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Theorem gsumzres 18133
 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 4729 . . . . . . . 8 (𝐴𝑉 → (𝐴𝑊) ∈ V)
42, 3syl 17 . . . . . . 7 (𝜑 → (𝐴𝑊) ∈ V)
5 gsumzcl.0 . . . . . . . 8 0 = (0g𝐺)
65gsumz 17197 . . . . . . 7 ((𝐺 ∈ Mnd ∧ (𝐴𝑊) ∈ V) → (𝐺 Σg (𝑘 ∈ (𝐴𝑊) ↦ 0 )) = 0 )
71, 4, 6syl2anc 691 . . . . . 6 (𝜑 → (𝐺 Σg (𝑘 ∈ (𝐴𝑊) ↦ 0 )) = 0 )
85gsumz 17197 . . . . . . 7 ((𝐺 ∈ Mnd ∧ 𝐴𝑉) → (𝐺 Σg (𝑘𝐴0 )) = 0 )
91, 2, 8syl2anc 691 . . . . . 6 (𝜑 → (𝐺 Σg (𝑘𝐴0 )) = 0 )
107, 9eqtr4d 2647 . . . . 5 (𝜑 → (𝐺 Σg (𝑘 ∈ (𝐴𝑊) ↦ 0 )) = (𝐺 Σg (𝑘𝐴0 )))
1110adantr 480 . . . 4 ((𝜑 ∧ (𝐹 supp 0 ) = ∅) → (𝐺 Σg (𝑘 ∈ (𝐴𝑊) ↦ 0 )) = (𝐺 Σg (𝑘𝐴0 )))
12 resres 5329 . . . . . . . 8 ((𝐹𝐴) ↾ 𝑊) = (𝐹 ↾ (𝐴𝑊))
13 gsumzcl.f . . . . . . . . . 10 (𝜑𝐹:𝐴𝐵)
14 ffn 5958 . . . . . . . . . 10 (𝐹:𝐴𝐵𝐹 Fn 𝐴)
15 fnresdm 5914 . . . . . . . . . 10 (𝐹 Fn 𝐴 → (𝐹𝐴) = 𝐹)
1613, 14, 153syl 18 . . . . . . . . 9 (𝜑 → (𝐹𝐴) = 𝐹)
1716reseq1d 5316 . . . . . . . 8 (𝜑 → ((𝐹𝐴) ↾ 𝑊) = (𝐹𝑊))
1812, 17syl5eqr 2658 . . . . . . 7 (𝜑 → (𝐹 ↾ (𝐴𝑊)) = (𝐹𝑊))
1918adantr 480 . . . . . 6 ((𝜑 ∧ (𝐹 supp 0 ) = ∅) → (𝐹 ↾ (𝐴𝑊)) = (𝐹𝑊))
20 fvex 6113 . . . . . . . . . . 11 (0g𝐺) ∈ V
215, 20eqeltri 2684 . . . . . . . . . 10 0 ∈ V
2221a1i 11 . . . . . . . . 9 (𝜑0 ∈ V)
23 ssid 3587 . . . . . . . . . 10 (𝐹 supp 0 ) ⊆ (𝐹 supp 0 )
2423a1i 11 . . . . . . . . 9 (𝜑 → (𝐹 supp 0 ) ⊆ (𝐹 supp 0 ))
2513, 2, 22, 24gsumcllem 18132 . . . . . . . 8 ((𝜑 ∧ (𝐹 supp 0 ) = ∅) → 𝐹 = (𝑘𝐴0 ))
2625reseq1d 5316 . . . . . . 7 ((𝜑 ∧ (𝐹 supp 0 ) = ∅) → (𝐹 ↾ (𝐴𝑊)) = ((𝑘𝐴0 ) ↾ (𝐴𝑊)))
27 inss1 3795 . . . . . . . 8 (𝐴𝑊) ⊆ 𝐴
28 resmpt 5369 . . . . . . . 8 ((𝐴𝑊) ⊆ 𝐴 → ((𝑘𝐴0 ) ↾ (𝐴𝑊)) = (𝑘 ∈ (𝐴𝑊) ↦ 0 ))
2927, 28ax-mp 5 . . . . . . 7 ((𝑘𝐴0 ) ↾ (𝐴𝑊)) = (𝑘 ∈ (𝐴𝑊) ↦ 0 )
3026, 29syl6eq 2660 . . . . . 6 ((𝜑 ∧ (𝐹 supp 0 ) = ∅) → (𝐹 ↾ (𝐴𝑊)) = (𝑘 ∈ (𝐴𝑊) ↦ 0 ))
3119, 30eqtr3d 2646 . . . . 5 ((𝜑 ∧ (𝐹 supp 0 ) = ∅) → (𝐹𝑊) = (𝑘 ∈ (𝐴𝑊) ↦ 0 ))
3231oveq2d 6565 . . . 4 ((𝜑 ∧ (𝐹 supp 0 ) = ∅) → (𝐺 Σg (𝐹𝑊)) = (𝐺 Σg (𝑘 ∈ (𝐴𝑊) ↦ 0 )))
3325oveq2d 6565 . . . 4 ((𝜑 ∧ (𝐹 supp 0 ) = ∅) → (𝐺 Σg 𝐹) = (𝐺 Σg (𝑘𝐴0 )))
3411, 32, 333eqtr4d 2654 . . 3 ((𝜑 ∧ (𝐹 supp 0 ) = ∅) → (𝐺 Σg (𝐹𝑊)) = (𝐺 Σg 𝐹))
3534ex 449 . 2 (𝜑 → ((𝐹 supp 0 ) = ∅ → (𝐺 Σg (𝐹𝑊)) = (𝐺 Σg 𝐹)))
36 f1ofo 6057 . . . . . . . . . . . 12 (𝑓:(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ) → 𝑓:(1...(#‘(𝐹 supp 0 )))–onto→(𝐹 supp 0 ))
37 forn 6031 . . . . . . . . . . . 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 761 . . . . . . . . . 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 480 . . . . . . . . . 10 ((𝜑 ∧ ((#‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (𝐹 supp 0 ) ⊆ 𝑊)
4239, 41eqsstrd 3602 . . . . . . . . 9 ((𝜑 ∧ ((#‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → ran 𝑓𝑊)
43 cores 5555 . . . . . . . . 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 12671 . . . . . . 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 6105 . . . . . 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 2610 . . . . . . 7 (+g𝐺) = (+g𝐺)
49 gsumzcl.z . . . . . . 7 𝑍 = (Cntz‘𝐺)
501adantr 480 . . . . . . 7 ((𝜑 ∧ ((#‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → 𝐺 ∈ Mnd)
514adantr 480 . . . . . . 7 ((𝜑 ∧ ((#‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (𝐴𝑊) ∈ V)
5213adantr 480 . . . . . . . . 9 ((𝜑 ∧ ((#‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → 𝐹:𝐴𝐵)
53 fssres 5983 . . . . . . . . 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 5943 . . . . . . . . 9 (𝜑 → ((𝐹 ↾ (𝐴𝑊)):(𝐴𝑊)⟶𝐵 ↔ (𝐹𝑊):(𝐴𝑊)⟶𝐵))
5655biimpa 500 . . . . . . . 8 ((𝜑 ∧ (𝐹 ↾ (𝐴𝑊)):(𝐴𝑊)⟶𝐵) → (𝐹𝑊):(𝐴𝑊)⟶𝐵)
5754, 56syldan 486 . . . . . . 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 5342 . . . . . . . . . 10 (𝐹𝑊) ⊆ 𝐹
60 rnss 5275 . . . . . . . . . 10 ((𝐹𝑊) ⊆ 𝐹 → ran (𝐹𝑊) ⊆ ran 𝐹)
6159, 60ax-mp 5 . . . . . . . . 9 ran (𝐹𝑊) ⊆ ran 𝐹
6249cntzidss 17593 . . . . . . . . 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 480 . . . . . . 7 ((𝜑 ∧ ((#‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → ran (𝐹𝑊) ⊆ (𝑍‘ran (𝐹𝑊)))
65 simprl 790 . . . . . . 7 ((𝜑 ∧ ((#‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (#‘(𝐹 supp 0 )) ∈ ℕ)
66 f1of1 6049 . . . . . . . . 9 (𝑓:(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ) → 𝑓:(1...(#‘(𝐹 supp 0 )))–1-1→(𝐹 supp 0 ))
6766ad2antll 761 . . . . . . . 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 7195 . . . . . . . . . . 11 (𝐹 supp 0 ) ⊆ dom 𝐹
69 fdm 5964 . . . . . . . . . . . 12 (𝐹:𝐴𝐵 → dom 𝐹 = 𝐴)
7013, 69syl 17 . . . . . . . . . . 11 (𝜑 → dom 𝐹 = 𝐴)
7168, 70syl5sseq 3616 . . . . . . . . . 10 (𝜑 → (𝐹 supp 0 ) ⊆ 𝐴)
7271, 40ssind 3799 . . . . . . . . 9 (𝜑 → (𝐹 supp 0 ) ⊆ (𝐴𝑊))
7372adantr 480 . . . . . . . 8 ((𝜑 ∧ ((#‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (𝐹 supp 0 ) ⊆ (𝐴𝑊))
74 f1ss 6019 . . . . . . . 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 691 . . . . . . 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 6394 . . . . . . . . . . . . 13 ((𝐹:𝐴𝐵𝐴𝑉) → 𝐹 ∈ V)
7713, 2, 76syl2anc 691 . . . . . . . . . . . 12 (𝜑𝐹 ∈ V)
78 ressuppss 7201 . . . . . . . . . . . 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 3590 . . . . . . . . . . 11 (ran 𝑓 = (𝐹 supp 0 ) → (((𝐹𝑊) supp 0 ) ⊆ ran 𝑓 ↔ ((𝐹𝑊) supp 0 ) ⊆ (𝐹 supp 0 )))
8179, 80syl5ibr 235 . . . . . . . . . 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 481 . . . . . . . 8 (((#‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 )) → (𝜑 → ((𝐹𝑊) supp 0 ) ⊆ ran 𝑓))
8483impcom 445 . . . . . . 7 ((𝜑 ∧ ((#‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → ((𝐹𝑊) supp 0 ) ⊆ ran 𝑓)
85 eqid 2610 . . . . . . 7 (((𝐹𝑊) ∘ 𝑓) supp 0 ) = (((𝐹𝑊) ∘ 𝑓) supp 0 )
8647, 5, 48, 49, 50, 51, 57, 64, 65, 75, 84, 85gsumval3 18131 . . . . . 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 480 . . . . . . 7 ((𝜑 ∧ ((#‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → 𝐴𝑉)
8858adantr 480 . . . . . . 7 ((𝜑 ∧ ((#‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → ran 𝐹 ⊆ (𝑍‘ran 𝐹))
8971adantr 480 . . . . . . . 8 ((𝜑 ∧ ((#‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (𝐹 supp 0 ) ⊆ 𝐴)
90 f1ss 6019 . . . . . . . 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 691 . . . . . . 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 3617 . . . . . . 7 ((𝜑 ∧ ((#‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (𝐹 supp 0 ) ⊆ ran 𝑓)
93 eqid 2610 . . . . . . 7 ((𝐹𝑓) supp 0 ) = ((𝐹𝑓) supp 0 )
9447, 5, 48, 49, 50, 87, 52, 88, 65, 91, 92, 93gsumval3 18131 . . . . . 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 2654 . . . . 5 ((𝜑 ∧ ((#‘(𝐹 supp 0 )) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ))) → (𝐺 Σg (𝐹𝑊)) = (𝐺 Σg 𝐹))
9695expr 641 . . . 4 ((𝜑 ∧ (#‘(𝐹 supp 0 )) ∈ ℕ) → (𝑓:(1...(#‘(𝐹 supp 0 )))–1-1-onto→(𝐹 supp 0 ) → (𝐺 Σg (𝐹𝑊)) = (𝐺 Σg 𝐹)))
9796exlimdv 1848 . . 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 627 . 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 8164 . . . 4 (𝐹 finSupp 0 → (Fun 𝐹 ∧ (𝐹 supp 0 ) ∈ Fin))
101100simprd 478 . . 3 (𝐹 finSupp 0 → (𝐹 supp 0 ) ∈ Fin)
102 fz1f1o 14288 . . 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 395 1 (𝜑 → (𝐺 Σg (𝐹𝑊)) = (𝐺 Σg 𝐹))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∨ wo 382   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977  Vcvv 3173   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874   class class class wbr 4583   ↦ cmpt 4643  dom cdm 5038  ran crn 5039   ↾ cres 5040   ∘ ccom 5042  Fun wfun 5798   Fn wfn 5799  ⟶wf 5800  –1-1→wf1 5801  –onto→wfo 5802  –1-1-onto→wf1o 5803  ‘cfv 5804  (class class class)co 6549   supp csupp 7182  Fincfn 7841   finSupp cfsupp 8158  1c1 9816  ℕcn 10897  ...cfz 12197  seqcseq 12663  #chash 12979  Basecbs 15695  +gcplusg 15768  0gc0g 15923   Σg cgsu 15924  Mndcmnd 17117  Cntzccntz 17571 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-supp 7183  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fsupp 8159  df-oi 8298  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-seq 12664  df-hash 12980  df-0g 15925  df-gsum 15926  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-cntz 17573 This theorem is referenced by:  gsumres  18137  gsumzsplit  18150  gsumpt  18184  dmdprdsplitlem  18259  dpjidcl  18280  mplcoe5  19289
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