Step | Hyp | Ref
| Expression |
1 | | gsumzsplit.b |
. . 3
⊢ 𝐵 = (Base‘𝐺) |
2 | | gsumzsplit.0 |
. . 3
⊢ 0 =
(0g‘𝐺) |
3 | | gsumzsplit.p |
. . 3
⊢ + =
(+g‘𝐺) |
4 | | gsumzsplit.z |
. . 3
⊢ 𝑍 = (Cntz‘𝐺) |
5 | | gsumzsplit.g |
. . 3
⊢ (𝜑 → 𝐺 ∈ Mnd) |
6 | | gsumzsplit.a |
. . 3
⊢ (𝜑 → 𝐴 ∈ 𝑉) |
7 | | gsumzsplit.f |
. . . 4
⊢ (𝜑 → 𝐹:𝐴⟶𝐵) |
8 | | fvex 6113 |
. . . . . 6
⊢
(0g‘𝐺) ∈ V |
9 | 2, 8 | eqeltri 2684 |
. . . . 5
⊢ 0 ∈
V |
10 | 9 | a1i 11 |
. . . 4
⊢ (𝜑 → 0 ∈ V) |
11 | | gsumzsplit.w |
. . . 4
⊢ (𝜑 → 𝐹 finSupp 0 ) |
12 | 7, 6, 10, 11 | fsuppmptif 8188 |
. . 3
⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 )) finSupp 0
) |
13 | 7, 6, 10, 11 | fsuppmptif 8188 |
. . 3
⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 )) finSupp 0
) |
14 | 1 | submacs 17188 |
. . . . 5
⊢ (𝐺 ∈ Mnd →
(SubMnd‘𝐺) ∈
(ACS‘𝐵)) |
15 | | acsmre 16136 |
. . . . 5
⊢
((SubMnd‘𝐺)
∈ (ACS‘𝐵) →
(SubMnd‘𝐺) ∈
(Moore‘𝐵)) |
16 | 5, 14, 15 | 3syl 18 |
. . . 4
⊢ (𝜑 → (SubMnd‘𝐺) ∈ (Moore‘𝐵)) |
17 | | frn 5966 |
. . . . 5
⊢ (𝐹:𝐴⟶𝐵 → ran 𝐹 ⊆ 𝐵) |
18 | 7, 17 | syl 17 |
. . . 4
⊢ (𝜑 → ran 𝐹 ⊆ 𝐵) |
19 | | eqid 2610 |
. . . . 5
⊢
(mrCls‘(SubMnd‘𝐺)) = (mrCls‘(SubMnd‘𝐺)) |
20 | 19 | mrccl 16094 |
. . . 4
⊢
(((SubMnd‘𝐺)
∈ (Moore‘𝐵)
∧ ran 𝐹 ⊆ 𝐵) →
((mrCls‘(SubMnd‘𝐺))‘ran 𝐹) ∈ (SubMnd‘𝐺)) |
21 | 16, 18, 20 | syl2anc 691 |
. . 3
⊢ (𝜑 →
((mrCls‘(SubMnd‘𝐺))‘ran 𝐹) ∈ (SubMnd‘𝐺)) |
22 | | gsumzsplit.c |
. . . . 5
⊢ (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹)) |
23 | | eqid 2610 |
. . . . . 6
⊢ (𝐺 ↾s
((mrCls‘(SubMnd‘𝐺))‘ran 𝐹)) = (𝐺 ↾s
((mrCls‘(SubMnd‘𝐺))‘ran 𝐹)) |
24 | 4, 19, 23 | cntzspan 18070 |
. . . . 5
⊢ ((𝐺 ∈ Mnd ∧ ran 𝐹 ⊆ (𝑍‘ran 𝐹)) → (𝐺 ↾s
((mrCls‘(SubMnd‘𝐺))‘ran 𝐹)) ∈ CMnd) |
25 | 5, 22, 24 | syl2anc 691 |
. . . 4
⊢ (𝜑 → (𝐺 ↾s
((mrCls‘(SubMnd‘𝐺))‘ran 𝐹)) ∈ CMnd) |
26 | 23, 4 | submcmn2 18067 |
. . . . 5
⊢
(((mrCls‘(SubMnd‘𝐺))‘ran 𝐹) ∈ (SubMnd‘𝐺) → ((𝐺 ↾s
((mrCls‘(SubMnd‘𝐺))‘ran 𝐹)) ∈ CMnd ↔
((mrCls‘(SubMnd‘𝐺))‘ran 𝐹) ⊆ (𝑍‘((mrCls‘(SubMnd‘𝐺))‘ran 𝐹)))) |
27 | 21, 26 | syl 17 |
. . . 4
⊢ (𝜑 → ((𝐺 ↾s
((mrCls‘(SubMnd‘𝐺))‘ran 𝐹)) ∈ CMnd ↔
((mrCls‘(SubMnd‘𝐺))‘ran 𝐹) ⊆ (𝑍‘((mrCls‘(SubMnd‘𝐺))‘ran 𝐹)))) |
28 | 25, 27 | mpbid 221 |
. . 3
⊢ (𝜑 →
((mrCls‘(SubMnd‘𝐺))‘ran 𝐹) ⊆ (𝑍‘((mrCls‘(SubMnd‘𝐺))‘ran 𝐹))) |
29 | 16, 19, 18 | mrcssidd 16108 |
. . . . . . 7
⊢ (𝜑 → ran 𝐹 ⊆ ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹)) |
30 | 29 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ran 𝐹 ⊆ ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹)) |
31 | | ffn 5958 |
. . . . . . . 8
⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴) |
32 | 7, 31 | syl 17 |
. . . . . . 7
⊢ (𝜑 → 𝐹 Fn 𝐴) |
33 | | fnfvelrn 6264 |
. . . . . . 7
⊢ ((𝐹 Fn 𝐴 ∧ 𝑘 ∈ 𝐴) → (𝐹‘𝑘) ∈ ran 𝐹) |
34 | 32, 33 | sylan 487 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐹‘𝑘) ∈ ran 𝐹) |
35 | 30, 34 | sseldd 3569 |
. . . . 5
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐹‘𝑘) ∈ ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹)) |
36 | 2 | subm0cl 17175 |
. . . . . . 7
⊢
(((mrCls‘(SubMnd‘𝐺))‘ran 𝐹) ∈ (SubMnd‘𝐺) → 0 ∈
((mrCls‘(SubMnd‘𝐺))‘ran 𝐹)) |
37 | 21, 36 | syl 17 |
. . . . . 6
⊢ (𝜑 → 0 ∈
((mrCls‘(SubMnd‘𝐺))‘ran 𝐹)) |
38 | 37 | adantr 480 |
. . . . 5
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 0 ∈
((mrCls‘(SubMnd‘𝐺))‘ran 𝐹)) |
39 | 35, 38 | ifcld 4081 |
. . . 4
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 ) ∈
((mrCls‘(SubMnd‘𝐺))‘ran 𝐹)) |
40 | | eqid 2610 |
. . . 4
⊢ (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 )) = (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 )) |
41 | 39, 40 | fmptd 6292 |
. . 3
⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 )):𝐴⟶((mrCls‘(SubMnd‘𝐺))‘ran 𝐹)) |
42 | 35, 38 | ifcld 4081 |
. . . 4
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 ) ∈
((mrCls‘(SubMnd‘𝐺))‘ran 𝐹)) |
43 | | eqid 2610 |
. . . 4
⊢ (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 )) = (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 )) |
44 | 42, 43 | fmptd 6292 |
. . 3
⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 )):𝐴⟶((mrCls‘(SubMnd‘𝐺))‘ran 𝐹)) |
45 | 1, 2, 3, 4, 5, 6, 12, 13, 21, 28, 41, 44 | gsumzadd 18145 |
. 2
⊢ (𝜑 → (𝐺 Σg ((𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 ))
∘𝑓 + (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 )))) = ((𝐺 Σg (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 ))) + (𝐺 Σg (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 ))))) |
46 | 7 | feqmptd 6159 |
. . . . 5
⊢ (𝜑 → 𝐹 = (𝑘 ∈ 𝐴 ↦ (𝐹‘𝑘))) |
47 | | iftrue 4042 |
. . . . . . . . . 10
⊢ (𝑘 ∈ 𝐶 → if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 ) = (𝐹‘𝑘)) |
48 | 47 | adantl 481 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ 𝑘 ∈ 𝐶) → if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 ) = (𝐹‘𝑘)) |
49 | | gsumzsplit.i |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (𝐶 ∩ 𝐷) = ∅) |
50 | | noel 3878 |
. . . . . . . . . . . . . . . 16
⊢ ¬
𝑘 ∈
∅ |
51 | | eleq2 2677 |
. . . . . . . . . . . . . . . 16
⊢ ((𝐶 ∩ 𝐷) = ∅ → (𝑘 ∈ (𝐶 ∩ 𝐷) ↔ 𝑘 ∈ ∅)) |
52 | 50, 51 | mtbiri 316 |
. . . . . . . . . . . . . . 15
⊢ ((𝐶 ∩ 𝐷) = ∅ → ¬ 𝑘 ∈ (𝐶 ∩ 𝐷)) |
53 | 49, 52 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ¬ 𝑘 ∈ (𝐶 ∩ 𝐷)) |
54 | 53 | adantr 480 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ¬ 𝑘 ∈ (𝐶 ∩ 𝐷)) |
55 | | elin 3758 |
. . . . . . . . . . . . 13
⊢ (𝑘 ∈ (𝐶 ∩ 𝐷) ↔ (𝑘 ∈ 𝐶 ∧ 𝑘 ∈ 𝐷)) |
56 | 54, 55 | sylnib 317 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ¬ (𝑘 ∈ 𝐶 ∧ 𝑘 ∈ 𝐷)) |
57 | | imnan 437 |
. . . . . . . . . . . 12
⊢ ((𝑘 ∈ 𝐶 → ¬ 𝑘 ∈ 𝐷) ↔ ¬ (𝑘 ∈ 𝐶 ∧ 𝑘 ∈ 𝐷)) |
58 | 56, 57 | sylibr 223 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝑘 ∈ 𝐶 → ¬ 𝑘 ∈ 𝐷)) |
59 | 58 | imp 444 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ 𝑘 ∈ 𝐶) → ¬ 𝑘 ∈ 𝐷) |
60 | 59 | iffalsed 4047 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ 𝑘 ∈ 𝐶) → if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 ) = 0 ) |
61 | 48, 60 | oveq12d 6567 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ 𝑘 ∈ 𝐶) → (if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 ) + if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 )) = ((𝐹‘𝑘) + 0 )) |
62 | 7 | ffvelrnda 6267 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐹‘𝑘) ∈ 𝐵) |
63 | 1, 3, 2 | mndrid 17135 |
. . . . . . . . . . 11
⊢ ((𝐺 ∈ Mnd ∧ (𝐹‘𝑘) ∈ 𝐵) → ((𝐹‘𝑘) + 0 ) = (𝐹‘𝑘)) |
64 | 5, 63 | sylan 487 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝐹‘𝑘) ∈ 𝐵) → ((𝐹‘𝑘) + 0 ) = (𝐹‘𝑘)) |
65 | 62, 64 | syldan 486 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ((𝐹‘𝑘) + 0 ) = (𝐹‘𝑘)) |
66 | 65 | adantr 480 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ 𝑘 ∈ 𝐶) → ((𝐹‘𝑘) + 0 ) = (𝐹‘𝑘)) |
67 | 61, 66 | eqtrd 2644 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ 𝑘 ∈ 𝐶) → (if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 ) + if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 )) = (𝐹‘𝑘)) |
68 | 58 | con2d 128 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝑘 ∈ 𝐷 → ¬ 𝑘 ∈ 𝐶)) |
69 | 68 | imp 444 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ 𝑘 ∈ 𝐷) → ¬ 𝑘 ∈ 𝐶) |
70 | 69 | iffalsed 4047 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ 𝑘 ∈ 𝐷) → if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 ) = 0 ) |
71 | | iftrue 4042 |
. . . . . . . . . 10
⊢ (𝑘 ∈ 𝐷 → if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 ) = (𝐹‘𝑘)) |
72 | 71 | adantl 481 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ 𝑘 ∈ 𝐷) → if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 ) = (𝐹‘𝑘)) |
73 | 70, 72 | oveq12d 6567 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ 𝑘 ∈ 𝐷) → (if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 ) + if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 )) = ( 0 + (𝐹‘𝑘))) |
74 | 1, 3, 2 | mndlid 17134 |
. . . . . . . . . . 11
⊢ ((𝐺 ∈ Mnd ∧ (𝐹‘𝑘) ∈ 𝐵) → ( 0 + (𝐹‘𝑘)) = (𝐹‘𝑘)) |
75 | 5, 74 | sylan 487 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝐹‘𝑘) ∈ 𝐵) → ( 0 + (𝐹‘𝑘)) = (𝐹‘𝑘)) |
76 | 62, 75 | syldan 486 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ( 0 + (𝐹‘𝑘)) = (𝐹‘𝑘)) |
77 | 76 | adantr 480 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ 𝑘 ∈ 𝐷) → ( 0 + (𝐹‘𝑘)) = (𝐹‘𝑘)) |
78 | 73, 77 | eqtrd 2644 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ 𝑘 ∈ 𝐷) → (if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 ) + if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 )) = (𝐹‘𝑘)) |
79 | | gsumzsplit.u |
. . . . . . . . . 10
⊢ (𝜑 → 𝐴 = (𝐶 ∪ 𝐷)) |
80 | 79 | eleq2d 2673 |
. . . . . . . . 9
⊢ (𝜑 → (𝑘 ∈ 𝐴 ↔ 𝑘 ∈ (𝐶 ∪ 𝐷))) |
81 | | elun 3715 |
. . . . . . . . 9
⊢ (𝑘 ∈ (𝐶 ∪ 𝐷) ↔ (𝑘 ∈ 𝐶 ∨ 𝑘 ∈ 𝐷)) |
82 | 80, 81 | syl6bb 275 |
. . . . . . . 8
⊢ (𝜑 → (𝑘 ∈ 𝐴 ↔ (𝑘 ∈ 𝐶 ∨ 𝑘 ∈ 𝐷))) |
83 | 82 | biimpa 500 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝑘 ∈ 𝐶 ∨ 𝑘 ∈ 𝐷)) |
84 | 67, 78, 83 | mpjaodan 823 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 ) + if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 )) = (𝐹‘𝑘)) |
85 | 84 | mpteq2dva 4672 |
. . . . 5
⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ (if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 ) + if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 ))) = (𝑘 ∈ 𝐴 ↦ (𝐹‘𝑘))) |
86 | 46, 85 | eqtr4d 2647 |
. . . 4
⊢ (𝜑 → 𝐹 = (𝑘 ∈ 𝐴 ↦ (if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 ) + if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 )))) |
87 | 1, 2 | mndidcl 17131 |
. . . . . . . 8
⊢ (𝐺 ∈ Mnd → 0 ∈ 𝐵) |
88 | 5, 87 | syl 17 |
. . . . . . 7
⊢ (𝜑 → 0 ∈ 𝐵) |
89 | 88 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 0 ∈ 𝐵) |
90 | 62, 89 | ifcld 4081 |
. . . . 5
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 ) ∈ 𝐵) |
91 | 62, 89 | ifcld 4081 |
. . . . 5
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 ) ∈ 𝐵) |
92 | | eqidd 2611 |
. . . . 5
⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 )) = (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 ))) |
93 | | eqidd 2611 |
. . . . 5
⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 )) = (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 ))) |
94 | 6, 90, 91, 92, 93 | offval2 6812 |
. . . 4
⊢ (𝜑 → ((𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 ))
∘𝑓 + (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 ))) = (𝑘 ∈ 𝐴 ↦ (if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 ) + if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 )))) |
95 | 86, 94 | eqtr4d 2647 |
. . 3
⊢ (𝜑 → 𝐹 = ((𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 ))
∘𝑓 + (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 )))) |
96 | 95 | oveq2d 6565 |
. 2
⊢ (𝜑 → (𝐺 Σg 𝐹) = (𝐺 Σg ((𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 ))
∘𝑓 + (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 ))))) |
97 | 46 | reseq1d 5316 |
. . . . . 6
⊢ (𝜑 → (𝐹 ↾ 𝐶) = ((𝑘 ∈ 𝐴 ↦ (𝐹‘𝑘)) ↾ 𝐶)) |
98 | | ssun1 3738 |
. . . . . . . 8
⊢ 𝐶 ⊆ (𝐶 ∪ 𝐷) |
99 | 98, 79 | syl5sseqr 3617 |
. . . . . . 7
⊢ (𝜑 → 𝐶 ⊆ 𝐴) |
100 | 47 | mpteq2ia 4668 |
. . . . . . . 8
⊢ (𝑘 ∈ 𝐶 ↦ if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 )) = (𝑘 ∈ 𝐶 ↦ (𝐹‘𝑘)) |
101 | | resmpt 5369 |
. . . . . . . 8
⊢ (𝐶 ⊆ 𝐴 → ((𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 )) ↾ 𝐶) = (𝑘 ∈ 𝐶 ↦ if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 ))) |
102 | | resmpt 5369 |
. . . . . . . 8
⊢ (𝐶 ⊆ 𝐴 → ((𝑘 ∈ 𝐴 ↦ (𝐹‘𝑘)) ↾ 𝐶) = (𝑘 ∈ 𝐶 ↦ (𝐹‘𝑘))) |
103 | 100, 101,
102 | 3eqtr4a 2670 |
. . . . . . 7
⊢ (𝐶 ⊆ 𝐴 → ((𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 )) ↾ 𝐶) = ((𝑘 ∈ 𝐴 ↦ (𝐹‘𝑘)) ↾ 𝐶)) |
104 | 99, 103 | syl 17 |
. . . . . 6
⊢ (𝜑 → ((𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 )) ↾ 𝐶) = ((𝑘 ∈ 𝐴 ↦ (𝐹‘𝑘)) ↾ 𝐶)) |
105 | 97, 104 | eqtr4d 2647 |
. . . . 5
⊢ (𝜑 → (𝐹 ↾ 𝐶) = ((𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 )) ↾ 𝐶)) |
106 | 105 | oveq2d 6565 |
. . . 4
⊢ (𝜑 → (𝐺 Σg (𝐹 ↾ 𝐶)) = (𝐺 Σg ((𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 )) ↾ 𝐶))) |
107 | 90, 40 | fmptd 6292 |
. . . . 5
⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 )):𝐴⟶𝐵) |
108 | | frn 5966 |
. . . . . . 7
⊢ ((𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 )):𝐴⟶((mrCls‘(SubMnd‘𝐺))‘ran 𝐹) → ran (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 )) ⊆
((mrCls‘(SubMnd‘𝐺))‘ran 𝐹)) |
109 | 41, 108 | syl 17 |
. . . . . 6
⊢ (𝜑 → ran (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 )) ⊆
((mrCls‘(SubMnd‘𝐺))‘ran 𝐹)) |
110 | 4 | cntzidss 17593 |
. . . . . 6
⊢
((((mrCls‘(SubMnd‘𝐺))‘ran 𝐹) ⊆ (𝑍‘((mrCls‘(SubMnd‘𝐺))‘ran 𝐹)) ∧ ran (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 )) ⊆
((mrCls‘(SubMnd‘𝐺))‘ran 𝐹)) → ran (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 )) ⊆ (𝑍‘ran (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 )))) |
111 | 28, 109, 110 | syl2anc 691 |
. . . . 5
⊢ (𝜑 → ran (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 )) ⊆ (𝑍‘ran (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 )))) |
112 | | eldifn 3695 |
. . . . . . . 8
⊢ (𝑘 ∈ (𝐴 ∖ 𝐶) → ¬ 𝑘 ∈ 𝐶) |
113 | 112 | adantl 481 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ 𝐶)) → ¬ 𝑘 ∈ 𝐶) |
114 | 113 | iffalsed 4047 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ 𝐶)) → if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 ) = 0 ) |
115 | 114, 6 | suppss2 7216 |
. . . . 5
⊢ (𝜑 → ((𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 )) supp 0 ) ⊆ 𝐶) |
116 | 1, 2, 4, 5, 6, 107, 111, 115, 12 | gsumzres 18133 |
. . . 4
⊢ (𝜑 → (𝐺 Σg ((𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 )) ↾ 𝐶)) = (𝐺 Σg (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 )))) |
117 | 106, 116 | eqtrd 2644 |
. . 3
⊢ (𝜑 → (𝐺 Σg (𝐹 ↾ 𝐶)) = (𝐺 Σg (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 )))) |
118 | 46 | reseq1d 5316 |
. . . . . 6
⊢ (𝜑 → (𝐹 ↾ 𝐷) = ((𝑘 ∈ 𝐴 ↦ (𝐹‘𝑘)) ↾ 𝐷)) |
119 | | ssun2 3739 |
. . . . . . . 8
⊢ 𝐷 ⊆ (𝐶 ∪ 𝐷) |
120 | 119, 79 | syl5sseqr 3617 |
. . . . . . 7
⊢ (𝜑 → 𝐷 ⊆ 𝐴) |
121 | 71 | mpteq2ia 4668 |
. . . . . . . 8
⊢ (𝑘 ∈ 𝐷 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 )) = (𝑘 ∈ 𝐷 ↦ (𝐹‘𝑘)) |
122 | | resmpt 5369 |
. . . . . . . 8
⊢ (𝐷 ⊆ 𝐴 → ((𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 )) ↾ 𝐷) = (𝑘 ∈ 𝐷 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 ))) |
123 | | resmpt 5369 |
. . . . . . . 8
⊢ (𝐷 ⊆ 𝐴 → ((𝑘 ∈ 𝐴 ↦ (𝐹‘𝑘)) ↾ 𝐷) = (𝑘 ∈ 𝐷 ↦ (𝐹‘𝑘))) |
124 | 121, 122,
123 | 3eqtr4a 2670 |
. . . . . . 7
⊢ (𝐷 ⊆ 𝐴 → ((𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 )) ↾ 𝐷) = ((𝑘 ∈ 𝐴 ↦ (𝐹‘𝑘)) ↾ 𝐷)) |
125 | 120, 124 | syl 17 |
. . . . . 6
⊢ (𝜑 → ((𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 )) ↾ 𝐷) = ((𝑘 ∈ 𝐴 ↦ (𝐹‘𝑘)) ↾ 𝐷)) |
126 | 118, 125 | eqtr4d 2647 |
. . . . 5
⊢ (𝜑 → (𝐹 ↾ 𝐷) = ((𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 )) ↾ 𝐷)) |
127 | 126 | oveq2d 6565 |
. . . 4
⊢ (𝜑 → (𝐺 Σg (𝐹 ↾ 𝐷)) = (𝐺 Σg ((𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 )) ↾ 𝐷))) |
128 | 91, 43 | fmptd 6292 |
. . . . 5
⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 )):𝐴⟶𝐵) |
129 | | frn 5966 |
. . . . . . 7
⊢ ((𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 )):𝐴⟶((mrCls‘(SubMnd‘𝐺))‘ran 𝐹) → ran (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 )) ⊆
((mrCls‘(SubMnd‘𝐺))‘ran 𝐹)) |
130 | 44, 129 | syl 17 |
. . . . . 6
⊢ (𝜑 → ran (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 )) ⊆
((mrCls‘(SubMnd‘𝐺))‘ran 𝐹)) |
131 | 4 | cntzidss 17593 |
. . . . . 6
⊢
((((mrCls‘(SubMnd‘𝐺))‘ran 𝐹) ⊆ (𝑍‘((mrCls‘(SubMnd‘𝐺))‘ran 𝐹)) ∧ ran (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 )) ⊆
((mrCls‘(SubMnd‘𝐺))‘ran 𝐹)) → ran (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 )) ⊆ (𝑍‘ran (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 )))) |
132 | 28, 130, 131 | syl2anc 691 |
. . . . 5
⊢ (𝜑 → ran (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 )) ⊆ (𝑍‘ran (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 )))) |
133 | | eldifn 3695 |
. . . . . . . 8
⊢ (𝑘 ∈ (𝐴 ∖ 𝐷) → ¬ 𝑘 ∈ 𝐷) |
134 | 133 | adantl 481 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ 𝐷)) → ¬ 𝑘 ∈ 𝐷) |
135 | 134 | iffalsed 4047 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ 𝐷)) → if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 ) = 0 ) |
136 | 135, 6 | suppss2 7216 |
. . . . 5
⊢ (𝜑 → ((𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 )) supp 0 ) ⊆ 𝐷) |
137 | 1, 2, 4, 5, 6, 128, 132, 136, 13 | gsumzres 18133 |
. . . 4
⊢ (𝜑 → (𝐺 Σg ((𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 )) ↾ 𝐷)) = (𝐺 Σg (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 )))) |
138 | 127, 137 | eqtrd 2644 |
. . 3
⊢ (𝜑 → (𝐺 Σg (𝐹 ↾ 𝐷)) = (𝐺 Σg (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 )))) |
139 | 117, 138 | oveq12d 6567 |
. 2
⊢ (𝜑 → ((𝐺 Σg (𝐹 ↾ 𝐶)) + (𝐺 Σg (𝐹 ↾ 𝐷))) = ((𝐺 Σg (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐶, (𝐹‘𝑘), 0 ))) + (𝐺 Σg (𝑘 ∈ 𝐴 ↦ if(𝑘 ∈ 𝐷, (𝐹‘𝑘), 0 ))))) |
140 | 45, 96, 139 | 3eqtr4d 2654 |
1
⊢ (𝜑 → (𝐺 Σg 𝐹) = ((𝐺 Σg (𝐹 ↾ 𝐶)) + (𝐺 Σg (𝐹 ↾ 𝐷)))) |