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Theorem gsumconst 18983
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 1185 . . . . 5 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ 𝐴 = ∅) → 𝑋𝐵)
2 gsumconst.b . . . . . 6 𝐵 = (Base‘𝐺)
3 eqid 2818 . . . . . 6 (0g𝐺) = (0g𝐺)
4 gsumconst.m . . . . . 6 · = (.g𝐺)
52, 3, 4mulg0 18169 . . . . 5 (𝑋𝐵 → (0 · 𝑋) = (0g𝐺))
61, 5syl 17 . . . 4 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ 𝐴 = ∅) → (0 · 𝑋) = (0g𝐺))
7 fveq2 6663 . . . . . . 7 (𝐴 = ∅ → (♯‘𝐴) = (♯‘∅))
87adantl 482 . . . . . 6 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ 𝐴 = ∅) → (♯‘𝐴) = (♯‘∅))
9 hash0 13716 . . . . . 6 (♯‘∅) = 0
108, 9syl6eq 2869 . . . . 5 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ 𝐴 = ∅) → (♯‘𝐴) = 0)
1110oveq1d 7160 . . . 4 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ 𝐴 = ∅) → ((♯‘𝐴) · 𝑋) = (0 · 𝑋))
12 mpteq1 5145 . . . . . . . 8 (𝐴 = ∅ → (𝑘𝐴𝑋) = (𝑘 ∈ ∅ ↦ 𝑋))
1312adantl 482 . . . . . . 7 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ 𝐴 = ∅) → (𝑘𝐴𝑋) = (𝑘 ∈ ∅ ↦ 𝑋))
14 mpt0 6483 . . . . . . 7 (𝑘 ∈ ∅ ↦ 𝑋) = ∅
1513, 14syl6eq 2869 . . . . . 6 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ 𝐴 = ∅) → (𝑘𝐴𝑋) = ∅)
1615oveq2d 7161 . . . . 5 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ 𝐴 = ∅) → (𝐺 Σg (𝑘𝐴𝑋)) = (𝐺 Σg ∅))
173gsum0 17882 . . . . 5 (𝐺 Σg ∅) = (0g𝐺)
1816, 17syl6eq 2869 . . . 4 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ 𝐴 = ∅) → (𝐺 Σg (𝑘𝐴𝑋)) = (0g𝐺))
196, 11, 183eqtr4rd 2864 . . 3 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ 𝐴 = ∅) → (𝐺 Σg (𝑘𝐴𝑋)) = ((♯‘𝐴) · 𝑋))
2019ex 413 . 2 ((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) → (𝐴 = ∅ → (𝐺 Σg (𝑘𝐴𝑋)) = ((♯‘𝐴) · 𝑋)))
21 simprl 767 . . . . . . . 8 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (♯‘𝐴) ∈ ℕ)
22 nnuz 12269 . . . . . . . 8 ℕ = (ℤ‘1)
2321, 22eleqtrdi 2920 . . . . . . 7 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (♯‘𝐴) ∈ (ℤ‘1))
24 simpr 485 . . . . . . . . 9 ((((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ (1...(♯‘𝐴))) → 𝑥 ∈ (1...(♯‘𝐴)))
25 simpl3 1185 . . . . . . . . . 10 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → 𝑋𝐵)
2625adantr 481 . . . . . . . . 9 ((((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ (1...(♯‘𝐴))) → 𝑋𝐵)
27 eqid 2818 . . . . . . . . . 10 (𝑥 ∈ (1...(♯‘𝐴)) ↦ 𝑋) = (𝑥 ∈ (1...(♯‘𝐴)) ↦ 𝑋)
2827fvmpt2 6771 . . . . . . . . 9 ((𝑥 ∈ (1...(♯‘𝐴)) ∧ 𝑋𝐵) → ((𝑥 ∈ (1...(♯‘𝐴)) ↦ 𝑋)‘𝑥) = 𝑋)
2924, 26, 28syl2anc 584 . . . . . . . 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 6608 . . . . . . . . . . . . 13 (𝑓:(1...(♯‘𝐴))–1-1-onto𝐴𝑓:(1...(♯‘𝐴))⟶𝐴)
3130ad2antll 725 . . . . . . . . . . . 12 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → 𝑓:(1...(♯‘𝐴))⟶𝐴)
3231ffvelrnda 6843 . . . . . . . . . . 11 ((((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ (1...(♯‘𝐴))) → (𝑓𝑥) ∈ 𝐴)
3331feqmptd 6726 . . . . . . . . . . 11 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → 𝑓 = (𝑥 ∈ (1...(♯‘𝐴)) ↦ (𝑓𝑥)))
34 eqidd 2819 . . . . . . . . . . 11 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (𝑘𝐴𝑋) = (𝑘𝐴𝑋))
35 eqidd 2819 . . . . . . . . . . 11 (𝑘 = (𝑓𝑥) → 𝑋 = 𝑋)
3632, 33, 34, 35fmptco 6883 . . . . . . . . . 10 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → ((𝑘𝐴𝑋) ∘ 𝑓) = (𝑥 ∈ (1...(♯‘𝐴)) ↦ 𝑋))
3736fveq1d 6665 . . . . . . . . 9 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (((𝑘𝐴𝑋) ∘ 𝑓)‘𝑥) = ((𝑥 ∈ (1...(♯‘𝐴)) ↦ 𝑋)‘𝑥))
3837adantr 481 . . . . . . . 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 12924 . . . . . . . . 9 (𝑥 ∈ (1...(♯‘𝐴)) → 𝑥 ∈ ℕ)
40 fvconst2g 6956 . . . . . . . . 9 ((𝑋𝐵𝑥 ∈ ℕ) → ((ℕ × {𝑋})‘𝑥) = 𝑋)
4125, 39, 40syl2an 595 . . . . . . . 8 ((((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ (1...(♯‘𝐴))) → ((ℕ × {𝑋})‘𝑥) = 𝑋)
4229, 38, 413eqtr4d 2863 . . . . . . 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 13382 . . . . . 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 2818 . . . . . . 7 (+g𝐺) = (+g𝐺)
45 eqid 2818 . . . . . . 7 (Cntz‘𝐺) = (Cntz‘𝐺)
46 simpl1 1183 . . . . . . 7 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → 𝐺 ∈ Mnd)
47 simpl2 1184 . . . . . . 7 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → 𝐴 ∈ Fin)
4825adantr 481 . . . . . . . 8 ((((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑘𝐴) → 𝑋𝐵)
4948fmpttd 6871 . . . . . . 7 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (𝑘𝐴𝑋):𝐴𝐵)
50 eqidd 2819 . . . . . . . . . 10 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (𝑋(+g𝐺)𝑋) = (𝑋(+g𝐺)𝑋))
512, 44, 45elcntzsn 18393 . . . . . . . . . . 11 (𝑋𝐵 → (𝑋 ∈ ((Cntz‘𝐺)‘{𝑋}) ↔ (𝑋𝐵 ∧ (𝑋(+g𝐺)𝑋) = (𝑋(+g𝐺)𝑋))))
5225, 51syl 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))))
5325, 50, 52mpbir2and 709 . . . . . . . . 9 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → 𝑋 ∈ ((Cntz‘𝐺)‘{𝑋}))
5453snssd 4734 . . . . . . . 8 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → {𝑋} ⊆ ((Cntz‘𝐺)‘{𝑋}))
55 snidg 4589 . . . . . . . . . . . 12 (𝑋𝐵𝑋 ∈ {𝑋})
5625, 55syl 17 . . . . . . . . . . 11 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → 𝑋 ∈ {𝑋})
5756adantr 481 . . . . . . . . . 10 ((((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑘𝐴) → 𝑋 ∈ {𝑋})
5857fmpttd 6871 . . . . . . . . 9 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (𝑘𝐴𝑋):𝐴⟶{𝑋})
5958frnd 6514 . . . . . . . 8 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → ran (𝑘𝐴𝑋) ⊆ {𝑋})
6045cntzidss 18406 . . . . . . . 8 (({𝑋} ⊆ ((Cntz‘𝐺)‘{𝑋}) ∧ ran (𝑘𝐴𝑋) ⊆ {𝑋}) → ran (𝑘𝐴𝑋) ⊆ ((Cntz‘𝐺)‘ran (𝑘𝐴𝑋)))
6154, 59, 60syl2anc 584 . . . . . . 7 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → ran (𝑘𝐴𝑋) ⊆ ((Cntz‘𝐺)‘ran (𝑘𝐴𝑋)))
62 f1of1 6607 . . . . . . . 8 (𝑓:(1...(♯‘𝐴))–1-1-onto𝐴𝑓:(1...(♯‘𝐴))–1-1𝐴)
6362ad2antll 725 . . . . . . 7 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → 𝑓:(1...(♯‘𝐴))–1-1𝐴)
64 suppssdm 7832 . . . . . . . . 9 ((𝑘𝐴𝑋) supp (0g𝐺)) ⊆ dom (𝑘𝐴𝑋)
65 eqid 2818 . . . . . . . . . . 11 (𝑘𝐴𝑋) = (𝑘𝐴𝑋)
6665dmmptss 6088 . . . . . . . . . 10 dom (𝑘𝐴𝑋) ⊆ 𝐴
6766a1i 11 . . . . . . . . 9 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → dom (𝑘𝐴𝑋) ⊆ 𝐴)
6864, 67sstrid 3975 . . . . . . . 8 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → ((𝑘𝐴𝑋) supp (0g𝐺)) ⊆ 𝐴)
69 f1ofo 6615 . . . . . . . . . 10 (𝑓:(1...(♯‘𝐴))–1-1-onto𝐴𝑓:(1...(♯‘𝐴))–onto𝐴)
70 forn 6586 . . . . . . . . . 10 (𝑓:(1...(♯‘𝐴))–onto𝐴 → ran 𝑓 = 𝐴)
7169, 70syl 17 . . . . . . . . 9 (𝑓:(1...(♯‘𝐴))–1-1-onto𝐴 → ran 𝑓 = 𝐴)
7271ad2antll 725 . . . . . . . 8 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → ran 𝑓 = 𝐴)
7368, 72sseqtrrd 4005 . . . . . . 7 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → ((𝑘𝐴𝑋) supp (0g𝐺)) ⊆ ran 𝑓)
74 eqid 2818 . . . . . . 7 (((𝑘𝐴𝑋) ∘ 𝑓) supp (0g𝐺)) = (((𝑘𝐴𝑋) ∘ 𝑓) supp (0g𝐺))
752, 3, 44, 45, 46, 47, 49, 61, 21, 63, 73, 74gsumval3 18956 . . . . . 6 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (𝐺 Σg (𝑘𝐴𝑋)) = (seq1((+g𝐺), ((𝑘𝐴𝑋) ∘ 𝑓))‘(♯‘𝐴)))
76 eqid 2818 . . . . . . . 8 seq1((+g𝐺), (ℕ × {𝑋})) = seq1((+g𝐺), (ℕ × {𝑋}))
772, 44, 4, 76mulgnn 18170 . . . . . . 7 (((♯‘𝐴) ∈ ℕ ∧ 𝑋𝐵) → ((♯‘𝐴) · 𝑋) = (seq1((+g𝐺), (ℕ × {𝑋}))‘(♯‘𝐴)))
7821, 25, 77syl2anc 584 . . . . . 6 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → ((♯‘𝐴) · 𝑋) = (seq1((+g𝐺), (ℕ × {𝑋}))‘(♯‘𝐴)))
7943, 75, 783eqtr4d 2863 . . . . 5 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (𝐺 Σg (𝑘𝐴𝑋)) = ((♯‘𝐴) · 𝑋))
8079expr 457 . . . 4 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ (♯‘𝐴) ∈ ℕ) → (𝑓:(1...(♯‘𝐴))–1-1-onto𝐴 → (𝐺 Σg (𝑘𝐴𝑋)) = ((♯‘𝐴) · 𝑋)))
8180exlimdv 1925 . . 3 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ (♯‘𝐴) ∈ ℕ) → (∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴 → (𝐺 Σg (𝑘𝐴𝑋)) = ((♯‘𝐴) · 𝑋)))
8281expimpd 454 . 2 ((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) → (((♯‘𝐴) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) → (𝐺 Σg (𝑘𝐴𝑋)) = ((♯‘𝐴) · 𝑋)))
83 fz1f1o 15055 . . 3 (𝐴 ∈ Fin → (𝐴 = ∅ ∨ ((♯‘𝐴) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)))
84833ad2ant2 1126 . 2 ((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) → (𝐴 = ∅ ∨ ((♯‘𝐴) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)))
8520, 82, 84mpjaod 854 1 ((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) → (𝐺 Σg (𝑘𝐴𝑋)) = ((♯‘𝐴) · 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  wo 841  w3a 1079   = wceq 1528  wex 1771  wcel 2105  wss 3933  c0 4288  {csn 4557  cmpt 5137   × cxp 5546  dom cdm 5548  ran crn 5549  ccom 5552  wf 6344  1-1wf1 6345  ontowfo 6346  1-1-ontowf1o 6347  cfv 6348  (class class class)co 7145   supp csupp 7819  Fincfn 8497  0cc0 10525  1c1 10526  cn 11626  cuz 12231  ...cfz 12880  seqcseq 13357  chash 13678  Basecbs 16471  +gcplusg 16553  0gc0g 16701   Σg cgsu 16702  Mndcmnd 17899  .gcmg 18162  Cntzccntz 18383
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-se 5508  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-isom 6357  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-supp 7820  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-oadd 8095  df-er 8278  df-en 8498  df-dom 8499  df-sdom 8500  df-fin 8501  df-oi 8962  df-card 9356  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-nn 11627  df-n0 11886  df-z 11970  df-uz 12232  df-fz 12881  df-fzo 13022  df-seq 13358  df-hash 13679  df-0g 16703  df-gsum 16704  df-mgm 17840  df-sgrp 17889  df-mnd 17900  df-mulg 18163  df-cntz 18385
This theorem is referenced by:  gsumconstf  18984  mdetdiagid  21137  chpscmat  21378  chp0mat  21382  chpidmat  21383  tmdgsum2  22632  amgmlem  25494  lgseisenlem4  25881
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