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Theorem gsumconst 19535
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 1192 . . . . 5 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ 𝐴 = ∅) → 𝑋𝐵)
2 gsumconst.b . . . . . 6 𝐵 = (Base‘𝐺)
3 eqid 2738 . . . . . 6 (0g𝐺) = (0g𝐺)
4 gsumconst.m . . . . . 6 · = (.g𝐺)
52, 3, 4mulg0 18707 . . . . 5 (𝑋𝐵 → (0 · 𝑋) = (0g𝐺))
61, 5syl 17 . . . 4 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ 𝐴 = ∅) → (0 · 𝑋) = (0g𝐺))
7 fveq2 6774 . . . . . . 7 (𝐴 = ∅ → (♯‘𝐴) = (♯‘∅))
87adantl 482 . . . . . 6 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ 𝐴 = ∅) → (♯‘𝐴) = (♯‘∅))
9 hash0 14082 . . . . . 6 (♯‘∅) = 0
108, 9eqtrdi 2794 . . . . 5 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ 𝐴 = ∅) → (♯‘𝐴) = 0)
1110oveq1d 7290 . . . 4 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ 𝐴 = ∅) → ((♯‘𝐴) · 𝑋) = (0 · 𝑋))
12 mpteq1 5167 . . . . . . . 8 (𝐴 = ∅ → (𝑘𝐴𝑋) = (𝑘 ∈ ∅ ↦ 𝑋))
1312adantl 482 . . . . . . 7 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ 𝐴 = ∅) → (𝑘𝐴𝑋) = (𝑘 ∈ ∅ ↦ 𝑋))
14 mpt0 6575 . . . . . . 7 (𝑘 ∈ ∅ ↦ 𝑋) = ∅
1513, 14eqtrdi 2794 . . . . . 6 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ 𝐴 = ∅) → (𝑘𝐴𝑋) = ∅)
1615oveq2d 7291 . . . . 5 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ 𝐴 = ∅) → (𝐺 Σg (𝑘𝐴𝑋)) = (𝐺 Σg ∅))
173gsum0 18368 . . . . 5 (𝐺 Σg ∅) = (0g𝐺)
1816, 17eqtrdi 2794 . . . 4 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ 𝐴 = ∅) → (𝐺 Σg (𝑘𝐴𝑋)) = (0g𝐺))
196, 11, 183eqtr4rd 2789 . . 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 768 . . . . . . . 8 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (♯‘𝐴) ∈ ℕ)
22 nnuz 12621 . . . . . . . 8 ℕ = (ℤ‘1)
2321, 22eleqtrdi 2849 . . . . . . 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 1192 . . . . . . . . . 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 2738 . . . . . . . . . 10 (𝑥 ∈ (1...(♯‘𝐴)) ↦ 𝑋) = (𝑥 ∈ (1...(♯‘𝐴)) ↦ 𝑋)
2827fvmpt2 6886 . . . . . . . . 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 6716 . . . . . . . . . . . . 13 (𝑓:(1...(♯‘𝐴))–1-1-onto𝐴𝑓:(1...(♯‘𝐴))⟶𝐴)
3130ad2antll 726 . . . . . . . . . . . 12 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → 𝑓:(1...(♯‘𝐴))⟶𝐴)
3231ffvelrnda 6961 . . . . . . . . . . 11 ((((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ (1...(♯‘𝐴))) → (𝑓𝑥) ∈ 𝐴)
3331feqmptd 6837 . . . . . . . . . . 11 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → 𝑓 = (𝑥 ∈ (1...(♯‘𝐴)) ↦ (𝑓𝑥)))
34 eqidd 2739 . . . . . . . . . . 11 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (𝑘𝐴𝑋) = (𝑘𝐴𝑋))
35 eqidd 2739 . . . . . . . . . . 11 (𝑘 = (𝑓𝑥) → 𝑋 = 𝑋)
3632, 33, 34, 35fmptco 7001 . . . . . . . . . 10 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → ((𝑘𝐴𝑋) ∘ 𝑓) = (𝑥 ∈ (1...(♯‘𝐴)) ↦ 𝑋))
3736fveq1d 6776 . . . . . . . . 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 13285 . . . . . . . . 9 (𝑥 ∈ (1...(♯‘𝐴)) → 𝑥 ∈ ℕ)
40 fvconst2g 7077 . . . . . . . . 9 ((𝑋𝐵𝑥 ∈ ℕ) → ((ℕ × {𝑋})‘𝑥) = 𝑋)
4125, 39, 40syl2an 596 . . . . . . . 8 ((((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ (1...(♯‘𝐴))) → ((ℕ × {𝑋})‘𝑥) = 𝑋)
4229, 38, 413eqtr4d 2788 . . . . . . 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 13747 . . . . . 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 2738 . . . . . . 7 (+g𝐺) = (+g𝐺)
45 eqid 2738 . . . . . . 7 (Cntz‘𝐺) = (Cntz‘𝐺)
46 simpl1 1190 . . . . . . 7 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → 𝐺 ∈ Mnd)
47 simpl2 1191 . . . . . . 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 6989 . . . . . . 7 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (𝑘𝐴𝑋):𝐴𝐵)
50 eqidd 2739 . . . . . . . . . 10 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (𝑋(+g𝐺)𝑋) = (𝑋(+g𝐺)𝑋))
512, 44, 45elcntzsn 18931 . . . . . . . . . . 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 710 . . . . . . . . 9 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → 𝑋 ∈ ((Cntz‘𝐺)‘{𝑋}))
5453snssd 4742 . . . . . . . 8 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → {𝑋} ⊆ ((Cntz‘𝐺)‘{𝑋}))
55 snidg 4595 . . . . . . . . . . . 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 6989 . . . . . . . . 9 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (𝑘𝐴𝑋):𝐴⟶{𝑋})
5958frnd 6608 . . . . . . . 8 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → ran (𝑘𝐴𝑋) ⊆ {𝑋})
6045cntzidss 18944 . . . . . . . 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 6715 . . . . . . . 8 (𝑓:(1...(♯‘𝐴))–1-1-onto𝐴𝑓:(1...(♯‘𝐴))–1-1𝐴)
6362ad2antll 726 . . . . . . 7 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → 𝑓:(1...(♯‘𝐴))–1-1𝐴)
64 suppssdm 7993 . . . . . . . . 9 ((𝑘𝐴𝑋) supp (0g𝐺)) ⊆ dom (𝑘𝐴𝑋)
65 eqid 2738 . . . . . . . . . . 11 (𝑘𝐴𝑋) = (𝑘𝐴𝑋)
6665dmmptss 6144 . . . . . . . . . 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 3932 . . . . . . . 8 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → ((𝑘𝐴𝑋) supp (0g𝐺)) ⊆ 𝐴)
69 f1ofo 6723 . . . . . . . . . 10 (𝑓:(1...(♯‘𝐴))–1-1-onto𝐴𝑓:(1...(♯‘𝐴))–onto𝐴)
70 forn 6691 . . . . . . . . . 10 (𝑓:(1...(♯‘𝐴))–onto𝐴 → ran 𝑓 = 𝐴)
7169, 70syl 17 . . . . . . . . 9 (𝑓:(1...(♯‘𝐴))–1-1-onto𝐴 → ran 𝑓 = 𝐴)
7271ad2antll 726 . . . . . . . 8 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → ran 𝑓 = 𝐴)
7368, 72sseqtrrd 3962 . . . . . . 7 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → ((𝑘𝐴𝑋) supp (0g𝐺)) ⊆ ran 𝑓)
74 eqid 2738 . . . . . . 7 (((𝑘𝐴𝑋) ∘ 𝑓) supp (0g𝐺)) = (((𝑘𝐴𝑋) ∘ 𝑓) supp (0g𝐺))
752, 3, 44, 45, 46, 47, 49, 61, 21, 63, 73, 74gsumval3 19508 . . . . . 6 (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (𝐺 Σg (𝑘𝐴𝑋)) = (seq1((+g𝐺), ((𝑘𝐴𝑋) ∘ 𝑓))‘(♯‘𝐴)))
76 eqid 2738 . . . . . . . 8 seq1((+g𝐺), (ℕ × {𝑋})) = seq1((+g𝐺), (ℕ × {𝑋}))
772, 44, 4, 76mulgnn 18708 . . . . . . 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 2788 . . . . 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 1936 . . 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 15422 . . 3 (𝐴 ∈ Fin → (𝐴 = ∅ ∨ ((♯‘𝐴) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)))
84833ad2ant2 1133 . 2 ((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) → (𝐴 = ∅ ∨ ((♯‘𝐴) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)))
8520, 82, 84mpjaod 857 1 ((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋𝐵) → (𝐺 Σg (𝑘𝐴𝑋)) = ((♯‘𝐴) · 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  wo 844  w3a 1086   = wceq 1539  wex 1782  wcel 2106  wss 3887  c0 4256  {csn 4561  cmpt 5157   × cxp 5587  dom cdm 5589  ran crn 5590  ccom 5593  wf 6429  1-1wf1 6430  ontowfo 6431  1-1-ontowf1o 6432  cfv 6433  (class class class)co 7275   supp csupp 7977  Fincfn 8733  0cc0 10871  1c1 10872  cn 11973  cuz 12582  ...cfz 13239  seqcseq 13721  chash 14044  Basecbs 16912  +gcplusg 16962  0gc0g 17150   Σg cgsu 17151  Mndcmnd 18385  .gcmg 18700  Cntzccntz 18921
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-se 5545  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-isom 6442  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-supp 7978  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-er 8498  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-oi 9269  df-card 9697  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-nn 11974  df-n0 12234  df-z 12320  df-uz 12583  df-fz 13240  df-fzo 13383  df-seq 13722  df-hash 14045  df-0g 17152  df-gsum 17153  df-mgm 18326  df-sgrp 18375  df-mnd 18386  df-mulg 18701  df-cntz 18923
This theorem is referenced by:  gsumconstf  19536  mdetdiagid  21749  chpscmat  21991  chp0mat  21995  chpidmat  21996  tmdgsum2  23247  amgmlem  26139  lgseisenlem4  26526  gsumhashmul  31316
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