Theorem gsumconst 18157

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 1059	. . . . 5 ⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵) ∧ 𝐴 = ∅) → 𝑋 ∈ 𝐵)
2		gsumconst.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐺)
3		eqid 2610	. . . . . 6 ⊢ (0g‘𝐺) = (0g‘𝐺)
4		gsumconst.m	. . . . . 6 ⊢ · = (.g‘𝐺)
5	2, 3, 4	mulg0 17369	. . . . 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 6103	. . . . . . 7 ⊢ (𝐴 = ∅ → (#‘𝐴) = (#‘∅))
8	7	adantl 481	. . . . . 6 ⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵) ∧ 𝐴 = ∅) → (#‘𝐴) = (#‘∅))
9		hash0 13019	. . . . . 6 ⊢ (#‘∅) = 0
10	8, 9	syl6eq 2660	. . . . 5 ⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵) ∧ 𝐴 = ∅) → (#‘𝐴) = 0)
11	10	oveq1d 6564	. . . 4 ⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵) ∧ 𝐴 = ∅) → ((#‘𝐴) · 𝑋) = (0 · 𝑋))
12		mpteq1 4665	. . . . . . . 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 5934	. . . . . . 7 ⊢ (𝑘 ∈ ∅ ↦ 𝑋) = ∅
15	13, 14	syl6eq 2660	. . . . . 6 ⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵) ∧ 𝐴 = ∅) → (𝑘 ∈ 𝐴 ↦ 𝑋) = ∅)
16	15	oveq2d 6565	. . . . 5 ⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵) ∧ 𝐴 = ∅) → (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 𝑋)) = (𝐺 Σg ∅))
17	3	gsum0 17101	. . . . 5 ⊢ (𝐺 Σg ∅) = (0g‘𝐺)
18	16, 17	syl6eq 2660	. . . 4 ⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵) ∧ 𝐴 = ∅) → (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 𝑋)) = (0g‘𝐺))
19	6, 11, 18	3eqtr4rd 2655	. . 3 ⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵) ∧ 𝐴 = ∅) → (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 𝑋)) = ((#‘𝐴) · 𝑋))
20	19	ex 449	. 2 ⊢ ((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵) → (𝐴 = ∅ → (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 𝑋)) = ((#‘𝐴) · 𝑋)))
21		simprl 790	. . . . . . . 8 ⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵) ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto→𝐴)) → (#‘𝐴) ∈ ℕ)
22		nnuz 11599	. . . . . . . 8 ⊢ ℕ = (ℤ≥‘1)
23	21, 22	syl6eleq 2698	. . . . . . 7 ⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵) ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto→𝐴)) → (#‘𝐴) ∈ (ℤ≥‘1))
24		simpr 476	. . . . . . . . 9 ⊢ ((((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵) ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto→𝐴)) ∧ 𝑥 ∈ (1...(#‘𝐴))) → 𝑥 ∈ (1...(#‘𝐴)))
25		simpl3 1059	. . . . . . . . . 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 2610	. . . . . . . . . 10 ⊢ (𝑥 ∈ (1...(#‘𝐴)) ↦ 𝑋) = (𝑥 ∈ (1...(#‘𝐴)) ↦ 𝑋)
28	27	fvmpt2 6200	. . . . . . . . 9 ⊢ ((𝑥 ∈ (1...(#‘𝐴)) ∧ 𝑋 ∈ 𝐵) → ((𝑥 ∈ (1...(#‘𝐴)) ↦ 𝑋)‘𝑥) = 𝑋)
29	24, 26, 28	syl2anc 691	. . . . . . . 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 6050	. . . . . . . . . . . . 13 ⊢ (𝑓:(1...(#‘𝐴))–1-1-onto→𝐴 → 𝑓:(1...(#‘𝐴))⟶𝐴)
31	30	ad2antll 761	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵) ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto→𝐴)) → 𝑓:(1...(#‘𝐴))⟶𝐴)
32	31	ffvelrnda 6267	. . . . . . . . . . 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 6159	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵) ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto→𝐴)) → 𝑓 = (𝑥 ∈ (1...(#‘𝐴)) ↦ (𝑓‘𝑥)))
34		eqidd 2611	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵) ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto→𝐴)) → (𝑘 ∈ 𝐴 ↦ 𝑋) = (𝑘 ∈ 𝐴 ↦ 𝑋))
35		eqidd 2611	. . . . . . . . . . 11 ⊢ (𝑘 = (𝑓‘𝑥) → 𝑋 = 𝑋)
36	32, 33, 34, 35	fmptco 6303	. . . . . . . . . 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 6105	. . . . . . . . 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 12241	. . . . . . . . 9 ⊢ (𝑥 ∈ (1...(#‘𝐴)) → 𝑥 ∈ ℕ)
40		fvconst2g 6372	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑥 ∈ ℕ) → ((ℕ × {𝑋})‘𝑥) = 𝑋)
41	25, 39, 40	syl2an 493	. . . . . . . 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 2654	. . . . . . 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 12687	. . . . . 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 2610	. . . . . . 7 ⊢ (+g‘𝐺) = (+g‘𝐺)
45		eqid 2610	. . . . . . 7 ⊢ (Cntz‘𝐺) = (Cntz‘𝐺)
46		simpl1 1057	. . . . . . 7 ⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵) ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto→𝐴)) → 𝐺 ∈ Mnd)
47		simpl2 1058	. . . . . . 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		eqid 2610	. . . . . . . 8 ⊢ (𝑘 ∈ 𝐴 ↦ 𝑋) = (𝑘 ∈ 𝐴 ↦ 𝑋)
50	48, 49	fmptd 6292	. . . . . . 7 ⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵) ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto→𝐴)) → (𝑘 ∈ 𝐴 ↦ 𝑋):𝐴⟶𝐵)
51		eqidd 2611	. . . . . . . . . 10 ⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵) ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto→𝐴)) → (𝑋(+g‘𝐺)𝑋) = (𝑋(+g‘𝐺)𝑋))
52	2, 44, 45	elcntzsn 17581	. . . . . . . . . . 11 ⊢ (𝑋 ∈ 𝐵 → (𝑋 ∈ ((Cntz‘𝐺)‘{𝑋}) ↔ (𝑋 ∈ 𝐵 ∧ (𝑋(+g‘𝐺)𝑋) = (𝑋(+g‘𝐺)𝑋))))
53	25, 52	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))))
54	25, 51, 53	mpbir2and 959	. . . . . . . . 9 ⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵) ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto→𝐴)) → 𝑋 ∈ ((Cntz‘𝐺)‘{𝑋}))
55	54	snssd 4281	. . . . . . . 8 ⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵) ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto→𝐴)) → {𝑋} ⊆ ((Cntz‘𝐺)‘{𝑋}))
56		snidg 4153	. . . . . . . . . . . 12 ⊢ (𝑋 ∈ 𝐵 → 𝑋 ∈ {𝑋})
57	25, 56	syl 17	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵) ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto→𝐴)) → 𝑋 ∈ {𝑋})
58	57	adantr 480	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵) ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto→𝐴)) ∧ 𝑘 ∈ 𝐴) → 𝑋 ∈ {𝑋})
59	58, 49	fmptd 6292	. . . . . . . . 9 ⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵) ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto→𝐴)) → (𝑘 ∈ 𝐴 ↦ 𝑋):𝐴⟶{𝑋})
60		frn 5966	. . . . . . . . 9 ⊢ ((𝑘 ∈ 𝐴 ↦ 𝑋):𝐴⟶{𝑋} → ran (𝑘 ∈ 𝐴 ↦ 𝑋) ⊆ {𝑋})
61	59, 60	syl 17	. . . . . . . 8 ⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵) ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto→𝐴)) → ran (𝑘 ∈ 𝐴 ↦ 𝑋) ⊆ {𝑋})
62	45	cntzidss 17593	. . . . . . . 8 ⊢ (({𝑋} ⊆ ((Cntz‘𝐺)‘{𝑋}) ∧ ran (𝑘 ∈ 𝐴 ↦ 𝑋) ⊆ {𝑋}) → ran (𝑘 ∈ 𝐴 ↦ 𝑋) ⊆ ((Cntz‘𝐺)‘ran (𝑘 ∈ 𝐴 ↦ 𝑋)))
63	55, 61, 62	syl2anc 691	. . . . . . 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)))
64		f1of1 6049	. . . . . . . 8 ⊢ (𝑓:(1...(#‘𝐴))–1-1-onto→𝐴 → 𝑓:(1...(#‘𝐴))–1-1→𝐴)
65	64	ad2antll 761	. . . . . . 7 ⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵) ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto→𝐴)) → 𝑓:(1...(#‘𝐴))–1-1→𝐴)
66		suppssdm 7195	. . . . . . . . 9 ⊢ ((𝑘 ∈ 𝐴 ↦ 𝑋) supp (0g‘𝐺)) ⊆ dom (𝑘 ∈ 𝐴 ↦ 𝑋)
67	49	dmmptss 5548	. . . . . . . . . 10 ⊢ dom (𝑘 ∈ 𝐴 ↦ 𝑋) ⊆ 𝐴
68	67	a1i 11	. . . . . . . . 9 ⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵) ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto→𝐴)) → dom (𝑘 ∈ 𝐴 ↦ 𝑋) ⊆ 𝐴)
69	66, 68	syl5ss 3579	. . . . . . . 8 ⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵) ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto→𝐴)) → ((𝑘 ∈ 𝐴 ↦ 𝑋) supp (0g‘𝐺)) ⊆ 𝐴)
70		f1ofo 6057	. . . . . . . . . 10 ⊢ (𝑓:(1...(#‘𝐴))–1-1-onto→𝐴 → 𝑓:(1...(#‘𝐴))–onto→𝐴)
71		forn 6031	. . . . . . . . . 10 ⊢ (𝑓:(1...(#‘𝐴))–onto→𝐴 → ran 𝑓 = 𝐴)
72	70, 71	syl 17	. . . . . . . . 9 ⊢ (𝑓:(1...(#‘𝐴))–1-1-onto→𝐴 → ran 𝑓 = 𝐴)
73	72	ad2antll 761	. . . . . . . 8 ⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵) ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto→𝐴)) → ran 𝑓 = 𝐴)
74	69, 73	sseqtr4d 3605	. . . . . . 7 ⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵) ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto→𝐴)) → ((𝑘 ∈ 𝐴 ↦ 𝑋) supp (0g‘𝐺)) ⊆ ran 𝑓)
75		eqid 2610	. . . . . . 7 ⊢ (((𝑘 ∈ 𝐴 ↦ 𝑋) ∘ 𝑓) supp (0g‘𝐺)) = (((𝑘 ∈ 𝐴 ↦ 𝑋) ∘ 𝑓) supp (0g‘𝐺))
76	2, 3, 44, 45, 46, 47, 50, 63, 21, 65, 74, 75	gsumval3 18131	. . . . . 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)))
77		eqid 2610	. . . . . . . 8 ⊢ seq1((+g‘𝐺), (ℕ × {𝑋})) = seq1((+g‘𝐺), (ℕ × {𝑋}))
78	2, 44, 4, 77	mulgnn 17370	. . . . . . 7 ⊢ (((#‘𝐴) ∈ ℕ ∧ 𝑋 ∈ 𝐵) → ((#‘𝐴) · 𝑋) = (seq1((+g‘𝐺), (ℕ × {𝑋}))‘(#‘𝐴)))
79	21, 25, 78	syl2anc 691	. . . . . 6 ⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵) ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto→𝐴)) → ((#‘𝐴) · 𝑋) = (seq1((+g‘𝐺), (ℕ × {𝑋}))‘(#‘𝐴)))
80	43, 76, 79	3eqtr4d 2654	. . . . 5 ⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵) ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto→𝐴)) → (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 𝑋)) = ((#‘𝐴) · 𝑋))
81	80	expr 641	. . . 4 ⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵) ∧ (#‘𝐴) ∈ ℕ) → (𝑓:(1...(#‘𝐴))–1-1-onto→𝐴 → (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 𝑋)) = ((#‘𝐴) · 𝑋)))
82	81	exlimdv 1848	. . 3 ⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵) ∧ (#‘𝐴) ∈ ℕ) → (∃𝑓 𝑓:(1...(#‘𝐴))–1-1-onto→𝐴 → (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 𝑋)) = ((#‘𝐴) · 𝑋)))
83	82	expimpd 627	. 2 ⊢ ((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵) → (((#‘𝐴) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(#‘𝐴))–1-1-onto→𝐴) → (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 𝑋)) = ((#‘𝐴) · 𝑋)))
84		fz1f1o 14288	. . 3 ⊢ (𝐴 ∈ Fin → (𝐴 = ∅ ∨ ((#‘𝐴) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(#‘𝐴))–1-1-onto→𝐴)))
85	84	3ad2ant2 1076	. 2 ⊢ ((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵) → (𝐴 = ∅ ∨ ((#‘𝐴) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(#‘𝐴))–1-1-onto→𝐴)))
86	20, 83, 85	mpjaod 395	1 ⊢ ((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵) → (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 𝑋)) = ((#‘𝐴) · 𝑋))

Syntax hints:   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   ∧ w3a 1031   = wceq 1475  ∃wex 1695   ∈ wcel 1977   ⊆ wss 3540  ∅c0 3874  {csn 4125   ↦ cmpt 4643   × cxp 5036  dom cdm 5038  ran crn 5039   ∘ ccom 5042  ⟶wf 5800  –1-1→wf1 5801  –onto→wfo 5802  –1-1-onto→wf1o 5803  ‘cfv 5804  (class class class)co 6549   supp csupp 7182  Fincfn 7841  0cc0 9815  1c1 9816  ℕcn 10897  ℤ_≥cuz 11563  ...cfz 12197  seqcseq 12663  #chash 12979  Basecbs 15695  +_gcplusg 15768  0_gc0g 15923   Σ_g cgsu 15924  Mndcmnd 17117  ._gcmg 17363  Cntzccntz 17571

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-supp 7183  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-oi 8298  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-seq 12664  df-hash 12980  df-0g 15925  df-gsum 15926  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-mulg 17364  df-cntz 17573

This theorem is referenced by:  gsumconstf  18158  mdetdiagid  20225  chpscmat  20466  chp0mat  20470  chpidmat  20471  tmdgsum2  21710  amgmlem  24516  lgseisenlem4  24903