Step | Hyp | Ref
| Expression |
1 | | gsumzmhm.h |
. . . . . . 7
⊢ (𝜑 → 𝐻 ∈ Mnd) |
2 | | gsumzmhm.a |
. . . . . . 7
⊢ (𝜑 → 𝐴 ∈ 𝑉) |
3 | | eqid 2737 |
. . . . . . . 8
⊢
(0g‘𝐻) = (0g‘𝐻) |
4 | 3 | gsumz 18262 |
. . . . . . 7
⊢ ((𝐻 ∈ Mnd ∧ 𝐴 ∈ 𝑉) → (𝐻 Σg (𝑘 ∈ 𝐴 ↦ (0g‘𝐻))) = (0g‘𝐻)) |
5 | 1, 2, 4 | syl2anc 587 |
. . . . . 6
⊢ (𝜑 → (𝐻 Σg (𝑘 ∈ 𝐴 ↦ (0g‘𝐻))) = (0g‘𝐻)) |
6 | 5 | adantr 484 |
. . . . 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 18226 |
. . . . . . 7
⊢ (𝐾 ∈ (𝐺 MndHom 𝐻) → (𝐾‘ 0 ) =
(0g‘𝐻)) |
10 | 7, 9 | syl 17 |
. . . . . 6
⊢ (𝜑 → (𝐾‘ 0 ) =
(0g‘𝐻)) |
11 | 10 | adantr 484 |
. . . . 5
⊢ ((𝜑 ∧ (◡𝐹 “ (V ∖ { 0 })) = ∅) →
(𝐾‘ 0 ) =
(0g‘𝐻)) |
12 | 6, 11 | eqtr4d 2780 |
. . . 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 18188 |
. . . . . . . . 9
⊢ (𝐺 ∈ Mnd → 0 ∈ 𝐵) |
16 | 13, 15 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → 0 ∈ 𝐵) |
17 | 16 | ad2antrr 726 |
. . . . . . 7
⊢ (((𝜑 ∧ (◡𝐹 “ (V ∖ { 0 })) = ∅) ∧ 𝑘 ∈ 𝐴) → 0 ∈ 𝐵) |
18 | | gsumzmhm.f |
. . . . . . . 8
⊢ (𝜑 → 𝐹:𝐴⟶𝐵) |
19 | 8 | fvexi 6731 |
. . . . . . . . 9
⊢ 0 ∈
V |
20 | 19 | a1i 11 |
. . . . . . . 8
⊢ (𝜑 → 0 ∈ V) |
21 | 18, 2 | fexd 7043 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐹 ∈ V) |
22 | | suppimacnv 7916 |
. . . . . . . . . 10
⊢ ((𝐹 ∈ V ∧ 0 ∈ V)
→ (𝐹 supp 0 ) = (◡𝐹 “ (V ∖ { 0 }))) |
23 | 21, 20, 22 | syl2anc 587 |
. . . . . . . . 9
⊢ (𝜑 → (𝐹 supp 0 ) = (◡𝐹 “ (V ∖ { 0 }))) |
24 | | ssid 3923 |
. . . . . . . . 9
⊢ (◡𝐹 “ (V ∖ { 0 })) ⊆ (◡𝐹 “ (V ∖ { 0 })) |
25 | 23, 24 | eqsstrdi 3955 |
. . . . . . . 8
⊢ (𝜑 → (𝐹 supp 0 ) ⊆ (◡𝐹 “ (V ∖ { 0 }))) |
26 | 18, 2, 20, 25 | gsumcllem 19293 |
. . . . . . 7
⊢ ((𝜑 ∧ (◡𝐹 “ (V ∖ { 0 })) = ∅) →
𝐹 = (𝑘 ∈ 𝐴 ↦ 0 )) |
27 | | eqid 2737 |
. . . . . . . . . . 11
⊢
(Base‘𝐻) =
(Base‘𝐻) |
28 | 14, 27 | mhmf 18223 |
. . . . . . . . . 10
⊢ (𝐾 ∈ (𝐺 MndHom 𝐻) → 𝐾:𝐵⟶(Base‘𝐻)) |
29 | 7, 28 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → 𝐾:𝐵⟶(Base‘𝐻)) |
30 | 29 | feqmptd 6780 |
. . . . . . . 8
⊢ (𝜑 → 𝐾 = (𝑥 ∈ 𝐵 ↦ (𝐾‘𝑥))) |
31 | 30 | adantr 484 |
. . . . . . 7
⊢ ((𝜑 ∧ (◡𝐹 “ (V ∖ { 0 })) = ∅) →
𝐾 = (𝑥 ∈ 𝐵 ↦ (𝐾‘𝑥))) |
32 | | fveq2 6717 |
. . . . . . 7
⊢ (𝑥 = 0 → (𝐾‘𝑥) = (𝐾‘ 0 )) |
33 | 17, 26, 31, 32 | fmptco 6944 |
. . . . . 6
⊢ ((𝜑 ∧ (◡𝐹 “ (V ∖ { 0 })) = ∅) →
(𝐾 ∘ 𝐹) = (𝑘 ∈ 𝐴 ↦ (𝐾‘ 0 ))) |
34 | 10 | mpteq2dv 5151 |
. . . . . . 7
⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ (𝐾‘ 0 )) = (𝑘 ∈ 𝐴 ↦ (0g‘𝐻))) |
35 | 34 | adantr 484 |
. . . . . 6
⊢ ((𝜑 ∧ (◡𝐹 “ (V ∖ { 0 })) = ∅) →
(𝑘 ∈ 𝐴 ↦ (𝐾‘ 0 )) = (𝑘 ∈ 𝐴 ↦ (0g‘𝐻))) |
36 | 33, 35 | eqtrd 2777 |
. . . . 5
⊢ ((𝜑 ∧ (◡𝐹 “ (V ∖ { 0 })) = ∅) →
(𝐾 ∘ 𝐹) = (𝑘 ∈ 𝐴 ↦ (0g‘𝐻))) |
37 | 36 | oveq2d 7229 |
. . . 4
⊢ ((𝜑 ∧ (◡𝐹 “ (V ∖ { 0 })) = ∅) →
(𝐻
Σg (𝐾 ∘ 𝐹)) = (𝐻 Σg (𝑘 ∈ 𝐴 ↦ (0g‘𝐻)))) |
38 | 26 | oveq2d 7229 |
. . . . . 6
⊢ ((𝜑 ∧ (◡𝐹 “ (V ∖ { 0 })) = ∅) →
(𝐺
Σg 𝐹) = (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 0 ))) |
39 | 8 | gsumz 18262 |
. . . . . . . 8
⊢ ((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝑉) → (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 0 )) = 0 ) |
40 | 13, 2, 39 | syl2anc 587 |
. . . . . . 7
⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 0 )) = 0 ) |
41 | 40 | adantr 484 |
. . . . . 6
⊢ ((𝜑 ∧ (◡𝐹 “ (V ∖ { 0 })) = ∅) →
(𝐺
Σg (𝑘 ∈ 𝐴 ↦ 0 )) = 0 ) |
42 | 38, 41 | eqtrd 2777 |
. . . . 5
⊢ ((𝜑 ∧ (◡𝐹 “ (V ∖ { 0 })) = ∅) →
(𝐺
Σg 𝐹) = 0 ) |
43 | 42 | fveq2d 6721 |
. . . 4
⊢ ((𝜑 ∧ (◡𝐹 “ (V ∖ { 0 })) = ∅) →
(𝐾‘(𝐺 Σg 𝐹)) = (𝐾‘ 0 )) |
44 | 12, 37, 43 | 3eqtr4d 2787 |
. . 3
⊢ ((𝜑 ∧ (◡𝐹 “ (V ∖ { 0 })) = ∅) →
(𝐻
Σg (𝐾 ∘ 𝐹)) = (𝐾‘(𝐺 Σg 𝐹))) |
45 | 44 | ex 416 |
. 2
⊢ (𝜑 → ((◡𝐹 “ (V ∖ { 0 })) = ∅ →
(𝐻
Σg (𝐾 ∘ 𝐹)) = (𝐾‘(𝐺 Σg 𝐹)))) |
46 | 13 | adantr 484 |
. . . . . . . 8
⊢ ((𝜑 ∧ ((♯‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧
𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → 𝐺 ∈ Mnd) |
47 | | eqid 2737 |
. . . . . . . . . 10
⊢
(+g‘𝐺) = (+g‘𝐺) |
48 | 14, 47 | mndcl 18181 |
. . . . . . . . 9
⊢ ((𝐺 ∈ Mnd ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥(+g‘𝐺)𝑦) ∈ 𝐵) |
49 | 48 | 3expb 1122 |
. . . . . . . 8
⊢ ((𝐺 ∈ Mnd ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐺)𝑦) ∈ 𝐵) |
50 | 46, 49 | sylan 583 |
. . . . . . 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) |
51 | | f1of1 6660 |
. . . . . . . . . . . 12
⊢ (𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })) → 𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1→(◡𝐹 “ (V ∖ { 0
}))) |
52 | 51 | ad2antll 729 |
. . . . . . . . . . 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
}))) |
53 | | cnvimass 5949 |
. . . . . . . . . . . 12
⊢ (◡𝐹 “ (V ∖ { 0 })) ⊆ dom 𝐹 |
54 | 18 | adantr 484 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ ((♯‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧
𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → 𝐹:𝐴⟶𝐵) |
55 | 53, 54 | fssdm 6565 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ ((♯‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧
𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → (◡𝐹 “ (V ∖ { 0 })) ⊆ 𝐴) |
56 | | f1ss 6621 |
. . . . . . . . . . 11
⊢ ((𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1→(◡𝐹 “ (V ∖ { 0 })) ∧
(◡𝐹 “ (V ∖ { 0 })) ⊆ 𝐴) → 𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1→𝐴) |
57 | 52, 55, 56 | syl2anc 587 |
. . . . . . . . . 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) |
58 | | f1f 6615 |
. . . . . . . . . 10
⊢ (𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1→𝐴 → 𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))⟶𝐴) |
59 | 57, 58 | 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) |
60 | | fco 6569 |
. . . . . . . . 9
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))⟶𝐴) → (𝐹 ∘ 𝑓):(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))⟶𝐵) |
61 | 18, 59, 60 | syl2an2r 685 |
. . . . . . . 8
⊢ ((𝜑 ∧ ((♯‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧
𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → (𝐹 ∘ 𝑓):(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))⟶𝐵) |
62 | 61 | ffvelrnda 6904 |
. . . . . . 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) |
63 | | simprl 771 |
. . . . . . . 8
⊢ ((𝜑 ∧ ((♯‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧
𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) →
(♯‘(◡𝐹 “ (V ∖ { 0 }))) ∈
ℕ) |
64 | | nnuz 12477 |
. . . . . . . 8
⊢ ℕ =
(ℤ≥‘1) |
65 | 63, 64 | eleqtrdi 2848 |
. . . . . . 7
⊢ ((𝜑 ∧ ((♯‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧
𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) →
(♯‘(◡𝐹 “ (V ∖ { 0 }))) ∈
(ℤ≥‘1)) |
66 | 7 | adantr 484 |
. . . . . . . 8
⊢ ((𝜑 ∧ ((♯‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧
𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → 𝐾 ∈ (𝐺 MndHom 𝐻)) |
67 | | eqid 2737 |
. . . . . . . . . 10
⊢
(+g‘𝐻) = (+g‘𝐻) |
68 | 14, 47, 67 | mhmlin 18225 |
. . . . . . . . 9
⊢ ((𝐾 ∈ (𝐺 MndHom 𝐻) ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝐾‘(𝑥(+g‘𝐺)𝑦)) = ((𝐾‘𝑥)(+g‘𝐻)(𝐾‘𝑦))) |
69 | 68 | 3expb 1122 |
. . . . . . . 8
⊢ ((𝐾 ∈ (𝐺 MndHom 𝐻) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐾‘(𝑥(+g‘𝐺)𝑦)) = ((𝐾‘𝑥)(+g‘𝐻)(𝐾‘𝑦))) |
70 | 66, 69 | sylan 583 |
. . . . . . 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))) |
71 | | coass 6129 |
. . . . . . . . 9
⊢ ((𝐾 ∘ 𝐹) ∘ 𝑓) = (𝐾 ∘ (𝐹 ∘ 𝑓)) |
72 | 71 | fveq1i 6718 |
. . . . . . . 8
⊢ (((𝐾 ∘ 𝐹) ∘ 𝑓)‘𝑥) = ((𝐾 ∘ (𝐹 ∘ 𝑓))‘𝑥) |
73 | | fvco3 6810 |
. . . . . . . . 9
⊢ (((𝐹 ∘ 𝑓):(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))⟶𝐵 ∧ 𝑥 ∈ (1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))) → ((𝐾 ∘ (𝐹 ∘ 𝑓))‘𝑥) = (𝐾‘((𝐹 ∘ 𝑓)‘𝑥))) |
74 | 61, 73 | sylan 583 |
. . . . . . . 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))) |
75 | 72, 74 | eqtr2id 2791 |
. . . . . . 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)) |
76 | 50, 62, 65, 70, 75 | seqhomo 13623 |
. . . . . 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 }))))) |
77 | | gsumzmhm.z |
. . . . . . . 8
⊢ 𝑍 = (Cntz‘𝐺) |
78 | 2 | adantr 484 |
. . . . . . . 8
⊢ ((𝜑 ∧ ((♯‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧
𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → 𝐴 ∈ 𝑉) |
79 | | gsumzmhm.c |
. . . . . . . . 9
⊢ (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹)) |
80 | 79 | adantr 484 |
. . . . . . . 8
⊢ ((𝜑 ∧ ((♯‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧
𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → ran 𝐹 ⊆ (𝑍‘ran 𝐹)) |
81 | 25 | adantr 484 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ((♯‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧
𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → (𝐹 supp 0 ) ⊆ (◡𝐹 “ (V ∖ { 0 }))) |
82 | | f1ofo 6668 |
. . . . . . . . . . 11
⊢ (𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })) → 𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–onto→(◡𝐹 “ (V ∖ { 0 }))) |
83 | | forn 6636 |
. . . . . . . . . . 11
⊢ (𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–onto→(◡𝐹 “ (V ∖ { 0 })) → ran 𝑓 = (◡𝐹 “ (V ∖ { 0 }))) |
84 | 82, 83 | syl 17 |
. . . . . . . . . 10
⊢ (𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })) → ran 𝑓 = (◡𝐹 “ (V ∖ { 0 }))) |
85 | 84 | ad2antll 729 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ((♯‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧
𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → ran 𝑓 = (◡𝐹 “ (V ∖ { 0 }))) |
86 | 81, 85 | sseqtrrd 3942 |
. . . . . . . 8
⊢ ((𝜑 ∧ ((♯‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧
𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → (𝐹 supp 0 ) ⊆ ran 𝑓) |
87 | | eqid 2737 |
. . . . . . . 8
⊢ ((𝐹 ∘ 𝑓) supp 0 ) = ((𝐹 ∘ 𝑓) supp 0 ) |
88 | 14, 8, 47, 77, 46, 78, 54, 80, 63, 57, 86, 87 | gsumval3 19292 |
. . . . . . 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 }))))) |
89 | 88 | fveq2d 6721 |
. . . . . 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 })))))) |
90 | | eqid 2737 |
. . . . . . 7
⊢
(Cntz‘𝐻) =
(Cntz‘𝐻) |
91 | 1 | adantr 484 |
. . . . . . 7
⊢ ((𝜑 ∧ ((♯‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧
𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → 𝐻 ∈ Mnd) |
92 | | fco 6569 |
. . . . . . . 8
⊢ ((𝐾:𝐵⟶(Base‘𝐻) ∧ 𝐹:𝐴⟶𝐵) → (𝐾 ∘ 𝐹):𝐴⟶(Base‘𝐻)) |
93 | 29, 54, 92 | syl2an2r 685 |
. . . . . . 7
⊢ ((𝜑 ∧ ((♯‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧
𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → (𝐾 ∘ 𝐹):𝐴⟶(Base‘𝐻)) |
94 | 77, 90 | cntzmhm2 18734 |
. . . . . . . . 9
⊢ ((𝐾 ∈ (𝐺 MndHom 𝐻) ∧ ran 𝐹 ⊆ (𝑍‘ran 𝐹)) → (𝐾 “ ran 𝐹) ⊆ ((Cntz‘𝐻)‘(𝐾 “ ran 𝐹))) |
95 | 7, 80, 94 | syl2an2r 685 |
. . . . . . . 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))) |
96 | | rnco2 6117 |
. . . . . . . 8
⊢ ran
(𝐾 ∘ 𝐹) = (𝐾 “ ran 𝐹) |
97 | 96 | fveq2i 6720 |
. . . . . . . 8
⊢
((Cntz‘𝐻)‘ran (𝐾 ∘ 𝐹)) = ((Cntz‘𝐻)‘(𝐾 “ ran 𝐹)) |
98 | 95, 96, 97 | 3sstr4g 3946 |
. . . . . . 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))) |
99 | | eldifi 4041 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ (𝐴 ∖ (◡𝐹 “ (V ∖ { 0 }))) → 𝑥 ∈ 𝐴) |
100 | | fvco3 6810 |
. . . . . . . . . . 11
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → ((𝐾 ∘ 𝐹)‘𝑥) = (𝐾‘(𝐹‘𝑥))) |
101 | 54, 99, 100 | syl2an 599 |
. . . . . . . . . 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))) |
102 | 19 | a1i 11 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ ((♯‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧
𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → 0 ∈
V) |
103 | 54, 81, 78, 102 | suppssr 7938 |
. . . . . . . . . . 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 ) |
104 | 103 | fveq2d 6721 |
. . . . . . . . . 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 )) |
105 | 10 | ad2antrr 726 |
. . . . . . . . . 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)) |
106 | 101, 104,
105 | 3eqtrd 2781 |
. . . . . . . . 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)) |
107 | 93, 106 | suppss 7936 |
. . . . . . . 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 }))) |
108 | 107, 85 | sseqtrrd 3942 |
. . . . . . 7
⊢ ((𝜑 ∧ ((♯‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧
𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → ((𝐾 ∘ 𝐹) supp (0g‘𝐻)) ⊆ ran 𝑓) |
109 | | eqid 2737 |
. . . . . . 7
⊢ (((𝐾 ∘ 𝐹) ∘ 𝑓) supp (0g‘𝐻)) = (((𝐾 ∘ 𝐹) ∘ 𝑓) supp (0g‘𝐻)) |
110 | 27, 3, 67, 90, 91, 78, 93, 98, 63, 57, 108, 109 | gsumval3 19292 |
. . . . . 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 }))))) |
111 | 76, 89, 110 | 3eqtr4rd 2788 |
. . . . 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))) |
112 | 111 | expr 460 |
. . . 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)))) |
113 | 112 | exlimdv 1941 |
. . 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)))) |
114 | 113 | expimpd 457 |
. 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)))) |
115 | | gsumzmhm.w |
. . . . 5
⊢ (𝜑 → 𝐹 finSupp 0 ) |
116 | 115 | fsuppimpd 8992 |
. . . 4
⊢ (𝜑 → (𝐹 supp 0 ) ∈
Fin) |
117 | 23, 116 | eqeltrrd 2839 |
. . 3
⊢ (𝜑 → (◡𝐹 “ (V ∖ { 0 })) ∈
Fin) |
118 | | fz1f1o 15274 |
. . 3
⊢ ((◡𝐹 “ (V ∖ { 0 })) ∈ Fin →
((◡𝐹 “ (V ∖ { 0 })) = ∅ ∨
((♯‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧
∃𝑓 𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 }))))) |
119 | 117, 118 | syl 17 |
. 2
⊢ (𝜑 → ((◡𝐹 “ (V ∖ { 0 })) = ∅ ∨
((♯‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧
∃𝑓 𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 }))))) |
120 | 45, 114, 119 | mpjaod 860 |
1
⊢ (𝜑 → (𝐻 Σg (𝐾 ∘ 𝐹)) = (𝐾‘(𝐺 Σg 𝐹))) |