Theorem gsumfzz 13067

Description: Value of a group sum over the zero element. (Contributed by Mario Carneiro, 7-Dec-2014.) (Revised by Jim Kingdon, 15-Aug-2025.)

Hypothesis

Ref	Expression
gsumz.z	⊢ 0 = (0g‘𝐺)

Assertion

Ref	Expression
gsumfzz	⊢ ((𝐺 ∈ Mnd ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐺 Σg (𝑘 ∈ (𝑀...𝑁) ↦ 0 )) = 0 )

Distinct variable groups:   0 ,𝑘   𝑘,𝐺   𝑘,𝑀   𝑘,𝑁

Proof of Theorem gsumfzz

Dummy variables 𝑤 𝑢 𝑣 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2193	. . . . 5 ⊢ (Base‘𝐺) = (Base‘𝐺)
2		gsumz.z	. . . . 5 ⊢ 0 = (0g‘𝐺)
3		eqid 2193	. . . . 5 ⊢ (+g‘𝐺) = (+g‘𝐺)
4		simp1 999	. . . . 5 ⊢ ((𝐺 ∈ Mnd ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 𝐺 ∈ Mnd)
5		simp2 1000	. . . . 5 ⊢ ((𝐺 ∈ Mnd ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 𝑀 ∈ ℤ)
6		simp3 1001	. . . . 5 ⊢ ((𝐺 ∈ Mnd ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 𝑁 ∈ ℤ)
7	1, 2	mndidcl 13011	. . . . . . . 8 ⊢ (𝐺 ∈ Mnd → 0 ∈ (Base‘𝐺))
8	4, 7	syl 14	. . . . . . 7 ⊢ ((𝐺 ∈ Mnd ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 0 ∈ (Base‘𝐺))
9	8	adantr 276	. . . . . 6 ⊢ (((𝐺 ∈ Mnd ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑘 ∈ (𝑀...𝑁)) → 0 ∈ (Base‘𝐺))
10	9	fmpttd 5713	. . . . 5 ⊢ ((𝐺 ∈ Mnd ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑘 ∈ (𝑀...𝑁) ↦ 0 ):(𝑀...𝑁)⟶(Base‘𝐺))
11	1, 2, 3, 4, 5, 6, 10	gsumfzval 12974	. . . 4 ⊢ ((𝐺 ∈ Mnd ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐺 Σg (𝑘 ∈ (𝑀...𝑁) ↦ 0 )) = if(𝑁 < 𝑀, 0 , (seq𝑀((+g‘𝐺), (𝑘 ∈ (𝑀...𝑁) ↦ 0 ))‘𝑁)))
12	11	adantr 276	. . 3 ⊢ (((𝐺 ∈ Mnd ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑁 < 𝑀) → (𝐺 Σg (𝑘 ∈ (𝑀...𝑁) ↦ 0 )) = if(𝑁 < 𝑀, 0 , (seq𝑀((+g‘𝐺), (𝑘 ∈ (𝑀...𝑁) ↦ 0 ))‘𝑁)))
13		simpr 110	. . . 4 ⊢ (((𝐺 ∈ Mnd ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑁 < 𝑀) → 𝑁 < 𝑀)
14	13	iftrued 3564	. . 3 ⊢ (((𝐺 ∈ Mnd ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑁 < 𝑀) → if(𝑁 < 𝑀, 0 , (seq𝑀((+g‘𝐺), (𝑘 ∈ (𝑀...𝑁) ↦ 0 ))‘𝑁)) = 0 )
15	12, 14	eqtrd 2226	. 2 ⊢ (((𝐺 ∈ Mnd ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑁 < 𝑀) → (𝐺 Σg (𝑘 ∈ (𝑀...𝑁) ↦ 0 )) = 0 )
16	11	adantr 276	. . 3 ⊢ (((𝐺 ∈ Mnd ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ 𝑁 < 𝑀) → (𝐺 Σg (𝑘 ∈ (𝑀...𝑁) ↦ 0 )) = if(𝑁 < 𝑀, 0 , (seq𝑀((+g‘𝐺), (𝑘 ∈ (𝑀...𝑁) ↦ 0 ))‘𝑁)))
17		simpr 110	. . . 4 ⊢ (((𝐺 ∈ Mnd ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ 𝑁 < 𝑀) → ¬ 𝑁 < 𝑀)
18	17	iffalsed 3567	. . 3 ⊢ (((𝐺 ∈ Mnd ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ 𝑁 < 𝑀) → if(𝑁 < 𝑀, 0 , (seq𝑀((+g‘𝐺), (𝑘 ∈ (𝑀...𝑁) ↦ 0 ))‘𝑁)) = (seq𝑀((+g‘𝐺), (𝑘 ∈ (𝑀...𝑁) ↦ 0 ))‘𝑁))
19	5	adantr 276	. . . . . 6 ⊢ (((𝐺 ∈ Mnd ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ 𝑁 < 𝑀) → 𝑀 ∈ ℤ)
20	6	adantr 276	. . . . . 6 ⊢ (((𝐺 ∈ Mnd ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ 𝑁 < 𝑀) → 𝑁 ∈ ℤ)
21	5	zred 9439	. . . . . . . 8 ⊢ ((𝐺 ∈ Mnd ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 𝑀 ∈ ℝ)
22	6	zred 9439	. . . . . . . 8 ⊢ ((𝐺 ∈ Mnd ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 𝑁 ∈ ℝ)
23	21, 22	lenltd 8137	. . . . . . 7 ⊢ ((𝐺 ∈ Mnd ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ≤ 𝑁 ↔ ¬ 𝑁 < 𝑀))
24	23	biimpar 297	. . . . . 6 ⊢ (((𝐺 ∈ Mnd ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ 𝑁 < 𝑀) → 𝑀 ≤ 𝑁)
25		eluz2 9598	. . . . . 6 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) ↔ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁))
26	19, 20, 24, 25	syl3anbrc 1183	. . . . 5 ⊢ (((𝐺 ∈ Mnd ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ 𝑁 < 𝑀) → 𝑁 ∈ (ℤ≥‘𝑀))
27		eluzfz2 10098	. . . . 5 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ (𝑀...𝑁))
28	26, 27	syl 14	. . . 4 ⊢ (((𝐺 ∈ Mnd ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ 𝑁 < 𝑀) → 𝑁 ∈ (𝑀...𝑁))
29	4	adantr 276	. . . 4 ⊢ (((𝐺 ∈ Mnd ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ 𝑁 < 𝑀) → 𝐺 ∈ Mnd)
30		fveqeq2 5563	. . . . . 6 ⊢ (𝑤 = 𝑀 → ((seq𝑀((+g‘𝐺), (𝑘 ∈ (𝑀...𝑁) ↦ 0 ))‘𝑤) = 0 ↔ (seq𝑀((+g‘𝐺), (𝑘 ∈ (𝑀...𝑁) ↦ 0 ))‘𝑀) = 0 ))
31	30	imbi2d 230	. . . . 5 ⊢ (𝑤 = 𝑀 → ((𝐺 ∈ Mnd → (seq𝑀((+g‘𝐺), (𝑘 ∈ (𝑀...𝑁) ↦ 0 ))‘𝑤) = 0 ) ↔ (𝐺 ∈ Mnd → (seq𝑀((+g‘𝐺), (𝑘 ∈ (𝑀...𝑁) ↦ 0 ))‘𝑀) = 0 )))
32		fveqeq2 5563	. . . . . 6 ⊢ (𝑤 = 𝑦 → ((seq𝑀((+g‘𝐺), (𝑘 ∈ (𝑀...𝑁) ↦ 0 ))‘𝑤) = 0 ↔ (seq𝑀((+g‘𝐺), (𝑘 ∈ (𝑀...𝑁) ↦ 0 ))‘𝑦) = 0 ))
33	32	imbi2d 230	. . . . 5 ⊢ (𝑤 = 𝑦 → ((𝐺 ∈ Mnd → (seq𝑀((+g‘𝐺), (𝑘 ∈ (𝑀...𝑁) ↦ 0 ))‘𝑤) = 0 ) ↔ (𝐺 ∈ Mnd → (seq𝑀((+g‘𝐺), (𝑘 ∈ (𝑀...𝑁) ↦ 0 ))‘𝑦) = 0 )))
34		fveqeq2 5563	. . . . . 6 ⊢ (𝑤 = (𝑦 + 1) → ((seq𝑀((+g‘𝐺), (𝑘 ∈ (𝑀...𝑁) ↦ 0 ))‘𝑤) = 0 ↔ (seq𝑀((+g‘𝐺), (𝑘 ∈ (𝑀...𝑁) ↦ 0 ))‘(𝑦 + 1)) = 0 ))
35	34	imbi2d 230	. . . . 5 ⊢ (𝑤 = (𝑦 + 1) → ((𝐺 ∈ Mnd → (seq𝑀((+g‘𝐺), (𝑘 ∈ (𝑀...𝑁) ↦ 0 ))‘𝑤) = 0 ) ↔ (𝐺 ∈ Mnd → (seq𝑀((+g‘𝐺), (𝑘 ∈ (𝑀...𝑁) ↦ 0 ))‘(𝑦 + 1)) = 0 )))
36		fveqeq2 5563	. . . . . 6 ⊢ (𝑤 = 𝑁 → ((seq𝑀((+g‘𝐺), (𝑘 ∈ (𝑀...𝑁) ↦ 0 ))‘𝑤) = 0 ↔ (seq𝑀((+g‘𝐺), (𝑘 ∈ (𝑀...𝑁) ↦ 0 ))‘𝑁) = 0 ))
37	36	imbi2d 230	. . . . 5 ⊢ (𝑤 = 𝑁 → ((𝐺 ∈ Mnd → (seq𝑀((+g‘𝐺), (𝑘 ∈ (𝑀...𝑁) ↦ 0 ))‘𝑤) = 0 ) ↔ (𝐺 ∈ Mnd → (seq𝑀((+g‘𝐺), (𝑘 ∈ (𝑀...𝑁) ↦ 0 ))‘𝑁) = 0 )))
38		eluzel2 9597	. . . . . . . . 9 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ ℤ)
39	38	adantr 276	. . . . . . . 8 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ 𝐺 ∈ Mnd) → 𝑀 ∈ ℤ)
40	39	adantr 276	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ (ℤ≥‘𝑀) ∧ 𝐺 ∈ Mnd) ∧ 𝑢 ∈ (ℤ≥‘𝑀)) → 𝑀 ∈ ℤ)
41		eluzelz 9601	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ ℤ)
42	41	ad2antrr 488	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ (ℤ≥‘𝑀) ∧ 𝐺 ∈ Mnd) ∧ 𝑢 ∈ (ℤ≥‘𝑀)) → 𝑁 ∈ ℤ)
43	40, 42	fzfigd 10502	. . . . . . . . . 10 ⊢ (((𝑁 ∈ (ℤ≥‘𝑀) ∧ 𝐺 ∈ Mnd) ∧ 𝑢 ∈ (ℤ≥‘𝑀)) → (𝑀...𝑁) ∈ Fin)
44	43	mptexd 5785	. . . . . . . . 9 ⊢ (((𝑁 ∈ (ℤ≥‘𝑀) ∧ 𝐺 ∈ Mnd) ∧ 𝑢 ∈ (ℤ≥‘𝑀)) → (𝑘 ∈ (𝑀...𝑁) ↦ 0 ) ∈ V)
45		vex 2763	. . . . . . . . 9 ⊢ 𝑢 ∈ V
46		fvexg 5573	. . . . . . . . 9 ⊢ (((𝑘 ∈ (𝑀...𝑁) ↦ 0 ) ∈ V ∧ 𝑢 ∈ V) → ((𝑘 ∈ (𝑀...𝑁) ↦ 0 )‘𝑢) ∈ V)
47	44, 45, 46	sylancl 413	. . . . . . . 8 ⊢ (((𝑁 ∈ (ℤ≥‘𝑀) ∧ 𝐺 ∈ Mnd) ∧ 𝑢 ∈ (ℤ≥‘𝑀)) → ((𝑘 ∈ (𝑀...𝑁) ↦ 0 )‘𝑢) ∈ V)
48		plusgslid 12730	. . . . . . . . . . 11 ⊢ (+g = Slot (+g‘ndx) ∧ (+g‘ndx) ∈ ℕ)
49	48	slotex 12645	. . . . . . . . . 10 ⊢ (𝐺 ∈ Mnd → (+g‘𝐺) ∈ V)
50	49	ad2antlr 489	. . . . . . . . 9 ⊢ (((𝑁 ∈ (ℤ≥‘𝑀) ∧ 𝐺 ∈ Mnd) ∧ (𝑢 ∈ V ∧ 𝑣 ∈ V)) → (+g‘𝐺) ∈ V)
51		simprr 531	. . . . . . . . 9 ⊢ (((𝑁 ∈ (ℤ≥‘𝑀) ∧ 𝐺 ∈ Mnd) ∧ (𝑢 ∈ V ∧ 𝑣 ∈ V)) → 𝑣 ∈ V)
52		ovexg 5952	. . . . . . . . 9 ⊢ ((𝑢 ∈ V ∧ (+g‘𝐺) ∈ V ∧ 𝑣 ∈ V) → (𝑢(+g‘𝐺)𝑣) ∈ V)
53	45, 50, 51, 52	mp3an2i 1353	. . . . . . . 8 ⊢ (((𝑁 ∈ (ℤ≥‘𝑀) ∧ 𝐺 ∈ Mnd) ∧ (𝑢 ∈ V ∧ 𝑣 ∈ V)) → (𝑢(+g‘𝐺)𝑣) ∈ V)
54	39, 47, 53	seq3-1 10533	. . . . . . 7 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ 𝐺 ∈ Mnd) → (seq𝑀((+g‘𝐺), (𝑘 ∈ (𝑀...𝑁) ↦ 0 ))‘𝑀) = ((𝑘 ∈ (𝑀...𝑁) ↦ 0 )‘𝑀))
55		eqid 2193	. . . . . . . 8 ⊢ (𝑘 ∈ (𝑀...𝑁) ↦ 0 ) = (𝑘 ∈ (𝑀...𝑁) ↦ 0 )
56		eqidd 2194	. . . . . . . 8 ⊢ (𝑘 = 𝑀 → 0 = 0 )
57		eluzfz1 10097	. . . . . . . . 9 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ (𝑀...𝑁))
58	57	adantr 276	. . . . . . . 8 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ 𝐺 ∈ Mnd) → 𝑀 ∈ (𝑀...𝑁))
59	7	adantl 277	. . . . . . . 8 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ 𝐺 ∈ Mnd) → 0 ∈ (Base‘𝐺))
60	55, 56, 58, 59	fvmptd3 5651	. . . . . . 7 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ 𝐺 ∈ Mnd) → ((𝑘 ∈ (𝑀...𝑁) ↦ 0 )‘𝑀) = 0 )
61	54, 60	eqtrd 2226	. . . . . 6 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ 𝐺 ∈ Mnd) → (seq𝑀((+g‘𝐺), (𝑘 ∈ (𝑀...𝑁) ↦ 0 ))‘𝑀) = 0 )
62	61	ex 115	. . . . 5 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝐺 ∈ Mnd → (seq𝑀((+g‘𝐺), (𝑘 ∈ (𝑀...𝑁) ↦ 0 ))‘𝑀) = 0 ))
63		elfzouz 10217	. . . . . . . . . . 11 ⊢ (𝑦 ∈ (𝑀..^𝑁) → 𝑦 ∈ (ℤ≥‘𝑀))
64	63	adantr 276	. . . . . . . . . 10 ⊢ ((𝑦 ∈ (𝑀..^𝑁) ∧ 𝐺 ∈ Mnd) → 𝑦 ∈ (ℤ≥‘𝑀))
65		elfzouz2 10228	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ (𝑀..^𝑁) → 𝑁 ∈ (ℤ≥‘𝑦))
66		uztrn 9609	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ (ℤ≥‘𝑦) ∧ 𝑦 ∈ (ℤ≥‘𝑀)) → 𝑁 ∈ (ℤ≥‘𝑀))
67	65, 63, 66	syl2anc 411	. . . . . . . . . . 11 ⊢ (𝑦 ∈ (𝑀..^𝑁) → 𝑁 ∈ (ℤ≥‘𝑀))
68	67, 47	sylanl1 402	. . . . . . . . . 10 ⊢ (((𝑦 ∈ (𝑀..^𝑁) ∧ 𝐺 ∈ Mnd) ∧ 𝑢 ∈ (ℤ≥‘𝑀)) → ((𝑘 ∈ (𝑀...𝑁) ↦ 0 )‘𝑢) ∈ V)
69	67, 53	sylanl1 402	. . . . . . . . . 10 ⊢ (((𝑦 ∈ (𝑀..^𝑁) ∧ 𝐺 ∈ Mnd) ∧ (𝑢 ∈ V ∧ 𝑣 ∈ V)) → (𝑢(+g‘𝐺)𝑣) ∈ V)
70	64, 68, 69	seq3p1 10536	. . . . . . . . 9 ⊢ ((𝑦 ∈ (𝑀..^𝑁) ∧ 𝐺 ∈ Mnd) → (seq𝑀((+g‘𝐺), (𝑘 ∈ (𝑀...𝑁) ↦ 0 ))‘(𝑦 + 1)) = ((seq𝑀((+g‘𝐺), (𝑘 ∈ (𝑀...𝑁) ↦ 0 ))‘𝑦)(+g‘𝐺)((𝑘 ∈ (𝑀...𝑁) ↦ 0 )‘(𝑦 + 1))))
71	70	adantr 276	. . . . . . . 8 ⊢ (((𝑦 ∈ (𝑀..^𝑁) ∧ 𝐺 ∈ Mnd) ∧ (seq𝑀((+g‘𝐺), (𝑘 ∈ (𝑀...𝑁) ↦ 0 ))‘𝑦) = 0 ) → (seq𝑀((+g‘𝐺), (𝑘 ∈ (𝑀...𝑁) ↦ 0 ))‘(𝑦 + 1)) = ((seq𝑀((+g‘𝐺), (𝑘 ∈ (𝑀...𝑁) ↦ 0 ))‘𝑦)(+g‘𝐺)((𝑘 ∈ (𝑀...𝑁) ↦ 0 )‘(𝑦 + 1))))
72		simpr 110	. . . . . . . . 9 ⊢ (((𝑦 ∈ (𝑀..^𝑁) ∧ 𝐺 ∈ Mnd) ∧ (seq𝑀((+g‘𝐺), (𝑘 ∈ (𝑀...𝑁) ↦ 0 ))‘𝑦) = 0 ) → (seq𝑀((+g‘𝐺), (𝑘 ∈ (𝑀...𝑁) ↦ 0 ))‘𝑦) = 0 )
73		eqidd 2194	. . . . . . . . . . 11 ⊢ (𝑘 = (𝑦 + 1) → 0 = 0 )
74		fzofzp1 10294	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ (𝑀..^𝑁) → (𝑦 + 1) ∈ (𝑀...𝑁))
75	74	adantr 276	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ (𝑀..^𝑁) ∧ 𝐺 ∈ Mnd) → (𝑦 + 1) ∈ (𝑀...𝑁))
76	7	adantl 277	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ (𝑀..^𝑁) ∧ 𝐺 ∈ Mnd) → 0 ∈ (Base‘𝐺))
77	55, 73, 75, 76	fvmptd3 5651	. . . . . . . . . 10 ⊢ ((𝑦 ∈ (𝑀..^𝑁) ∧ 𝐺 ∈ Mnd) → ((𝑘 ∈ (𝑀...𝑁) ↦ 0 )‘(𝑦 + 1)) = 0 )
78	77	adantr 276	. . . . . . . . 9 ⊢ (((𝑦 ∈ (𝑀..^𝑁) ∧ 𝐺 ∈ Mnd) ∧ (seq𝑀((+g‘𝐺), (𝑘 ∈ (𝑀...𝑁) ↦ 0 ))‘𝑦) = 0 ) → ((𝑘 ∈ (𝑀...𝑁) ↦ 0 )‘(𝑦 + 1)) = 0 )
79	72, 78	oveq12d 5936	. . . . . . . 8 ⊢ (((𝑦 ∈ (𝑀..^𝑁) ∧ 𝐺 ∈ Mnd) ∧ (seq𝑀((+g‘𝐺), (𝑘 ∈ (𝑀...𝑁) ↦ 0 ))‘𝑦) = 0 ) → ((seq𝑀((+g‘𝐺), (𝑘 ∈ (𝑀...𝑁) ↦ 0 ))‘𝑦)(+g‘𝐺)((𝑘 ∈ (𝑀...𝑁) ↦ 0 )‘(𝑦 + 1))) = ( 0 (+g‘𝐺) 0 ))
80	1, 3, 2	mndlid 13016	. . . . . . . . . 10 ⊢ ((𝐺 ∈ Mnd ∧ 0 ∈ (Base‘𝐺)) → ( 0 (+g‘𝐺) 0 ) = 0 )
81	7, 80	mpdan 421	. . . . . . . . 9 ⊢ (𝐺 ∈ Mnd → ( 0 (+g‘𝐺) 0 ) = 0 )
82	81	ad2antlr 489	. . . . . . . 8 ⊢ (((𝑦 ∈ (𝑀..^𝑁) ∧ 𝐺 ∈ Mnd) ∧ (seq𝑀((+g‘𝐺), (𝑘 ∈ (𝑀...𝑁) ↦ 0 ))‘𝑦) = 0 ) → ( 0 (+g‘𝐺) 0 ) = 0 )
83	71, 79, 82	3eqtrd 2230	. . . . . . 7 ⊢ (((𝑦 ∈ (𝑀..^𝑁) ∧ 𝐺 ∈ Mnd) ∧ (seq𝑀((+g‘𝐺), (𝑘 ∈ (𝑀...𝑁) ↦ 0 ))‘𝑦) = 0 ) → (seq𝑀((+g‘𝐺), (𝑘 ∈ (𝑀...𝑁) ↦ 0 ))‘(𝑦 + 1)) = 0 )
84	83	exp31 364	. . . . . 6 ⊢ (𝑦 ∈ (𝑀..^𝑁) → (𝐺 ∈ Mnd → ((seq𝑀((+g‘𝐺), (𝑘 ∈ (𝑀...𝑁) ↦ 0 ))‘𝑦) = 0 → (seq𝑀((+g‘𝐺), (𝑘 ∈ (𝑀...𝑁) ↦ 0 ))‘(𝑦 + 1)) = 0 )))
85	84	a2d 26	. . . . 5 ⊢ (𝑦 ∈ (𝑀..^𝑁) → ((𝐺 ∈ Mnd → (seq𝑀((+g‘𝐺), (𝑘 ∈ (𝑀...𝑁) ↦ 0 ))‘𝑦) = 0 ) → (𝐺 ∈ Mnd → (seq𝑀((+g‘𝐺), (𝑘 ∈ (𝑀...𝑁) ↦ 0 ))‘(𝑦 + 1)) = 0 )))
86	31, 33, 35, 37, 62, 85	fzind2 10306	. . . 4 ⊢ (𝑁 ∈ (𝑀...𝑁) → (𝐺 ∈ Mnd → (seq𝑀((+g‘𝐺), (𝑘 ∈ (𝑀...𝑁) ↦ 0 ))‘𝑁) = 0 ))
87	28, 29, 86	sylc 62	. . 3 ⊢ (((𝐺 ∈ Mnd ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ 𝑁 < 𝑀) → (seq𝑀((+g‘𝐺), (𝑘 ∈ (𝑀...𝑁) ↦ 0 ))‘𝑁) = 0 )
88	16, 18, 87	3eqtrd 2230	. 2 ⊢ (((𝐺 ∈ Mnd ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ 𝑁 < 𝑀) → (𝐺 Σg (𝑘 ∈ (𝑀...𝑁) ↦ 0 )) = 0 )
89		zdclt 9394	. . . 4 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) → DECID 𝑁 < 𝑀)
90	6, 5, 89	syl2anc 411	. . 3 ⊢ ((𝐺 ∈ Mnd ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → DECID 𝑁 < 𝑀)
91		exmiddc 837	. . 3 ⊢ (DECID 𝑁 < 𝑀 → (𝑁 < 𝑀 ∨ ¬ 𝑁 < 𝑀))
92	90, 91	syl 14	. 2 ⊢ ((𝐺 ∈ Mnd ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 < 𝑀 ∨ ¬ 𝑁 < 𝑀))
93	15, 88, 92	mpjaodan 799	1 ⊢ ((𝐺 ∈ Mnd ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐺 Σg (𝑘 ∈ (𝑀...𝑁) ↦ 0 )) = 0 )

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ∨ wo 709  DECID wdc 835   ∧ w3a 980   = wceq 1364   ∈ wcel 2164  Vcvv 2760  ifcif 3557   class class class wbr 4029   ↦ cmpt 4090  ‘cfv 5254  (class class class)co 5918  Fincfn 6794  1c1 7873   + caddc 7875   < clt 8054   ≤ cle 8055  ℤcz 9317  ℤ_≥cuz 9592  ...cfz 10074  ..^cfzo 10208  seqcseq 10518  Basecbs 12618  +_gcplusg 12695  0_gc0g 12867   Σ_g cgsu 12868  Mndcmnd 12997

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-addcom 7972  ax-addass 7974  ax-distr 7976  ax-i2m1 7977  ax-0lt1 7978  ax-0id 7980  ax-rnegex 7981  ax-cnre 7983  ax-pre-ltirr 7984  ax-pre-ltwlin 7985  ax-pre-lttrn 7986  ax-pre-apti 7987  ax-pre-ltadd 7988

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-if 3558  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-id 4324  df-iord 4397  df-on 4399  df-ilim 4400  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-recs 6358  df-frec 6444  df-1o 6469  df-er 6587  df-en 6795  df-fin 6797  df-pnf 8056  df-mnf 8057  df-xr 8058  df-ltxr 8059  df-le 8060  df-sub 8192  df-neg 8193  df-inn 8983  df-2 9041  df-n0 9241  df-z 9318  df-uz 9593  df-fz 10075  df-fzo 10209  df-seqfrec 10519  df-ndx 12621  df-slot 12622  df-base 12624  df-plusg 12708  df-0g 12869  df-igsum 12870  df-mgm 12939  df-sgrp 12985  df-mnd 12998

This theorem is referenced by: (None)