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Theorem gsumval2 17888
 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 2798 . . . 4 (0g𝐺) = (0g𝐺)
3 gsumval2.p . . . 4 + = (+g𝐺)
4 eqid 2798 . . . 4 {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)} = {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}
5 gsumval2.g . . . . 5 (𝜑𝐺𝑉)
65adantr 484 . . . 4 ((𝜑 ∧ ran 𝐹 ⊆ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → 𝐺𝑉)
7 ovexd 7170 . . . 4 ((𝜑 ∧ ran 𝐹 ⊆ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → (𝑀...𝑁) ∈ V)
8 gsumval2.f . . . . . . 7 (𝜑𝐹:(𝑀...𝑁)⟶𝐵)
98ffnd 6488 . . . . . 6 (𝜑𝐹 Fn (𝑀...𝑁))
109adantr 484 . . . . 5 ((𝜑 ∧ ran 𝐹 ⊆ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → 𝐹 Fn (𝑀...𝑁))
11 simpr 488 . . . . 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 6328 . . . . 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 586 . . . 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 17885 . . 3 ((𝜑 ∧ ran 𝐹 ⊆ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → (𝐺 Σg 𝐹) = (0g𝐺))
15 simpl 486 . . . . . . . . 9 (((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦) → (𝑥 + 𝑦) = 𝑦)
1615ralimi 3128 . . . . . . . 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 4004 . . . . . 6 {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)} ⊆ {𝑥𝐵 ∣ ∀𝑦𝐵 (𝑥 + 𝑦) = 𝑦}
19 fvex 6658 . . . . . . . 8 (0g𝐺) ∈ V
2019snid 4561 . . . . . . 7 (0g𝐺) ∈ {(0g𝐺)}
218fdmd 6497 . . . . . . . . . . . . 13 (𝜑 → dom 𝐹 = (𝑀...𝑁))
22 gsumval2.n . . . . . . . . . . . . . 14 (𝜑𝑁 ∈ (ℤ𝑀))
23 eluzfz1 12909 . . . . . . . . . . . . . 14 (𝑁 ∈ (ℤ𝑀) → 𝑀 ∈ (𝑀...𝑁))
24 ne0i 4250 . . . . . . . . . . . . . 14 (𝑀 ∈ (𝑀...𝑁) → (𝑀...𝑁) ≠ ∅)
2522, 23, 243syl 18 . . . . . . . . . . . . 13 (𝜑 → (𝑀...𝑁) ≠ ∅)
2621, 25eqnetrd 3054 . . . . . . . . . . . 12 (𝜑 → dom 𝐹 ≠ ∅)
27 dm0rn0 5759 . . . . . . . . . . . . 13 (dom 𝐹 = ∅ ↔ ran 𝐹 = ∅)
2827necon3bii 3039 . . . . . . . . . . . 12 (dom 𝐹 ≠ ∅ ↔ ran 𝐹 ≠ ∅)
2926, 28sylib 221 . . . . . . . . . . 11 (𝜑 → ran 𝐹 ≠ ∅)
3029adantr 484 . . . . . . . . . 10 ((𝜑 ∧ ran 𝐹 ⊆ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → ran 𝐹 ≠ ∅)
31 ssn0 4308 . . . . . . . . . 10 ((ran 𝐹 ⊆ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)} ∧ ran 𝐹 ≠ ∅) → {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)} ≠ ∅)
3211, 30, 31syl2anc 587 . . . . . . . . 9 ((𝜑 ∧ ran 𝐹 ⊆ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)} ≠ ∅)
3332neneqd 2992 . . . . . . . 8 ((𝜑 ∧ ran 𝐹 ⊆ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → ¬ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)} = ∅)
341, 2, 3, 4mgmidsssn0 17874 . . . . . . . . . . 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 4719 . . . . . . . . . 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 221 . . . . . . . . 9 (𝜑 → ({𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)} = ∅ ∨ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)} = {(0g𝐺)}))
3837orcanai 1000 . . . . . . . 8 ((𝜑 ∧ ¬ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)} = ∅) → {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)} = {(0g𝐺)})
3933, 38syldan 594 . . . . . . 7 ((𝜑 ∧ ran 𝐹 ⊆ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)} = {(0g𝐺)})
4020, 39eleqtrrid 2897 . . . . . 6 ((𝜑 ∧ ran 𝐹 ⊆ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → (0g𝐺) ∈ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)})
4118, 40sseldi 3913 . . . . 5 ((𝜑 ∧ ran 𝐹 ⊆ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → (0g𝐺) ∈ {𝑥𝐵 ∣ ∀𝑦𝐵 (𝑥 + 𝑦) = 𝑦})
42 oveq1 7142 . . . . . . . . 9 (𝑥 = (0g𝐺) → (𝑥 + 𝑦) = ((0g𝐺) + 𝑦))
4342eqeq1d 2800 . . . . . . . 8 (𝑥 = (0g𝐺) → ((𝑥 + 𝑦) = 𝑦 ↔ ((0g𝐺) + 𝑦) = 𝑦))
4443ralbidv 3162 . . . . . . 7 (𝑥 = (0g𝐺) → (∀𝑦𝐵 (𝑥 + 𝑦) = 𝑦 ↔ ∀𝑦𝐵 ((0g𝐺) + 𝑦) = 𝑦))
4544elrab 3628 . . . . . 6 ((0g𝐺) ∈ {𝑥𝐵 ∣ ∀𝑦𝐵 (𝑥 + 𝑦) = 𝑦} ↔ ((0g𝐺) ∈ 𝐵 ∧ ∀𝑦𝐵 ((0g𝐺) + 𝑦) = 𝑦))
46 oveq2 7143 . . . . . . . 8 (𝑦 = (0g𝐺) → ((0g𝐺) + 𝑦) = ((0g𝐺) + (0g𝐺)))
47 id 22 . . . . . . . 8 (𝑦 = (0g𝐺) → 𝑦 = (0g𝐺))
4846, 47eqeq12d 2814 . . . . . . 7 (𝑦 = (0g𝐺) → (((0g𝐺) + 𝑦) = 𝑦 ↔ ((0g𝐺) + (0g𝐺)) = (0g𝐺)))
4948rspcva 3569 . . . . . 6 (((0g𝐺) ∈ 𝐵 ∧ ∀𝑦𝐵 ((0g𝐺) + 𝑦) = 𝑦) → ((0g𝐺) + (0g𝐺)) = (0g𝐺))
5045, 49sylbi 220 . . . . 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 484 . . . 4 ((𝜑 ∧ ran 𝐹 ⊆ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → 𝑁 ∈ (ℤ𝑀))
5335ad2antrr 725 . . . . . 6 (((𝜑 ∧ ran 𝐹 ⊆ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) ∧ 𝑧 ∈ (𝑀...𝑁)) → {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)} ⊆ {(0g𝐺)})
5413ffvelrnda 6828 . . . . . 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 3916 . . . . 5 (((𝜑 ∧ ran 𝐹 ⊆ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) ∧ 𝑧 ∈ (𝑀...𝑁)) → (𝐹𝑧) ∈ {(0g𝐺)})
56 elsni 4542 . . . . 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 13410 . . 3 ((𝜑 ∧ ran 𝐹 ⊆ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → (seq𝑀( + , 𝐹)‘𝑁) = (0g𝐺))
5914, 58eqtr4d 2836 . 2 ((𝜑 ∧ ran 𝐹 ⊆ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → (𝐺 Σg 𝐹) = (seq𝑀( + , 𝐹)‘𝑁))
605adantr 484 . . 3 ((𝜑 ∧ ¬ ran 𝐹 ⊆ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → 𝐺𝑉)
6122adantr 484 . . 3 ((𝜑 ∧ ¬ ran 𝐹 ⊆ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → 𝑁 ∈ (ℤ𝑀))
628adantr 484 . . 3 ((𝜑 ∧ ¬ ran 𝐹 ⊆ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → 𝐹:(𝑀...𝑁)⟶𝐵)
63 simpr 488 . . 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 17887 . 2 ((𝜑 ∧ ¬ ran 𝐹 ⊆ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → (𝐺 Σg 𝐹) = (seq𝑀( + , 𝐹)‘𝑁))
6559, 64pm2.61dan 812 1 (𝜑 → (𝐺 Σg 𝐹) = (seq𝑀( + , 𝐹)‘𝑁))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   ∨ wo 844   = wceq 1538   ∈ wcel 2111   ≠ wne 2987  ∀wral 3106  {crab 3110  Vcvv 3441   ⊆ wss 3881  ∅c0 4243  {csn 4525  dom cdm 5519  ran crn 5520   Fn wfn 6319  ⟶wf 6320  ‘cfv 6324  (class class class)co 7135  ℤ≥cuz 12231  ...cfz 12885  seqcseq 13364  Basecbs 16475  +gcplusg 16557  0gc0g 16705   Σg cgsu 16706 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7561  df-1st 7671  df-2nd 7672  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-er 8272  df-en 8493  df-dom 8494  df-sdom 8495  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-nn 11626  df-n0 11886  df-z 11970  df-uz 12232  df-fz 12886  df-seq 13365  df-0g 16707  df-gsum 16708 This theorem is referenced by:  gsumsplit1r  17889  gsumprval  17890  gsumwsubmcl  17993  gsumws1  17994  gsumsgrpccat  17996  gsumccatOLD  17997  gsumwmhm  18002  mulgnngsum  18225  gsumval3  19020  gsummptfzcl  19082  gsumncl  31920  gsumnunsn  31921
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