MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  gsumval2 Structured version   Visualization version   GIF version

Theorem gsumval2 18619
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 2726 . . . 4 (0g𝐺) = (0g𝐺)
3 gsumval2.p . . . 4 + = (+g𝐺)
4 eqid 2726 . . . 4 {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)} = {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}
5 gsumval2.g . . . . 5 (𝜑𝐺𝑉)
65adantr 480 . . . 4 ((𝜑 ∧ ran 𝐹 ⊆ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → 𝐺𝑉)
7 ovexd 7440 . . . 4 ((𝜑 ∧ ran 𝐹 ⊆ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → (𝑀...𝑁) ∈ V)
8 gsumval2.f . . . . . . 7 (𝜑𝐹:(𝑀...𝑁)⟶𝐵)
98ffnd 6712 . . . . . 6 (𝜑𝐹 Fn (𝑀...𝑁))
109adantr 480 . . . . 5 ((𝜑 ∧ ran 𝐹 ⊆ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → 𝐹 Fn (𝑀...𝑁))
11 simpr 484 . . . . 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 6541 . . . . 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 582 . . . 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 18616 . . 3 ((𝜑 ∧ ran 𝐹 ⊆ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → (𝐺 Σg 𝐹) = (0g𝐺))
15 simpl 482 . . . . . . . . 9 (((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦) → (𝑥 + 𝑦) = 𝑦)
1615ralimi 3077 . . . . . . . 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 4069 . . . . . 6 {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)} ⊆ {𝑥𝐵 ∣ ∀𝑦𝐵 (𝑥 + 𝑦) = 𝑦}
19 fvex 6898 . . . . . . . 8 (0g𝐺) ∈ V
2019snid 4659 . . . . . . 7 (0g𝐺) ∈ {(0g𝐺)}
218fdmd 6722 . . . . . . . . . . . . 13 (𝜑 → dom 𝐹 = (𝑀...𝑁))
22 gsumval2.n . . . . . . . . . . . . . 14 (𝜑𝑁 ∈ (ℤ𝑀))
23 eluzfz1 13514 . . . . . . . . . . . . . 14 (𝑁 ∈ (ℤ𝑀) → 𝑀 ∈ (𝑀...𝑁))
24 ne0i 4329 . . . . . . . . . . . . . 14 (𝑀 ∈ (𝑀...𝑁) → (𝑀...𝑁) ≠ ∅)
2522, 23, 243syl 18 . . . . . . . . . . . . 13 (𝜑 → (𝑀...𝑁) ≠ ∅)
2621, 25eqnetrd 3002 . . . . . . . . . . . 12 (𝜑 → dom 𝐹 ≠ ∅)
27 dm0rn0 5918 . . . . . . . . . . . . 13 (dom 𝐹 = ∅ ↔ ran 𝐹 = ∅)
2827necon3bii 2987 . . . . . . . . . . . 12 (dom 𝐹 ≠ ∅ ↔ ran 𝐹 ≠ ∅)
2926, 28sylib 217 . . . . . . . . . . 11 (𝜑 → ran 𝐹 ≠ ∅)
3029adantr 480 . . . . . . . . . 10 ((𝜑 ∧ ran 𝐹 ⊆ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → ran 𝐹 ≠ ∅)
31 ssn0 4395 . . . . . . . . . 10 ((ran 𝐹 ⊆ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)} ∧ ran 𝐹 ≠ ∅) → {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)} ≠ ∅)
3211, 30, 31syl2anc 583 . . . . . . . . 9 ((𝜑 ∧ ran 𝐹 ⊆ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)} ≠ ∅)
3332neneqd 2939 . . . . . . . 8 ((𝜑 ∧ ran 𝐹 ⊆ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → ¬ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)} = ∅)
341, 2, 3, 4mgmidsssn0 18605 . . . . . . . . . . 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 4824 . . . . . . . . . 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 999 . . . . . . . 8 ((𝜑 ∧ ¬ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)} = ∅) → {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)} = {(0g𝐺)})
3933, 38syldan 590 . . . . . . 7 ((𝜑 ∧ ran 𝐹 ⊆ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)} = {(0g𝐺)})
4020, 39eleqtrrid 2834 . . . . . 6 ((𝜑 ∧ ran 𝐹 ⊆ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → (0g𝐺) ∈ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)})
4118, 40sselid 3975 . . . . 5 ((𝜑 ∧ ran 𝐹 ⊆ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → (0g𝐺) ∈ {𝑥𝐵 ∣ ∀𝑦𝐵 (𝑥 + 𝑦) = 𝑦})
42 oveq1 7412 . . . . . . . . 9 (𝑥 = (0g𝐺) → (𝑥 + 𝑦) = ((0g𝐺) + 𝑦))
4342eqeq1d 2728 . . . . . . . 8 (𝑥 = (0g𝐺) → ((𝑥 + 𝑦) = 𝑦 ↔ ((0g𝐺) + 𝑦) = 𝑦))
4443ralbidv 3171 . . . . . . 7 (𝑥 = (0g𝐺) → (∀𝑦𝐵 (𝑥 + 𝑦) = 𝑦 ↔ ∀𝑦𝐵 ((0g𝐺) + 𝑦) = 𝑦))
4544elrab 3678 . . . . . 6 ((0g𝐺) ∈ {𝑥𝐵 ∣ ∀𝑦𝐵 (𝑥 + 𝑦) = 𝑦} ↔ ((0g𝐺) ∈ 𝐵 ∧ ∀𝑦𝐵 ((0g𝐺) + 𝑦) = 𝑦))
46 oveq2 7413 . . . . . . . 8 (𝑦 = (0g𝐺) → ((0g𝐺) + 𝑦) = ((0g𝐺) + (0g𝐺)))
47 id 22 . . . . . . . 8 (𝑦 = (0g𝐺) → 𝑦 = (0g𝐺))
4846, 47eqeq12d 2742 . . . . . . 7 (𝑦 = (0g𝐺) → (((0g𝐺) + 𝑦) = 𝑦 ↔ ((0g𝐺) + (0g𝐺)) = (0g𝐺)))
4948rspcva 3604 . . . . . 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 480 . . . 4 ((𝜑 ∧ ran 𝐹 ⊆ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → 𝑁 ∈ (ℤ𝑀))
5335ad2antrr 723 . . . . . 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 7080 . . . . . 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 3978 . . . . 5 (((𝜑 ∧ ran 𝐹 ⊆ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) ∧ 𝑧 ∈ (𝑀...𝑁)) → (𝐹𝑧) ∈ {(0g𝐺)})
56 elsni 4640 . . . . 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 14017 . . 3 ((𝜑 ∧ ran 𝐹 ⊆ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → (seq𝑀( + , 𝐹)‘𝑁) = (0g𝐺))
5914, 58eqtr4d 2769 . 2 ((𝜑 ∧ ran 𝐹 ⊆ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → (𝐺 Σg 𝐹) = (seq𝑀( + , 𝐹)‘𝑁))
605adantr 480 . . 3 ((𝜑 ∧ ¬ ran 𝐹 ⊆ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → 𝐺𝑉)
6122adantr 480 . . 3 ((𝜑 ∧ ¬ ran 𝐹 ⊆ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → 𝑁 ∈ (ℤ𝑀))
628adantr 480 . . 3 ((𝜑 ∧ ¬ ran 𝐹 ⊆ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → 𝐹:(𝑀...𝑁)⟶𝐵)
63 simpr 484 . . 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 18618 . 2 ((𝜑 ∧ ¬ ran 𝐹 ⊆ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → (𝐺 Σg 𝐹) = (seq𝑀( + , 𝐹)‘𝑁))
6559, 64pm2.61dan 810 1 (𝜑 → (𝐺 Σg 𝐹) = (seq𝑀( + , 𝐹)‘𝑁))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  wo 844   = wceq 1533  wcel 2098  wne 2934  wral 3055  {crab 3426  Vcvv 3468  wss 3943  c0 4317  {csn 4623  dom cdm 5669  ran crn 5670   Fn wfn 6532  wf 6533  cfv 6537  (class class class)co 7405  cuz 12826  ...cfz 13490  seqcseq 13972  Basecbs 17153  +gcplusg 17206  0gc0g 17394   Σg cgsu 17395
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-nel 3041  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6294  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7853  df-1st 7974  df-2nd 7975  df-frecs 8267  df-wrecs 8298  df-recs 8372  df-rdg 8411  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-nn 12217  df-n0 12477  df-z 12563  df-uz 12827  df-fz 13491  df-seq 13973  df-0g 17396  df-gsum 17397
This theorem is referenced by:  gsumsplit1r  18620  gsumprval  18621  gsumwsubmcl  18762  gsumws1  18763  gsumsgrpccat  18765  gsumwmhm  18770  mulgnngsum  19006  gsumval3  19827  gsummptfzcl  19889  gsumncl  34081  gsumnunsn  34082
  Copyright terms: Public domain W3C validator