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Theorem gsumzres 18238
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 4766 . . . . . . . 8 (𝐴𝑉 → (𝐴𝑊) ∈ V)
42, 3syl 17 . . . . . . 7 (𝜑 → (𝐴𝑊) ∈ V)
5 gsumzcl.0 . . . . . . . 8 0 = (0g𝐺)
65gsumz 17302 . . . . . . 7 ((𝐺 ∈ Mnd ∧ (𝐴𝑊) ∈ V) → (𝐺 Σg (𝑘 ∈ (𝐴𝑊) ↦ 0 )) = 0 )
71, 4, 6syl2anc 692 . . . . . 6 (𝜑 → (𝐺 Σg (𝑘 ∈ (𝐴𝑊) ↦ 0 )) = 0 )
85gsumz 17302 . . . . . . 7 ((𝐺 ∈ Mnd ∧ 𝐴𝑉) → (𝐺 Σg (𝑘𝐴0 )) = 0 )
91, 2, 8syl2anc 692 . . . . . 6 (𝜑 → (𝐺 Σg (𝑘𝐴0 )) = 0 )
107, 9eqtr4d 2658 . . . . 5 (𝜑 → (𝐺 Σg (𝑘 ∈ (𝐴𝑊) ↦ 0 )) = (𝐺 Σg (𝑘𝐴0 )))
1110adantr 481 . . . 4 ((𝜑 ∧ (𝐹 supp 0 ) = ∅) → (𝐺 Σg (𝑘 ∈ (𝐴𝑊) ↦ 0 )) = (𝐺 Σg (𝑘𝐴0 )))
12 resres 5373 . . . . . . . 8 ((𝐹𝐴) ↾ 𝑊) = (𝐹 ↾ (𝐴𝑊))
13 gsumzcl.f . . . . . . . . . 10 (𝜑𝐹:𝐴𝐵)
14 ffn 6007 . . . . . . . . . 10 (𝐹:𝐴𝐵𝐹 Fn 𝐴)
15 fnresdm 5963 . . . . . . . . . 10 (𝐹 Fn 𝐴 → (𝐹𝐴) = 𝐹)
1613, 14, 153syl 18 . . . . . . . . 9 (𝜑 → (𝐹𝐴) = 𝐹)
1716reseq1d 5360 . . . . . . . 8 (𝜑 → ((𝐹𝐴) ↾ 𝑊) = (𝐹𝑊))
1812, 17syl5eqr 2669 . . . . . . 7 (𝜑 → (𝐹 ↾ (𝐴𝑊)) = (𝐹𝑊))
1918adantr 481 . . . . . 6 ((𝜑 ∧ (𝐹 supp 0 ) = ∅) → (𝐹 ↾ (𝐴𝑊)) = (𝐹𝑊))
20 fvex 6163 . . . . . . . . . . 11 (0g𝐺) ∈ V
215, 20eqeltri 2694 . . . . . . . . . 10 0 ∈ V
2221a1i 11 . . . . . . . . 9 (𝜑0 ∈ V)
23 ssid 3608 . . . . . . . . . 10 (𝐹 supp 0 ) ⊆ (𝐹 supp 0 )
2423a1i 11 . . . . . . . . 9 (𝜑 → (𝐹 supp 0 ) ⊆ (𝐹 supp 0 ))
2513, 2, 22, 24gsumcllem 18237 . . . . . . . 8 ((𝜑 ∧ (𝐹 supp 0 ) = ∅) → 𝐹 = (𝑘𝐴0 ))
2625reseq1d 5360 . . . . . . 7 ((𝜑 ∧ (𝐹 supp 0 ) = ∅) → (𝐹 ↾ (𝐴𝑊)) = ((𝑘𝐴0 ) ↾ (𝐴𝑊)))
27 inss1 3816 . . . . . . . 8 (𝐴𝑊) ⊆ 𝐴
28 resmpt 5413 . . . . . . . 8 ((𝐴𝑊) ⊆ 𝐴 → ((𝑘𝐴0 ) ↾ (𝐴𝑊)) = (𝑘 ∈ (𝐴𝑊) ↦ 0 ))
2927, 28ax-mp 5 . . . . . . 7 ((𝑘𝐴0 ) ↾ (𝐴𝑊)) = (𝑘 ∈ (𝐴𝑊) ↦ 0 )
3026, 29syl6eq 2671 . . . . . 6 ((𝜑 ∧ (𝐹 supp 0 ) = ∅) → (𝐹 ↾ (𝐴𝑊)) = (𝑘 ∈ (𝐴𝑊) ↦ 0 ))
3119, 30eqtr3d 2657 . . . . 5 ((𝜑 ∧ (𝐹 supp 0 ) = ∅) → (𝐹𝑊) = (𝑘 ∈ (𝐴𝑊) ↦ 0 ))
3231oveq2d 6626 . . . 4 ((𝜑 ∧ (𝐹 supp 0 ) = ∅) → (𝐺 Σg (𝐹𝑊)) = (𝐺 Σg (𝑘 ∈ (𝐴𝑊) ↦ 0 )))
3325oveq2d 6626 . . . 4 ((𝜑 ∧ (𝐹 supp 0 ) = ∅) → (𝐺 Σg 𝐹) = (𝐺 Σg (𝑘𝐴0 )))
3411, 32, 333eqtr4d 2665 . . 3 ((𝜑 ∧ (𝐹 supp 0 ) = ∅) → (𝐺 Σg (𝐹𝑊)) = (𝐺 Σg 𝐹))
3534ex 450 . 2 (𝜑 → ((𝐹 supp 0 ) = ∅ → (𝐺 Σg (𝐹𝑊)) = (𝐺 Σg 𝐹)))
36 f1ofo 6106 . . . . . . . . . . . 12 (𝑓:(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ) → 𝑓:(1...(#‘(𝐹 supp 0 )))–onto→(𝐹 supp 0 ))
37 forn 6080 . . . . . . . . . . . 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 764 . . . . . . . . . 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 481 . . . . . . . . . 10 ((𝜑 ∧ ((#‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (𝐹 supp 0 ) ⊆ 𝑊)
4239, 41eqsstrd 3623 . . . . . . . . 9 ((𝜑 ∧ ((#‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → ran 𝑓𝑊)
43 cores 5602 . . . . . . . . 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 12756 . . . . . . 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 6155 . . . . . 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 2621 . . . . . . 7 (+g𝐺) = (+g𝐺)
49 gsumzcl.z . . . . . . 7 𝑍 = (Cntz‘𝐺)
501adantr 481 . . . . . . 7 ((𝜑 ∧ ((#‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → 𝐺 ∈ Mnd)
514adantr 481 . . . . . . 7 ((𝜑 ∧ ((#‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (𝐴𝑊) ∈ V)
5213adantr 481 . . . . . . . . 9 ((𝜑 ∧ ((#‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → 𝐹:𝐴𝐵)
53 fssres 6032 . . . . . . . . 9 ((𝐹:𝐴𝐵 ∧ (𝐴𝑊) ⊆ 𝐴) → (𝐹 ↾ (𝐴𝑊)):(𝐴𝑊)⟶𝐵)
5452, 27, 53sylancl 693 . . . . . . . 8 ((𝜑 ∧ ((#‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (𝐹 ↾ (𝐴𝑊)):(𝐴𝑊)⟶𝐵)
5518feq1d 5992 . . . . . . . . 9 (𝜑 → ((𝐹 ↾ (𝐴𝑊)):(𝐴𝑊)⟶𝐵 ↔ (𝐹𝑊):(𝐴𝑊)⟶𝐵))
5655biimpa 501 . . . . . . . 8 ((𝜑 ∧ (𝐹 ↾ (𝐴𝑊)):(𝐴𝑊)⟶𝐵) → (𝐹𝑊):(𝐴𝑊)⟶𝐵)
5754, 56syldan 487 . . . . . . 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 5386 . . . . . . . . . 10 (𝐹𝑊) ⊆ 𝐹
60 rnss 5319 . . . . . . . . . 10 ((𝐹𝑊) ⊆ 𝐹 → ran (𝐹𝑊) ⊆ ran 𝐹)
6159, 60ax-mp 5 . . . . . . . . 9 ran (𝐹𝑊) ⊆ ran 𝐹
6249cntzidss 17698 . . . . . . . . 9 ((ran 𝐹 ⊆ (𝑍‘ran 𝐹) ∧ ran (𝐹𝑊) ⊆ ran 𝐹) → ran (𝐹𝑊) ⊆ (𝑍‘ran (𝐹𝑊)))
6358, 61, 62sylancl 693 . . . . . . . 8 (𝜑 → ran (𝐹𝑊) ⊆ (𝑍‘ran (𝐹𝑊)))
6463adantr 481 . . . . . . 7 ((𝜑 ∧ ((#‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → ran (𝐹𝑊) ⊆ (𝑍‘ran (𝐹𝑊)))
65 simprl 793 . . . . . . 7 ((𝜑 ∧ ((#‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (#‘(𝐹 supp 0 )) ∈ ℕ)
66 f1of1 6098 . . . . . . . . 9 (𝑓:(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ) → 𝑓:(1...(#‘(𝐹 supp 0 )))–1-1→(𝐹 supp 0 ))
6766ad2antll 764 . . . . . . . 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 7260 . . . . . . . . . . 11 (𝐹 supp 0 ) ⊆ dom 𝐹
69 fdm 6013 . . . . . . . . . . . 12 (𝐹:𝐴𝐵 → dom 𝐹 = 𝐴)
7013, 69syl 17 . . . . . . . . . . 11 (𝜑 → dom 𝐹 = 𝐴)
7168, 70syl5sseq 3637 . . . . . . . . . 10 (𝜑 → (𝐹 supp 0 ) ⊆ 𝐴)
7271, 40ssind 3820 . . . . . . . . 9 (𝜑 → (𝐹 supp 0 ) ⊆ (𝐴𝑊))
7372adantr 481 . . . . . . . 8 ((𝜑 ∧ ((#‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (𝐹 supp 0 ) ⊆ (𝐴𝑊))
74 f1ss 6068 . . . . . . . 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 692 . . . . . . 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 6450 . . . . . . . . . . . . 13 ((𝐹:𝐴𝐵𝐴𝑉) → 𝐹 ∈ V)
7713, 2, 76syl2anc 692 . . . . . . . . . . . 12 (𝜑𝐹 ∈ V)
78 ressuppss 7266 . . . . . . . . . . . 12 ((𝐹 ∈ V ∧ 0 ∈ V) → ((𝐹𝑊) supp 0 ) ⊆ (𝐹 supp 0 ))
7977, 21, 78sylancl 693 . . . . . . . . . . 11 (𝜑 → ((𝐹𝑊) supp 0 ) ⊆ (𝐹 supp 0 ))
80 sseq2 3611 . . . . . . . . . . 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 482 . . . . . . . 8 (((#‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 )) → (𝜑 → ((𝐹𝑊) supp 0 ) ⊆ ran 𝑓))
8483impcom 446 . . . . . . 7 ((𝜑 ∧ ((#‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → ((𝐹𝑊) supp 0 ) ⊆ ran 𝑓)
85 eqid 2621 . . . . . . 7 (((𝐹𝑊) ∘ 𝑓) supp 0 ) = (((𝐹𝑊) ∘ 𝑓) supp 0 )
8647, 5, 48, 49, 50, 51, 57, 64, 65, 75, 84, 85gsumval3 18236 . . . . . 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 481 . . . . . . 7 ((𝜑 ∧ ((#‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → 𝐴𝑉)
8858adantr 481 . . . . . . 7 ((𝜑 ∧ ((#‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → ran 𝐹 ⊆ (𝑍‘ran 𝐹))
8971adantr 481 . . . . . . . 8 ((𝜑 ∧ ((#‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (𝐹 supp 0 ) ⊆ 𝐴)
90 f1ss 6068 . . . . . . . 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 692 . . . . . . 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 3638 . . . . . . 7 ((𝜑 ∧ ((#‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (𝐹 supp 0 ) ⊆ ran 𝑓)
93 eqid 2621 . . . . . . 7 ((𝐹𝑓) supp 0 ) = ((𝐹𝑓) supp 0 )
9447, 5, 48, 49, 50, 87, 52, 88, 65, 91, 92, 93gsumval3 18236 . . . . . 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 2665 . . . . 5 ((𝜑 ∧ ((#‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (𝐺 Σg (𝐹𝑊)) = (𝐺 Σg 𝐹))
9695expr 642 . . . 4 ((𝜑 ∧ (#‘(𝐹 supp 0 )) ∈ ℕ) → (𝑓:(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ) → (𝐺 Σg (𝐹𝑊)) = (𝐺 Σg 𝐹)))
9796exlimdv 1858 . . 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 628 . 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 8232 . . . 4 (𝐹 finSupp 0 → (Fun 𝐹 ∧ (𝐹 supp 0 ) ∈ Fin))
101100simprd 479 . . 3 (𝐹 finSupp 0 → (𝐹 supp 0 ) ∈ Fin)
102 fz1f1o 14381 . . 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 396 1 (𝜑 → (𝐺 Σg (𝐹𝑊)) = (𝐺 Σg 𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 383  wa 384   = wceq 1480  wex 1701  wcel 1987  Vcvv 3189  cin 3558  wss 3559  c0 3896   class class class wbr 4618  cmpt 4678  dom cdm 5079  ran crn 5080  cres 5081  ccom 5083  Fun wfun 5846   Fn wfn 5847  wf 5848  1-1wf1 5849  ontowfo 5850  1-1-ontowf1o 5851  cfv 5852  (class class class)co 6610   supp csupp 7247  Fincfn 7906   finSupp cfsupp 8226  1c1 9888  cn 10971  ...cfz 12275  seqcseq 12748  #chash 13064  Basecbs 15788  +gcplusg 15869  0gc0g 16028   Σg cgsu 16029  Mndcmnd 17222  Cntzccntz 17676
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-cnex 9943  ax-resscn 9944  ax-1cn 9945  ax-icn 9946  ax-addcl 9947  ax-addrcl 9948  ax-mulcl 9949  ax-mulrcl 9950  ax-mulcom 9951  ax-addass 9952  ax-mulass 9953  ax-distr 9954  ax-i2m1 9955  ax-1ne0 9956  ax-1rid 9957  ax-rnegex 9958  ax-rrecex 9959  ax-cnre 9960  ax-pre-lttri 9961  ax-pre-lttrn 9962  ax-pre-ltadd 9963  ax-pre-mulgt0 9964
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-se 5039  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-isom 5861  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-om 7020  df-1st 7120  df-2nd 7121  df-supp 7248  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-1o 7512  df-oadd 7516  df-er 7694  df-en 7907  df-dom 7908  df-sdom 7909  df-fin 7910  df-fsupp 8227  df-oi 8366  df-card 8716  df-pnf 10027  df-mnf 10028  df-xr 10029  df-ltxr 10030  df-le 10031  df-sub 10219  df-neg 10220  df-nn 10972  df-n0 11244  df-z 11329  df-uz 11639  df-fz 12276  df-fzo 12414  df-seq 12749  df-hash 13065  df-0g 16030  df-gsum 16031  df-mgm 17170  df-sgrp 17212  df-mnd 17223  df-cntz 17678
This theorem is referenced by:  gsumres  18242  gsumzsplit  18255  gsumpt  18289  dmdprdsplitlem  18364  dpjidcl  18385  mplcoe5  19396
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