Step | Hyp | Ref
| Expression |
1 | | gsumzmhm.h |
. . . . . . 7
⊢ (𝜑 → 𝐻 ∈ Mnd) |
2 | | gsumzmhm.a |
. . . . . . 7
⊢ (𝜑 → 𝐴 ∈ 𝑉) |
3 | | eqid 2818 |
. . . . . . . 8
⊢
(0g‘𝐻) = (0g‘𝐻) |
4 | 3 | gsumz 17988 |
. . . . . . 7
⊢ ((𝐻 ∈ Mnd ∧ 𝐴 ∈ 𝑉) → (𝐻 Σg (𝑘 ∈ 𝐴 ↦ (0g‘𝐻))) = (0g‘𝐻)) |
5 | 1, 2, 4 | syl2anc 584 |
. . . . . 6
⊢ (𝜑 → (𝐻 Σg (𝑘 ∈ 𝐴 ↦ (0g‘𝐻))) = (0g‘𝐻)) |
6 | 5 | adantr 481 |
. . . . 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 17952 |
. . . . . . 7
⊢ (𝐾 ∈ (𝐺 MndHom 𝐻) → (𝐾‘ 0 ) =
(0g‘𝐻)) |
10 | 7, 9 | syl 17 |
. . . . . 6
⊢ (𝜑 → (𝐾‘ 0 ) =
(0g‘𝐻)) |
11 | 10 | adantr 481 |
. . . . 5
⊢ ((𝜑 ∧ (◡𝐹 “ (V ∖ { 0 })) = ∅) →
(𝐾‘ 0 ) =
(0g‘𝐻)) |
12 | 6, 11 | eqtr4d 2856 |
. . . 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 17914 |
. . . . . . . . 9
⊢ (𝐺 ∈ Mnd → 0 ∈ 𝐵) |
16 | 13, 15 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → 0 ∈ 𝐵) |
17 | 16 | ad2antrr 722 |
. . . . . . 7
⊢ (((𝜑 ∧ (◡𝐹 “ (V ∖ { 0 })) = ∅) ∧ 𝑘 ∈ 𝐴) → 0 ∈ 𝐵) |
18 | | gsumzmhm.f |
. . . . . . . 8
⊢ (𝜑 → 𝐹:𝐴⟶𝐵) |
19 | 8 | fvexi 6677 |
. . . . . . . . 9
⊢ 0 ∈
V |
20 | 19 | a1i 11 |
. . . . . . . 8
⊢ (𝜑 → 0 ∈ V) |
21 | | fex 6980 |
. . . . . . . . . . 11
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → 𝐹 ∈ V) |
22 | 18, 2, 21 | syl2anc 584 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐹 ∈ V) |
23 | | suppimacnv 7830 |
. . . . . . . . . 10
⊢ ((𝐹 ∈ V ∧ 0 ∈ V)
→ (𝐹 supp 0 ) = (◡𝐹 “ (V ∖ { 0 }))) |
24 | 22, 20, 23 | syl2anc 584 |
. . . . . . . . 9
⊢ (𝜑 → (𝐹 supp 0 ) = (◡𝐹 “ (V ∖ { 0 }))) |
25 | | ssid 3986 |
. . . . . . . . 9
⊢ (◡𝐹 “ (V ∖ { 0 })) ⊆ (◡𝐹 “ (V ∖ { 0 })) |
26 | 24, 25 | eqsstrdi 4018 |
. . . . . . . 8
⊢ (𝜑 → (𝐹 supp 0 ) ⊆ (◡𝐹 “ (V ∖ { 0 }))) |
27 | 18, 2, 20, 26 | gsumcllem 18957 |
. . . . . . 7
⊢ ((𝜑 ∧ (◡𝐹 “ (V ∖ { 0 })) = ∅) →
𝐹 = (𝑘 ∈ 𝐴 ↦ 0 )) |
28 | | eqid 2818 |
. . . . . . . . . . 11
⊢
(Base‘𝐻) =
(Base‘𝐻) |
29 | 14, 28 | mhmf 17949 |
. . . . . . . . . 10
⊢ (𝐾 ∈ (𝐺 MndHom 𝐻) → 𝐾:𝐵⟶(Base‘𝐻)) |
30 | 7, 29 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → 𝐾:𝐵⟶(Base‘𝐻)) |
31 | 30 | feqmptd 6726 |
. . . . . . . 8
⊢ (𝜑 → 𝐾 = (𝑥 ∈ 𝐵 ↦ (𝐾‘𝑥))) |
32 | 31 | adantr 481 |
. . . . . . 7
⊢ ((𝜑 ∧ (◡𝐹 “ (V ∖ { 0 })) = ∅) →
𝐾 = (𝑥 ∈ 𝐵 ↦ (𝐾‘𝑥))) |
33 | | fveq2 6663 |
. . . . . . 7
⊢ (𝑥 = 0 → (𝐾‘𝑥) = (𝐾‘ 0 )) |
34 | 17, 27, 32, 33 | fmptco 6883 |
. . . . . 6
⊢ ((𝜑 ∧ (◡𝐹 “ (V ∖ { 0 })) = ∅) →
(𝐾 ∘ 𝐹) = (𝑘 ∈ 𝐴 ↦ (𝐾‘ 0 ))) |
35 | 10 | mpteq2dv 5153 |
. . . . . . 7
⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ (𝐾‘ 0 )) = (𝑘 ∈ 𝐴 ↦ (0g‘𝐻))) |
36 | 35 | adantr 481 |
. . . . . 6
⊢ ((𝜑 ∧ (◡𝐹 “ (V ∖ { 0 })) = ∅) →
(𝑘 ∈ 𝐴 ↦ (𝐾‘ 0 )) = (𝑘 ∈ 𝐴 ↦ (0g‘𝐻))) |
37 | 34, 36 | eqtrd 2853 |
. . . . 5
⊢ ((𝜑 ∧ (◡𝐹 “ (V ∖ { 0 })) = ∅) →
(𝐾 ∘ 𝐹) = (𝑘 ∈ 𝐴 ↦ (0g‘𝐻))) |
38 | 37 | oveq2d 7161 |
. . . 4
⊢ ((𝜑 ∧ (◡𝐹 “ (V ∖ { 0 })) = ∅) →
(𝐻
Σg (𝐾 ∘ 𝐹)) = (𝐻 Σg (𝑘 ∈ 𝐴 ↦ (0g‘𝐻)))) |
39 | 27 | oveq2d 7161 |
. . . . . 6
⊢ ((𝜑 ∧ (◡𝐹 “ (V ∖ { 0 })) = ∅) →
(𝐺
Σg 𝐹) = (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 0 ))) |
40 | 8 | gsumz 17988 |
. . . . . . . 8
⊢ ((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝑉) → (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 0 )) = 0 ) |
41 | 13, 2, 40 | syl2anc 584 |
. . . . . . 7
⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 0 )) = 0 ) |
42 | 41 | adantr 481 |
. . . . . 6
⊢ ((𝜑 ∧ (◡𝐹 “ (V ∖ { 0 })) = ∅) →
(𝐺
Σg (𝑘 ∈ 𝐴 ↦ 0 )) = 0 ) |
43 | 39, 42 | eqtrd 2853 |
. . . . 5
⊢ ((𝜑 ∧ (◡𝐹 “ (V ∖ { 0 })) = ∅) →
(𝐺
Σg 𝐹) = 0 ) |
44 | 43 | fveq2d 6667 |
. . . 4
⊢ ((𝜑 ∧ (◡𝐹 “ (V ∖ { 0 })) = ∅) →
(𝐾‘(𝐺 Σg 𝐹)) = (𝐾‘ 0 )) |
45 | 12, 38, 44 | 3eqtr4d 2863 |
. . 3
⊢ ((𝜑 ∧ (◡𝐹 “ (V ∖ { 0 })) = ∅) →
(𝐻
Σg (𝐾 ∘ 𝐹)) = (𝐾‘(𝐺 Σg 𝐹))) |
46 | 45 | ex 413 |
. 2
⊢ (𝜑 → ((◡𝐹 “ (V ∖ { 0 })) = ∅ →
(𝐻
Σg (𝐾 ∘ 𝐹)) = (𝐾‘(𝐺 Σg 𝐹)))) |
47 | 13 | adantr 481 |
. . . . . . . 8
⊢ ((𝜑 ∧ ((♯‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧
𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → 𝐺 ∈ Mnd) |
48 | | eqid 2818 |
. . . . . . . . . 10
⊢
(+g‘𝐺) = (+g‘𝐺) |
49 | 14, 48 | mndcl 17907 |
. . . . . . . . 9
⊢ ((𝐺 ∈ Mnd ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥(+g‘𝐺)𝑦) ∈ 𝐵) |
50 | 49 | 3expb 1112 |
. . . . . . . 8
⊢ ((𝐺 ∈ Mnd ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐺)𝑦) ∈ 𝐵) |
51 | 47, 50 | sylan 580 |
. . . . . . 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) |
52 | | f1of1 6607 |
. . . . . . . . . . . 12
⊢ (𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })) → 𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1→(◡𝐹 “ (V ∖ { 0
}))) |
53 | 52 | ad2antll 725 |
. . . . . . . . . . 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
}))) |
54 | | cnvimass 5942 |
. . . . . . . . . . . 12
⊢ (◡𝐹 “ (V ∖ { 0 })) ⊆ dom 𝐹 |
55 | 18 | adantr 481 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ ((♯‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧
𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → 𝐹:𝐴⟶𝐵) |
56 | 54, 55 | fssdm 6523 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ ((♯‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧
𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → (◡𝐹 “ (V ∖ { 0 })) ⊆ 𝐴) |
57 | | f1ss 6573 |
. . . . . . . . . . 11
⊢ ((𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1→(◡𝐹 “ (V ∖ { 0 })) ∧
(◡𝐹 “ (V ∖ { 0 })) ⊆ 𝐴) → 𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1→𝐴) |
58 | 53, 56, 57 | syl2anc 584 |
. . . . . . . . . 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) |
59 | | f1f 6568 |
. . . . . . . . . 10
⊢ (𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1→𝐴 → 𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))⟶𝐴) |
60 | 58, 59 | 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) |
61 | | fco 6524 |
. . . . . . . . 9
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))⟶𝐴) → (𝐹 ∘ 𝑓):(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))⟶𝐵) |
62 | 18, 60, 61 | syl2an2r 681 |
. . . . . . . 8
⊢ ((𝜑 ∧ ((♯‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧
𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → (𝐹 ∘ 𝑓):(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))⟶𝐵) |
63 | 62 | ffvelrnda 6843 |
. . . . . . 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) |
64 | | simprl 767 |
. . . . . . . 8
⊢ ((𝜑 ∧ ((♯‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧
𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) →
(♯‘(◡𝐹 “ (V ∖ { 0 }))) ∈
ℕ) |
65 | | nnuz 12269 |
. . . . . . . 8
⊢ ℕ =
(ℤ≥‘1) |
66 | 64, 65 | eleqtrdi 2920 |
. . . . . . 7
⊢ ((𝜑 ∧ ((♯‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧
𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) →
(♯‘(◡𝐹 “ (V ∖ { 0 }))) ∈
(ℤ≥‘1)) |
67 | 7 | adantr 481 |
. . . . . . . 8
⊢ ((𝜑 ∧ ((♯‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧
𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → 𝐾 ∈ (𝐺 MndHom 𝐻)) |
68 | | eqid 2818 |
. . . . . . . . . 10
⊢
(+g‘𝐻) = (+g‘𝐻) |
69 | 14, 48, 68 | mhmlin 17951 |
. . . . . . . . 9
⊢ ((𝐾 ∈ (𝐺 MndHom 𝐻) ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝐾‘(𝑥(+g‘𝐺)𝑦)) = ((𝐾‘𝑥)(+g‘𝐻)(𝐾‘𝑦))) |
70 | 69 | 3expb 1112 |
. . . . . . . 8
⊢ ((𝐾 ∈ (𝐺 MndHom 𝐻) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐾‘(𝑥(+g‘𝐺)𝑦)) = ((𝐾‘𝑥)(+g‘𝐻)(𝐾‘𝑦))) |
71 | 67, 70 | sylan 580 |
. . . . . . 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))) |
72 | | coass 6111 |
. . . . . . . . 9
⊢ ((𝐾 ∘ 𝐹) ∘ 𝑓) = (𝐾 ∘ (𝐹 ∘ 𝑓)) |
73 | 72 | fveq1i 6664 |
. . . . . . . 8
⊢ (((𝐾 ∘ 𝐹) ∘ 𝑓)‘𝑥) = ((𝐾 ∘ (𝐹 ∘ 𝑓))‘𝑥) |
74 | | fvco3 6753 |
. . . . . . . . 9
⊢ (((𝐹 ∘ 𝑓):(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))⟶𝐵 ∧ 𝑥 ∈ (1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))) → ((𝐾 ∘ (𝐹 ∘ 𝑓))‘𝑥) = (𝐾‘((𝐹 ∘ 𝑓)‘𝑥))) |
75 | 62, 74 | sylan 580 |
. . . . . . . 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))) |
76 | 73, 75 | syl5req 2866 |
. . . . . . 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)) |
77 | 51, 63, 66, 71, 76 | seqhomo 13405 |
. . . . . 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 }))))) |
78 | | gsumzmhm.z |
. . . . . . . 8
⊢ 𝑍 = (Cntz‘𝐺) |
79 | 2 | adantr 481 |
. . . . . . . 8
⊢ ((𝜑 ∧ ((♯‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧
𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → 𝐴 ∈ 𝑉) |
80 | | gsumzmhm.c |
. . . . . . . . 9
⊢ (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹)) |
81 | 80 | adantr 481 |
. . . . . . . 8
⊢ ((𝜑 ∧ ((♯‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧
𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → ran 𝐹 ⊆ (𝑍‘ran 𝐹)) |
82 | 26 | adantr 481 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ((♯‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧
𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → (𝐹 supp 0 ) ⊆ (◡𝐹 “ (V ∖ { 0 }))) |
83 | | f1ofo 6615 |
. . . . . . . . . . 11
⊢ (𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })) → 𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–onto→(◡𝐹 “ (V ∖ { 0 }))) |
84 | | forn 6586 |
. . . . . . . . . . 11
⊢ (𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–onto→(◡𝐹 “ (V ∖ { 0 })) → ran 𝑓 = (◡𝐹 “ (V ∖ { 0 }))) |
85 | 83, 84 | syl 17 |
. . . . . . . . . 10
⊢ (𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })) → ran 𝑓 = (◡𝐹 “ (V ∖ { 0 }))) |
86 | 85 | ad2antll 725 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ((♯‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧
𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → ran 𝑓 = (◡𝐹 “ (V ∖ { 0 }))) |
87 | 82, 86 | sseqtrrd 4005 |
. . . . . . . 8
⊢ ((𝜑 ∧ ((♯‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧
𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → (𝐹 supp 0 ) ⊆ ran 𝑓) |
88 | | eqid 2818 |
. . . . . . . 8
⊢ ((𝐹 ∘ 𝑓) supp 0 ) = ((𝐹 ∘ 𝑓) supp 0 ) |
89 | 14, 8, 48, 78, 47, 79, 55, 81, 64, 58, 87, 88 | gsumval3 18956 |
. . . . . . 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 }))))) |
90 | 89 | fveq2d 6667 |
. . . . . 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 })))))) |
91 | | eqid 2818 |
. . . . . . 7
⊢
(Cntz‘𝐻) =
(Cntz‘𝐻) |
92 | 1 | adantr 481 |
. . . . . . 7
⊢ ((𝜑 ∧ ((♯‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧
𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → 𝐻 ∈ Mnd) |
93 | | fco 6524 |
. . . . . . . 8
⊢ ((𝐾:𝐵⟶(Base‘𝐻) ∧ 𝐹:𝐴⟶𝐵) → (𝐾 ∘ 𝐹):𝐴⟶(Base‘𝐻)) |
94 | 30, 55, 93 | syl2an2r 681 |
. . . . . . 7
⊢ ((𝜑 ∧ ((♯‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧
𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → (𝐾 ∘ 𝐹):𝐴⟶(Base‘𝐻)) |
95 | 78, 91 | cntzmhm2 18408 |
. . . . . . . . 9
⊢ ((𝐾 ∈ (𝐺 MndHom 𝐻) ∧ ran 𝐹 ⊆ (𝑍‘ran 𝐹)) → (𝐾 “ ran 𝐹) ⊆ ((Cntz‘𝐻)‘(𝐾 “ ran 𝐹))) |
96 | 7, 81, 95 | syl2an2r 681 |
. . . . . . . 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))) |
97 | | rnco2 6099 |
. . . . . . . 8
⊢ ran
(𝐾 ∘ 𝐹) = (𝐾 “ ran 𝐹) |
98 | 97 | fveq2i 6666 |
. . . . . . . 8
⊢
((Cntz‘𝐻)‘ran (𝐾 ∘ 𝐹)) = ((Cntz‘𝐻)‘(𝐾 “ ran 𝐹)) |
99 | 96, 97, 98 | 3sstr4g 4009 |
. . . . . . 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))) |
100 | | eldifi 4100 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ (𝐴 ∖ (◡𝐹 “ (V ∖ { 0 }))) → 𝑥 ∈ 𝐴) |
101 | | fvco3 6753 |
. . . . . . . . . . 11
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → ((𝐾 ∘ 𝐹)‘𝑥) = (𝐾‘(𝐹‘𝑥))) |
102 | 55, 100, 101 | syl2an 595 |
. . . . . . . . . 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))) |
103 | 19 | a1i 11 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ ((♯‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧
𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → 0 ∈
V) |
104 | 55, 82, 79, 103 | suppssr 7850 |
. . . . . . . . . . 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 ) |
105 | 104 | fveq2d 6667 |
. . . . . . . . . 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 )) |
106 | 10 | ad2antrr 722 |
. . . . . . . . . 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)) |
107 | 102, 105,
106 | 3eqtrd 2857 |
. . . . . . . . 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)) |
108 | 94, 107 | suppss 7849 |
. . . . . . . 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 }))) |
109 | 108, 86 | sseqtrrd 4005 |
. . . . . . 7
⊢ ((𝜑 ∧ ((♯‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧
𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → ((𝐾 ∘ 𝐹) supp (0g‘𝐻)) ⊆ ran 𝑓) |
110 | | eqid 2818 |
. . . . . . 7
⊢ (((𝐾 ∘ 𝐹) ∘ 𝑓) supp (0g‘𝐻)) = (((𝐾 ∘ 𝐹) ∘ 𝑓) supp (0g‘𝐻)) |
111 | 28, 3, 68, 91, 92, 79, 94, 99, 64, 58, 109, 110 | gsumval3 18956 |
. . . . . 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 }))))) |
112 | 77, 90, 111 | 3eqtr4rd 2864 |
. . . . 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))) |
113 | 112 | expr 457 |
. . . 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)))) |
114 | 113 | exlimdv 1925 |
. . 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)))) |
115 | 114 | expimpd 454 |
. 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)))) |
116 | | gsumzmhm.w |
. . . . 5
⊢ (𝜑 → 𝐹 finSupp 0 ) |
117 | 116 | fsuppimpd 8828 |
. . . 4
⊢ (𝜑 → (𝐹 supp 0 ) ∈
Fin) |
118 | 24, 117 | eqeltrrd 2911 |
. . 3
⊢ (𝜑 → (◡𝐹 “ (V ∖ { 0 })) ∈
Fin) |
119 | | fz1f1o 15055 |
. . 3
⊢ ((◡𝐹 “ (V ∖ { 0 })) ∈ Fin →
((◡𝐹 “ (V ∖ { 0 })) = ∅ ∨
((♯‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧
∃𝑓 𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 }))))) |
120 | 118, 119 | syl 17 |
. 2
⊢ (𝜑 → ((◡𝐹 “ (V ∖ { 0 })) = ∅ ∨
((♯‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧
∃𝑓 𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 }))))) |
121 | 46, 115, 120 | mpjaod 854 |
1
⊢ (𝜑 → (𝐻 Σg (𝐾 ∘ 𝐹)) = (𝐾‘(𝐺 Σg 𝐹))) |