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Theorem gsumconst 18157
 Description: Sum of a constant series. (Contributed by Mario Carneiro, 19-Dec-2014.) (Revised by Mario Carneiro, 24-Apr-2016.)
Hypotheses
Ref Expression
gsumconst.b 𝐵 = (Base‘𝐺)
gsumconst.m · = (.g𝐺)
Assertion
Ref Expression
gsumconst ((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) → (𝐺 Σg (𝑘𝐴𝑋)) = ((#‘𝐴) · 𝑋))
Distinct variable groups:   𝐴,𝑘   𝐵,𝑘   𝑘,𝐺   𝑘,𝑋
Allowed substitution hint:   · (𝑘)

Proof of Theorem gsumconst
Dummy variables 𝑓 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl3 1059 . . . . 5 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ 𝐴 = ∅) → 𝑋𝐵)
2 gsumconst.b . . . . . 6 𝐵 = (Base‘𝐺)
3 eqid 2610 . . . . . 6 (0g𝐺) = (0g𝐺)
4 gsumconst.m . . . . . 6 · = (.g𝐺)
52, 3, 4mulg0 17369 . . . . 5 (𝑋𝐵 → (0 · 𝑋) = (0g𝐺))
61, 5syl 17 . . . 4 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ 𝐴 = ∅) → (0 · 𝑋) = (0g𝐺))
7 fveq2 6103 . . . . . . 7 (𝐴 = ∅ → (#‘𝐴) = (#‘∅))
87adantl 481 . . . . . 6 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ 𝐴 = ∅) → (#‘𝐴) = (#‘∅))
9 hash0 13019 . . . . . 6 (#‘∅) = 0
108, 9syl6eq 2660 . . . . 5 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ 𝐴 = ∅) → (#‘𝐴) = 0)
1110oveq1d 6564 . . . 4 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ 𝐴 = ∅) → ((#‘𝐴) · 𝑋) = (0 · 𝑋))
12 mpteq1 4665 . . . . . . . 8 (𝐴 = ∅ → (𝑘𝐴𝑋) = (𝑘 ∈ ∅ ↦ 𝑋))
1312adantl 481 . . . . . . 7 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ 𝐴 = ∅) → (𝑘𝐴𝑋) = (𝑘 ∈ ∅ ↦ 𝑋))
14 mpt0 5934 . . . . . . 7 (𝑘 ∈ ∅ ↦ 𝑋) = ∅
1513, 14syl6eq 2660 . . . . . 6 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ 𝐴 = ∅) → (𝑘𝐴𝑋) = ∅)
1615oveq2d 6565 . . . . 5 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ 𝐴 = ∅) → (𝐺 Σg (𝑘𝐴𝑋)) = (𝐺 Σg ∅))
173gsum0 17101 . . . . 5 (𝐺 Σg ∅) = (0g𝐺)
1816, 17syl6eq 2660 . . . 4 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ 𝐴 = ∅) → (𝐺 Σg (𝑘𝐴𝑋)) = (0g𝐺))
196, 11, 183eqtr4rd 2655 . . 3 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ 𝐴 = ∅) → (𝐺 Σg (𝑘𝐴𝑋)) = ((#‘𝐴) · 𝑋))
2019ex 449 . 2 ((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) → (𝐴 = ∅ → (𝐺 Σg (𝑘𝐴𝑋)) = ((#‘𝐴) · 𝑋)))
21 simprl 790 . . . . . . . 8 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) → (#‘𝐴) ∈ ℕ)
22 nnuz 11599 . . . . . . . 8 ℕ = (ℤ‘1)
2321, 22syl6eleq 2698 . . . . . . 7 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) → (#‘𝐴) ∈ (ℤ‘1))
24 simpr 476 . . . . . . . . 9 ((((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ (1...(#‘𝐴))) → 𝑥 ∈ (1...(#‘𝐴)))
25 simpl3 1059 . . . . . . . . . 10 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) → 𝑋𝐵)
2625adantr 480 . . . . . . . . 9 ((((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ (1...(#‘𝐴))) → 𝑋𝐵)
27 eqid 2610 . . . . . . . . . 10 (𝑥 ∈ (1...(#‘𝐴)) ↦ 𝑋) = (𝑥 ∈ (1...(#‘𝐴)) ↦ 𝑋)
2827fvmpt2 6200 . . . . . . . . 9 ((𝑥 ∈ (1...(#‘𝐴)) ∧ 𝑋𝐵) → ((𝑥 ∈ (1...(#‘𝐴)) ↦ 𝑋)‘𝑥) = 𝑋)
2924, 26, 28syl2anc 691 . . . . . . . 8 ((((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ (1...(#‘𝐴))) → ((𝑥 ∈ (1...(#‘𝐴)) ↦ 𝑋)‘𝑥) = 𝑋)
30 f1of 6050 . . . . . . . . . . . . 13 (𝑓:(1...(#‘𝐴))–1-1-onto𝐴𝑓:(1...(#‘𝐴))⟶𝐴)
3130ad2antll 761 . . . . . . . . . . . 12 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) → 𝑓:(1...(#‘𝐴))⟶𝐴)
3231ffvelrnda 6267 . . . . . . . . . . 11 ((((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ (1...(#‘𝐴))) → (𝑓𝑥) ∈ 𝐴)
3331feqmptd 6159 . . . . . . . . . . 11 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) → 𝑓 = (𝑥 ∈ (1...(#‘𝐴)) ↦ (𝑓𝑥)))
34 eqidd 2611 . . . . . . . . . . 11 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) → (𝑘𝐴𝑋) = (𝑘𝐴𝑋))
35 eqidd 2611 . . . . . . . . . . 11 (𝑘 = (𝑓𝑥) → 𝑋 = 𝑋)
3632, 33, 34, 35fmptco 6303 . . . . . . . . . 10 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) → ((𝑘𝐴𝑋) ∘ 𝑓) = (𝑥 ∈ (1...(#‘𝐴)) ↦ 𝑋))
3736fveq1d 6105 . . . . . . . . 9 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) → (((𝑘𝐴𝑋) ∘ 𝑓)‘𝑥) = ((𝑥 ∈ (1...(#‘𝐴)) ↦ 𝑋)‘𝑥))
3837adantr 480 . . . . . . . 8 ((((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ (1...(#‘𝐴))) → (((𝑘𝐴𝑋) ∘ 𝑓)‘𝑥) = ((𝑥 ∈ (1...(#‘𝐴)) ↦ 𝑋)‘𝑥))
39 elfznn 12241 . . . . . . . . 9 (𝑥 ∈ (1...(#‘𝐴)) → 𝑥 ∈ ℕ)
40 fvconst2g 6372 . . . . . . . . 9 ((𝑋𝐵𝑥 ∈ ℕ) → ((ℕ × {𝑋})‘𝑥) = 𝑋)
4125, 39, 40syl2an 493 . . . . . . . 8 ((((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ (1...(#‘𝐴))) → ((ℕ × {𝑋})‘𝑥) = 𝑋)
4229, 38, 413eqtr4d 2654 . . . . . . 7 ((((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ (1...(#‘𝐴))) → (((𝑘𝐴𝑋) ∘ 𝑓)‘𝑥) = ((ℕ × {𝑋})‘𝑥))
4323, 42seqfveq 12687 . . . . . 6 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) → (seq1((+g𝐺), ((𝑘𝐴𝑋) ∘ 𝑓))‘(#‘𝐴)) = (seq1((+g𝐺), (ℕ × {𝑋}))‘(#‘𝐴)))
44 eqid 2610 . . . . . . 7 (+g𝐺) = (+g𝐺)
45 eqid 2610 . . . . . . 7 (Cntz‘𝐺) = (Cntz‘𝐺)
46 simpl1 1057 . . . . . . 7 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) → 𝐺 ∈ Mnd)
47 simpl2 1058 . . . . . . 7 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) → 𝐴 ∈ Fin)
4825adantr 480 . . . . . . . 8 ((((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) ∧ 𝑘𝐴) → 𝑋𝐵)
49 eqid 2610 . . . . . . . 8 (𝑘𝐴𝑋) = (𝑘𝐴𝑋)
5048, 49fmptd 6292 . . . . . . 7 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) → (𝑘𝐴𝑋):𝐴𝐵)
51 eqidd 2611 . . . . . . . . . 10 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) → (𝑋(+g𝐺)𝑋) = (𝑋(+g𝐺)𝑋))
522, 44, 45elcntzsn 17581 . . . . . . . . . . 11 (𝑋𝐵 → (𝑋 ∈ ((Cntz‘𝐺)‘{𝑋}) ↔ (𝑋𝐵 ∧ (𝑋(+g𝐺)𝑋) = (𝑋(+g𝐺)𝑋))))
5325, 52syl 17 . . . . . . . . . 10 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) → (𝑋 ∈ ((Cntz‘𝐺)‘{𝑋}) ↔ (𝑋𝐵 ∧ (𝑋(+g𝐺)𝑋) = (𝑋(+g𝐺)𝑋))))
5425, 51, 53mpbir2and 959 . . . . . . . . 9 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) → 𝑋 ∈ ((Cntz‘𝐺)‘{𝑋}))
5554snssd 4281 . . . . . . . 8 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) → {𝑋} ⊆ ((Cntz‘𝐺)‘{𝑋}))
56 snidg 4153 . . . . . . . . . . . 12 (𝑋𝐵𝑋 ∈ {𝑋})
5725, 56syl 17 . . . . . . . . . . 11 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) → 𝑋 ∈ {𝑋})
5857adantr 480 . . . . . . . . . 10 ((((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) ∧ 𝑘𝐴) → 𝑋 ∈ {𝑋})
5958, 49fmptd 6292 . . . . . . . . 9 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) → (𝑘𝐴𝑋):𝐴⟶{𝑋})
60 frn 5966 . . . . . . . . 9 ((𝑘𝐴𝑋):𝐴⟶{𝑋} → ran (𝑘𝐴𝑋) ⊆ {𝑋})
6159, 60syl 17 . . . . . . . 8 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) → ran (𝑘𝐴𝑋) ⊆ {𝑋})
6245cntzidss 17593 . . . . . . . 8 (({𝑋} ⊆ ((Cntz‘𝐺)‘{𝑋}) ∧ ran (𝑘𝐴𝑋) ⊆ {𝑋}) → ran (𝑘𝐴𝑋) ⊆ ((Cntz‘𝐺)‘ran (𝑘𝐴𝑋)))
6355, 61, 62syl2anc 691 . . . . . . 7 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) → ran (𝑘𝐴𝑋) ⊆ ((Cntz‘𝐺)‘ran (𝑘𝐴𝑋)))
64 f1of1 6049 . . . . . . . 8 (𝑓:(1...(#‘𝐴))–1-1-onto𝐴𝑓:(1...(#‘𝐴))–1-1𝐴)
6564ad2antll 761 . . . . . . 7 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) → 𝑓:(1...(#‘𝐴))–1-1𝐴)
66 suppssdm 7195 . . . . . . . . 9 ((𝑘𝐴𝑋) supp (0g𝐺)) ⊆ dom (𝑘𝐴𝑋)
6749dmmptss 5548 . . . . . . . . . 10 dom (𝑘𝐴𝑋) ⊆ 𝐴
6867a1i 11 . . . . . . . . 9 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) → dom (𝑘𝐴𝑋) ⊆ 𝐴)
6966, 68syl5ss 3579 . . . . . . . 8 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) → ((𝑘𝐴𝑋) supp (0g𝐺)) ⊆ 𝐴)
70 f1ofo 6057 . . . . . . . . . 10 (𝑓:(1...(#‘𝐴))–1-1-onto𝐴𝑓:(1...(#‘𝐴))–onto𝐴)
71 forn 6031 . . . . . . . . . 10 (𝑓:(1...(#‘𝐴))–onto𝐴 → ran 𝑓 = 𝐴)
7270, 71syl 17 . . . . . . . . 9 (𝑓:(1...(#‘𝐴))–1-1-onto𝐴 → ran 𝑓 = 𝐴)
7372ad2antll 761 . . . . . . . 8 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) → ran 𝑓 = 𝐴)
7469, 73sseqtr4d 3605 . . . . . . 7 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) → ((𝑘𝐴𝑋) supp (0g𝐺)) ⊆ ran 𝑓)
75 eqid 2610 . . . . . . 7 (((𝑘𝐴𝑋) ∘ 𝑓) supp (0g𝐺)) = (((𝑘𝐴𝑋) ∘ 𝑓) supp (0g𝐺))
762, 3, 44, 45, 46, 47, 50, 63, 21, 65, 74, 75gsumval3 18131 . . . . . 6 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) → (𝐺 Σg (𝑘𝐴𝑋)) = (seq1((+g𝐺), ((𝑘𝐴𝑋) ∘ 𝑓))‘(#‘𝐴)))
77 eqid 2610 . . . . . . . 8 seq1((+g𝐺), (ℕ × {𝑋})) = seq1((+g𝐺), (ℕ × {𝑋}))
782, 44, 4, 77mulgnn 17370 . . . . . . 7 (((#‘𝐴) ∈ ℕ ∧ 𝑋𝐵) → ((#‘𝐴) · 𝑋) = (seq1((+g𝐺), (ℕ × {𝑋}))‘(#‘𝐴)))
7921, 25, 78syl2anc 691 . . . . . 6 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) → ((#‘𝐴) · 𝑋) = (seq1((+g𝐺), (ℕ × {𝑋}))‘(#‘𝐴)))
8043, 76, 793eqtr4d 2654 . . . . 5 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) → (𝐺 Σg (𝑘𝐴𝑋)) = ((#‘𝐴) · 𝑋))
8180expr 641 . . . 4 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ (#‘𝐴) ∈ ℕ) → (𝑓:(1...(#‘𝐴))–1-1-onto𝐴 → (𝐺 Σg (𝑘𝐴𝑋)) = ((#‘𝐴) · 𝑋)))
8281exlimdv 1848 . . 3 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ (#‘𝐴) ∈ ℕ) → (∃𝑓 𝑓:(1...(#‘𝐴))–1-1-onto𝐴 → (𝐺 Σg (𝑘𝐴𝑋)) = ((#‘𝐴) · 𝑋)))
8382expimpd 627 . 2 ((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) → (((#‘𝐴) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(#‘𝐴))–1-1-onto𝐴) → (𝐺 Σg (𝑘𝐴𝑋)) = ((#‘𝐴) · 𝑋)))
84 fz1f1o 14288 . . 3 (𝐴 ∈ Fin → (𝐴 = ∅ ∨ ((#‘𝐴) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)))
85843ad2ant2 1076 . 2 ((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) → (𝐴 = ∅ ∨ ((#‘𝐴) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)))
8620, 83, 85mpjaod 395 1 ((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) → (𝐺 Σg (𝑘𝐴𝑋)) = ((#‘𝐴) · 𝑋))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   ∧ w3a 1031   = wceq 1475  ∃wex 1695   ∈ wcel 1977   ⊆ wss 3540  ∅c0 3874  {csn 4125   ↦ cmpt 4643   × cxp 5036  dom cdm 5038  ran crn 5039   ∘ ccom 5042  ⟶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  0cc0 9815  1c1 9816  ℕcn 10897  ℤ≥cuz 11563  ...cfz 12197  seqcseq 12663  #chash 12979  Basecbs 15695  +gcplusg 15768  0gc0g 15923   Σg cgsu 15924  Mndcmnd 17117  .gcmg 17363  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-inf2 8421  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-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-mulg 17364  df-cntz 17573 This theorem is referenced by:  gsumconstf  18158  mdetdiagid  20225  chpscmat  20466  chp0mat  20470  chpidmat  20471  tmdgsum2  21710  amgmlem  24516  lgseisenlem4  24903
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