Metamath Proof Explorer

Theorem gsumpt

Description: Sum of a family that is nonzero at at most one point. (Contributed by Stefan O'Rear, 7-Feb-2015) (Revised by Mario Carneiro, 25-Apr-2016) (Revised by AV, 6-Jun-2019)

Ref Expression
Hypotheses gsumpt.b ⊢ 𝐵 = ( Base ‘ 𝐺 )
gsumpt.z ⊢ 0 = ( 0g ‘ 𝐺 )
gsumpt.g ⊢ ( 𝜑 → 𝐺 ∈ Mnd )
gsumpt.a ⊢ ( 𝜑 → 𝐴 ∈ 𝑉 )
gsumpt.x ⊢ ( 𝜑 → 𝑋 ∈ 𝐴 )
gsumpt.f ⊢ ( 𝜑 → 𝐹 : 𝐴 ⟶ 𝐵 )
gsumpt.s ⊢ ( 𝜑 → ( 𝐹 supp 0 ) ⊆ { 𝑋 } )
Assertion gsumpt ( 𝜑 → ( 𝐺 Σg 𝐹 ) = ( 𝐹 ‘ 𝑋 ) )

Proof

Step Hyp Ref Expression
1 gsumpt.b ⊢ 𝐵 = ( Base ‘ 𝐺 )
2 gsumpt.z ⊢ 0 = ( 0g ‘ 𝐺 )
3 gsumpt.g ⊢ ( 𝜑 → 𝐺 ∈ Mnd )
4 gsumpt.a ⊢ ( 𝜑 → 𝐴 ∈ 𝑉 )
5 gsumpt.x ⊢ ( 𝜑 → 𝑋 ∈ 𝐴 )
6 gsumpt.f ⊢ ( 𝜑 → 𝐹 : 𝐴 ⟶ 𝐵 )
7 gsumpt.s ⊢ ( 𝜑 → ( 𝐹 supp 0 ) ⊆ { 𝑋 } )
8 5 snssd ⊢ ( 𝜑 → { 𝑋 } ⊆ 𝐴 )
9 6 8 feqresmpt ⊢ ( 𝜑 → ( 𝐹 ↾ { 𝑋 } ) = ( 𝑎 ∈ { 𝑋 } ↦ ( 𝐹 ‘ 𝑎 ) ) )
10 9 oveq2d ⊢ ( 𝜑 → ( 𝐺 Σg ( 𝐹 ↾ { 𝑋 } ) ) = ( 𝐺 Σg ( 𝑎 ∈ { 𝑋 } ↦ ( 𝐹 ‘ 𝑎 ) ) ) )
11 eqid ⊢ ( Cntz ‘ 𝐺 ) = ( Cntz ‘ 𝐺 )
12 6 5 ffvelrnd ⊢ ( 𝜑 → ( 𝐹 ‘ 𝑋 ) ∈ 𝐵 )
13 eqidd ⊢ ( 𝜑 → ( ( 𝐹 ‘ 𝑋 ) ( +g ‘ 𝐺 ) ( 𝐹 ‘ 𝑋 ) ) = ( ( 𝐹 ‘ 𝑋 ) ( +g ‘ 𝐺 ) ( 𝐹 ‘ 𝑋 ) ) )
14 eqid ⊢ ( +g ‘ 𝐺 ) = ( +g ‘ 𝐺 )
15 1 14 11 elcntzsn ⊢ ( ( 𝐹 ‘ 𝑋 ) ∈ 𝐵 → ( ( 𝐹 ‘ 𝑋 ) ∈ ( ( Cntz ‘ 𝐺 ) ‘ { ( 𝐹 ‘ 𝑋 ) } ) ↔ ( ( 𝐹 ‘ 𝑋 ) ∈ 𝐵 ∧ ( ( 𝐹 ‘ 𝑋 ) ( +g ‘ 𝐺 ) ( 𝐹 ‘ 𝑋 ) ) = ( ( 𝐹 ‘ 𝑋 ) ( +g ‘ 𝐺 ) ( 𝐹 ‘ 𝑋 ) ) ) ) )
16 12 15 syl ⊢ ( 𝜑 → ( ( 𝐹 ‘ 𝑋 ) ∈ ( ( Cntz ‘ 𝐺 ) ‘ { ( 𝐹 ‘ 𝑋 ) } ) ↔ ( ( 𝐹 ‘ 𝑋 ) ∈ 𝐵 ∧ ( ( 𝐹 ‘ 𝑋 ) ( +g ‘ 𝐺 ) ( 𝐹 ‘ 𝑋 ) ) = ( ( 𝐹 ‘ 𝑋 ) ( +g ‘ 𝐺 ) ( 𝐹 ‘ 𝑋 ) ) ) ) )
17 12 13 16 mpbir2and ⊢ ( 𝜑 → ( 𝐹 ‘ 𝑋 ) ∈ ( ( Cntz ‘ 𝐺 ) ‘ { ( 𝐹 ‘ 𝑋 ) } ) )
18 17 snssd ⊢ ( 𝜑 → { ( 𝐹 ‘ 𝑋 ) } ⊆ ( ( Cntz ‘ 𝐺 ) ‘ { ( 𝐹 ‘ 𝑋 ) } ) )
19 eqid ⊢ ( mrCls ‘ ( SubMnd ‘ 𝐺 ) ) = ( mrCls ‘ ( SubMnd ‘ 𝐺 ) )
20 eqid ⊢ ( 𝐺 ↾s ( ( mrCls ‘ ( SubMnd ‘ 𝐺 ) ) ‘ { ( 𝐹 ‘ 𝑋 ) } ) ) = ( 𝐺 ↾s ( ( mrCls ‘ ( SubMnd ‘ 𝐺 ) ) ‘ { ( 𝐹 ‘ 𝑋 ) } ) )
21 11 19 20 cntzspan ⊢ ( ( 𝐺 ∈ Mnd ∧ { ( 𝐹 ‘ 𝑋 ) } ⊆ ( ( Cntz ‘ 𝐺 ) ‘ { ( 𝐹 ‘ 𝑋 ) } ) ) → ( 𝐺 ↾s ( ( mrCls ‘ ( SubMnd ‘ 𝐺 ) ) ‘ { ( 𝐹 ‘ 𝑋 ) } ) ) ∈ CMnd )
22 3 18 21 syl2anc ⊢ ( 𝜑 → ( 𝐺 ↾s ( ( mrCls ‘ ( SubMnd ‘ 𝐺 ) ) ‘ { ( 𝐹 ‘ 𝑋 ) } ) ) ∈ CMnd )
23 1 submacs ⊢ ( 𝐺 ∈ Mnd → ( SubMnd ‘ 𝐺 ) ∈ ( ACS ‘ 𝐵 ) )
24 acsmre ⊢ ( ( SubMnd ‘ 𝐺 ) ∈ ( ACS ‘ 𝐵 ) → ( SubMnd ‘ 𝐺 ) ∈ ( Moore ‘ 𝐵 ) )
25 3 23 24 3syl ⊢ ( 𝜑 → ( SubMnd ‘ 𝐺 ) ∈ ( Moore ‘ 𝐵 ) )
26 12 snssd ⊢ ( 𝜑 → { ( 𝐹 ‘ 𝑋 ) } ⊆ 𝐵 )
27 19 mrccl ⊢ ( ( ( SubMnd ‘ 𝐺 ) ∈ ( Moore ‘ 𝐵 ) ∧ { ( 𝐹 ‘ 𝑋 ) } ⊆ 𝐵 ) → ( ( mrCls ‘ ( SubMnd ‘ 𝐺 ) ) ‘ { ( 𝐹 ‘ 𝑋 ) } ) ∈ ( SubMnd ‘ 𝐺 ) )
28 25 26 27 syl2anc ⊢ ( 𝜑 → ( ( mrCls ‘ ( SubMnd ‘ 𝐺 ) ) ‘ { ( 𝐹 ‘ 𝑋 ) } ) ∈ ( SubMnd ‘ 𝐺 ) )
29 20 11 submcmn2 ⊢ ( ( ( mrCls ‘ ( SubMnd ‘ 𝐺 ) ) ‘ { ( 𝐹 ‘ 𝑋 ) } ) ∈ ( SubMnd ‘ 𝐺 ) → ( ( 𝐺 ↾s ( ( mrCls ‘ ( SubMnd ‘ 𝐺 ) ) ‘ { ( 𝐹 ‘ 𝑋 ) } ) ) ∈ CMnd ↔ ( ( mrCls ‘ ( SubMnd ‘ 𝐺 ) ) ‘ { ( 𝐹 ‘ 𝑋 ) } ) ⊆ ( ( Cntz ‘ 𝐺 ) ‘ ( ( mrCls ‘ ( SubMnd ‘ 𝐺 ) ) ‘ { ( 𝐹 ‘ 𝑋 ) } ) ) ) )
30 28 29 syl ⊢ ( 𝜑 → ( ( 𝐺 ↾s ( ( mrCls ‘ ( SubMnd ‘ 𝐺 ) ) ‘ { ( 𝐹 ‘ 𝑋 ) } ) ) ∈ CMnd ↔ ( ( mrCls ‘ ( SubMnd ‘ 𝐺 ) ) ‘ { ( 𝐹 ‘ 𝑋 ) } ) ⊆ ( ( Cntz ‘ 𝐺 ) ‘ ( ( mrCls ‘ ( SubMnd ‘ 𝐺 ) ) ‘ { ( 𝐹 ‘ 𝑋 ) } ) ) ) )
31 22 30 mpbid ⊢ ( 𝜑 → ( ( mrCls ‘ ( SubMnd ‘ 𝐺 ) ) ‘ { ( 𝐹 ‘ 𝑋 ) } ) ⊆ ( ( Cntz ‘ 𝐺 ) ‘ ( ( mrCls ‘ ( SubMnd ‘ 𝐺 ) ) ‘ { ( 𝐹 ‘ 𝑋 ) } ) ) )
32 6 ffnd ⊢ ( 𝜑 → 𝐹 Fn 𝐴 )
33 simpr ⊢ ( ( ( 𝜑 ∧ 𝑎 ∈ 𝐴 ) ∧ 𝑎 = 𝑋 ) → 𝑎 = 𝑋 )
34 33 fveq2d ⊢ ( ( ( 𝜑 ∧ 𝑎 ∈ 𝐴 ) ∧ 𝑎 = 𝑋 ) → ( 𝐹 ‘ 𝑎 ) = ( 𝐹 ‘ 𝑋 ) )
35 25 19 26 mrcssidd ⊢ ( 𝜑 → { ( 𝐹 ‘ 𝑋 ) } ⊆ ( ( mrCls ‘ ( SubMnd ‘ 𝐺 ) ) ‘ { ( 𝐹 ‘ 𝑋 ) } ) )
36 fvex ⊢ ( 𝐹 ‘ 𝑋 ) ∈ V
37 36 snss ⊢ ( ( 𝐹 ‘ 𝑋 ) ∈ ( ( mrCls ‘ ( SubMnd ‘ 𝐺 ) ) ‘ { ( 𝐹 ‘ 𝑋 ) } ) ↔ { ( 𝐹 ‘ 𝑋 ) } ⊆ ( ( mrCls ‘ ( SubMnd ‘ 𝐺 ) ) ‘ { ( 𝐹 ‘ 𝑋 ) } ) )
38 35 37 sylibr ⊢ ( 𝜑 → ( 𝐹 ‘ 𝑋 ) ∈ ( ( mrCls ‘ ( SubMnd ‘ 𝐺 ) ) ‘ { ( 𝐹 ‘ 𝑋 ) } ) )
39 38 ad2antrr ⊢ ( ( ( 𝜑 ∧ 𝑎 ∈ 𝐴 ) ∧ 𝑎 = 𝑋 ) → ( 𝐹 ‘ 𝑋 ) ∈ ( ( mrCls ‘ ( SubMnd ‘ 𝐺 ) ) ‘ { ( 𝐹 ‘ 𝑋 ) } ) )
40 34 39 eqeltrd ⊢ ( ( ( 𝜑 ∧ 𝑎 ∈ 𝐴 ) ∧ 𝑎 = 𝑋 ) → ( 𝐹 ‘ 𝑎 ) ∈ ( ( mrCls ‘ ( SubMnd ‘ 𝐺 ) ) ‘ { ( 𝐹 ‘ 𝑋 ) } ) )
41 eldifsn ⊢ ( 𝑎 ∈ ( 𝐴 ∖ { 𝑋 } ) ↔ ( 𝑎 ∈ 𝐴 ∧ 𝑎 ≠ 𝑋 ) )
42 2 fvexi ⊢ 0 ∈ V
43 42 a1i ⊢ ( 𝜑 → 0 ∈ V )
44 6 7 4 43 suppssr ⊢ ( ( 𝜑 ∧ 𝑎 ∈ ( 𝐴 ∖ { 𝑋 } ) ) → ( 𝐹 ‘ 𝑎 ) = 0 )
45 41 44 sylan2br ⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ 𝐴 ∧ 𝑎 ≠ 𝑋 ) ) → ( 𝐹 ‘ 𝑎 ) = 0 )
46 2 subm0cl ⊢ ( ( ( mrCls ‘ ( SubMnd ‘ 𝐺 ) ) ‘ { ( 𝐹 ‘ 𝑋 ) } ) ∈ ( SubMnd ‘ 𝐺 ) → 0 ∈ ( ( mrCls ‘ ( SubMnd ‘ 𝐺 ) ) ‘ { ( 𝐹 ‘ 𝑋 ) } ) )
47 28 46 syl ⊢ ( 𝜑 → 0 ∈ ( ( mrCls ‘ ( SubMnd ‘ 𝐺 ) ) ‘ { ( 𝐹 ‘ 𝑋 ) } ) )
48 47 adantr ⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ 𝐴 ∧ 𝑎 ≠ 𝑋 ) ) → 0 ∈ ( ( mrCls ‘ ( SubMnd ‘ 𝐺 ) ) ‘ { ( 𝐹 ‘ 𝑋 ) } ) )
49 45 48 eqeltrd ⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ 𝐴 ∧ 𝑎 ≠ 𝑋 ) ) → ( 𝐹 ‘ 𝑎 ) ∈ ( ( mrCls ‘ ( SubMnd ‘ 𝐺 ) ) ‘ { ( 𝐹 ‘ 𝑋 ) } ) )
50 49 anassrs ⊢ ( ( ( 𝜑 ∧ 𝑎 ∈ 𝐴 ) ∧ 𝑎 ≠ 𝑋 ) → ( 𝐹 ‘ 𝑎 ) ∈ ( ( mrCls ‘ ( SubMnd ‘ 𝐺 ) ) ‘ { ( 𝐹 ‘ 𝑋 ) } ) )
51 40 50 pm2.61dane ⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝐴 ) → ( 𝐹 ‘ 𝑎 ) ∈ ( ( mrCls ‘ ( SubMnd ‘ 𝐺 ) ) ‘ { ( 𝐹 ‘ 𝑋 ) } ) )
52 51 ralrimiva ⊢ ( 𝜑 → ∀ 𝑎 ∈ 𝐴 ( 𝐹 ‘ 𝑎 ) ∈ ( ( mrCls ‘ ( SubMnd ‘ 𝐺 ) ) ‘ { ( 𝐹 ‘ 𝑋 ) } ) )
53 ffnfv ⊢ ( 𝐹 : 𝐴 ⟶ ( ( mrCls ‘ ( SubMnd ‘ 𝐺 ) ) ‘ { ( 𝐹 ‘ 𝑋 ) } ) ↔ ( 𝐹 Fn 𝐴 ∧ ∀ 𝑎 ∈ 𝐴 ( 𝐹 ‘ 𝑎 ) ∈ ( ( mrCls ‘ ( SubMnd ‘ 𝐺 ) ) ‘ { ( 𝐹 ‘ 𝑋 ) } ) ) )
54 32 52 53 sylanbrc ⊢ ( 𝜑 → 𝐹 : 𝐴 ⟶ ( ( mrCls ‘ ( SubMnd ‘ 𝐺 ) ) ‘ { ( 𝐹 ‘ 𝑋 ) } ) )
55 54 frnd ⊢ ( 𝜑 → ran 𝐹 ⊆ ( ( mrCls ‘ ( SubMnd ‘ 𝐺 ) ) ‘ { ( 𝐹 ‘ 𝑋 ) } ) )
56 11 cntzidss ⊢ ( ( ( ( mrCls ‘ ( SubMnd ‘ 𝐺 ) ) ‘ { ( 𝐹 ‘ 𝑋 ) } ) ⊆ ( ( Cntz ‘ 𝐺 ) ‘ ( ( mrCls ‘ ( SubMnd ‘ 𝐺 ) ) ‘ { ( 𝐹 ‘ 𝑋 ) } ) ) ∧ ran 𝐹 ⊆ ( ( mrCls ‘ ( SubMnd ‘ 𝐺 ) ) ‘ { ( 𝐹 ‘ 𝑋 ) } ) ) → ran 𝐹 ⊆ ( ( Cntz ‘ 𝐺 ) ‘ ran 𝐹 ) )
57 31 55 56 syl2anc ⊢ ( 𝜑 → ran 𝐹 ⊆ ( ( Cntz ‘ 𝐺 ) ‘ ran 𝐹 ) )
58 6 ffund ⊢ ( 𝜑 → Fun 𝐹 )
59 snfi ⊢ { 𝑋 } ∈ Fin
60 ssfi ⊢ ( ( { 𝑋 } ∈ Fin ∧ ( 𝐹 supp 0 ) ⊆ { 𝑋 } ) → ( 𝐹 supp 0 ) ∈ Fin )
61 59 7 60 sylancr ⊢ ( 𝜑 → ( 𝐹 supp 0 ) ∈ Fin )
62 6 4 fexd ⊢ ( 𝜑 → 𝐹 ∈ V )
63 isfsupp ⊢ ( ( 𝐹 ∈ V ∧ 0 ∈ V ) → ( 𝐹 finSupp 0 ↔ ( Fun 𝐹 ∧ ( 𝐹 supp 0 ) ∈ Fin ) ) )
64 62 43 63 syl2anc ⊢ ( 𝜑 → ( 𝐹 finSupp 0 ↔ ( Fun 𝐹 ∧ ( 𝐹 supp 0 ) ∈ Fin ) ) )
65 58 61 64 mpbir2and ⊢ ( 𝜑 → 𝐹 finSupp 0 )
66 1 2 11 3 4 6 57 7 65 gsumzres ⊢ ( 𝜑 → ( 𝐺 Σg ( 𝐹 ↾ { 𝑋 } ) ) = ( 𝐺 Σg 𝐹 ) )
67 fveq2 ⊢ ( 𝑎 = 𝑋 → ( 𝐹 ‘ 𝑎 ) = ( 𝐹 ‘ 𝑋 ) )
68 1 67 gsumsn ⊢ ( ( 𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐴 ∧ ( 𝐹 ‘ 𝑋 ) ∈ 𝐵 ) → ( 𝐺 Σg ( 𝑎 ∈ { 𝑋 } ↦ ( 𝐹 ‘ 𝑎 ) ) ) = ( 𝐹 ‘ 𝑋 ) )
69 3 5 12 68 syl3anc ⊢ ( 𝜑 → ( 𝐺 Σg ( 𝑎 ∈ { 𝑋 } ↦ ( 𝐹 ‘ 𝑎 ) ) ) = ( 𝐹 ‘ 𝑋 ) )
70 10 66 69 3eqtr3d ⊢ ( 𝜑 → ( 𝐺 Σg 𝐹 ) = ( 𝐹 ‘ 𝑋 ) )