Metamath Proof Explorer

Theorem gsumfsum

Description: Relate a group sum on CCfld to a finite sum on the complex numbers. (Contributed by Mario Carneiro, 28-Dec-2014)

Ref Expression
Hypotheses gsumfsum.1 ⊢ ( 𝜑 → 𝐴 ∈ Fin )
gsumfsum.2 ⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝐵 ∈ ℂ )
Assertion gsumfsum ( 𝜑 → ( ℂfld Σg ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ) = Σ 𝑘 ∈ 𝐴 𝐵 )

Proof

Step Hyp Ref Expression
1 gsumfsum.1 ⊢ ( 𝜑 → 𝐴 ∈ Fin )
2 gsumfsum.2 ⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝐵 ∈ ℂ )
3 mpteq1 ⊢ ( 𝐴 = ∅ → ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) = ( 𝑘 ∈ ∅ ↦ 𝐵 ) )
4 mpt0 ⊢ ( 𝑘 ∈ ∅ ↦ 𝐵 ) = ∅
5 3 4 eqtrdi ⊢ ( 𝐴 = ∅ → ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) = ∅ )
6 5 oveq2d ⊢ ( 𝐴 = ∅ → ( ℂfld Σg ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ) = ( ℂfld Σg ∅ ) )
7 cnfld0 ⊢ 0 = ( 0g ‘ ℂfld )
8 7 gsum0 ⊢ ( ℂfld Σg ∅ ) = 0
9 sum0 ⊢ Σ 𝑘 ∈ ∅ 𝐵 = 0
10 8 9 eqtr4i ⊢ ( ℂfld Σg ∅ ) = Σ 𝑘 ∈ ∅ 𝐵
11 6 10 eqtrdi ⊢ ( 𝐴 = ∅ → ( ℂfld Σg ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ) = Σ 𝑘 ∈ ∅ 𝐵 )
12 sumeq1 ⊢ ( 𝐴 = ∅ → Σ 𝑘 ∈ 𝐴 𝐵 = Σ 𝑘 ∈ ∅ 𝐵 )
13 11 12 eqtr4d ⊢ ( 𝐴 = ∅ → ( ℂfld Σg ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ) = Σ 𝑘 ∈ 𝐴 𝐵 )
14 13 a1i ⊢ ( 𝜑 → ( 𝐴 = ∅ → ( ℂfld Σg ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ) = Σ 𝑘 ∈ 𝐴 𝐵 ) )
15 cnfldbas ⊢ ℂ = ( Base ‘ ℂfld )
16 cnfldadd ⊢ + = ( +g ‘ ℂfld )
17 eqid ⊢ ( Cntz ‘ ℂfld ) = ( Cntz ‘ ℂfld )
18 cnring ⊢ ℂfld ∈ Ring
19 ringmnd ⊢ ( ℂfld ∈ Ring → ℂfld ∈ Mnd )
20 18 19 mp1i ⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) → ℂfld ∈ Mnd )
21 1 adantr ⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) → 𝐴 ∈ Fin )
22 2 fmpttd ⊢ ( 𝜑 → ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) : 𝐴 ⟶ ℂ )
23 22 adantr ⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) → ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) : 𝐴 ⟶ ℂ )
24 ringcmn ⊢ ( ℂfld ∈ Ring → ℂfld ∈ CMnd )
25 18 24 mp1i ⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) → ℂfld ∈ CMnd )
26 15 17 25 23 cntzcmnf ⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) → ran ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ⊆ ( ( Cntz ‘ ℂfld ) ‘ ran ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ) )
27 simprl ⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) → ( ♯ ‘ 𝐴 ) ∈ ℕ )
28 simprr ⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) → 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 )
29 f1of1 ⊢ ( 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 → 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1→ 𝐴 )
30 28 29 syl ⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) → 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1→ 𝐴 )
31 suppssdm ⊢ ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) supp 0 ) ⊆ dom ( 𝑘 ∈ 𝐴 ↦ 𝐵 )
32 31 23 fssdm ⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) → ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) supp 0 ) ⊆ 𝐴 )
33 f1ofo ⊢ ( 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 → 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –onto→ 𝐴 )
34 forn ⊢ ( 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –onto→ 𝐴 → ran 𝑓 = 𝐴 )
35 28 33 34 3syl ⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) → ran 𝑓 = 𝐴 )
36 32 35 sseqtrrd ⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) → ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) supp 0 ) ⊆ ran 𝑓 )
37 eqid ⊢ ( ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ∘ 𝑓 ) supp 0 ) = ( ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ∘ 𝑓 ) supp 0 )
38 15 7 16 17 20 21 23 26 27 30 36 37 gsumval3 ⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) → ( ℂfld Σg ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ) = ( seq 1 ( + , ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ∘ 𝑓 ) ) ‘ ( ♯ ‘ 𝐴 ) ) )
39 sumfc ⊢ Σ 𝑥 ∈ 𝐴 ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ‘ 𝑥 ) = Σ 𝑘 ∈ 𝐴 𝐵
40 fveq2 ⊢ ( 𝑥 = ( 𝑓 ‘ 𝑛 ) → ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ‘ 𝑥 ) = ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ‘ ( 𝑓 ‘ 𝑛 ) ) )
41 23 ffvelrnda ⊢ ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) ∧ 𝑥 ∈ 𝐴 ) → ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ‘ 𝑥 ) ∈ ℂ )
42 f1of ⊢ ( 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 → 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) ⟶ 𝐴 )
43 28 42 syl ⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) → 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) ⟶ 𝐴 )
44 fvco3 ⊢ ( ( 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) ⟶ 𝐴 ∧ 𝑛 ∈ ( 1 ... ( ♯ ‘ 𝐴 ) ) ) → ( ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ∘ 𝑓 ) ‘ 𝑛 ) = ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ‘ ( 𝑓 ‘ 𝑛 ) ) )
45 43 44 sylan ⊢ ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) ∧ 𝑛 ∈ ( 1 ... ( ♯ ‘ 𝐴 ) ) ) → ( ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ∘ 𝑓 ) ‘ 𝑛 ) = ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ‘ ( 𝑓 ‘ 𝑛 ) ) )
46 40 27 28 41 45 fsum ⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) → Σ 𝑥 ∈ 𝐴 ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ‘ 𝑥 ) = ( seq 1 ( + , ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ∘ 𝑓 ) ) ‘ ( ♯ ‘ 𝐴 ) ) )
47 39 46 eqtr3id ⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) → Σ 𝑘 ∈ 𝐴 𝐵 = ( seq 1 ( + , ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ∘ 𝑓 ) ) ‘ ( ♯ ‘ 𝐴 ) ) )
48 38 47 eqtr4d ⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) → ( ℂfld Σg ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ) = Σ 𝑘 ∈ 𝐴 𝐵 )
49 48 expr ⊢ ( ( 𝜑 ∧ ( ♯ ‘ 𝐴 ) ∈ ℕ ) → ( 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 → ( ℂfld Σg ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ) = Σ 𝑘 ∈ 𝐴 𝐵 ) )
50 49 exlimdv ⊢ ( ( 𝜑 ∧ ( ♯ ‘ 𝐴 ) ∈ ℕ ) → ( ∃ 𝑓 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 → ( ℂfld Σg ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ) = Σ 𝑘 ∈ 𝐴 𝐵 ) )
51 50 expimpd ⊢ ( 𝜑 → ( ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ ∃ 𝑓 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) → ( ℂfld Σg ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ) = Σ 𝑘 ∈ 𝐴 𝐵 ) )
52 fz1f1o ⊢ ( 𝐴 ∈ Fin → ( 𝐴 = ∅ ∨ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ ∃ 𝑓 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) )
53 1 52 syl ⊢ ( 𝜑 → ( 𝐴 = ∅ ∨ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ ∃ 𝑓 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) )
54 14 51 53 mpjaod ⊢ ( 𝜑 → ( ℂfld Σg ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ) = Σ 𝑘 ∈ 𝐴 𝐵 )