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Theorem gsumzres 18517
Description: Extend a finite group sum by padding outside with zeroes. (Contributed by Mario Carneiro, 24-Apr-2016.) (Revised by AV, 31-May-2019.)
Hypotheses
Ref Expression
gsumzcl.b 𝐵 = (Base‘𝐺)
gsumzcl.0 0 = (0g𝐺)
gsumzcl.z 𝑍 = (Cntz‘𝐺)
gsumzcl.g (𝜑𝐺 ∈ Mnd)
gsumzcl.a (𝜑𝐴𝑉)
gsumzcl.f (𝜑𝐹:𝐴𝐵)
gsumzcl.c (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹))
gsumzres.s (𝜑 → (𝐹 supp 0 ) ⊆ 𝑊)
gsumzres.w (𝜑𝐹 finSupp 0 )
Assertion
Ref Expression
gsumzres (𝜑 → (𝐺 Σg (𝐹𝑊)) = (𝐺 Σg 𝐹))

Proof of Theorem gsumzres
Dummy variables 𝑓 𝑘 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 gsumzcl.g . . . . . . 7 (𝜑𝐺 ∈ Mnd)
2 gsumzcl.a . . . . . . . 8 (𝜑𝐴𝑉)
3 inex1g 4935 . . . . . . . 8 (𝐴𝑉 → (𝐴𝑊) ∈ V)
42, 3syl 17 . . . . . . 7 (𝜑 → (𝐴𝑊) ∈ V)
5 gsumzcl.0 . . . . . . . 8 0 = (0g𝐺)
65gsumz 17582 . . . . . . 7 ((𝐺 ∈ Mnd ∧ (𝐴𝑊) ∈ V) → (𝐺 Σg (𝑘 ∈ (𝐴𝑊) ↦ 0 )) = 0 )
71, 4, 6syl2anc 573 . . . . . 6 (𝜑 → (𝐺 Σg (𝑘 ∈ (𝐴𝑊) ↦ 0 )) = 0 )
85gsumz 17582 . . . . . . 7 ((𝐺 ∈ Mnd ∧ 𝐴𝑉) → (𝐺 Σg (𝑘𝐴0 )) = 0 )
91, 2, 8syl2anc 573 . . . . . 6 (𝜑 → (𝐺 Σg (𝑘𝐴0 )) = 0 )
107, 9eqtr4d 2808 . . . . 5 (𝜑 → (𝐺 Σg (𝑘 ∈ (𝐴𝑊) ↦ 0 )) = (𝐺 Σg (𝑘𝐴0 )))
1110adantr 466 . . . 4 ((𝜑 ∧ (𝐹 supp 0 ) = ∅) → (𝐺 Σg (𝑘 ∈ (𝐴𝑊) ↦ 0 )) = (𝐺 Σg (𝑘𝐴0 )))
12 resres 5550 . . . . . . . 8 ((𝐹𝐴) ↾ 𝑊) = (𝐹 ↾ (𝐴𝑊))
13 gsumzcl.f . . . . . . . . . 10 (𝜑𝐹:𝐴𝐵)
14 ffn 6185 . . . . . . . . . 10 (𝐹:𝐴𝐵𝐹 Fn 𝐴)
15 fnresdm 6140 . . . . . . . . . 10 (𝐹 Fn 𝐴 → (𝐹𝐴) = 𝐹)
1613, 14, 153syl 18 . . . . . . . . 9 (𝜑 → (𝐹𝐴) = 𝐹)
1716reseq1d 5533 . . . . . . . 8 (𝜑 → ((𝐹𝐴) ↾ 𝑊) = (𝐹𝑊))
1812, 17syl5eqr 2819 . . . . . . 7 (𝜑 → (𝐹 ↾ (𝐴𝑊)) = (𝐹𝑊))
1918adantr 466 . . . . . 6 ((𝜑 ∧ (𝐹 supp 0 ) = ∅) → (𝐹 ↾ (𝐴𝑊)) = (𝐹𝑊))
20 fvex 6342 . . . . . . . . . . 11 (0g𝐺) ∈ V
215, 20eqeltri 2846 . . . . . . . . . 10 0 ∈ V
2221a1i 11 . . . . . . . . 9 (𝜑0 ∈ V)
23 ssid 3773 . . . . . . . . . 10 (𝐹 supp 0 ) ⊆ (𝐹 supp 0 )
2423a1i 11 . . . . . . . . 9 (𝜑 → (𝐹 supp 0 ) ⊆ (𝐹 supp 0 ))
2513, 2, 22, 24gsumcllem 18516 . . . . . . . 8 ((𝜑 ∧ (𝐹 supp 0 ) = ∅) → 𝐹 = (𝑘𝐴0 ))
2625reseq1d 5533 . . . . . . 7 ((𝜑 ∧ (𝐹 supp 0 ) = ∅) → (𝐹 ↾ (𝐴𝑊)) = ((𝑘𝐴0 ) ↾ (𝐴𝑊)))
27 inss1 3981 . . . . . . . 8 (𝐴𝑊) ⊆ 𝐴
28 resmpt 5590 . . . . . . . 8 ((𝐴𝑊) ⊆ 𝐴 → ((𝑘𝐴0 ) ↾ (𝐴𝑊)) = (𝑘 ∈ (𝐴𝑊) ↦ 0 ))
2927, 28ax-mp 5 . . . . . . 7 ((𝑘𝐴0 ) ↾ (𝐴𝑊)) = (𝑘 ∈ (𝐴𝑊) ↦ 0 )
3026, 29syl6eq 2821 . . . . . 6 ((𝜑 ∧ (𝐹 supp 0 ) = ∅) → (𝐹 ↾ (𝐴𝑊)) = (𝑘 ∈ (𝐴𝑊) ↦ 0 ))
3119, 30eqtr3d 2807 . . . . 5 ((𝜑 ∧ (𝐹 supp 0 ) = ∅) → (𝐹𝑊) = (𝑘 ∈ (𝐴𝑊) ↦ 0 ))
3231oveq2d 6809 . . . 4 ((𝜑 ∧ (𝐹 supp 0 ) = ∅) → (𝐺 Σg (𝐹𝑊)) = (𝐺 Σg (𝑘 ∈ (𝐴𝑊) ↦ 0 )))
3325oveq2d 6809 . . . 4 ((𝜑 ∧ (𝐹 supp 0 ) = ∅) → (𝐺 Σg 𝐹) = (𝐺 Σg (𝑘𝐴0 )))
3411, 32, 333eqtr4d 2815 . . 3 ((𝜑 ∧ (𝐹 supp 0 ) = ∅) → (𝐺 Σg (𝐹𝑊)) = (𝐺 Σg 𝐹))
3534ex 397 . 2 (𝜑 → ((𝐹 supp 0 ) = ∅ → (𝐺 Σg (𝐹𝑊)) = (𝐺 Σg 𝐹)))
36 f1ofo 6285 . . . . . . . . . . . 12 (𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ) → 𝑓:(1...(♯‘(𝐹 supp 0 )))–onto→(𝐹 supp 0 ))
37 forn 6259 . . . . . . . . . . . 12 (𝑓:(1...(♯‘(𝐹 supp 0 )))–onto→(𝐹 supp 0 ) → ran 𝑓 = (𝐹 supp 0 ))
3836, 37syl 17 . . . . . . . . . . 11 (𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ) → ran 𝑓 = (𝐹 supp 0 ))
3938ad2antll 708 . . . . . . . . . 10 ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → ran 𝑓 = (𝐹 supp 0 ))
40 gsumzres.s . . . . . . . . . . 11 (𝜑 → (𝐹 supp 0 ) ⊆ 𝑊)
4140adantr 466 . . . . . . . . . 10 ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (𝐹 supp 0 ) ⊆ 𝑊)
4239, 41eqsstrd 3788 . . . . . . . . 9 ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → ran 𝑓𝑊)
43 cores 5782 . . . . . . . . 9 (ran 𝑓𝑊 → ((𝐹𝑊) ∘ 𝑓) = (𝐹𝑓))
4442, 43syl 17 . . . . . . . 8 ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → ((𝐹𝑊) ∘ 𝑓) = (𝐹𝑓))
4544seqeq3d 13016 . . . . . . 7 ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → seq1((+g𝐺), ((𝐹𝑊) ∘ 𝑓)) = seq1((+g𝐺), (𝐹𝑓)))
4645fveq1d 6334 . . . . . 6 ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (seq1((+g𝐺), ((𝐹𝑊) ∘ 𝑓))‘(♯‘(𝐹 supp 0 ))) = (seq1((+g𝐺), (𝐹𝑓))‘(♯‘(𝐹 supp 0 ))))
47 gsumzcl.b . . . . . . 7 𝐵 = (Base‘𝐺)
48 eqid 2771 . . . . . . 7 (+g𝐺) = (+g𝐺)
49 gsumzcl.z . . . . . . 7 𝑍 = (Cntz‘𝐺)
501adantr 466 . . . . . . 7 ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → 𝐺 ∈ Mnd)
514adantr 466 . . . . . . 7 ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (𝐴𝑊) ∈ V)
5213adantr 466 . . . . . . . . 9 ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → 𝐹:𝐴𝐵)
53 fssres 6210 . . . . . . . . 9 ((𝐹:𝐴𝐵 ∧ (𝐴𝑊) ⊆ 𝐴) → (𝐹 ↾ (𝐴𝑊)):(𝐴𝑊)⟶𝐵)
5452, 27, 53sylancl 574 . . . . . . . 8 ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (𝐹 ↾ (𝐴𝑊)):(𝐴𝑊)⟶𝐵)
5518feq1d 6170 . . . . . . . . 9 (𝜑 → ((𝐹 ↾ (𝐴𝑊)):(𝐴𝑊)⟶𝐵 ↔ (𝐹𝑊):(𝐴𝑊)⟶𝐵))
5655biimpa 462 . . . . . . . 8 ((𝜑 ∧ (𝐹 ↾ (𝐴𝑊)):(𝐴𝑊)⟶𝐵) → (𝐹𝑊):(𝐴𝑊)⟶𝐵)
5754, 56syldan 579 . . . . . . 7 ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (𝐹𝑊):(𝐴𝑊)⟶𝐵)
58 gsumzcl.c . . . . . . . . 9 (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹))
59 resss 5563 . . . . . . . . . 10 (𝐹𝑊) ⊆ 𝐹
60 rnss 5492 . . . . . . . . . 10 ((𝐹𝑊) ⊆ 𝐹 → ran (𝐹𝑊) ⊆ ran 𝐹)
6159, 60ax-mp 5 . . . . . . . . 9 ran (𝐹𝑊) ⊆ ran 𝐹
6249cntzidss 17977 . . . . . . . . 9 ((ran 𝐹 ⊆ (𝑍‘ran 𝐹) ∧ ran (𝐹𝑊) ⊆ ran 𝐹) → ran (𝐹𝑊) ⊆ (𝑍‘ran (𝐹𝑊)))
6358, 61, 62sylancl 574 . . . . . . . 8 (𝜑 → ran (𝐹𝑊) ⊆ (𝑍‘ran (𝐹𝑊)))
6463adantr 466 . . . . . . 7 ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → ran (𝐹𝑊) ⊆ (𝑍‘ran (𝐹𝑊)))
65 simprl 754 . . . . . . 7 ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (♯‘(𝐹 supp 0 )) ∈ ℕ)
66 f1of1 6277 . . . . . . . . 9 (𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ) → 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1→(𝐹 supp 0 ))
6766ad2antll 708 . . . . . . . 8 ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1→(𝐹 supp 0 ))
68 suppssdm 7459 . . . . . . . . . . 11 (𝐹 supp 0 ) ⊆ dom 𝐹
69 fdm 6191 . . . . . . . . . . . 12 (𝐹:𝐴𝐵 → dom 𝐹 = 𝐴)
7013, 69syl 17 . . . . . . . . . . 11 (𝜑 → dom 𝐹 = 𝐴)
7168, 70syl5sseq 3802 . . . . . . . . . 10 (𝜑 → (𝐹 supp 0 ) ⊆ 𝐴)
7271, 40ssind 3985 . . . . . . . . 9 (𝜑 → (𝐹 supp 0 ) ⊆ (𝐴𝑊))
7372adantr 466 . . . . . . . 8 ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (𝐹 supp 0 ) ⊆ (𝐴𝑊))
74 f1ss 6246 . . . . . . . 8 ((𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1→(𝐹 supp 0 ) ∧ (𝐹 supp 0 ) ⊆ (𝐴𝑊)) → 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1→(𝐴𝑊))
7567, 73, 74syl2anc 573 . . . . . . 7 ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1→(𝐴𝑊))
76 fex 6633 . . . . . . . . . . . . 13 ((𝐹:𝐴𝐵𝐴𝑉) → 𝐹 ∈ V)
7713, 2, 76syl2anc 573 . . . . . . . . . . . 12 (𝜑𝐹 ∈ V)
78 ressuppss 7465 . . . . . . . . . . . 12 ((𝐹 ∈ V ∧ 0 ∈ V) → ((𝐹𝑊) supp 0 ) ⊆ (𝐹 supp 0 ))
7977, 21, 78sylancl 574 . . . . . . . . . . 11 (𝜑 → ((𝐹𝑊) supp 0 ) ⊆ (𝐹 supp 0 ))
80 sseq2 3776 . . . . . . . . . . 11 (ran 𝑓 = (𝐹 supp 0 ) → (((𝐹𝑊) supp 0 ) ⊆ ran 𝑓 ↔ ((𝐹𝑊) supp 0 ) ⊆ (𝐹 supp 0 )))
8179, 80syl5ibr 236 . . . . . . . . . 10 (ran 𝑓 = (𝐹 supp 0 ) → (𝜑 → ((𝐹𝑊) supp 0 ) ⊆ ran 𝑓))
8236, 37, 813syl 18 . . . . . . . . 9 (𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ) → (𝜑 → ((𝐹𝑊) supp 0 ) ⊆ ran 𝑓))
8382adantl 467 . . . . . . . 8 (((♯‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 )) → (𝜑 → ((𝐹𝑊) supp 0 ) ⊆ ran 𝑓))
8483impcom 394 . . . . . . 7 ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → ((𝐹𝑊) supp 0 ) ⊆ ran 𝑓)
85 eqid 2771 . . . . . . 7 (((𝐹𝑊) ∘ 𝑓) supp 0 ) = (((𝐹𝑊) ∘ 𝑓) supp 0 )
8647, 5, 48, 49, 50, 51, 57, 64, 65, 75, 84, 85gsumval3 18515 . . . . . 6 ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (𝐺 Σg (𝐹𝑊)) = (seq1((+g𝐺), ((𝐹𝑊) ∘ 𝑓))‘(♯‘(𝐹 supp 0 ))))
872adantr 466 . . . . . . 7 ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → 𝐴𝑉)
8858adantr 466 . . . . . . 7 ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → ran 𝐹 ⊆ (𝑍‘ran 𝐹))
8971adantr 466 . . . . . . . 8 ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (𝐹 supp 0 ) ⊆ 𝐴)
90 f1ss 6246 . . . . . . . 8 ((𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1→(𝐹 supp 0 ) ∧ (𝐹 supp 0 ) ⊆ 𝐴) → 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1𝐴)
9167, 89, 90syl2anc 573 . . . . . . 7 ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1𝐴)
9223, 39syl5sseqr 3803 . . . . . . 7 ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (𝐹 supp 0 ) ⊆ ran 𝑓)
93 eqid 2771 . . . . . . 7 ((𝐹𝑓) supp 0 ) = ((𝐹𝑓) supp 0 )
9447, 5, 48, 49, 50, 87, 52, 88, 65, 91, 92, 93gsumval3 18515 . . . . . 6 ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (𝐺 Σg 𝐹) = (seq1((+g𝐺), (𝐹𝑓))‘(♯‘(𝐹 supp 0 ))))
9546, 86, 943eqtr4d 2815 . . . . 5 ((𝜑 ∧ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (𝐺 Σg (𝐹𝑊)) = (𝐺 Σg 𝐹))
9695expr 444 . . . 4 ((𝜑 ∧ (♯‘(𝐹 supp 0 )) ∈ ℕ) → (𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ) → (𝐺 Σg (𝐹𝑊)) = (𝐺 Σg 𝐹)))
9796exlimdv 2013 . . 3 ((𝜑 ∧ (♯‘(𝐹 supp 0 )) ∈ ℕ) → (∃𝑓 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ) → (𝐺 Σg (𝐹𝑊)) = (𝐺 Σg 𝐹)))
9897expimpd 441 . 2 (𝜑 → (((♯‘(𝐹 supp 0 )) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 )) → (𝐺 Σg (𝐹𝑊)) = (𝐺 Σg 𝐹)))
99 gsumzres.w . . 3 (𝜑𝐹 finSupp 0 )
100 fsuppimp 8437 . . . 4 (𝐹 finSupp 0 → (Fun 𝐹 ∧ (𝐹 supp 0 ) ∈ Fin))
101100simprd 483 . . 3 (𝐹 finSupp 0 → (𝐹 supp 0 ) ∈ Fin)
102 fz1f1o 14649 . . 3 ((𝐹 supp 0 ) ∈ Fin → ((𝐹 supp 0 ) = ∅ ∨ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))))
10399, 101, 1023syl 18 . 2 (𝜑 → ((𝐹 supp 0 ) = ∅ ∨ ((♯‘(𝐹 supp 0 )) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))))
10435, 98, 103mpjaod 849 1 (𝜑 → (𝐺 Σg (𝐹𝑊)) = (𝐺 Σg 𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  wo 836   = wceq 1631  wex 1852  wcel 2145  Vcvv 3351  cin 3722  wss 3723  c0 4063   class class class wbr 4786  cmpt 4863  dom cdm 5249  ran crn 5250  cres 5251  ccom 5253  Fun wfun 6025   Fn wfn 6026  wf 6027  1-1wf1 6028  ontowfo 6029  1-1-ontowf1o 6030  cfv 6031  (class class class)co 6793   supp csupp 7446  Fincfn 8109   finSupp cfsupp 8431  1c1 10139  cn 11222  ...cfz 12533  seqcseq 13008  chash 13321  Basecbs 16064  +gcplusg 16149  0gc0g 16308   Σg cgsu 16309  Mndcmnd 17502  Cntzccntz 17955
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-cnex 10194  ax-resscn 10195  ax-1cn 10196  ax-icn 10197  ax-addcl 10198  ax-addrcl 10199  ax-mulcl 10200  ax-mulrcl 10201  ax-mulcom 10202  ax-addass 10203  ax-mulass 10204  ax-distr 10205  ax-i2m1 10206  ax-1ne0 10207  ax-1rid 10208  ax-rnegex 10209  ax-rrecex 10210  ax-cnre 10211  ax-pre-lttri 10212  ax-pre-lttrn 10213  ax-pre-ltadd 10214  ax-pre-mulgt0 10215
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-se 5209  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-isom 6040  df-riota 6754  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-om 7213  df-1st 7315  df-2nd 7316  df-supp 7447  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-1o 7713  df-oadd 7717  df-er 7896  df-en 8110  df-dom 8111  df-sdom 8112  df-fin 8113  df-fsupp 8432  df-oi 8571  df-card 8965  df-pnf 10278  df-mnf 10279  df-xr 10280  df-ltxr 10281  df-le 10282  df-sub 10470  df-neg 10471  df-nn 11223  df-n0 11495  df-z 11580  df-uz 11889  df-fz 12534  df-fzo 12674  df-seq 13009  df-hash 13322  df-0g 16310  df-gsum 16311  df-mgm 17450  df-sgrp 17492  df-mnd 17503  df-cntz 17957
This theorem is referenced by:  gsumres  18521  gsumzsplit  18534  gsumpt  18568  dmdprdsplitlem  18644  dpjidcl  18665  mplcoe5  19683
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