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Theorem gsumval2 18541
Description: Value of the group sum operation over a finite set of sequential integers. (Contributed by Mario Carneiro, 7-Dec-2014.)
Hypotheses
Ref Expression
gsumval2.b 𝐵 = (Base‘𝐺)
gsumval2.p + = (+g𝐺)
gsumval2.g (𝜑𝐺𝑉)
gsumval2.n (𝜑𝑁 ∈ (ℤ𝑀))
gsumval2.f (𝜑𝐹:(𝑀...𝑁)⟶𝐵)
Assertion
Ref Expression
gsumval2 (𝜑 → (𝐺 Σg 𝐹) = (seq𝑀( + , 𝐹)‘𝑁))

Proof of Theorem gsumval2
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 gsumval2.b . . . 4 𝐵 = (Base‘𝐺)
2 eqid 2736 . . . 4 (0g𝐺) = (0g𝐺)
3 gsumval2.p . . . 4 + = (+g𝐺)
4 eqid 2736 . . . 4 {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)} = {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}
5 gsumval2.g . . . . 5 (𝜑𝐺𝑉)
65adantr 481 . . . 4 ((𝜑 ∧ ran 𝐹 ⊆ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → 𝐺𝑉)
7 ovexd 7392 . . . 4 ((𝜑 ∧ ran 𝐹 ⊆ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → (𝑀...𝑁) ∈ V)
8 gsumval2.f . . . . . . 7 (𝜑𝐹:(𝑀...𝑁)⟶𝐵)
98ffnd 6669 . . . . . 6 (𝜑𝐹 Fn (𝑀...𝑁))
109adantr 481 . . . . 5 ((𝜑 ∧ ran 𝐹 ⊆ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → 𝐹 Fn (𝑀...𝑁))
11 simpr 485 . . . . 5 ((𝜑 ∧ ran 𝐹 ⊆ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → ran 𝐹 ⊆ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)})
12 df-f 6500 . . . . 5 (𝐹:(𝑀...𝑁)⟶{𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)} ↔ (𝐹 Fn (𝑀...𝑁) ∧ ran 𝐹 ⊆ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}))
1310, 11, 12sylanbrc 583 . . . 4 ((𝜑 ∧ ran 𝐹 ⊆ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → 𝐹:(𝑀...𝑁)⟶{𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)})
141, 2, 3, 4, 6, 7, 13gsumval1 18538 . . 3 ((𝜑 ∧ ran 𝐹 ⊆ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → (𝐺 Σg 𝐹) = (0g𝐺))
15 simpl 483 . . . . . . . . 9 (((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦) → (𝑥 + 𝑦) = 𝑦)
1615ralimi 3086 . . . . . . . 8 (∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦) → ∀𝑦𝐵 (𝑥 + 𝑦) = 𝑦)
1716a1i 11 . . . . . . 7 (𝑥𝐵 → (∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦) → ∀𝑦𝐵 (𝑥 + 𝑦) = 𝑦))
1817ss2rabi 4034 . . . . . 6 {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)} ⊆ {𝑥𝐵 ∣ ∀𝑦𝐵 (𝑥 + 𝑦) = 𝑦}
19 fvex 6855 . . . . . . . 8 (0g𝐺) ∈ V
2019snid 4622 . . . . . . 7 (0g𝐺) ∈ {(0g𝐺)}
218fdmd 6679 . . . . . . . . . . . . 13 (𝜑 → dom 𝐹 = (𝑀...𝑁))
22 gsumval2.n . . . . . . . . . . . . . 14 (𝜑𝑁 ∈ (ℤ𝑀))
23 eluzfz1 13448 . . . . . . . . . . . . . 14 (𝑁 ∈ (ℤ𝑀) → 𝑀 ∈ (𝑀...𝑁))
24 ne0i 4294 . . . . . . . . . . . . . 14 (𝑀 ∈ (𝑀...𝑁) → (𝑀...𝑁) ≠ ∅)
2522, 23, 243syl 18 . . . . . . . . . . . . 13 (𝜑 → (𝑀...𝑁) ≠ ∅)
2621, 25eqnetrd 3011 . . . . . . . . . . . 12 (𝜑 → dom 𝐹 ≠ ∅)
27 dm0rn0 5880 . . . . . . . . . . . . 13 (dom 𝐹 = ∅ ↔ ran 𝐹 = ∅)
2827necon3bii 2996 . . . . . . . . . . . 12 (dom 𝐹 ≠ ∅ ↔ ran 𝐹 ≠ ∅)
2926, 28sylib 217 . . . . . . . . . . 11 (𝜑 → ran 𝐹 ≠ ∅)
3029adantr 481 . . . . . . . . . 10 ((𝜑 ∧ ran 𝐹 ⊆ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → ran 𝐹 ≠ ∅)
31 ssn0 4360 . . . . . . . . . 10 ((ran 𝐹 ⊆ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)} ∧ ran 𝐹 ≠ ∅) → {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)} ≠ ∅)
3211, 30, 31syl2anc 584 . . . . . . . . 9 ((𝜑 ∧ ran 𝐹 ⊆ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)} ≠ ∅)
3332neneqd 2948 . . . . . . . 8 ((𝜑 ∧ ran 𝐹 ⊆ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → ¬ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)} = ∅)
341, 2, 3, 4mgmidsssn0 18527 . . . . . . . . . . 11 (𝐺𝑉 → {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)} ⊆ {(0g𝐺)})
355, 34syl 17 . . . . . . . . . 10 (𝜑 → {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)} ⊆ {(0g𝐺)})
36 sssn 4786 . . . . . . . . . 10 ({𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)} ⊆ {(0g𝐺)} ↔ ({𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)} = ∅ ∨ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)} = {(0g𝐺)}))
3735, 36sylib 217 . . . . . . . . 9 (𝜑 → ({𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)} = ∅ ∨ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)} = {(0g𝐺)}))
3837orcanai 1001 . . . . . . . 8 ((𝜑 ∧ ¬ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)} = ∅) → {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)} = {(0g𝐺)})
3933, 38syldan 591 . . . . . . 7 ((𝜑 ∧ ran 𝐹 ⊆ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)} = {(0g𝐺)})
4020, 39eleqtrrid 2845 . . . . . 6 ((𝜑 ∧ ran 𝐹 ⊆ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → (0g𝐺) ∈ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)})
4118, 40sselid 3942 . . . . 5 ((𝜑 ∧ ran 𝐹 ⊆ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → (0g𝐺) ∈ {𝑥𝐵 ∣ ∀𝑦𝐵 (𝑥 + 𝑦) = 𝑦})
42 oveq1 7364 . . . . . . . . 9 (𝑥 = (0g𝐺) → (𝑥 + 𝑦) = ((0g𝐺) + 𝑦))
4342eqeq1d 2738 . . . . . . . 8 (𝑥 = (0g𝐺) → ((𝑥 + 𝑦) = 𝑦 ↔ ((0g𝐺) + 𝑦) = 𝑦))
4443ralbidv 3174 . . . . . . 7 (𝑥 = (0g𝐺) → (∀𝑦𝐵 (𝑥 + 𝑦) = 𝑦 ↔ ∀𝑦𝐵 ((0g𝐺) + 𝑦) = 𝑦))
4544elrab 3645 . . . . . 6 ((0g𝐺) ∈ {𝑥𝐵 ∣ ∀𝑦𝐵 (𝑥 + 𝑦) = 𝑦} ↔ ((0g𝐺) ∈ 𝐵 ∧ ∀𝑦𝐵 ((0g𝐺) + 𝑦) = 𝑦))
46 oveq2 7365 . . . . . . . 8 (𝑦 = (0g𝐺) → ((0g𝐺) + 𝑦) = ((0g𝐺) + (0g𝐺)))
47 id 22 . . . . . . . 8 (𝑦 = (0g𝐺) → 𝑦 = (0g𝐺))
4846, 47eqeq12d 2752 . . . . . . 7 (𝑦 = (0g𝐺) → (((0g𝐺) + 𝑦) = 𝑦 ↔ ((0g𝐺) + (0g𝐺)) = (0g𝐺)))
4948rspcva 3579 . . . . . 6 (((0g𝐺) ∈ 𝐵 ∧ ∀𝑦𝐵 ((0g𝐺) + 𝑦) = 𝑦) → ((0g𝐺) + (0g𝐺)) = (0g𝐺))
5045, 49sylbi 216 . . . . 5 ((0g𝐺) ∈ {𝑥𝐵 ∣ ∀𝑦𝐵 (𝑥 + 𝑦) = 𝑦} → ((0g𝐺) + (0g𝐺)) = (0g𝐺))
5141, 50syl 17 . . . 4 ((𝜑 ∧ ran 𝐹 ⊆ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → ((0g𝐺) + (0g𝐺)) = (0g𝐺))
5222adantr 481 . . . 4 ((𝜑 ∧ ran 𝐹 ⊆ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → 𝑁 ∈ (ℤ𝑀))
5335ad2antrr 724 . . . . . 6 (((𝜑 ∧ ran 𝐹 ⊆ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) ∧ 𝑧 ∈ (𝑀...𝑁)) → {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)} ⊆ {(0g𝐺)})
5413ffvelcdmda 7035 . . . . . 6 (((𝜑 ∧ ran 𝐹 ⊆ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) ∧ 𝑧 ∈ (𝑀...𝑁)) → (𝐹𝑧) ∈ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)})
5553, 54sseldd 3945 . . . . 5 (((𝜑 ∧ ran 𝐹 ⊆ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) ∧ 𝑧 ∈ (𝑀...𝑁)) → (𝐹𝑧) ∈ {(0g𝐺)})
56 elsni 4603 . . . . 5 ((𝐹𝑧) ∈ {(0g𝐺)} → (𝐹𝑧) = (0g𝐺))
5755, 56syl 17 . . . 4 (((𝜑 ∧ ran 𝐹 ⊆ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) ∧ 𝑧 ∈ (𝑀...𝑁)) → (𝐹𝑧) = (0g𝐺))
5851, 52, 57seqid3 13952 . . 3 ((𝜑 ∧ ran 𝐹 ⊆ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → (seq𝑀( + , 𝐹)‘𝑁) = (0g𝐺))
5914, 58eqtr4d 2779 . 2 ((𝜑 ∧ ran 𝐹 ⊆ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → (𝐺 Σg 𝐹) = (seq𝑀( + , 𝐹)‘𝑁))
605adantr 481 . . 3 ((𝜑 ∧ ¬ ran 𝐹 ⊆ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → 𝐺𝑉)
6122adantr 481 . . 3 ((𝜑 ∧ ¬ ran 𝐹 ⊆ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → 𝑁 ∈ (ℤ𝑀))
628adantr 481 . . 3 ((𝜑 ∧ ¬ ran 𝐹 ⊆ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → 𝐹:(𝑀...𝑁)⟶𝐵)
63 simpr 485 . . 3 ((𝜑 ∧ ¬ ran 𝐹 ⊆ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → ¬ ran 𝐹 ⊆ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)})
641, 3, 60, 61, 62, 4, 63gsumval2a 18540 . 2 ((𝜑 ∧ ¬ ran 𝐹 ⊆ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → (𝐺 Σg 𝐹) = (seq𝑀( + , 𝐹)‘𝑁))
6559, 64pm2.61dan 811 1 (𝜑 → (𝐺 Σg 𝐹) = (seq𝑀( + , 𝐹)‘𝑁))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  wo 845   = wceq 1541  wcel 2106  wne 2943  wral 3064  {crab 3407  Vcvv 3445  wss 3910  c0 4282  {csn 4586  dom cdm 5633  ran crn 5634   Fn wfn 6491  wf 6492  cfv 6496  (class class class)co 7357  cuz 12763  ...cfz 13424  seqcseq 13906  Basecbs 17083  +gcplusg 17133  0gc0g 17321   Σg cgsu 17322
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-n0 12414  df-z 12500  df-uz 12764  df-fz 13425  df-seq 13907  df-0g 17323  df-gsum 17324
This theorem is referenced by:  gsumsplit1r  18542  gsumprval  18543  gsumwsubmcl  18647  gsumws1  18648  gsumsgrpccat  18650  gsumwmhm  18655  mulgnngsum  18881  gsumval3  19684  gsummptfzcl  19746  gsumncl  33152  gsumnunsn  33153
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