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Theorem gsumfzz 13057
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 = (0g𝐺)
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.
StepHypRef Expression
1 eqid 2193 . . . . 5 (Base‘𝐺) = (Base‘𝐺)
2 gsumz.z . . . . 5 0 = (0g𝐺)
3 eqid 2193 . . . . 5 (+g𝐺) = (+g𝐺)
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)
71, 2mndidcl 13001 . . . . . . . 8 (𝐺 ∈ Mnd → 0 ∈ (Base‘𝐺))
84, 7syl 14 . . . . . . 7 ((𝐺 ∈ Mnd ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 0 ∈ (Base‘𝐺))
98adantr 276 . . . . . 6 (((𝐺 ∈ Mnd ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑘 ∈ (𝑀...𝑁)) → 0 ∈ (Base‘𝐺))
109fmpttd 5705 . . . . 5 ((𝐺 ∈ Mnd ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑘 ∈ (𝑀...𝑁) ↦ 0 ):(𝑀...𝑁)⟶(Base‘𝐺))
111, 2, 3, 4, 5, 6, 10gsumfzval 12964 . . . 4 ((𝐺 ∈ Mnd ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐺 Σg (𝑘 ∈ (𝑀...𝑁) ↦ 0 )) = if(𝑁 < 𝑀, 0 , (seq𝑀((+g𝐺), (𝑘 ∈ (𝑀...𝑁) ↦ 0 ))‘𝑁)))
1211adantr 276 . . 3 (((𝐺 ∈ Mnd ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑁 < 𝑀) → (𝐺 Σg (𝑘 ∈ (𝑀...𝑁) ↦ 0 )) = if(𝑁 < 𝑀, 0 , (seq𝑀((+g𝐺), (𝑘 ∈ (𝑀...𝑁) ↦ 0 ))‘𝑁)))
13 simpr 110 . . . 4 (((𝐺 ∈ Mnd ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑁 < 𝑀) → 𝑁 < 𝑀)
1413iftrued 3564 . . 3 (((𝐺 ∈ Mnd ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑁 < 𝑀) → if(𝑁 < 𝑀, 0 , (seq𝑀((+g𝐺), (𝑘 ∈ (𝑀...𝑁) ↦ 0 ))‘𝑁)) = 0 )
1512, 14eqtrd 2226 . 2 (((𝐺 ∈ Mnd ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑁 < 𝑀) → (𝐺 Σg (𝑘 ∈ (𝑀...𝑁) ↦ 0 )) = 0 )
1611adantr 276 . . 3 (((𝐺 ∈ Mnd ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ 𝑁 < 𝑀) → (𝐺 Σg (𝑘 ∈ (𝑀...𝑁) ↦ 0 )) = if(𝑁 < 𝑀, 0 , (seq𝑀((+g𝐺), (𝑘 ∈ (𝑀...𝑁) ↦ 0 ))‘𝑁)))
17 simpr 110 . . . 4 (((𝐺 ∈ Mnd ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ 𝑁 < 𝑀) → ¬ 𝑁 < 𝑀)
1817iffalsed 3567 . . 3 (((𝐺 ∈ Mnd ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ 𝑁 < 𝑀) → if(𝑁 < 𝑀, 0 , (seq𝑀((+g𝐺), (𝑘 ∈ (𝑀...𝑁) ↦ 0 ))‘𝑁)) = (seq𝑀((+g𝐺), (𝑘 ∈ (𝑀...𝑁) ↦ 0 ))‘𝑁))
195adantr 276 . . . . . 6 (((𝐺 ∈ Mnd ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ 𝑁 < 𝑀) → 𝑀 ∈ ℤ)
206adantr 276 . . . . . 6 (((𝐺 ∈ Mnd ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ 𝑁 < 𝑀) → 𝑁 ∈ ℤ)
215zred 9429 . . . . . . . 8 ((𝐺 ∈ Mnd ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 𝑀 ∈ ℝ)
226zred 9429 . . . . . . . 8 ((𝐺 ∈ Mnd ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 𝑁 ∈ ℝ)
2321, 22lenltd 8127 . . . . . . 7 ((𝐺 ∈ Mnd ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀𝑁 ↔ ¬ 𝑁 < 𝑀))
2423biimpar 297 . . . . . 6 (((𝐺 ∈ Mnd ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ 𝑁 < 𝑀) → 𝑀𝑁)
25 eluz2 9588 . . . . . 6 (𝑁 ∈ (ℤ𝑀) ↔ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀𝑁))
2619, 20, 24, 25syl3anbrc 1183 . . . . 5 (((𝐺 ∈ Mnd ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ 𝑁 < 𝑀) → 𝑁 ∈ (ℤ𝑀))
27 eluzfz2 10088 . . . . 5 (𝑁 ∈ (ℤ𝑀) → 𝑁 ∈ (𝑀...𝑁))
2826, 27syl 14 . . . 4 (((𝐺 ∈ Mnd ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ 𝑁 < 𝑀) → 𝑁 ∈ (𝑀...𝑁))
294adantr 276 . . . 4 (((𝐺 ∈ Mnd ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ 𝑁 < 𝑀) → 𝐺 ∈ Mnd)
30 fveqeq2 5555 . . . . . 6 (𝑤 = 𝑀 → ((seq𝑀((+g𝐺), (𝑘 ∈ (𝑀...𝑁) ↦ 0 ))‘𝑤) = 0 ↔ (seq𝑀((+g𝐺), (𝑘 ∈ (𝑀...𝑁) ↦ 0 ))‘𝑀) = 0 ))
3130imbi2d 230 . . . . 5 (𝑤 = 𝑀 → ((𝐺 ∈ Mnd → (seq𝑀((+g𝐺), (𝑘 ∈ (𝑀...𝑁) ↦ 0 ))‘𝑤) = 0 ) ↔ (𝐺 ∈ Mnd → (seq𝑀((+g𝐺), (𝑘 ∈ (𝑀...𝑁) ↦ 0 ))‘𝑀) = 0 )))
32 fveqeq2 5555 . . . . . 6 (𝑤 = 𝑦 → ((seq𝑀((+g𝐺), (𝑘 ∈ (𝑀...𝑁) ↦ 0 ))‘𝑤) = 0 ↔ (seq𝑀((+g𝐺), (𝑘 ∈ (𝑀...𝑁) ↦ 0 ))‘𝑦) = 0 ))
3332imbi2d 230 . . . . 5 (𝑤 = 𝑦 → ((𝐺 ∈ Mnd → (seq𝑀((+g𝐺), (𝑘 ∈ (𝑀...𝑁) ↦ 0 ))‘𝑤) = 0 ) ↔ (𝐺 ∈ Mnd → (seq𝑀((+g𝐺), (𝑘 ∈ (𝑀...𝑁) ↦ 0 ))‘𝑦) = 0 )))
34 fveqeq2 5555 . . . . . 6 (𝑤 = (𝑦 + 1) → ((seq𝑀((+g𝐺), (𝑘 ∈ (𝑀...𝑁) ↦ 0 ))‘𝑤) = 0 ↔ (seq𝑀((+g𝐺), (𝑘 ∈ (𝑀...𝑁) ↦ 0 ))‘(𝑦 + 1)) = 0 ))
3534imbi2d 230 . . . . 5 (𝑤 = (𝑦 + 1) → ((𝐺 ∈ Mnd → (seq𝑀((+g𝐺), (𝑘 ∈ (𝑀...𝑁) ↦ 0 ))‘𝑤) = 0 ) ↔ (𝐺 ∈ Mnd → (seq𝑀((+g𝐺), (𝑘 ∈ (𝑀...𝑁) ↦ 0 ))‘(𝑦 + 1)) = 0 )))
36 fveqeq2 5555 . . . . . 6 (𝑤 = 𝑁 → ((seq𝑀((+g𝐺), (𝑘 ∈ (𝑀...𝑁) ↦ 0 ))‘𝑤) = 0 ↔ (seq𝑀((+g𝐺), (𝑘 ∈ (𝑀...𝑁) ↦ 0 ))‘𝑁) = 0 ))
3736imbi2d 230 . . . . 5 (𝑤 = 𝑁 → ((𝐺 ∈ Mnd → (seq𝑀((+g𝐺), (𝑘 ∈ (𝑀...𝑁) ↦ 0 ))‘𝑤) = 0 ) ↔ (𝐺 ∈ Mnd → (seq𝑀((+g𝐺), (𝑘 ∈ (𝑀...𝑁) ↦ 0 ))‘𝑁) = 0 )))
38 eluzel2 9587 . . . . . . . . 9 (𝑁 ∈ (ℤ𝑀) → 𝑀 ∈ ℤ)
3938adantr 276 . . . . . . . 8 ((𝑁 ∈ (ℤ𝑀) ∧ 𝐺 ∈ Mnd) → 𝑀 ∈ ℤ)
4039adantr 276 . . . . . . . . . . 11 (((𝑁 ∈ (ℤ𝑀) ∧ 𝐺 ∈ Mnd) ∧ 𝑢 ∈ (ℤ𝑀)) → 𝑀 ∈ ℤ)
41 eluzelz 9591 . . . . . . . . . . . 12 (𝑁 ∈ (ℤ𝑀) → 𝑁 ∈ ℤ)
4241ad2antrr 488 . . . . . . . . . . 11 (((𝑁 ∈ (ℤ𝑀) ∧ 𝐺 ∈ Mnd) ∧ 𝑢 ∈ (ℤ𝑀)) → 𝑁 ∈ ℤ)
4340, 42fzfigd 10492 . . . . . . . . . 10 (((𝑁 ∈ (ℤ𝑀) ∧ 𝐺 ∈ Mnd) ∧ 𝑢 ∈ (ℤ𝑀)) → (𝑀...𝑁) ∈ Fin)
4443mptexd 5777 . . . . . . . . 9 (((𝑁 ∈ (ℤ𝑀) ∧ 𝐺 ∈ Mnd) ∧ 𝑢 ∈ (ℤ𝑀)) → (𝑘 ∈ (𝑀...𝑁) ↦ 0 ) ∈ V)
45 vex 2763 . . . . . . . . 9 𝑢 ∈ V
46 fvexg 5565 . . . . . . . . 9 (((𝑘 ∈ (𝑀...𝑁) ↦ 0 ) ∈ V ∧ 𝑢 ∈ V) → ((𝑘 ∈ (𝑀...𝑁) ↦ 0 )‘𝑢) ∈ V)
4744, 45, 46sylancl 413 . . . . . . . 8 (((𝑁 ∈ (ℤ𝑀) ∧ 𝐺 ∈ Mnd) ∧ 𝑢 ∈ (ℤ𝑀)) → ((𝑘 ∈ (𝑀...𝑁) ↦ 0 )‘𝑢) ∈ V)
48 plusgslid 12720 . . . . . . . . . . 11 (+g = Slot (+g‘ndx) ∧ (+g‘ndx) ∈ ℕ)
4948slotex 12635 . . . . . . . . . 10 (𝐺 ∈ Mnd → (+g𝐺) ∈ V)
5049ad2antlr 489 . . . . . . . . 9 (((𝑁 ∈ (ℤ𝑀) ∧ 𝐺 ∈ Mnd) ∧ (𝑢 ∈ V ∧ 𝑣 ∈ V)) → (+g𝐺) ∈ V)
51 simprr 531 . . . . . . . . 9 (((𝑁 ∈ (ℤ𝑀) ∧ 𝐺 ∈ Mnd) ∧ (𝑢 ∈ V ∧ 𝑣 ∈ V)) → 𝑣 ∈ V)
52 ovexg 5944 . . . . . . . . 9 ((𝑢 ∈ V ∧ (+g𝐺) ∈ V ∧ 𝑣 ∈ V) → (𝑢(+g𝐺)𝑣) ∈ V)
5345, 50, 51, 52mp3an2i 1353 . . . . . . . 8 (((𝑁 ∈ (ℤ𝑀) ∧ 𝐺 ∈ Mnd) ∧ (𝑢 ∈ V ∧ 𝑣 ∈ V)) → (𝑢(+g𝐺)𝑣) ∈ V)
5439, 47, 53seq3-1 10523 . . . . . . 7 ((𝑁 ∈ (ℤ𝑀) ∧ 𝐺 ∈ Mnd) → (seq𝑀((+g𝐺), (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 = 𝑀0 = 0 )
57 eluzfz1 10087 . . . . . . . . 9 (𝑁 ∈ (ℤ𝑀) → 𝑀 ∈ (𝑀...𝑁))
5857adantr 276 . . . . . . . 8 ((𝑁 ∈ (ℤ𝑀) ∧ 𝐺 ∈ Mnd) → 𝑀 ∈ (𝑀...𝑁))
597adantl 277 . . . . . . . 8 ((𝑁 ∈ (ℤ𝑀) ∧ 𝐺 ∈ Mnd) → 0 ∈ (Base‘𝐺))
6055, 56, 58, 59fvmptd3 5643 . . . . . . 7 ((𝑁 ∈ (ℤ𝑀) ∧ 𝐺 ∈ Mnd) → ((𝑘 ∈ (𝑀...𝑁) ↦ 0 )‘𝑀) = 0 )
6154, 60eqtrd 2226 . . . . . 6 ((𝑁 ∈ (ℤ𝑀) ∧ 𝐺 ∈ Mnd) → (seq𝑀((+g𝐺), (𝑘 ∈ (𝑀...𝑁) ↦ 0 ))‘𝑀) = 0 )
6261ex 115 . . . . 5 (𝑁 ∈ (ℤ𝑀) → (𝐺 ∈ Mnd → (seq𝑀((+g𝐺), (𝑘 ∈ (𝑀...𝑁) ↦ 0 ))‘𝑀) = 0 ))
63 elfzouz 10207 . . . . . . . . . . 11 (𝑦 ∈ (𝑀..^𝑁) → 𝑦 ∈ (ℤ𝑀))
6463adantr 276 . . . . . . . . . 10 ((𝑦 ∈ (𝑀..^𝑁) ∧ 𝐺 ∈ Mnd) → 𝑦 ∈ (ℤ𝑀))
65 elfzouz2 10218 . . . . . . . . . . . 12 (𝑦 ∈ (𝑀..^𝑁) → 𝑁 ∈ (ℤ𝑦))
66 uztrn 9599 . . . . . . . . . . . 12 ((𝑁 ∈ (ℤ𝑦) ∧ 𝑦 ∈ (ℤ𝑀)) → 𝑁 ∈ (ℤ𝑀))
6765, 63, 66syl2anc 411 . . . . . . . . . . 11 (𝑦 ∈ (𝑀..^𝑁) → 𝑁 ∈ (ℤ𝑀))
6867, 47sylanl1 402 . . . . . . . . . 10 (((𝑦 ∈ (𝑀..^𝑁) ∧ 𝐺 ∈ Mnd) ∧ 𝑢 ∈ (ℤ𝑀)) → ((𝑘 ∈ (𝑀...𝑁) ↦ 0 )‘𝑢) ∈ V)
6967, 53sylanl1 402 . . . . . . . . . 10 (((𝑦 ∈ (𝑀..^𝑁) ∧ 𝐺 ∈ Mnd) ∧ (𝑢 ∈ V ∧ 𝑣 ∈ V)) → (𝑢(+g𝐺)𝑣) ∈ V)
7064, 68, 69seq3p1 10526 . . . . . . . . 9 ((𝑦 ∈ (𝑀..^𝑁) ∧ 𝐺 ∈ Mnd) → (seq𝑀((+g𝐺), (𝑘 ∈ (𝑀...𝑁) ↦ 0 ))‘(𝑦 + 1)) = ((seq𝑀((+g𝐺), (𝑘 ∈ (𝑀...𝑁) ↦ 0 ))‘𝑦)(+g𝐺)((𝑘 ∈ (𝑀...𝑁) ↦ 0 )‘(𝑦 + 1))))
7170adantr 276 . . . . . . . 8 (((𝑦 ∈ (𝑀..^𝑁) ∧ 𝐺 ∈ Mnd) ∧ (seq𝑀((+g𝐺), (𝑘 ∈ (𝑀...𝑁) ↦ 0 ))‘𝑦) = 0 ) → (seq𝑀((+g𝐺), (𝑘 ∈ (𝑀...𝑁) ↦ 0 ))‘(𝑦 + 1)) = ((seq𝑀((+g𝐺), (𝑘 ∈ (𝑀...𝑁) ↦ 0 ))‘𝑦)(+g𝐺)((𝑘 ∈ (𝑀...𝑁) ↦ 0 )‘(𝑦 + 1))))
72 simpr 110 . . . . . . . . 9 (((𝑦 ∈ (𝑀..^𝑁) ∧ 𝐺 ∈ Mnd) ∧ (seq𝑀((+g𝐺), (𝑘 ∈ (𝑀...𝑁) ↦ 0 ))‘𝑦) = 0 ) → (seq𝑀((+g𝐺), (𝑘 ∈ (𝑀...𝑁) ↦ 0 ))‘𝑦) = 0 )
73 eqidd 2194 . . . . . . . . . . 11 (𝑘 = (𝑦 + 1) → 0 = 0 )
74 fzofzp1 10284 . . . . . . . . . . . 12 (𝑦 ∈ (𝑀..^𝑁) → (𝑦 + 1) ∈ (𝑀...𝑁))
7574adantr 276 . . . . . . . . . . 11 ((𝑦 ∈ (𝑀..^𝑁) ∧ 𝐺 ∈ Mnd) → (𝑦 + 1) ∈ (𝑀...𝑁))
767adantl 277 . . . . . . . . . . 11 ((𝑦 ∈ (𝑀..^𝑁) ∧ 𝐺 ∈ Mnd) → 0 ∈ (Base‘𝐺))
7755, 73, 75, 76fvmptd3 5643 . . . . . . . . . 10 ((𝑦 ∈ (𝑀..^𝑁) ∧ 𝐺 ∈ Mnd) → ((𝑘 ∈ (𝑀...𝑁) ↦ 0 )‘(𝑦 + 1)) = 0 )
7877adantr 276 . . . . . . . . 9 (((𝑦 ∈ (𝑀..^𝑁) ∧ 𝐺 ∈ Mnd) ∧ (seq𝑀((+g𝐺), (𝑘 ∈ (𝑀...𝑁) ↦ 0 ))‘𝑦) = 0 ) → ((𝑘 ∈ (𝑀...𝑁) ↦ 0 )‘(𝑦 + 1)) = 0 )
7972, 78oveq12d 5928 . . . . . . . 8 (((𝑦 ∈ (𝑀..^𝑁) ∧ 𝐺 ∈ Mnd) ∧ (seq𝑀((+g𝐺), (𝑘 ∈ (𝑀...𝑁) ↦ 0 ))‘𝑦) = 0 ) → ((seq𝑀((+g𝐺), (𝑘 ∈ (𝑀...𝑁) ↦ 0 ))‘𝑦)(+g𝐺)((𝑘 ∈ (𝑀...𝑁) ↦ 0 )‘(𝑦 + 1))) = ( 0 (+g𝐺) 0 ))
801, 3, 2mndlid 13006 . . . . . . . . . 10 ((𝐺 ∈ Mnd ∧ 0 ∈ (Base‘𝐺)) → ( 0 (+g𝐺) 0 ) = 0 )
817, 80mpdan 421 . . . . . . . . 9 (𝐺 ∈ Mnd → ( 0 (+g𝐺) 0 ) = 0 )
8281ad2antlr 489 . . . . . . . 8 (((𝑦 ∈ (𝑀..^𝑁) ∧ 𝐺 ∈ Mnd) ∧ (seq𝑀((+g𝐺), (𝑘 ∈ (𝑀...𝑁) ↦ 0 ))‘𝑦) = 0 ) → ( 0 (+g𝐺) 0 ) = 0 )
8371, 79, 823eqtrd 2230 . . . . . . 7 (((𝑦 ∈ (𝑀..^𝑁) ∧ 𝐺 ∈ Mnd) ∧ (seq𝑀((+g𝐺), (𝑘 ∈ (𝑀...𝑁) ↦ 0 ))‘𝑦) = 0 ) → (seq𝑀((+g𝐺), (𝑘 ∈ (𝑀...𝑁) ↦ 0 ))‘(𝑦 + 1)) = 0 )
8483exp31 364 . . . . . 6 (𝑦 ∈ (𝑀..^𝑁) → (𝐺 ∈ Mnd → ((seq𝑀((+g𝐺), (𝑘 ∈ (𝑀...𝑁) ↦ 0 ))‘𝑦) = 0 → (seq𝑀((+g𝐺), (𝑘 ∈ (𝑀...𝑁) ↦ 0 ))‘(𝑦 + 1)) = 0 )))
8584a2d 26 . . . . 5 (𝑦 ∈ (𝑀..^𝑁) → ((𝐺 ∈ Mnd → (seq𝑀((+g𝐺), (𝑘 ∈ (𝑀...𝑁) ↦ 0 ))‘𝑦) = 0 ) → (𝐺 ∈ Mnd → (seq𝑀((+g𝐺), (𝑘 ∈ (𝑀...𝑁) ↦ 0 ))‘(𝑦 + 1)) = 0 )))
8631, 33, 35, 37, 62, 85fzind2 10296 . . . 4 (𝑁 ∈ (𝑀...𝑁) → (𝐺 ∈ Mnd → (seq𝑀((+g𝐺), (𝑘 ∈ (𝑀...𝑁) ↦ 0 ))‘𝑁) = 0 ))
8728, 29, 86sylc 62 . . 3 (((𝐺 ∈ Mnd ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ 𝑁 < 𝑀) → (seq𝑀((+g𝐺), (𝑘 ∈ (𝑀...𝑁) ↦ 0 ))‘𝑁) = 0 )
8816, 18, 873eqtrd 2230 . 2 (((𝐺 ∈ Mnd ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ 𝑁 < 𝑀) → (𝐺 Σg (𝑘 ∈ (𝑀...𝑁) ↦ 0 )) = 0 )
89 zdclt 9384 . . . 4 ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) → DECID 𝑁 < 𝑀)
906, 5, 89syl2anc 411 . . 3 ((𝐺 ∈ Mnd ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → DECID 𝑁 < 𝑀)
91 exmiddc 837 . . 3 (DECID 𝑁 < 𝑀 → (𝑁 < 𝑀 ∨ ¬ 𝑁 < 𝑀))
9290, 91syl 14 . 2 ((𝐺 ∈ Mnd ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 < 𝑀 ∨ ¬ 𝑁 < 𝑀))
9315, 88, 92mpjaodan 799 1 ((𝐺 ∈ Mnd ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐺 Σg (𝑘 ∈ (𝑀...𝑁) ↦ 0 )) = 0 )
Colors of variables: wff set class
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 5246  (class class class)co 5910  Fincfn 6785  1c1 7863   + caddc 7865   < clt 8044  cle 8045  cz 9307  cuz 9582  ...cfz 10064  ..^cfzo 10198  seqcseq 10508  Basecbs 12608  +gcplusg 12685  0gc0g 12857   Σg cgsu 12858  Mndcmnd 12987
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 4462  ax-setind 4565  ax-iinf 4616  ax-cnex 7953  ax-resscn 7954  ax-1cn 7955  ax-1re 7956  ax-icn 7957  ax-addcl 7958  ax-addrcl 7959  ax-mulcl 7960  ax-addcom 7962  ax-addass 7964  ax-distr 7966  ax-i2m1 7967  ax-0lt1 7968  ax-0id 7970  ax-rnegex 7971  ax-cnre 7973  ax-pre-ltirr 7974  ax-pre-ltwlin 7975  ax-pre-lttrn 7976  ax-pre-apti 7977  ax-pre-ltadd 7978
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 4322  df-iord 4395  df-on 4397  df-ilim 4398  df-suc 4400  df-iom 4619  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-iota 5207  df-fun 5248  df-fn 5249  df-f 5250  df-f1 5251  df-fo 5252  df-f1o 5253  df-fv 5254  df-riota 5865  df-ov 5913  df-oprab 5914  df-mpo 5915  df-1st 6184  df-2nd 6185  df-recs 6349  df-frec 6435  df-1o 6460  df-er 6578  df-en 6786  df-fin 6788  df-pnf 8046  df-mnf 8047  df-xr 8048  df-ltxr 8049  df-le 8050  df-sub 8182  df-neg 8183  df-inn 8973  df-2 9031  df-n0 9231  df-z 9308  df-uz 9583  df-fz 10065  df-fzo 10199  df-seqfrec 10509  df-ndx 12611  df-slot 12612  df-base 12614  df-plusg 12698  df-0g 12859  df-igsum 12860  df-mgm 12929  df-sgrp 12975  df-mnd 12988
This theorem is referenced by: (None)
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