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Theorem gsumconst 18797
 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 1173 . . . . 5 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ 𝐴 = ∅) → 𝑋𝐵)
2 gsumconst.b . . . . . 6 𝐵 = (Base‘𝐺)
3 eqid 2772 . . . . . 6 (0g𝐺) = (0g𝐺)
4 gsumconst.m . . . . . 6 · = (.g𝐺)
52, 3, 4mulg0 18008 . . . . 5 (𝑋𝐵 → (0 · 𝑋) = (0g𝐺))
61, 5syl 17 . . . 4 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ 𝐴 = ∅) → (0 · 𝑋) = (0g𝐺))
7 fveq2 6493 . . . . . . 7 (𝐴 = ∅ → (♯‘𝐴) = (♯‘∅))
87adantl 474 . . . . . 6 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ 𝐴 = ∅) → (♯‘𝐴) = (♯‘∅))
9 hash0 13536 . . . . . 6 (♯‘∅) = 0
108, 9syl6eq 2824 . . . . 5 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ 𝐴 = ∅) → (♯‘𝐴) = 0)
1110oveq1d 6985 . . . 4 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ 𝐴 = ∅) → ((♯‘𝐴) · 𝑋) = (0 · 𝑋))
12 mpteq1 5009 . . . . . . . 8 (𝐴 = ∅ → (𝑘𝐴𝑋) = (𝑘 ∈ ∅ ↦ 𝑋))
1312adantl 474 . . . . . . 7 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ 𝐴 = ∅) → (𝑘𝐴𝑋) = (𝑘 ∈ ∅ ↦ 𝑋))
14 mpt0 6314 . . . . . . 7 (𝑘 ∈ ∅ ↦ 𝑋) = ∅
1513, 14syl6eq 2824 . . . . . 6 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ 𝐴 = ∅) → (𝑘𝐴𝑋) = ∅)
1615oveq2d 6986 . . . . 5 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ 𝐴 = ∅) → (𝐺 Σg (𝑘𝐴𝑋)) = (𝐺 Σg ∅))
173gsum0 17736 . . . . 5 (𝐺 Σg ∅) = (0g𝐺)
1816, 17syl6eq 2824 . . . 4 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ 𝐴 = ∅) → (𝐺 Σg (𝑘𝐴𝑋)) = (0g𝐺))
196, 11, 183eqtr4rd 2819 . . 3 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ 𝐴 = ∅) → (𝐺 Σg (𝑘𝐴𝑋)) = ((♯‘𝐴) · 𝑋))
2019ex 405 . 2 ((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) → (𝐴 = ∅ → (𝐺 Σg (𝑘𝐴𝑋)) = ((♯‘𝐴) · 𝑋)))
21 simprl 758 . . . . . . . 8 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (♯‘𝐴) ∈ ℕ)
22 nnuz 12088 . . . . . . . 8 ℕ = (ℤ‘1)
2321, 22syl6eleq 2870 . . . . . . 7 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (♯‘𝐴) ∈ (ℤ‘1))
24 simpr 477 . . . . . . . . 9 ((((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ (1...(♯‘𝐴))) → 𝑥 ∈ (1...(♯‘𝐴)))
25 simpl3 1173 . . . . . . . . . 10 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → 𝑋𝐵)
2625adantr 473 . . . . . . . . 9 ((((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ (1...(♯‘𝐴))) → 𝑋𝐵)
27 eqid 2772 . . . . . . . . . 10 (𝑥 ∈ (1...(♯‘𝐴)) ↦ 𝑋) = (𝑥 ∈ (1...(♯‘𝐴)) ↦ 𝑋)
2827fvmpt2 6599 . . . . . . . . 9 ((𝑥 ∈ (1...(♯‘𝐴)) ∧ 𝑋𝐵) → ((𝑥 ∈ (1...(♯‘𝐴)) ↦ 𝑋)‘𝑥) = 𝑋)
2924, 26, 28syl2anc 576 . . . . . . . 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 6438 . . . . . . . . . . . . 13 (𝑓:(1...(♯‘𝐴))–1-1-onto𝐴𝑓:(1...(♯‘𝐴))⟶𝐴)
3130ad2antll 716 . . . . . . . . . . . 12 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → 𝑓:(1...(♯‘𝐴))⟶𝐴)
3231ffvelrnda 6670 . . . . . . . . . . 11 ((((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ (1...(♯‘𝐴))) → (𝑓𝑥) ∈ 𝐴)
3331feqmptd 6556 . . . . . . . . . . 11 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → 𝑓 = (𝑥 ∈ (1...(♯‘𝐴)) ↦ (𝑓𝑥)))
34 eqidd 2773 . . . . . . . . . . 11 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (𝑘𝐴𝑋) = (𝑘𝐴𝑋))
35 eqidd 2773 . . . . . . . . . . 11 (𝑘 = (𝑓𝑥) → 𝑋 = 𝑋)
3632, 33, 34, 35fmptco 6708 . . . . . . . . . 10 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → ((𝑘𝐴𝑋) ∘ 𝑓) = (𝑥 ∈ (1...(♯‘𝐴)) ↦ 𝑋))
3736fveq1d 6495 . . . . . . . . 9 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (((𝑘𝐴𝑋) ∘ 𝑓)‘𝑥) = ((𝑥 ∈ (1...(♯‘𝐴)) ↦ 𝑋)‘𝑥))
3837adantr 473 . . . . . . . 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 12745 . . . . . . . . 9 (𝑥 ∈ (1...(♯‘𝐴)) → 𝑥 ∈ ℕ)
40 fvconst2g 6785 . . . . . . . . 9 ((𝑋𝐵𝑥 ∈ ℕ) → ((ℕ × {𝑋})‘𝑥) = 𝑋)
4125, 39, 40syl2an 586 . . . . . . . 8 ((((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ (1...(♯‘𝐴))) → ((ℕ × {𝑋})‘𝑥) = 𝑋)
4229, 38, 413eqtr4d 2818 . . . . . . 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 13202 . . . . . 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 2772 . . . . . . 7 (+g𝐺) = (+g𝐺)
45 eqid 2772 . . . . . . 7 (Cntz‘𝐺) = (Cntz‘𝐺)
46 simpl1 1171 . . . . . . 7 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → 𝐺 ∈ Mnd)
47 simpl2 1172 . . . . . . 7 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → 𝐴 ∈ Fin)
4825adantr 473 . . . . . . . 8 ((((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑘𝐴) → 𝑋𝐵)
4948fmpttd 6696 . . . . . . 7 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (𝑘𝐴𝑋):𝐴𝐵)
50 eqidd 2773 . . . . . . . . . 10 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (𝑋(+g𝐺)𝑋) = (𝑋(+g𝐺)𝑋))
512, 44, 45elcntzsn 18216 . . . . . . . . . . 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 700 . . . . . . . . 9 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → 𝑋 ∈ ((Cntz‘𝐺)‘{𝑋}))
5453snssd 4610 . . . . . . . 8 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → {𝑋} ⊆ ((Cntz‘𝐺)‘{𝑋}))
55 snidg 4465 . . . . . . . . . . . 12 (𝑋𝐵𝑋 ∈ {𝑋})
5625, 55syl 17 . . . . . . . . . . 11 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → 𝑋 ∈ {𝑋})
5756adantr 473 . . . . . . . . . 10 ((((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑘𝐴) → 𝑋 ∈ {𝑋})
5857fmpttd 6696 . . . . . . . . 9 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (𝑘𝐴𝑋):𝐴⟶{𝑋})
5958frnd 6345 . . . . . . . 8 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → ran (𝑘𝐴𝑋) ⊆ {𝑋})
6045cntzidss 18229 . . . . . . . 8 (({𝑋} ⊆ ((Cntz‘𝐺)‘{𝑋}) ∧ ran (𝑘𝐴𝑋) ⊆ {𝑋}) → ran (𝑘𝐴𝑋) ⊆ ((Cntz‘𝐺)‘ran (𝑘𝐴𝑋)))
6154, 59, 60syl2anc 576 . . . . . . 7 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → ran (𝑘𝐴𝑋) ⊆ ((Cntz‘𝐺)‘ran (𝑘𝐴𝑋)))
62 f1of1 6437 . . . . . . . 8 (𝑓:(1...(♯‘𝐴))–1-1-onto𝐴𝑓:(1...(♯‘𝐴))–1-1𝐴)
6362ad2antll 716 . . . . . . 7 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → 𝑓:(1...(♯‘𝐴))–1-1𝐴)
64 suppssdm 7639 . . . . . . . . 9 ((𝑘𝐴𝑋) supp (0g𝐺)) ⊆ dom (𝑘𝐴𝑋)
65 eqid 2772 . . . . . . . . . . 11 (𝑘𝐴𝑋) = (𝑘𝐴𝑋)
6665dmmptss 5928 . . . . . . . . . 10 dom (𝑘𝐴𝑋) ⊆ 𝐴
6766a1i 11 . . . . . . . . 9 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → dom (𝑘𝐴𝑋) ⊆ 𝐴)
6864, 67syl5ss 3865 . . . . . . . 8 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → ((𝑘𝐴𝑋) supp (0g𝐺)) ⊆ 𝐴)
69 f1ofo 6445 . . . . . . . . . 10 (𝑓:(1...(♯‘𝐴))–1-1-onto𝐴𝑓:(1...(♯‘𝐴))–onto𝐴)
70 forn 6416 . . . . . . . . . 10 (𝑓:(1...(♯‘𝐴))–onto𝐴 → ran 𝑓 = 𝐴)
7169, 70syl 17 . . . . . . . . 9 (𝑓:(1...(♯‘𝐴))–1-1-onto𝐴 → ran 𝑓 = 𝐴)
7271ad2antll 716 . . . . . . . 8 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → ran 𝑓 = 𝐴)
7368, 72sseqtr4d 3894 . . . . . . 7 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → ((𝑘𝐴𝑋) supp (0g𝐺)) ⊆ ran 𝑓)
74 eqid 2772 . . . . . . 7 (((𝑘𝐴𝑋) ∘ 𝑓) supp (0g𝐺)) = (((𝑘𝐴𝑋) ∘ 𝑓) supp (0g𝐺))
752, 3, 44, 45, 46, 47, 49, 61, 21, 63, 73, 74gsumval3 18771 . . . . . 6 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (𝐺 Σg (𝑘𝐴𝑋)) = (seq1((+g𝐺), ((𝑘𝐴𝑋) ∘ 𝑓))‘(♯‘𝐴)))
76 eqid 2772 . . . . . . . 8 seq1((+g𝐺), (ℕ × {𝑋})) = seq1((+g𝐺), (ℕ × {𝑋}))
772, 44, 4, 76mulgnn 18009 . . . . . . 7 (((♯‘𝐴) ∈ ℕ ∧ 𝑋𝐵) → ((♯‘𝐴) · 𝑋) = (seq1((+g𝐺), (ℕ × {𝑋}))‘(♯‘𝐴)))
7821, 25, 77syl2anc 576 . . . . . 6 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → ((♯‘𝐴) · 𝑋) = (seq1((+g𝐺), (ℕ × {𝑋}))‘(♯‘𝐴)))
7943, 75, 783eqtr4d 2818 . . . . 5 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (𝐺 Σg (𝑘𝐴𝑋)) = ((♯‘𝐴) · 𝑋))
8079expr 449 . . . 4 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ (♯‘𝐴) ∈ ℕ) → (𝑓:(1...(♯‘𝐴))–1-1-onto𝐴 → (𝐺 Σg (𝑘𝐴𝑋)) = ((♯‘𝐴) · 𝑋)))
8180exlimdv 1892 . . 3 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ (♯‘𝐴) ∈ ℕ) → (∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴 → (𝐺 Σg (𝑘𝐴𝑋)) = ((♯‘𝐴) · 𝑋)))
8281expimpd 446 . 2 ((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) → (((♯‘𝐴) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) → (𝐺 Σg (𝑘𝐴𝑋)) = ((♯‘𝐴) · 𝑋)))
83 fz1f1o 14917 . . 3 (𝐴 ∈ Fin → (𝐴 = ∅ ∨ ((♯‘𝐴) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)))
84833ad2ant2 1114 . 2 ((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) → (𝐴 = ∅ ∨ ((♯‘𝐴) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)))
8520, 82, 84mpjaod 846 1 ((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) → (𝐺 Σg (𝑘𝐴𝑋)) = ((♯‘𝐴) · 𝑋))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 387   ∨ wo 833   ∧ w3a 1068   = wceq 1507  ∃wex 1742   ∈ wcel 2048   ⊆ wss 3825  ∅c0 4173  {csn 4435   ↦ cmpt 5002   × cxp 5398  dom cdm 5400  ran crn 5401   ∘ ccom 5404  ⟶wf 6178  –1-1→wf1 6179  –onto→wfo 6180  –1-1-onto→wf1o 6181  ‘cfv 6182  (class class class)co 6970   supp csupp 7626  Fincfn 8298  0cc0 10327  1c1 10328  ℕcn 11431  ℤ≥cuz 12051  ...cfz 12701  seqcseq 13177  ♯chash 13498  Basecbs 16329  +gcplusg 16411  0gc0g 16559   Σg cgsu 16560  Mndcmnd 17752  .gcmg 18001  Cntzccntz 18206 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1964  ax-8 2050  ax-9 2057  ax-10 2077  ax-11 2091  ax-12 2104  ax-13 2299  ax-ext 2745  ax-rep 5043  ax-sep 5054  ax-nul 5061  ax-pow 5113  ax-pr 5180  ax-un 7273  ax-cnex 10383  ax-resscn 10384  ax-1cn 10385  ax-icn 10386  ax-addcl 10387  ax-addrcl 10388  ax-mulcl 10389  ax-mulrcl 10390  ax-mulcom 10391  ax-addass 10392  ax-mulass 10393  ax-distr 10394  ax-i2m1 10395  ax-1ne0 10396  ax-1rid 10397  ax-rnegex 10398  ax-rrecex 10399  ax-cnre 10400  ax-pre-lttri 10401  ax-pre-lttrn 10402  ax-pre-ltadd 10403  ax-pre-mulgt0 10404 This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2014  df-mo 2544  df-eu 2580  df-clab 2754  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ne 2962  df-nel 3068  df-ral 3087  df-rex 3088  df-reu 3089  df-rmo 3090  df-rab 3091  df-v 3411  df-sbc 3678  df-csb 3783  df-dif 3828  df-un 3830  df-in 3832  df-ss 3839  df-pss 3841  df-nul 4174  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-tp 4440  df-op 4442  df-uni 4707  df-int 4744  df-iun 4788  df-br 4924  df-opab 4986  df-mpt 5003  df-tr 5025  df-id 5305  df-eprel 5310  df-po 5319  df-so 5320  df-fr 5359  df-se 5360  df-we 5361  df-xp 5406  df-rel 5407  df-cnv 5408  df-co 5409  df-dm 5410  df-rn 5411  df-res 5412  df-ima 5413  df-pred 5980  df-ord 6026  df-on 6027  df-lim 6028  df-suc 6029  df-iota 6146  df-fun 6184  df-fn 6185  df-f 6186  df-f1 6187  df-fo 6188  df-f1o 6189  df-fv 6190  df-isom 6191  df-riota 6931  df-ov 6973  df-oprab 6974  df-mpo 6975  df-om 7391  df-1st 7494  df-2nd 7495  df-supp 7627  df-wrecs 7743  df-recs 7805  df-rdg 7843  df-1o 7897  df-oadd 7901  df-er 8081  df-en 8299  df-dom 8300  df-sdom 8301  df-fin 8302  df-oi 8761  df-card 9154  df-pnf 10468  df-mnf 10469  df-xr 10470  df-ltxr 10471  df-le 10472  df-sub 10664  df-neg 10665  df-nn 11432  df-n0 11701  df-z 11787  df-uz 12052  df-fz 12702  df-fzo 12843  df-seq 13178  df-hash 13499  df-0g 16561  df-gsum 16562  df-mgm 17700  df-sgrp 17742  df-mnd 17753  df-mulg 18002  df-cntz 18208 This theorem is referenced by:  gsumconstf  18798  mdetdiagid  20903  chpscmat  21144  chp0mat  21148  chpidmat  21149  tmdgsum2  22398  amgmlem  25259  lgseisenlem4  25646
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