Theorem gsumfzz 13639

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 2231	. . . . 5 ⊢ (Base‘𝐺) = (Base‘𝐺)
2		gsumz.z	. . . . 5 ⊢ 0 = (0g‘𝐺)
3		eqid 2231	. . . . 5 ⊢ (+g‘𝐺) = (+g‘𝐺)
4		simp1 1024	. . . . 5 ⊢ ((𝐺 ∈ Mnd ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 𝐺 ∈ Mnd)
5		simp2 1025	. . . . 5 ⊢ ((𝐺 ∈ Mnd ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 𝑀 ∈ ℤ)
6		simp3 1026	. . . . 5 ⊢ ((𝐺 ∈ Mnd ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 𝑁 ∈ ℤ)
7	1, 2	mndidcl 13574	. . . . . . . 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 5810	. . . . 5 ⊢ ((𝐺 ∈ Mnd ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑘 ∈ (𝑀...𝑁) ↦ 0 ):(𝑀...𝑁)⟶(Base‘𝐺))
11	1, 2, 3, 4, 5, 6, 10	gsumfzval 13535	. . . 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 3616	. . 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 2264	. 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 3619	. . 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 9645	. . . . . . . 8 ⊢ ((𝐺 ∈ Mnd ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 𝑀 ∈ ℝ)
22	6	zred 9645	. . . . . . . 8 ⊢ ((𝐺 ∈ Mnd ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 𝑁 ∈ ℝ)
23	21, 22	lenltd 8340	. . . . . . 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 9804	. . . . . 6 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) ↔ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁))
26	19, 20, 24, 25	syl3anbrc 1208	. . . . 5 ⊢ (((𝐺 ∈ Mnd ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ 𝑁 < 𝑀) → 𝑁 ∈ (ℤ≥‘𝑀))
27		eluzfz2 10310	. . . . 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 5657	. . . . . 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 5657	. . . . . 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 5657	. . . . . 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 5657	. . . . . 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 9803	. . . . . . . . 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 9808	. . . . . . . . . . . 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 10737	. . . . . . . . . 10 ⊢ (((𝑁 ∈ (ℤ≥‘𝑀) ∧ 𝐺 ∈ Mnd) ∧ 𝑢 ∈ (ℤ≥‘𝑀)) → (𝑀...𝑁) ∈ Fin)
44	43	mptexd 5891	. . . . . . . . 9 ⊢ (((𝑁 ∈ (ℤ≥‘𝑀) ∧ 𝐺 ∈ Mnd) ∧ 𝑢 ∈ (ℤ≥‘𝑀)) → (𝑘 ∈ (𝑀...𝑁) ↦ 0 ) ∈ V)
45		vex 2806	. . . . . . . . 9 ⊢ 𝑢 ∈ V
46		fvexg 5667	. . . . . . . . 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 13256	. . . . . . . . . . 11 ⊢ (+g = Slot (+g‘ndx) ∧ (+g‘ndx) ∈ ℕ)
49	48	slotex 13170	. . . . . . . . . 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 533	. . . . . . . . 9 ⊢ (((𝑁 ∈ (ℤ≥‘𝑀) ∧ 𝐺 ∈ Mnd) ∧ (𝑢 ∈ V ∧ 𝑣 ∈ V)) → 𝑣 ∈ V)
52		ovexg 6062	. . . . . . . . 9 ⊢ ((𝑢 ∈ V ∧ (+g‘𝐺) ∈ V ∧ 𝑣 ∈ V) → (𝑢(+g‘𝐺)𝑣) ∈ V)
53	45, 50, 51, 52	mp3an2i 1379	. . . . . . . 8 ⊢ (((𝑁 ∈ (ℤ≥‘𝑀) ∧ 𝐺 ∈ Mnd) ∧ (𝑢 ∈ V ∧ 𝑣 ∈ V)) → (𝑢(+g‘𝐺)𝑣) ∈ V)
54	39, 47, 53	seq3-1 10768	. . . . . . 7 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ 𝐺 ∈ Mnd) → (seq𝑀((+g‘𝐺), (𝑘 ∈ (𝑀...𝑁) ↦ 0 ))‘𝑀) = ((𝑘 ∈ (𝑀...𝑁) ↦ 0 )‘𝑀))
55		eqid 2231	. . . . . . . 8 ⊢ (𝑘 ∈ (𝑀...𝑁) ↦ 0 ) = (𝑘 ∈ (𝑀...𝑁) ↦ 0 )
56		eqidd 2232	. . . . . . . 8 ⊢ (𝑘 = 𝑀 → 0 = 0 )
57		eluzfz1 10309	. . . . . . . . 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 5749	. . . . . . 7 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ 𝐺 ∈ Mnd) → ((𝑘 ∈ (𝑀...𝑁) ↦ 0 )‘𝑀) = 0 )
61	54, 60	eqtrd 2264	. . . . . 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 10429	. . . . . . . . . . 11 ⊢ (𝑦 ∈ (𝑀..^𝑁) → 𝑦 ∈ (ℤ≥‘𝑀))
64	63	adantr 276	. . . . . . . . . 10 ⊢ ((𝑦 ∈ (𝑀..^𝑁) ∧ 𝐺 ∈ Mnd) → 𝑦 ∈ (ℤ≥‘𝑀))
65		elfzouz2 10440	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ (𝑀..^𝑁) → 𝑁 ∈ (ℤ≥‘𝑦))
66		uztrn 9816	. . . . . . . . . . . 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 10771	. . . . . . . . 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 2232	. . . . . . . . . . 11 ⊢ (𝑘 = (𝑦 + 1) → 0 = 0 )
74		fzofzp1 10516	. . . . . . . . . . . 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 5749	. . . . . . . . . 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 6046	. . . . . . . 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 13579	. . . . . . . . . 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 2268	. . . . . . 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 10529	. . . 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 2268	. 2 ⊢ (((𝐺 ∈ Mnd ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ 𝑁 < 𝑀) → (𝐺 Σg (𝑘 ∈ (𝑀...𝑁) ↦ 0 )) = 0 )
89		zdclt 9600	. . . 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 844	. . 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 806	1 ⊢ ((𝐺 ∈ Mnd ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐺 Σg (𝑘 ∈ (𝑀...𝑁) ↦ 0 )) = 0 )

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ∨ wo 716  DECID wdc 842   ∧ w3a 1005   = wceq 1398   ∈ wcel 2202  Vcvv 2803  ifcif 3607   class class class wbr 4093   ↦ cmpt 4155  ‘cfv 5333  (class class class)co 6028  Fincfn 6952  1c1 8076   + caddc 8078   < clt 8257   ≤ cle 8258  ℤcz 9522  ℤ_≥cuz 9798  ...cfz 10286  ..^cfzo 10420  seqcseq 10753  Basecbs 13143  +_gcplusg 13221  0_gc0g 13400   Σ_g cgsu 13401  Mndcmnd 13560

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692  ax-cnex 8166  ax-resscn 8167  ax-1cn 8168  ax-1re 8169  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-addcom 8175  ax-addass 8177  ax-distr 8179  ax-i2m1 8180  ax-0lt1 8181  ax-0id 8183  ax-rnegex 8184  ax-cnre 8186  ax-pre-ltirr 8187  ax-pre-ltwlin 8188  ax-pre-lttrn 8189  ax-pre-apti 8190  ax-pre-ltadd 8191

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-iord 4469  df-on 4471  df-ilim 4472  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-frec 6600  df-1o 6625  df-er 6745  df-en 6953  df-fin 6955  df-pnf 8259  df-mnf 8260  df-xr 8261  df-ltxr 8262  df-le 8263  df-sub 8395  df-neg 8396  df-inn 9187  df-2 9245  df-n0 9446  df-z 9523  df-uz 9799  df-fz 10287  df-fzo 10421  df-seqfrec 10754  df-ndx 13146  df-slot 13147  df-base 13149  df-plusg 13234  df-0g 13402  df-igsum 13403  df-mgm 13500  df-sgrp 13546  df-mnd 13561

This theorem is referenced by: (None)