Theorem gsumconst 19839

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 18979	. . . . 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 6817	. . . . . . 7 ⊢ (𝐴 = ∅ → (♯‘𝐴) = (♯‘∅))
8	7	adantl 481	. . . . . 6 ⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵) ∧ 𝐴 = ∅) → (♯‘𝐴) = (♯‘∅))
9		hash0 14266	. . . . . 6 ⊢ (♯‘∅) = 0
10	8, 9	eqtrdi 2781	. . . . 5 ⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵) ∧ 𝐴 = ∅) → (♯‘𝐴) = 0)
11	10	oveq1d 7356	. . . 4 ⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵) ∧ 𝐴 = ∅) → ((♯‘𝐴) · 𝑋) = (0 · 𝑋))
12		mpteq1 5178	. . . . . . . 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 6619	. . . . . . 7 ⊢ (𝑘 ∈ ∅ ↦ 𝑋) = ∅
15	13, 14	eqtrdi 2781	. . . . . 6 ⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵) ∧ 𝐴 = ∅) → (𝑘 ∈ 𝐴 ↦ 𝑋) = ∅)
16	15	oveq2d 7357	. . . . 5 ⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵) ∧ 𝐴 = ∅) → (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 𝑋)) = (𝐺 Σg ∅))
17	3	gsum0 18584	. . . . 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 12767	. . . . . . . 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 6935	. . . . . . . . 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 6759	. . . . . . . . . . . . 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 7012	. . . . . . . . . . 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 6885	. . . . . . . . . . 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 7057	. . . . . . . . . 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 6819	. . . . . . . . 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 13445	. . . . . . . . 9 ⊢ (𝑥 ∈ (1...(♯‘𝐴)) → 𝑥 ∈ ℕ)
40		fvconst2g 7131	. . . . . . . . 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 13925	. . . . . 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 7043	. . . . . . 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 19230	. . . . . . . . . . 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 4759	. . . . . . . 8 ⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → {𝑋} ⊆ ((Cntz‘𝐺)‘{𝑋}))
55		snidg 4611	. . . . . . . . . . . 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 7043	. . . . . . . . 9 ⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → (𝑘 ∈ 𝐴 ↦ 𝑋):𝐴⟶{𝑋})
59	58	frnd 6655	. . . . . . . 8 ⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → ran (𝑘 ∈ 𝐴 ↦ 𝑋) ⊆ {𝑋})
60	45	cntzidss 19245	. . . . . . . 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 6758	. . . . . . . 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 8102	. . . . . . . . 9 ⊢ ((𝑘 ∈ 𝐴 ↦ 𝑋) supp (0g‘𝐺)) ⊆ dom (𝑘 ∈ 𝐴 ↦ 𝑋)
65		eqid 2730	. . . . . . . . . . 11 ⊢ (𝑘 ∈ 𝐴 ↦ 𝑋) = (𝑘 ∈ 𝐴 ↦ 𝑋)
66	65	dmmptss 6185	. . . . . . . . . 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 3944	. . . . . . . 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 6766	. . . . . . . . . 10 ⊢ (𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴 → 𝑓:(1...(♯‘𝐴))–onto→𝐴)
70		forn 6734	. . . . . . . . . 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 3970	. . . . . . 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 19812	. . . . . 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 18980	. . . . . . 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 1934	. . 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 15609	. . 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 1541  ∃wex 1780   ∈ wcel 2110   ⊆ wss 3900  ∅c0 4281  {csn 4574   ↦ cmpt 5170   × cxp 5612  dom cdm 5614  ran crn 5615   ∘ ccom 5618  ⟶wf 6473  –1-1→wf1 6474  –onto→wfo 6475  –1-1-onto→wf1o 6476  ‘cfv 6477  (class class class)co 7341   supp csupp 8085  Fincfn 8864  0cc0 10998  1c1 10999  ℕcn 12117  ℤ_≥cuz 12724  ...cfz 13399  seqcseq 13900  ♯chash 14229  Basecbs 17112  +_gcplusg 17153  0_gc0g 17335   Σ_g cgsu 17336  Mndcmnd 18634  ._gcmg 18972  Cntzccntz 19220

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2112  ax-9 2120  ax-10 2143  ax-11 2159  ax-12 2179  ax-ext 2702  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7663  ax-cnex 11054  ax-resscn 11055  ax-1cn 11056  ax-icn 11057  ax-addcl 11058  ax-addrcl 11059  ax-mulcl 11060  ax-mulrcl 11061  ax-mulcom 11062  ax-addass 11063  ax-mulass 11064  ax-distr 11065  ax-i2m1 11066  ax-1ne0 11067  ax-1rid 11068  ax-rnegex 11069  ax-rrecex 11070  ax-cnre 11071  ax-pre-lttri 11072  ax-pre-lttrn 11073  ax-pre-ltadd 11074  ax-pre-mulgt0 11075

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2067  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 3344  df-reu 3345  df-rab 3394  df-v 3436  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-pss 3920  df-nul 4282  df-if 4474  df-pw 4550  df-sn 4575  df-pr 4577  df-op 4581  df-uni 4858  df-int 4896  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-se 5568  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6244  df-ord 6305  df-on 6306  df-lim 6307  df-suc 6308  df-iota 6433  df-fun 6479  df-fn 6480  df-f 6481  df-f1 6482  df-fo 6483  df-f1o 6484  df-fv 6485  df-isom 6486  df-riota 7298  df-ov 7344  df-oprab 7345  df-mpo 7346  df-om 7792  df-1st 7916  df-2nd 7917  df-supp 8086  df-frecs 8206  df-wrecs 8237  df-recs 8286  df-rdg 8324  df-1o 8380  df-er 8617  df-en 8865  df-dom 8866  df-sdom 8867  df-fin 8868  df-oi 9391  df-card 9824  df-pnf 11140  df-mnf 11141  df-xr 11142  df-ltxr 11143  df-le 11144  df-sub 11338  df-neg 11339  df-nn 12118  df-n0 12374  df-z 12461  df-uz 12725  df-fz 13400  df-fzo 13547  df-seq 13901  df-hash 14230  df-0g 17337  df-gsum 17338  df-mgm 18540  df-sgrp 18619  df-mnd 18635  df-mulg 18973  df-cntz 19222

This theorem is referenced by:  gsumconstf  19840  mdetdiagid  22508  chpscmat  22750  chp0mat  22754  chpidmat  22755  tmdgsum2  24004  amgmlem  26920  lgseisenlem4  27309  gsumhashmul  33031  gsumind  33300