Theorem gsumconst 19867

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 1195	. . . . 5 ⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵) ∧ 𝐴 = ∅) → 𝑋 ∈ 𝐵)
2		gsumconst.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐺)
3		eqid 2737	. . . . . 6 ⊢ (0g‘𝐺) = (0g‘𝐺)
4		gsumconst.m	. . . . . 6 ⊢ · = (.g‘𝐺)
5	2, 3, 4	mulg0 19008	. . . . 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 6832	. . . . . . 7 ⊢ (𝐴 = ∅ → (♯‘𝐴) = (♯‘∅))
8	7	adantl 481	. . . . . 6 ⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵) ∧ 𝐴 = ∅) → (♯‘𝐴) = (♯‘∅))
9		hash0 14291	. . . . . 6 ⊢ (♯‘∅) = 0
10	8, 9	eqtrdi 2788	. . . . 5 ⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵) ∧ 𝐴 = ∅) → (♯‘𝐴) = 0)
11	10	oveq1d 7373	. . . 4 ⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵) ∧ 𝐴 = ∅) → ((♯‘𝐴) · 𝑋) = (0 · 𝑋))
12		mpteq1 5175	. . . . . . . 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 6632	. . . . . . 7 ⊢ (𝑘 ∈ ∅ ↦ 𝑋) = ∅
15	13, 14	eqtrdi 2788	. . . . . 6 ⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵) ∧ 𝐴 = ∅) → (𝑘 ∈ 𝐴 ↦ 𝑋) = ∅)
16	15	oveq2d 7374	. . . . 5 ⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵) ∧ 𝐴 = ∅) → (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 𝑋)) = (𝐺 Σg ∅))
17	3	gsum0 18610	. . . . 5 ⊢ (𝐺 Σg ∅) = (0g‘𝐺)
18	16, 17	eqtrdi 2788	. . . 4 ⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵) ∧ 𝐴 = ∅) → (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 𝑋)) = (0g‘𝐺))
19	6, 11, 18	3eqtr4rd 2783	. . 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 771	. . . . . . . 8 ⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → (♯‘𝐴) ∈ ℕ)
22		nnuz 12791	. . . . . . . 8 ⊢ ℕ = (ℤ≥‘1)
23	21, 22	eleqtrdi 2847	. . . . . . 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 1195	. . . . . . . . . 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 2737	. . . . . . . . . 10 ⊢ (𝑥 ∈ (1...(♯‘𝐴)) ↦ 𝑋) = (𝑥 ∈ (1...(♯‘𝐴)) ↦ 𝑋)
28	27	fvmpt2 6951	. . . . . . . . 9 ⊢ ((𝑥 ∈ (1...(♯‘𝐴)) ∧ 𝑋 ∈ 𝐵) → ((𝑥 ∈ (1...(♯‘𝐴)) ↦ 𝑋)‘𝑥) = 𝑋)
29	24, 26, 28	syl2anc 585	. . . . . . . 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 6772	. . . . . . . . . . . . 13 ⊢ (𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴 → 𝑓:(1...(♯‘𝐴))⟶𝐴)
31	30	ad2antll 730	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → 𝑓:(1...(♯‘𝐴))⟶𝐴)
32	31	ffvelcdmda 7028	. . . . . . . . . . 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 6900	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → 𝑓 = (𝑥 ∈ (1...(♯‘𝐴)) ↦ (𝑓‘𝑥)))
34		eqidd 2738	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → (𝑘 ∈ 𝐴 ↦ 𝑋) = (𝑘 ∈ 𝐴 ↦ 𝑋))
35		eqidd 2738	. . . . . . . . . . 11 ⊢ (𝑘 = (𝑓‘𝑥) → 𝑋 = 𝑋)
36	32, 33, 34, 35	fmptco 7074	. . . . . . . . . 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 6834	. . . . . . . . 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 13470	. . . . . . . . 9 ⊢ (𝑥 ∈ (1...(♯‘𝐴)) → 𝑥 ∈ ℕ)
40		fvconst2g 7148	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑥 ∈ ℕ) → ((ℕ × {𝑋})‘𝑥) = 𝑋)
41	25, 39, 40	syl2an 597	. . . . . . . 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 2782	. . . . . . 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 13950	. . . . . 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 2737	. . . . . . 7 ⊢ (+g‘𝐺) = (+g‘𝐺)
45		eqid 2737	. . . . . . 7 ⊢ (Cntz‘𝐺) = (Cntz‘𝐺)
46		simpl1 1193	. . . . . . 7 ⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → 𝐺 ∈ Mnd)
47		simpl2 1194	. . . . . . 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 7059	. . . . . . 7 ⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → (𝑘 ∈ 𝐴 ↦ 𝑋):𝐴⟶𝐵)
50		eqidd 2738	. . . . . . . . . 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 19258	. . . . . . . . . . 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 714	. . . . . . . . 9 ⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → 𝑋 ∈ ((Cntz‘𝐺)‘{𝑋}))
54	53	snssd 4753	. . . . . . . 8 ⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → {𝑋} ⊆ ((Cntz‘𝐺)‘{𝑋}))
55		snidg 4605	. . . . . . . . . . . 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 7059	. . . . . . . . 9 ⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → (𝑘 ∈ 𝐴 ↦ 𝑋):𝐴⟶{𝑋})
59	58	frnd 6668	. . . . . . . 8 ⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → ran (𝑘 ∈ 𝐴 ↦ 𝑋) ⊆ {𝑋})
60	45	cntzidss 19273	. . . . . . . 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 585	. . . . . . 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 6771	. . . . . . . 8 ⊢ (𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴 → 𝑓:(1...(♯‘𝐴))–1-1→𝐴)
63	62	ad2antll 730	. . . . . . 7 ⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → 𝑓:(1...(♯‘𝐴))–1-1→𝐴)
64		suppssdm 8118	. . . . . . . . 9 ⊢ ((𝑘 ∈ 𝐴 ↦ 𝑋) supp (0g‘𝐺)) ⊆ dom (𝑘 ∈ 𝐴 ↦ 𝑋)
65		eqid 2737	. . . . . . . . . . 11 ⊢ (𝑘 ∈ 𝐴 ↦ 𝑋) = (𝑘 ∈ 𝐴 ↦ 𝑋)
66	65	dmmptss 6197	. . . . . . . . . 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 3934	. . . . . . . 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 6779	. . . . . . . . . 10 ⊢ (𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴 → 𝑓:(1...(♯‘𝐴))–onto→𝐴)
70		forn 6747	. . . . . . . . . 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 730	. . . . . . . 8 ⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → ran 𝑓 = 𝐴)
73	68, 72	sseqtrrd 3960	. . . . . . 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 2737	. . . . . . 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 19840	. . . . . 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 2737	. . . . . . . 8 ⊢ seq1((+g‘𝐺), (ℕ × {𝑋})) = seq1((+g‘𝐺), (ℕ × {𝑋}))
77	2, 44, 4, 76	mulgnn 19009	. . . . . . 7 ⊢ (((♯‘𝐴) ∈ ℕ ∧ 𝑋 ∈ 𝐵) → ((♯‘𝐴) · 𝑋) = (seq1((+g‘𝐺), (ℕ × {𝑋}))‘(♯‘𝐴)))
78	21, 25, 77	syl2anc 585	. . . . . 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 2782	. . . . 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 1935	. . 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 15634	. . 3 ⊢ (𝐴 ∈ Fin → (𝐴 = ∅ ∨ ((♯‘𝐴) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)))
84	83	3ad2ant2 1135	. 2 ⊢ ((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵) → (𝐴 = ∅ ∨ ((♯‘𝐴) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)))
85	20, 82, 84	mpjaod 861	1 ⊢ ((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵) → (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 𝑋)) = ((♯‘𝐴) · 𝑋))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 848   ∧ w3a 1087   = wceq 1542  ∃wex 1781   ∈ wcel 2114   ⊆ wss 3890  ∅c0 4274  {csn 4568   ↦ cmpt 5167   × cxp 5620  dom cdm 5622  ran crn 5623   ∘ ccom 5626  ⟶wf 6486  –1-1→wf1 6487  –onto→wfo 6488  –1-1-onto→wf1o 6489  ‘cfv 6490  (class class class)co 7358   supp csupp 8101  Fincfn 8884  0cc0 11027  1c1 11028  ℕcn 12146  ℤ_≥cuz 12752  ...cfz 13424  seqcseq 13925  ♯chash 14254  Basecbs 17137  +_gcplusg 17178  0_gc0g 17360   Σ_g cgsu 17361  Mndcmnd 18660  ._gcmg 19001  Cntzccntz 19248

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5300  ax-pr 5368  ax-un 7680  ax-cnex 11083  ax-resscn 11084  ax-1cn 11085  ax-icn 11086  ax-addcl 11087  ax-addrcl 11088  ax-mulcl 11089  ax-mulrcl 11090  ax-mulcom 11091  ax-addass 11092  ax-mulass 11093  ax-distr 11094  ax-i2m1 11095  ax-1ne0 11096  ax-1rid 11097  ax-rnegex 11098  ax-rrecex 11099  ax-cnre 11100  ax-pre-lttri 11101  ax-pre-lttrn 11102  ax-pre-ltadd 11103  ax-pre-mulgt0 11104

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-se 5576  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-pred 6257  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-isom 6499  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-1st 7933  df-2nd 7934  df-supp 8102  df-frecs 8222  df-wrecs 8253  df-recs 8302  df-rdg 8340  df-1o 8396  df-er 8634  df-en 8885  df-dom 8886  df-sdom 8887  df-fin 8888  df-oi 9416  df-card 9852  df-pnf 11169  df-mnf 11170  df-xr 11171  df-ltxr 11172  df-le 11173  df-sub 11367  df-neg 11368  df-nn 12147  df-n0 12403  df-z 12490  df-uz 12753  df-fz 13425  df-fzo 13572  df-seq 13926  df-hash 14255  df-0g 17362  df-gsum 17363  df-mgm 18566  df-sgrp 18645  df-mnd 18661  df-mulg 19002  df-cntz 19250

This theorem is referenced by:  gsumconstf  19868  mdetdiagid  22543  chpscmat  22785  chp0mat  22789  chpidmat  22790  tmdgsum2  24039  amgmlem  26940  lgseisenlem4  27329  gsumhashmul  33133  gsumind  33410