(
·𝑠 ‘𝐹)𝑧) ∘𝑓 · 𝐴)) = (𝑇 Σg (𝑥 ∈ 𝐼 ↦ (𝑦 · ((𝑧 ∘𝑓
·
𝐴)‘𝑥))))) |
255 | 194 | feqmptd 6159 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘(Scalar‘𝐹)) ∧ 𝑧 ∈ 𝐵)) → (𝑧 ∘𝑓 · 𝐴) = (𝑥 ∈ 𝐼 ↦ ((𝑧 ∘𝑓 · 𝐴)‘𝑥))) |
256 | 255 | oveq2d 6565 |
. . . . 5
⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘(Scalar‘𝐹)) ∧ 𝑧 ∈ 𝐵)) → (𝑇 Σg (𝑧 ∘𝑓
·
𝐴)) = (𝑇 Σg (𝑥 ∈ 𝐼 ↦ ((𝑧 ∘𝑓 · 𝐴)‘𝑥)))) |
257 | 256 | oveq2d 6565 |
. . . 4
⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘(Scalar‘𝐹)) ∧ 𝑧 ∈ 𝐵)) → (𝑦 · (𝑇 Σg (𝑧 ∘𝑓
·
𝐴))) = (𝑦 · (𝑇 Σg (𝑥 ∈ 𝐼 ↦ ((𝑧 ∘𝑓 · 𝐴)‘𝑥))))) |
258 | 199, 254,
257 | 3eqtr4d 2654 |
. . 3
⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘(Scalar‘𝐹)) ∧ 𝑧 ∈ 𝐵)) → (𝑇 Σg ((𝑦(
·𝑠 ‘𝐹)𝑧) ∘𝑓 · 𝐴)) = (𝑦 · (𝑇 Σg (𝑧 ∘𝑓
·
𝐴)))) |
259 | | oveq1 6556 |
. . . . . 6
⊢ (𝑥 = (𝑦( ·𝑠
‘𝐹)𝑧) → (𝑥 ∘𝑓 · 𝐴) = ((𝑦( ·𝑠
‘𝐹)𝑧) ∘𝑓 · 𝐴)) |
260 | 259 | oveq2d 6565 |
. . . . 5
⊢ (𝑥 = (𝑦( ·𝑠
‘𝐹)𝑧) → (𝑇 Σg (𝑥 ∘𝑓
·
𝐴)) = (𝑇 Σg ((𝑦(
·𝑠 ‘𝐹)𝑧) ∘𝑓 · 𝐴))) |
261 | | ovex 6577 |
. . . . 5
⊢ (𝑇 Σg
((𝑦(
·𝑠 ‘𝐹)𝑧) ∘𝑓 · 𝐴)) ∈ V |
262 | 260, 81, 261 | fvmpt 6191 |
. . . 4
⊢ ((𝑦(
·𝑠 ‘𝐹)𝑧) ∈ 𝐵 → (𝐸‘(𝑦( ·𝑠
‘𝐹)𝑧)) = (𝑇 Σg ((𝑦(
·𝑠 ‘𝐹)𝑧) ∘𝑓 · 𝐴))) |
263 | 204, 262 | syl 17 |
. . 3
⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘(Scalar‘𝐹)) ∧ 𝑧 ∈ 𝐵)) → (𝐸‘(𝑦( ·𝑠
‘𝐹)𝑧)) = (𝑇 Σg ((𝑦(
·𝑠 ‘𝐹)𝑧) ∘𝑓 · 𝐴))) |
264 | 183 | oveq2d 6565 |
. . . 4
⊢ (𝑧 ∈ 𝐵 → (𝑦 · (𝐸‘𝑧)) = (𝑦 · (𝑇 Σg (𝑧 ∘𝑓
·
𝐴)))) |
265 | 264 | ad2antll 761 |
. . 3
⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘(Scalar‘𝐹)) ∧ 𝑧 ∈ 𝐵)) → (𝑦 · (𝐸‘𝑧)) = (𝑦 · (𝑇 Σg (𝑧 ∘𝑓
·
𝐴)))) |
266 | 258, 263,
265 | 3eqtr4d 2654 |
. 2
⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘(Scalar‘𝐹)) ∧ 𝑧 ∈ 𝐵)) → (𝐸‘(𝑦( ·𝑠
‘𝐹)𝑧)) = (𝑦 · (𝐸‘𝑧))) |
267 | 1, 2, 3, 4, 5, 6, 15, 8, 18, 187, 266 | islmhmd 18860 |
1
⊢ (𝜑 → 𝐸 ∈ (𝐹 LMHom 𝑇)) |