Step | Hyp | Ref
| Expression |
1 | | gsumzmhm.h |
. . . . . . 7
⊢ (𝜑 → 𝐻 ∈ Mnd) |
2 | | gsumzmhm.a |
. . . . . . 7
⊢ (𝜑 → 𝐴 ∈ 𝑉) |
3 | | eqid 2610 |
. . . . . . . 8
⊢
(0g‘𝐻) = (0g‘𝐻) |
4 | 3 | gsumz 17197 |
. . . . . . 7
⊢ ((𝐻 ∈ Mnd ∧ 𝐴 ∈ 𝑉) → (𝐻 Σg (𝑘 ∈ 𝐴 ↦ (0g‘𝐻))) = (0g‘𝐻)) |
5 | 1, 2, 4 | syl2anc 691 |
. . . . . 6
⊢ (𝜑 → (𝐻 Σg (𝑘 ∈ 𝐴 ↦ (0g‘𝐻))) = (0g‘𝐻)) |
6 | 5 | adantr 480 |
. . . . 5
⊢ ((𝜑 ∧ (◡𝐹 “ (V ∖ { 0 })) = ∅) →
(𝐻
Σg (𝑘 ∈ 𝐴 ↦ (0g‘𝐻))) = (0g‘𝐻)) |
7 | | gsumzmhm.k |
. . . . . . 7
⊢ (𝜑 → 𝐾 ∈ (𝐺 MndHom 𝐻)) |
8 | | gsumzmhm.0 |
. . . . . . . 8
⊢ 0 =
(0g‘𝐺) |
9 | 8, 3 | mhm0 17166 |
. . . . . . 7
⊢ (𝐾 ∈ (𝐺 MndHom 𝐻) → (𝐾‘ 0 ) =
(0g‘𝐻)) |
10 | 7, 9 | syl 17 |
. . . . . 6
⊢ (𝜑 → (𝐾‘ 0 ) =
(0g‘𝐻)) |
11 | 10 | adantr 480 |
. . . . 5
⊢ ((𝜑 ∧ (◡𝐹 “ (V ∖ { 0 })) = ∅) →
(𝐾‘ 0 ) =
(0g‘𝐻)) |
12 | 6, 11 | eqtr4d 2647 |
. . . 4
⊢ ((𝜑 ∧ (◡𝐹 “ (V ∖ { 0 })) = ∅) →
(𝐻
Σg (𝑘 ∈ 𝐴 ↦ (0g‘𝐻))) = (𝐾‘ 0 )) |
13 | | gsumzmhm.g |
. . . . . . . . 9
⊢ (𝜑 → 𝐺 ∈ Mnd) |
14 | | gsumzmhm.b |
. . . . . . . . . 10
⊢ 𝐵 = (Base‘𝐺) |
15 | 14, 8 | mndidcl 17131 |
. . . . . . . . 9
⊢ (𝐺 ∈ Mnd → 0 ∈ 𝐵) |
16 | 13, 15 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → 0 ∈ 𝐵) |
17 | 16 | ad2antrr 758 |
. . . . . . 7
⊢ (((𝜑 ∧ (◡𝐹 “ (V ∖ { 0 })) = ∅) ∧ 𝑘 ∈ 𝐴) → 0 ∈ 𝐵) |
18 | | gsumzmhm.f |
. . . . . . . 8
⊢ (𝜑 → 𝐹:𝐴⟶𝐵) |
19 | | fvex 6113 |
. . . . . . . . . 10
⊢
(0g‘𝐺) ∈ V |
20 | 8, 19 | eqeltri 2684 |
. . . . . . . . 9
⊢ 0 ∈
V |
21 | 20 | a1i 11 |
. . . . . . . 8
⊢ (𝜑 → 0 ∈ V) |
22 | | fex 6394 |
. . . . . . . . . . 11
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → 𝐹 ∈ V) |
23 | 18, 2, 22 | syl2anc 691 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐹 ∈ V) |
24 | | suppimacnv 7193 |
. . . . . . . . . 10
⊢ ((𝐹 ∈ V ∧ 0 ∈ V)
→ (𝐹 supp 0 ) = (◡𝐹 “ (V ∖ { 0 }))) |
25 | 23, 21, 24 | syl2anc 691 |
. . . . . . . . 9
⊢ (𝜑 → (𝐹 supp 0 ) = (◡𝐹 “ (V ∖ { 0 }))) |
26 | | ssid 3587 |
. . . . . . . . 9
⊢ (◡𝐹 “ (V ∖ { 0 })) ⊆ (◡𝐹 “ (V ∖ { 0 })) |
27 | 25, 26 | syl6eqss 3618 |
. . . . . . . 8
⊢ (𝜑 → (𝐹 supp 0 ) ⊆ (◡𝐹 “ (V ∖ { 0 }))) |
28 | 18, 2, 21, 27 | gsumcllem 18132 |
. . . . . . 7
⊢ ((𝜑 ∧ (◡𝐹 “ (V ∖ { 0 })) = ∅) →
𝐹 = (𝑘 ∈ 𝐴 ↦ 0 )) |
29 | | eqid 2610 |
. . . . . . . . . . 11
⊢
(Base‘𝐻) =
(Base‘𝐻) |
30 | 14, 29 | mhmf 17163 |
. . . . . . . . . 10
⊢ (𝐾 ∈ (𝐺 MndHom 𝐻) → 𝐾:𝐵⟶(Base‘𝐻)) |
31 | 7, 30 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → 𝐾:𝐵⟶(Base‘𝐻)) |
32 | 31 | feqmptd 6159 |
. . . . . . . 8
⊢ (𝜑 → 𝐾 = (𝑥 ∈ 𝐵 ↦ (𝐾‘𝑥))) |
33 | 32 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ (◡𝐹 “ (V ∖ { 0 })) = ∅) →
𝐾 = (𝑥 ∈ 𝐵 ↦ (𝐾‘𝑥))) |
34 | | fveq2 6103 |
. . . . . . 7
⊢ (𝑥 = 0 → (𝐾‘𝑥) = (𝐾‘ 0 )) |
35 | 17, 28, 33, 34 | fmptco 6303 |
. . . . . 6
⊢ ((𝜑 ∧ (◡𝐹 “ (V ∖ { 0 })) = ∅) →
(𝐾 ∘ 𝐹) = (𝑘 ∈ 𝐴 ↦ (𝐾‘ 0 ))) |
36 | 10 | mpteq2dv 4673 |
. . . . . . 7
⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ (𝐾‘ 0 )) = (𝑘 ∈ 𝐴 ↦ (0g‘𝐻))) |
37 | 36 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ (◡𝐹 “ (V ∖ { 0 })) = ∅) →
(𝑘 ∈ 𝐴 ↦ (𝐾‘ 0 )) = (𝑘 ∈ 𝐴 ↦ (0g‘𝐻))) |
38 | 35, 37 | eqtrd 2644 |
. . . . 5
⊢ ((𝜑 ∧ (◡𝐹 “ (V ∖ { 0 })) = ∅) →
(𝐾 ∘ 𝐹) = (𝑘 ∈ 𝐴 ↦ (0g‘𝐻))) |
39 | 38 | oveq2d 6565 |
. . . 4
⊢ ((𝜑 ∧ (◡𝐹 “ (V ∖ { 0 })) = ∅) →
(𝐻
Σg (𝐾 ∘ 𝐹)) = (𝐻 Σg (𝑘 ∈ 𝐴 ↦ (0g‘𝐻)))) |
40 | 28 | oveq2d 6565 |
. . . . . 6
⊢ ((𝜑 ∧ (◡𝐹 “ (V ∖ { 0 })) = ∅) →
(𝐺
Σg 𝐹) = (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 0 ))) |
41 | 8 | gsumz 17197 |
. . . . . . . 8
⊢ ((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝑉) → (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 0 )) = 0 ) |
42 | 13, 2, 41 | syl2anc 691 |
. . . . . . 7
⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 0 )) = 0 ) |
43 | 42 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ (◡𝐹 “ (V ∖ { 0 })) = ∅) →
(𝐺
Σg (𝑘 ∈ 𝐴 ↦ 0 )) = 0 ) |
44 | 40, 43 | eqtrd 2644 |
. . . . 5
⊢ ((𝜑 ∧ (◡𝐹 “ (V ∖ { 0 })) = ∅) →
(𝐺
Σg 𝐹) = 0 ) |
45 | 44 | fveq2d 6107 |
. . . 4
⊢ ((𝜑 ∧ (◡𝐹 “ (V ∖ { 0 })) = ∅) →
(𝐾‘(𝐺 Σg 𝐹)) = (𝐾‘ 0 )) |
46 | 12, 39, 45 | 3eqtr4d 2654 |
. . 3
⊢ ((𝜑 ∧ (◡𝐹 “ (V ∖ { 0 })) = ∅) →
(𝐻
Σg (𝐾 ∘ 𝐹)) = (𝐾‘(𝐺 Σg 𝐹))) |
47 | 46 | ex 449 |
. 2
⊢ (𝜑 → ((◡𝐹 “ (V ∖ { 0 })) = ∅ →
(𝐻
Σg (𝐾 ∘ 𝐹)) = (𝐾‘(𝐺 Σg 𝐹)))) |
48 | 13 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ ((#‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧
𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → 𝐺 ∈ Mnd) |
49 | | eqid 2610 |
. . . . . . . . . 10
⊢
(+g‘𝐺) = (+g‘𝐺) |
50 | 14, 49 | mndcl 17124 |
. . . . . . . . 9
⊢ ((𝐺 ∈ Mnd ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥(+g‘𝐺)𝑦) ∈ 𝐵) |
51 | 50 | 3expb 1258 |
. . . . . . . 8
⊢ ((𝐺 ∈ Mnd ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐺)𝑦) ∈ 𝐵) |
52 | 48, 51 | sylan 487 |
. . . . . . 7
⊢ (((𝜑 ∧ ((#‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧
𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐺)𝑦) ∈ 𝐵) |
53 | 18 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ((#‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧
𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → 𝐹:𝐴⟶𝐵) |
54 | | f1of1 6049 |
. . . . . . . . . . . 12
⊢ (𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })) → 𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1→(◡𝐹 “ (V ∖ { 0
}))) |
55 | 54 | ad2antll 761 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ ((#‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧
𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → 𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1→(◡𝐹 “ (V ∖ { 0
}))) |
56 | | cnvimass 5404 |
. . . . . . . . . . . 12
⊢ (◡𝐹 “ (V ∖ { 0 })) ⊆ dom 𝐹 |
57 | | fdm 5964 |
. . . . . . . . . . . . 13
⊢ (𝐹:𝐴⟶𝐵 → dom 𝐹 = 𝐴) |
58 | 53, 57 | syl 17 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ ((#‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧
𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → dom 𝐹 = 𝐴) |
59 | 56, 58 | syl5sseq 3616 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ ((#‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧
𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → (◡𝐹 “ (V ∖ { 0 })) ⊆ 𝐴) |
60 | | f1ss 6019 |
. . . . . . . . . . 11
⊢ ((𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1→(◡𝐹 “ (V ∖ { 0 })) ∧
(◡𝐹 “ (V ∖ { 0 })) ⊆ 𝐴) → 𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1→𝐴) |
61 | 55, 59, 60 | syl2anc 691 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ ((#‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧
𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → 𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1→𝐴) |
62 | | f1f 6014 |
. . . . . . . . . 10
⊢ (𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1→𝐴 → 𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))⟶𝐴) |
63 | 61, 62 | syl 17 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ((#‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧
𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → 𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))⟶𝐴) |
64 | | fco 5971 |
. . . . . . . . 9
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))⟶𝐴) → (𝐹 ∘ 𝑓):(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))⟶𝐵) |
65 | 53, 63, 64 | syl2anc 691 |
. . . . . . . 8
⊢ ((𝜑 ∧ ((#‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧
𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → (𝐹 ∘ 𝑓):(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))⟶𝐵) |
66 | 65 | ffvelrnda 6267 |
. . . . . . 7
⊢ (((𝜑 ∧ ((#‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧
𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) ∧ 𝑥 ∈ (1...(#‘(◡𝐹 “ (V ∖ { 0 }))))) → ((𝐹 ∘ 𝑓)‘𝑥) ∈ 𝐵) |
67 | | simprl 790 |
. . . . . . . 8
⊢ ((𝜑 ∧ ((#‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧
𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) →
(#‘(◡𝐹 “ (V ∖ { 0 }))) ∈
ℕ) |
68 | | nnuz 11599 |
. . . . . . . 8
⊢ ℕ =
(ℤ≥‘1) |
69 | 67, 68 | syl6eleq 2698 |
. . . . . . 7
⊢ ((𝜑 ∧ ((#‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧
𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) →
(#‘(◡𝐹 “ (V ∖ { 0 }))) ∈
(ℤ≥‘1)) |
70 | 7 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ ((#‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧
𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → 𝐾 ∈ (𝐺 MndHom 𝐻)) |
71 | | eqid 2610 |
. . . . . . . . . 10
⊢
(+g‘𝐻) = (+g‘𝐻) |
72 | 14, 49, 71 | mhmlin 17165 |
. . . . . . . . 9
⊢ ((𝐾 ∈ (𝐺 MndHom 𝐻) ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝐾‘(𝑥(+g‘𝐺)𝑦)) = ((𝐾‘𝑥)(+g‘𝐻)(𝐾‘𝑦))) |
73 | 72 | 3expb 1258 |
. . . . . . . 8
⊢ ((𝐾 ∈ (𝐺 MndHom 𝐻) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐾‘(𝑥(+g‘𝐺)𝑦)) = ((𝐾‘𝑥)(+g‘𝐻)(𝐾‘𝑦))) |
74 | 70, 73 | sylan 487 |
. . . . . . 7
⊢ (((𝜑 ∧ ((#‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧
𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐾‘(𝑥(+g‘𝐺)𝑦)) = ((𝐾‘𝑥)(+g‘𝐻)(𝐾‘𝑦))) |
75 | | coass 5571 |
. . . . . . . . 9
⊢ ((𝐾 ∘ 𝐹) ∘ 𝑓) = (𝐾 ∘ (𝐹 ∘ 𝑓)) |
76 | 75 | fveq1i 6104 |
. . . . . . . 8
⊢ (((𝐾 ∘ 𝐹) ∘ 𝑓)‘𝑥) = ((𝐾 ∘ (𝐹 ∘ 𝑓))‘𝑥) |
77 | | fvco3 6185 |
. . . . . . . . 9
⊢ (((𝐹 ∘ 𝑓):(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))⟶𝐵 ∧ 𝑥 ∈ (1...(#‘(◡𝐹 “ (V ∖ { 0 }))))) → ((𝐾 ∘ (𝐹 ∘ 𝑓))‘𝑥) = (𝐾‘((𝐹 ∘ 𝑓)‘𝑥))) |
78 | 65, 77 | sylan 487 |
. . . . . . . 8
⊢ (((𝜑 ∧ ((#‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧
𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) ∧ 𝑥 ∈ (1...(#‘(◡𝐹 “ (V ∖ { 0 }))))) → ((𝐾 ∘ (𝐹 ∘ 𝑓))‘𝑥) = (𝐾‘((𝐹 ∘ 𝑓)‘𝑥))) |
79 | 76, 78 | syl5req 2657 |
. . . . . . 7
⊢ (((𝜑 ∧ ((#‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧
𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) ∧ 𝑥 ∈ (1...(#‘(◡𝐹 “ (V ∖ { 0 }))))) → (𝐾‘((𝐹 ∘ 𝑓)‘𝑥)) = (((𝐾 ∘ 𝐹) ∘ 𝑓)‘𝑥)) |
80 | 52, 66, 69, 74, 79 | seqhomo 12710 |
. . . . . 6
⊢ ((𝜑 ∧ ((#‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧
𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → (𝐾‘(seq1((+g‘𝐺), (𝐹 ∘ 𝑓))‘(#‘(◡𝐹 “ (V ∖ { 0 }))))) =
(seq1((+g‘𝐻), ((𝐾 ∘ 𝐹) ∘ 𝑓))‘(#‘(◡𝐹 “ (V ∖ { 0 }))))) |
81 | | gsumzmhm.z |
. . . . . . . 8
⊢ 𝑍 = (Cntz‘𝐺) |
82 | 2 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ ((#‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧
𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → 𝐴 ∈ 𝑉) |
83 | | gsumzmhm.c |
. . . . . . . . 9
⊢ (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹)) |
84 | 83 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ ((#‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧
𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → ran 𝐹 ⊆ (𝑍‘ran 𝐹)) |
85 | 27 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ((#‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧
𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → (𝐹 supp 0 ) ⊆ (◡𝐹 “ (V ∖ { 0 }))) |
86 | | f1ofo 6057 |
. . . . . . . . . . 11
⊢ (𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })) → 𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–onto→(◡𝐹 “ (V ∖ { 0 }))) |
87 | | forn 6031 |
. . . . . . . . . . 11
⊢ (𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–onto→(◡𝐹 “ (V ∖ { 0 })) → ran 𝑓 = (◡𝐹 “ (V ∖ { 0 }))) |
88 | 86, 87 | syl 17 |
. . . . . . . . . 10
⊢ (𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })) → ran 𝑓 = (◡𝐹 “ (V ∖ { 0 }))) |
89 | 88 | ad2antll 761 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ((#‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧
𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → ran 𝑓 = (◡𝐹 “ (V ∖ { 0 }))) |
90 | 85, 89 | sseqtr4d 3605 |
. . . . . . . 8
⊢ ((𝜑 ∧ ((#‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧
𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → (𝐹 supp 0 ) ⊆ ran 𝑓) |
91 | | eqid 2610 |
. . . . . . . 8
⊢ ((𝐹 ∘ 𝑓) supp 0 ) = ((𝐹 ∘ 𝑓) supp 0 ) |
92 | 14, 8, 49, 81, 48, 82, 53, 84, 67, 61, 90, 91 | gsumval3 18131 |
. . . . . . 7
⊢ ((𝜑 ∧ ((#‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧
𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → (𝐺 Σg
𝐹) =
(seq1((+g‘𝐺), (𝐹 ∘ 𝑓))‘(#‘(◡𝐹 “ (V ∖ { 0 }))))) |
93 | 92 | fveq2d 6107 |
. . . . . 6
⊢ ((𝜑 ∧ ((#‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧
𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → (𝐾‘(𝐺 Σg 𝐹)) = (𝐾‘(seq1((+g‘𝐺), (𝐹 ∘ 𝑓))‘(#‘(◡𝐹 “ (V ∖ { 0 })))))) |
94 | | eqid 2610 |
. . . . . . 7
⊢
(Cntz‘𝐻) =
(Cntz‘𝐻) |
95 | 1 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ ((#‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧
𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → 𝐻 ∈ Mnd) |
96 | 31 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ ((#‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧
𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → 𝐾:𝐵⟶(Base‘𝐻)) |
97 | | fco 5971 |
. . . . . . . 8
⊢ ((𝐾:𝐵⟶(Base‘𝐻) ∧ 𝐹:𝐴⟶𝐵) → (𝐾 ∘ 𝐹):𝐴⟶(Base‘𝐻)) |
98 | 96, 53, 97 | syl2anc 691 |
. . . . . . 7
⊢ ((𝜑 ∧ ((#‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧
𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → (𝐾 ∘ 𝐹):𝐴⟶(Base‘𝐻)) |
99 | 81, 94 | cntzmhm2 17595 |
. . . . . . . . 9
⊢ ((𝐾 ∈ (𝐺 MndHom 𝐻) ∧ ran 𝐹 ⊆ (𝑍‘ran 𝐹)) → (𝐾 “ ran 𝐹) ⊆ ((Cntz‘𝐻)‘(𝐾 “ ran 𝐹))) |
100 | 70, 84, 99 | syl2anc 691 |
. . . . . . . 8
⊢ ((𝜑 ∧ ((#‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧
𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → (𝐾 “ ran 𝐹) ⊆ ((Cntz‘𝐻)‘(𝐾 “ ran 𝐹))) |
101 | | rnco2 5559 |
. . . . . . . 8
⊢ ran
(𝐾 ∘ 𝐹) = (𝐾 “ ran 𝐹) |
102 | 101 | fveq2i 6106 |
. . . . . . . 8
⊢
((Cntz‘𝐻)‘ran (𝐾 ∘ 𝐹)) = ((Cntz‘𝐻)‘(𝐾 “ ran 𝐹)) |
103 | 100, 101,
102 | 3sstr4g 3609 |
. . . . . . 7
⊢ ((𝜑 ∧ ((#‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧
𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → ran (𝐾 ∘ 𝐹) ⊆ ((Cntz‘𝐻)‘ran (𝐾 ∘ 𝐹))) |
104 | | eldifi 3694 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ (𝐴 ∖ (◡𝐹 “ (V ∖ { 0 }))) → 𝑥 ∈ 𝐴) |
105 | | fvco3 6185 |
. . . . . . . . . . 11
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → ((𝐾 ∘ 𝐹)‘𝑥) = (𝐾‘(𝐹‘𝑥))) |
106 | 53, 104, 105 | syl2an 493 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ ((#‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧
𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) ∧ 𝑥 ∈ (𝐴 ∖ (◡𝐹 “ (V ∖ { 0 })))) → ((𝐾 ∘ 𝐹)‘𝑥) = (𝐾‘(𝐹‘𝑥))) |
107 | 20 | a1i 11 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ ((#‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧
𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → 0 ∈
V) |
108 | 53, 85, 82, 107 | suppssr 7213 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ ((#‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧
𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) ∧ 𝑥 ∈ (𝐴 ∖ (◡𝐹 “ (V ∖ { 0 })))) → (𝐹‘𝑥) = 0 ) |
109 | 108 | fveq2d 6107 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ ((#‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧
𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) ∧ 𝑥 ∈ (𝐴 ∖ (◡𝐹 “ (V ∖ { 0 })))) → (𝐾‘(𝐹‘𝑥)) = (𝐾‘ 0 )) |
110 | 10 | ad2antrr 758 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ ((#‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧
𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) ∧ 𝑥 ∈ (𝐴 ∖ (◡𝐹 “ (V ∖ { 0 })))) → (𝐾‘ 0 ) =
(0g‘𝐻)) |
111 | 106, 109,
110 | 3eqtrd 2648 |
. . . . . . . . 9
⊢ (((𝜑 ∧ ((#‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧
𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) ∧ 𝑥 ∈ (𝐴 ∖ (◡𝐹 “ (V ∖ { 0 })))) → ((𝐾 ∘ 𝐹)‘𝑥) = (0g‘𝐻)) |
112 | 98, 111 | suppss 7212 |
. . . . . . . 8
⊢ ((𝜑 ∧ ((#‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧
𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → ((𝐾 ∘ 𝐹) supp (0g‘𝐻)) ⊆ (◡𝐹 “ (V ∖ { 0 }))) |
113 | 112, 89 | sseqtr4d 3605 |
. . . . . . 7
⊢ ((𝜑 ∧ ((#‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧
𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → ((𝐾 ∘ 𝐹) supp (0g‘𝐻)) ⊆ ran 𝑓) |
114 | | eqid 2610 |
. . . . . . 7
⊢ (((𝐾 ∘ 𝐹) ∘ 𝑓) supp (0g‘𝐻)) = (((𝐾 ∘ 𝐹) ∘ 𝑓) supp (0g‘𝐻)) |
115 | 29, 3, 71, 94, 95, 82, 98, 103, 67, 61, 113, 114 | gsumval3 18131 |
. . . . . 6
⊢ ((𝜑 ∧ ((#‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧
𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → (𝐻 Σg
(𝐾 ∘ 𝐹)) =
(seq1((+g‘𝐻), ((𝐾 ∘ 𝐹) ∘ 𝑓))‘(#‘(◡𝐹 “ (V ∖ { 0 }))))) |
116 | 80, 93, 115 | 3eqtr4rd 2655 |
. . . . 5
⊢ ((𝜑 ∧ ((#‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧
𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → (𝐻 Σg
(𝐾 ∘ 𝐹)) = (𝐾‘(𝐺 Σg 𝐹))) |
117 | 116 | expr 641 |
. . . 4
⊢ ((𝜑 ∧ (#‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ)
→ (𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })) → (𝐻 Σg
(𝐾 ∘ 𝐹)) = (𝐾‘(𝐺 Σg 𝐹)))) |
118 | 117 | exlimdv 1848 |
. . 3
⊢ ((𝜑 ∧ (#‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ)
→ (∃𝑓 𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })) → (𝐻 Σg
(𝐾 ∘ 𝐹)) = (𝐾‘(𝐺 Σg 𝐹)))) |
119 | 118 | expimpd 627 |
. 2
⊢ (𝜑 → (((#‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧
∃𝑓 𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 }))) → (𝐻 Σg
(𝐾 ∘ 𝐹)) = (𝐾‘(𝐺 Σg 𝐹)))) |
120 | | gsumzmhm.w |
. . . . 5
⊢ (𝜑 → 𝐹 finSupp 0 ) |
121 | 120 | fsuppimpd 8165 |
. . . 4
⊢ (𝜑 → (𝐹 supp 0 ) ∈
Fin) |
122 | 25, 121 | eqeltrrd 2689 |
. . 3
⊢ (𝜑 → (◡𝐹 “ (V ∖ { 0 })) ∈
Fin) |
123 | | fz1f1o 14288 |
. . 3
⊢ ((◡𝐹 “ (V ∖ { 0 })) ∈ Fin →
((◡𝐹 “ (V ∖ { 0 })) = ∅ ∨
((#‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧
∃𝑓 𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 }))))) |
124 | 122, 123 | syl 17 |
. 2
⊢ (𝜑 → ((◡𝐹 “ (V ∖ { 0 })) = ∅ ∨
((#‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧
∃𝑓 𝑓:(1...(#‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 }))))) |
125 | 47, 119, 124 | mpjaod 395 |
1
⊢ (𝜑 → (𝐻 Σg (𝐾 ∘ 𝐹)) = (𝐾‘(𝐺 Σg 𝐹))) |