Theorem gsumconst 19870

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	⊢ · = (.g‘𝐺)

Assertion

Ref	Expression
gsumconst	⊢ ((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵) → (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 𝑋)) = ((♯‘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.

Step	Hyp	Ref	Expression
1		simpl3 1194	. . . . 5 ⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵) ∧ 𝐴 = ∅) → 𝑋 ∈ 𝐵)
2		gsumconst.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐺)
3		eqid 2730	. . . . . 6 ⊢ (0g‘𝐺) = (0g‘𝐺)
4		gsumconst.m	. . . . . 6 ⊢ · = (.g‘𝐺)
5	2, 3, 4	mulg0 19012	. . . . 5 ⊢ (𝑋 ∈ 𝐵 → (0 · 𝑋) = (0g‘𝐺))
6	1, 5	syl 17	. . . 4 ⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵) ∧ 𝐴 = ∅) → (0 · 𝑋) = (0g‘𝐺))
7		fveq2 6860	. . . . . . 7 ⊢ (𝐴 = ∅ → (♯‘𝐴) = (♯‘∅))
8	7	adantl 481	. . . . . 6 ⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵) ∧ 𝐴 = ∅) → (♯‘𝐴) = (♯‘∅))
9		hash0 14338	. . . . . 6 ⊢ (♯‘∅) = 0
10	8, 9	eqtrdi 2781	. . . . 5 ⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵) ∧ 𝐴 = ∅) → (♯‘𝐴) = 0)
11	10	oveq1d 7404	. . . 4 ⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵) ∧ 𝐴 = ∅) → ((♯‘𝐴) · 𝑋) = (0 · 𝑋))
12		mpteq1 5198	. . . . . . . 8 ⊢ (𝐴 = ∅ → (𝑘 ∈ 𝐴 ↦ 𝑋) = (𝑘 ∈ ∅ ↦ 𝑋))
13	12	adantl 481	. . . . . . 7 ⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵) ∧ 𝐴 = ∅) → (𝑘 ∈ 𝐴 ↦ 𝑋) = (𝑘 ∈ ∅ ↦ 𝑋))
14		mpt0 6662	. . . . . . 7 ⊢ (𝑘 ∈ ∅ ↦ 𝑋) = ∅
15	13, 14	eqtrdi 2781	. . . . . 6 ⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵) ∧ 𝐴 = ∅) → (𝑘 ∈ 𝐴 ↦ 𝑋) = ∅)
16	15	oveq2d 7405	. . . . 5 ⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵) ∧ 𝐴 = ∅) → (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 𝑋)) = (𝐺 Σg ∅))
17	3	gsum0 18617	. . . . 5 ⊢ (𝐺 Σg ∅) = (0g‘𝐺)
18	16, 17	eqtrdi 2781	. . . 4 ⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵) ∧ 𝐴 = ∅) → (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 𝑋)) = (0g‘𝐺))
19	6, 11, 18	3eqtr4rd 2776	. . 3 ⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵) ∧ 𝐴 = ∅) → (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 𝑋)) = ((♯‘𝐴) · 𝑋))
20	19	ex 412	. 2 ⊢ ((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵) → (𝐴 = ∅ → (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 𝑋)) = ((♯‘𝐴) · 𝑋)))
21		simprl 770	. . . . . . . 8 ⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → (♯‘𝐴) ∈ ℕ)
22		nnuz 12842	. . . . . . . 8 ⊢ ℕ = (ℤ≥‘1)
23	21, 22	eleqtrdi 2839	. . . . . . 7 ⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → (♯‘𝐴) ∈ (ℤ≥‘1))
24		simpr 484	. . . . . . . . 9 ⊢ ((((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑥 ∈ (1...(♯‘𝐴))) → 𝑥 ∈ (1...(♯‘𝐴)))
25		simpl3 1194	. . . . . . . . . 10 ⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → 𝑋 ∈ 𝐵)
26	25	adantr 480	. . . . . . . . 9 ⊢ ((((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑥 ∈ (1...(♯‘𝐴))) → 𝑋 ∈ 𝐵)
27		eqid 2730	. . . . . . . . . 10 ⊢ (𝑥 ∈ (1...(♯‘𝐴)) ↦ 𝑋) = (𝑥 ∈ (1...(♯‘𝐴)) ↦ 𝑋)
28	27	fvmpt2 6981	. . . . . . . . 9 ⊢ ((𝑥 ∈ (1...(♯‘𝐴)) ∧ 𝑋 ∈ 𝐵) → ((𝑥 ∈ (1...(♯‘𝐴)) ↦ 𝑋)‘𝑥) = 𝑋)
29	24, 26, 28	syl2anc 584	. . . . . . . 8 ⊢ ((((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑥 ∈ (1...(♯‘𝐴))) → ((𝑥 ∈ (1...(♯‘𝐴)) ↦ 𝑋)‘𝑥) = 𝑋)
30		f1of 6802	. . . . . . . . . . . . 13 ⊢ (𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴 → 𝑓:(1...(♯‘𝐴))⟶𝐴)
31	30	ad2antll 729	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → 𝑓:(1...(♯‘𝐴))⟶𝐴)
32	31	ffvelcdmda 7058	. . . . . . . . . . 11 ⊢ ((((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑥 ∈ (1...(♯‘𝐴))) → (𝑓‘𝑥) ∈ 𝐴)
33	31	feqmptd 6931	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → 𝑓 = (𝑥 ∈ (1...(♯‘𝐴)) ↦ (𝑓‘𝑥)))
34		eqidd 2731	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → (𝑘 ∈ 𝐴 ↦ 𝑋) = (𝑘 ∈ 𝐴 ↦ 𝑋))
35		eqidd 2731	. . . . . . . . . . 11 ⊢ (𝑘 = (𝑓‘𝑥) → 𝑋 = 𝑋)
36	32, 33, 34, 35	fmptco 7103	. . . . . . . . . 10 ⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → ((𝑘 ∈ 𝐴 ↦ 𝑋) ∘ 𝑓) = (𝑥 ∈ (1...(♯‘𝐴)) ↦ 𝑋))
37	36	fveq1d 6862	. . . . . . . . 9 ⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → (((𝑘 ∈ 𝐴 ↦ 𝑋) ∘ 𝑓)‘𝑥) = ((𝑥 ∈ (1...(♯‘𝐴)) ↦ 𝑋)‘𝑥))
38	37	adantr 480	. . . . . . . 8 ⊢ ((((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑥 ∈ (1...(♯‘𝐴))) → (((𝑘 ∈ 𝐴 ↦ 𝑋) ∘ 𝑓)‘𝑥) = ((𝑥 ∈ (1...(♯‘𝐴)) ↦ 𝑋)‘𝑥))
39		elfznn 13520	. . . . . . . . 9 ⊢ (𝑥 ∈ (1...(♯‘𝐴)) → 𝑥 ∈ ℕ)
40		fvconst2g 7178	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑥 ∈ ℕ) → ((ℕ × {𝑋})‘𝑥) = 𝑋)
41	25, 39, 40	syl2an 596	. . . . . . . 8 ⊢ ((((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑥 ∈ (1...(♯‘𝐴))) → ((ℕ × {𝑋})‘𝑥) = 𝑋)
42	29, 38, 41	3eqtr4d 2775	. . . . . . 7 ⊢ ((((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑥 ∈ (1...(♯‘𝐴))) → (((𝑘 ∈ 𝐴 ↦ 𝑋) ∘ 𝑓)‘𝑥) = ((ℕ × {𝑋})‘𝑥))
43	23, 42	seqfveq 13997	. . . . . 6 ⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → (seq1((+g‘𝐺), ((𝑘 ∈ 𝐴 ↦ 𝑋) ∘ 𝑓))‘(♯‘𝐴)) = (seq1((+g‘𝐺), (ℕ × {𝑋}))‘(♯‘𝐴)))
44		eqid 2730	. . . . . . 7 ⊢ (+g‘𝐺) = (+g‘𝐺)
45		eqid 2730	. . . . . . 7 ⊢ (Cntz‘𝐺) = (Cntz‘𝐺)
46		simpl1 1192	. . . . . . 7 ⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → 𝐺 ∈ Mnd)
47		simpl2 1193	. . . . . . 7 ⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → 𝐴 ∈ Fin)
48	25	adantr 480	. . . . . . . 8 ⊢ ((((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑘 ∈ 𝐴) → 𝑋 ∈ 𝐵)
49	48	fmpttd 7089	. . . . . . 7 ⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → (𝑘 ∈ 𝐴 ↦ 𝑋):𝐴⟶𝐵)
50		eqidd 2731	. . . . . . . . . 10 ⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → (𝑋(+g‘𝐺)𝑋) = (𝑋(+g‘𝐺)𝑋))
51	2, 44, 45	elcntzsn 19263	. . . . . . . . . . 11 ⊢ (𝑋 ∈ 𝐵 → (𝑋 ∈ ((Cntz‘𝐺)‘{𝑋}) ↔ (𝑋 ∈ 𝐵 ∧ (𝑋(+g‘𝐺)𝑋) = (𝑋(+g‘𝐺)𝑋))))
52	25, 51	syl 17	. . . . . . . . . 10 ⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → (𝑋 ∈ ((Cntz‘𝐺)‘{𝑋}) ↔ (𝑋 ∈ 𝐵 ∧ (𝑋(+g‘𝐺)𝑋) = (𝑋(+g‘𝐺)𝑋))))
53	25, 50, 52	mpbir2and 713	. . . . . . . . 9 ⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → 𝑋 ∈ ((Cntz‘𝐺)‘{𝑋}))
54	53	snssd 4775	. . . . . . . 8 ⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → {𝑋} ⊆ ((Cntz‘𝐺)‘{𝑋}))
55		snidg 4626	. . . . . . . . . . . 12 ⊢ (𝑋 ∈ 𝐵 → 𝑋 ∈ {𝑋})
56	25, 55	syl 17	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → 𝑋 ∈ {𝑋})
57	56	adantr 480	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑘 ∈ 𝐴) → 𝑋 ∈ {𝑋})
58	57	fmpttd 7089	. . . . . . . . 9 ⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → (𝑘 ∈ 𝐴 ↦ 𝑋):𝐴⟶{𝑋})
59	58	frnd 6698	. . . . . . . 8 ⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → ran (𝑘 ∈ 𝐴 ↦ 𝑋) ⊆ {𝑋})
60	45	cntzidss 19278	. . . . . . . 8 ⊢ (({𝑋} ⊆ ((Cntz‘𝐺)‘{𝑋}) ∧ ran (𝑘 ∈ 𝐴 ↦ 𝑋) ⊆ {𝑋}) → ran (𝑘 ∈ 𝐴 ↦ 𝑋) ⊆ ((Cntz‘𝐺)‘ran (𝑘 ∈ 𝐴 ↦ 𝑋)))
61	54, 59, 60	syl2anc 584	. . . . . . 7 ⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → ran (𝑘 ∈ 𝐴 ↦ 𝑋) ⊆ ((Cntz‘𝐺)‘ran (𝑘 ∈ 𝐴 ↦ 𝑋)))
62		f1of1 6801	. . . . . . . 8 ⊢ (𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴 → 𝑓:(1...(♯‘𝐴))–1-1→𝐴)
63	62	ad2antll 729	. . . . . . 7 ⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → 𝑓:(1...(♯‘𝐴))–1-1→𝐴)
64		suppssdm 8158	. . . . . . . . 9 ⊢ ((𝑘 ∈ 𝐴 ↦ 𝑋) supp (0g‘𝐺)) ⊆ dom (𝑘 ∈ 𝐴 ↦ 𝑋)
65		eqid 2730	. . . . . . . . . . 11 ⊢ (𝑘 ∈ 𝐴 ↦ 𝑋) = (𝑘 ∈ 𝐴 ↦ 𝑋)
66	65	dmmptss 6216	. . . . . . . . . 10 ⊢ dom (𝑘 ∈ 𝐴 ↦ 𝑋) ⊆ 𝐴
67	66	a1i 11	. . . . . . . . 9 ⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → dom (𝑘 ∈ 𝐴 ↦ 𝑋) ⊆ 𝐴)
68	64, 67	sstrid 3960	. . . . . . . 8 ⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → ((𝑘 ∈ 𝐴 ↦ 𝑋) supp (0g‘𝐺)) ⊆ 𝐴)
69		f1ofo 6809	. . . . . . . . . 10 ⊢ (𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴 → 𝑓:(1...(♯‘𝐴))–onto→𝐴)
70		forn 6777	. . . . . . . . . 10 ⊢ (𝑓:(1...(♯‘𝐴))–onto→𝐴 → ran 𝑓 = 𝐴)
71	69, 70	syl 17	. . . . . . . . 9 ⊢ (𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴 → ran 𝑓 = 𝐴)
72	71	ad2antll 729	. . . . . . . 8 ⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → ran 𝑓 = 𝐴)
73	68, 72	sseqtrrd 3986	. . . . . . 7 ⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → ((𝑘 ∈ 𝐴 ↦ 𝑋) supp (0g‘𝐺)) ⊆ ran 𝑓)
74		eqid 2730	. . . . . . 7 ⊢ (((𝑘 ∈ 𝐴 ↦ 𝑋) ∘ 𝑓) supp (0g‘𝐺)) = (((𝑘 ∈ 𝐴 ↦ 𝑋) ∘ 𝑓) supp (0g‘𝐺))
75	2, 3, 44, 45, 46, 47, 49, 61, 21, 63, 73, 74	gsumval3 19843	. . . . . 6 ⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 𝑋)) = (seq1((+g‘𝐺), ((𝑘 ∈ 𝐴 ↦ 𝑋) ∘ 𝑓))‘(♯‘𝐴)))
76		eqid 2730	. . . . . . . 8 ⊢ seq1((+g‘𝐺), (ℕ × {𝑋})) = seq1((+g‘𝐺), (ℕ × {𝑋}))
77	2, 44, 4, 76	mulgnn 19013	. . . . . . 7 ⊢ (((♯‘𝐴) ∈ ℕ ∧ 𝑋 ∈ 𝐵) → ((♯‘𝐴) · 𝑋) = (seq1((+g‘𝐺), (ℕ × {𝑋}))‘(♯‘𝐴)))
78	21, 25, 77	syl2anc 584	. . . . . 6 ⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → ((♯‘𝐴) · 𝑋) = (seq1((+g‘𝐺), (ℕ × {𝑋}))‘(♯‘𝐴)))
79	43, 75, 78	3eqtr4d 2775	. . . . 5 ⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 𝑋)) = ((♯‘𝐴) · 𝑋))
80	79	expr 456	. . . 4 ⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵) ∧ (♯‘𝐴) ∈ ℕ) → (𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴 → (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 𝑋)) = ((♯‘𝐴) · 𝑋)))
81	80	exlimdv 1933	. . 3 ⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵) ∧ (♯‘𝐴) ∈ ℕ) → (∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴 → (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 𝑋)) = ((♯‘𝐴) · 𝑋)))
82	81	expimpd 453	. 2 ⊢ ((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵) → (((♯‘𝐴) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) → (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 𝑋)) = ((♯‘𝐴) · 𝑋)))
83		fz1f1o 15682	. . 3 ⊢ (𝐴 ∈ Fin → (𝐴 = ∅ ∨ ((♯‘𝐴) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)))
84	83	3ad2ant2 1134	. 2 ⊢ ((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵) → (𝐴 = ∅ ∨ ((♯‘𝐴) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)))
85	20, 82, 84	mpjaod 860	1 ⊢ ((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵) → (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 𝑋)) = ((♯‘𝐴) · 𝑋))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   ∧ w3a 1086   = wceq 1540  ∃wex 1779   ∈ wcel 2109   ⊆ wss 3916  ∅c0 4298  {csn 4591   ↦ cmpt 5190   × cxp 5638  dom cdm 5640  ran crn 5641   ∘ ccom 5644  ⟶wf 6509  –1-1→wf1 6510  –onto→wfo 6511  –1-1-onto→wf1o 6512  ‘cfv 6513  (class class class)co 7389   supp csupp 8141  Fincfn 8920  0cc0 11074  1c1 11075  ℕcn 12187  ℤ_≥cuz 12799  ...cfz 13474  seqcseq 13972  ♯chash 14301  Basecbs 17185  +_gcplusg 17226  0_gc0g 17408   Σ_g cgsu 17409  Mndcmnd 18667  ._gcmg 19005  Cntzccntz 19253

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5236  ax-sep 5253  ax-nul 5263  ax-pow 5322  ax-pr 5389  ax-un 7713  ax-cnex 11130  ax-resscn 11131  ax-1cn 11132  ax-icn 11133  ax-addcl 11134  ax-addrcl 11135  ax-mulcl 11136  ax-mulrcl 11137  ax-mulcom 11138  ax-addass 11139  ax-mulass 11140  ax-distr 11141  ax-i2m1 11142  ax-1ne0 11143  ax-1rid 11144  ax-rnegex 11145  ax-rrecex 11146  ax-cnre 11147  ax-pre-lttri 11148  ax-pre-lttrn 11149  ax-pre-ltadd 11150  ax-pre-mulgt0 11151

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3756  df-csb 3865  df-dif 3919  df-un 3921  df-in 3923  df-ss 3933  df-pss 3936  df-nul 4299  df-if 4491  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-int 4913  df-iun 4959  df-br 5110  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5535  df-eprel 5540  df-po 5548  df-so 5549  df-fr 5593  df-se 5594  df-we 5595  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-res 5652  df-ima 5653  df-pred 6276  df-ord 6337  df-on 6338  df-lim 6339  df-suc 6340  df-iota 6466  df-fun 6515  df-fn 6516  df-f 6517  df-f1 6518  df-fo 6519  df-f1o 6520  df-fv 6521  df-isom 6522  df-riota 7346  df-ov 7392  df-oprab 7393  df-mpo 7394  df-om 7845  df-1st 7970  df-2nd 7971  df-supp 8142  df-frecs 8262  df-wrecs 8293  df-recs 8342  df-rdg 8380  df-1o 8436  df-er 8673  df-en 8921  df-dom 8922  df-sdom 8923  df-fin 8924  df-oi 9469  df-card 9898  df-pnf 11216  df-mnf 11217  df-xr 11218  df-ltxr 11219  df-le 11220  df-sub 11413  df-neg 11414  df-nn 12188  df-n0 12449  df-z 12536  df-uz 12800  df-fz 13475  df-fzo 13622  df-seq 13973  df-hash 14302  df-0g 17410  df-gsum 17411  df-mgm 18573  df-sgrp 18652  df-mnd 18668  df-mulg 19006  df-cntz 19255

This theorem is referenced by:  gsumconstf  19871  mdetdiagid  22493  chpscmat  22735  chp0mat  22739  chpidmat  22740  tmdgsum2  23989  amgmlem  26906  lgseisenlem4  27295  gsumhashmul  33007