Theorem gsumconst 19966

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 1192	. . . . 5 ⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵) ∧ 𝐴 = ∅) → 𝑋 ∈ 𝐵)
2		gsumconst.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐺)
3		eqid 2734	. . . . . 6 ⊢ (0g‘𝐺) = (0g‘𝐺)
4		gsumconst.m	. . . . . 6 ⊢ · = (.g‘𝐺)
5	2, 3, 4	mulg0 19104	. . . . 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 6906	. . . . . . 7 ⊢ (𝐴 = ∅ → (♯‘𝐴) = (♯‘∅))
8	7	adantl 481	. . . . . 6 ⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵) ∧ 𝐴 = ∅) → (♯‘𝐴) = (♯‘∅))
9		hash0 14402	. . . . . 6 ⊢ (♯‘∅) = 0
10	8, 9	eqtrdi 2790	. . . . 5 ⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵) ∧ 𝐴 = ∅) → (♯‘𝐴) = 0)
11	10	oveq1d 7445	. . . 4 ⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵) ∧ 𝐴 = ∅) → ((♯‘𝐴) · 𝑋) = (0 · 𝑋))
12		mpteq1 5240	. . . . . . . 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 6710	. . . . . . 7 ⊢ (𝑘 ∈ ∅ ↦ 𝑋) = ∅
15	13, 14	eqtrdi 2790	. . . . . 6 ⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵) ∧ 𝐴 = ∅) → (𝑘 ∈ 𝐴 ↦ 𝑋) = ∅)
16	15	oveq2d 7446	. . . . 5 ⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵) ∧ 𝐴 = ∅) → (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 𝑋)) = (𝐺 Σg ∅))
17	3	gsum0 18709	. . . . 5 ⊢ (𝐺 Σg ∅) = (0g‘𝐺)
18	16, 17	eqtrdi 2790	. . . 4 ⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵) ∧ 𝐴 = ∅) → (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 𝑋)) = (0g‘𝐺))
19	6, 11, 18	3eqtr4rd 2785	. . 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 12918	. . . . . . . 8 ⊢ ℕ = (ℤ≥‘1)
23	21, 22	eleqtrdi 2848	. . . . . . 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 1192	. . . . . . . . . 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 2734	. . . . . . . . . 10 ⊢ (𝑥 ∈ (1...(♯‘𝐴)) ↦ 𝑋) = (𝑥 ∈ (1...(♯‘𝐴)) ↦ 𝑋)
28	27	fvmpt2 7026	. . . . . . . . 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 6848	. . . . . . . . . . . . 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 7103	. . . . . . . . . . 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 6976	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → 𝑓 = (𝑥 ∈ (1...(♯‘𝐴)) ↦ (𝑓‘𝑥)))
34		eqidd 2735	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → (𝑘 ∈ 𝐴 ↦ 𝑋) = (𝑘 ∈ 𝐴 ↦ 𝑋))
35		eqidd 2735	. . . . . . . . . . 11 ⊢ (𝑘 = (𝑓‘𝑥) → 𝑋 = 𝑋)
36	32, 33, 34, 35	fmptco 7148	. . . . . . . . . 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 6908	. . . . . . . . 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 13589	. . . . . . . . 9 ⊢ (𝑥 ∈ (1...(♯‘𝐴)) → 𝑥 ∈ ℕ)
40		fvconst2g 7221	. . . . . . . . 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 2784	. . . . . . 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 14063	. . . . . 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 2734	. . . . . . 7 ⊢ (+g‘𝐺) = (+g‘𝐺)
45		eqid 2734	. . . . . . 7 ⊢ (Cntz‘𝐺) = (Cntz‘𝐺)
46		simpl1 1190	. . . . . . 7 ⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → 𝐺 ∈ Mnd)
47		simpl2 1191	. . . . . . 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 7134	. . . . . . 7 ⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → (𝑘 ∈ 𝐴 ↦ 𝑋):𝐴⟶𝐵)
50		eqidd 2735	. . . . . . . . . 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 19355	. . . . . . . . . . 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 4813	. . . . . . . 8 ⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → {𝑋} ⊆ ((Cntz‘𝐺)‘{𝑋}))
55		snidg 4664	. . . . . . . . . . . 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 7134	. . . . . . . . 9 ⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → (𝑘 ∈ 𝐴 ↦ 𝑋):𝐴⟶{𝑋})
59	58	frnd 6744	. . . . . . . 8 ⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐵) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → ran (𝑘 ∈ 𝐴 ↦ 𝑋) ⊆ {𝑋})
60	45	cntzidss 19370	. . . . . . . 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 6847	. . . . . . . 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 8200	. . . . . . . . 9 ⊢ ((𝑘 ∈ 𝐴 ↦ 𝑋) supp (0g‘𝐺)) ⊆ dom (𝑘 ∈ 𝐴 ↦ 𝑋)
65		eqid 2734	. . . . . . . . . . 11 ⊢ (𝑘 ∈ 𝐴 ↦ 𝑋) = (𝑘 ∈ 𝐴 ↦ 𝑋)
66	65	dmmptss 6262	. . . . . . . . . 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 4006	. . . . . . . 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 6855	. . . . . . . . . 10 ⊢ (𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴 → 𝑓:(1...(♯‘𝐴))–onto→𝐴)
70		forn 6823	. . . . . . . . . 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 4036	. . . . . . 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 2734	. . . . . . 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 19939	. . . . . 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 2734	. . . . . . . 8 ⊢ seq1((+g‘𝐺), (ℕ × {𝑋})) = seq1((+g‘𝐺), (ℕ × {𝑋}))
77	2, 44, 4, 76	mulgnn 19105	. . . . . . 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 2784	. . . . 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 1930	. . 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 15742	. . 3 ⊢ (𝐴 ∈ Fin → (𝐴 = ∅ ∨ ((♯‘𝐴) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)))
84	83	3ad2ant2 1133	. 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 1536  ∃wex 1775   ∈ wcel 2105   ⊆ wss 3962  ∅c0 4338  {csn 4630   ↦ cmpt 5230   × cxp 5686  dom cdm 5688  ran crn 5689   ∘ ccom 5692  ⟶wf 6558  –1-1→wf1 6559  –onto→wfo 6560  –1-1-onto→wf1o 6561  ‘cfv 6562  (class class class)co 7430   supp csupp 8183  Fincfn 8983  0cc0 11152  1c1 11153  ℕcn 12263  ℤ_≥cuz 12875  ...cfz 13543  seqcseq 14038  ♯chash 14365  Basecbs 17244  +_gcplusg 17297  0_gc0g 17485   Σ_g cgsu 17486  Mndcmnd 18759  ._gcmg 19097  Cntzccntz 19345

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-se 5641  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-isom 6571  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-1st 8012  df-2nd 8013  df-supp 8184  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-er 8743  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-oi 9547  df-card 9976  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-nn 12264  df-n0 12524  df-z 12611  df-uz 12876  df-fz 13544  df-fzo 13691  df-seq 14039  df-hash 14366  df-0g 17487  df-gsum 17488  df-mgm 18665  df-sgrp 18744  df-mnd 18760  df-mulg 19098  df-cntz 19347

This theorem is referenced by:  gsumconstf  19967  mdetdiagid  22621  chpscmat  22863  chp0mat  22867  chpidmat  22868  tmdgsum2  24119  amgmlem  27047  lgseisenlem4  27436  gsumhashmul  33046