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Theorem gsumval2 17327
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 2651 . . . 4 (0g𝐺) = (0g𝐺)
3 gsumval2.p . . . 4 + = (+g𝐺)
4 eqid 2651 . . . 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 6720 . . . 4 ((𝜑 ∧ ran 𝐹 ⊆ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → (𝑀...𝑁) ∈ V)
8 gsumval2.f . . . . . . 7 (𝜑𝐹:(𝑀...𝑁)⟶𝐵)
9 ffn 6083 . . . . . . 7 (𝐹:(𝑀...𝑁)⟶𝐵𝐹 Fn (𝑀...𝑁))
108, 9syl 17 . . . . . 6 (𝜑𝐹 Fn (𝑀...𝑁))
1110adantr 480 . . . . 5 ((𝜑 ∧ ran 𝐹 ⊆ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → 𝐹 Fn (𝑀...𝑁))
12 simpr 476 . . . . 5 ((𝜑 ∧ ran 𝐹 ⊆ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → ran 𝐹 ⊆ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)})
13 df-f 5930 . . . . 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)}))
1411, 12, 13sylanbrc 699 . . . 4 ((𝜑 ∧ ran 𝐹 ⊆ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → 𝐹:(𝑀...𝑁)⟶{𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)})
151, 2, 3, 4, 6, 7, 14gsumval1 17324 . . 3 ((𝜑 ∧ ran 𝐹 ⊆ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → (𝐺 Σg 𝐹) = (0g𝐺))
16 simpl 472 . . . . . . . . 9 (((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦) → (𝑥 + 𝑦) = 𝑦)
1716ralimi 2981 . . . . . . . 8 (∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦) → ∀𝑦𝐵 (𝑥 + 𝑦) = 𝑦)
1817a1i 11 . . . . . . 7 (𝑥𝐵 → (∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦) → ∀𝑦𝐵 (𝑥 + 𝑦) = 𝑦))
1918ss2rabi 3717 . . . . . 6 {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)} ⊆ {𝑥𝐵 ∣ ∀𝑦𝐵 (𝑥 + 𝑦) = 𝑦}
20 fvex 6239 . . . . . . . 8 (0g𝐺) ∈ V
2120snid 4241 . . . . . . 7 (0g𝐺) ∈ {(0g𝐺)}
22 fdm 6089 . . . . . . . . . . . . . 14 (𝐹:(𝑀...𝑁)⟶𝐵 → dom 𝐹 = (𝑀...𝑁))
238, 22syl 17 . . . . . . . . . . . . 13 (𝜑 → dom 𝐹 = (𝑀...𝑁))
24 gsumval2.n . . . . . . . . . . . . . 14 (𝜑𝑁 ∈ (ℤ𝑀))
25 eluzfz1 12386 . . . . . . . . . . . . . 14 (𝑁 ∈ (ℤ𝑀) → 𝑀 ∈ (𝑀...𝑁))
26 ne0i 3954 . . . . . . . . . . . . . 14 (𝑀 ∈ (𝑀...𝑁) → (𝑀...𝑁) ≠ ∅)
2724, 25, 263syl 18 . . . . . . . . . . . . 13 (𝜑 → (𝑀...𝑁) ≠ ∅)
2823, 27eqnetrd 2890 . . . . . . . . . . . 12 (𝜑 → dom 𝐹 ≠ ∅)
29 dm0rn0 5374 . . . . . . . . . . . . 13 (dom 𝐹 = ∅ ↔ ran 𝐹 = ∅)
3029necon3bii 2875 . . . . . . . . . . . 12 (dom 𝐹 ≠ ∅ ↔ ran 𝐹 ≠ ∅)
3128, 30sylib 208 . . . . . . . . . . 11 (𝜑 → ran 𝐹 ≠ ∅)
3231adantr 480 . . . . . . . . . 10 ((𝜑 ∧ ran 𝐹 ⊆ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → ran 𝐹 ≠ ∅)
33 ssn0 4009 . . . . . . . . . 10 ((ran 𝐹 ⊆ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)} ∧ ran 𝐹 ≠ ∅) → {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)} ≠ ∅)
3412, 32, 33syl2anc 694 . . . . . . . . 9 ((𝜑 ∧ ran 𝐹 ⊆ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)} ≠ ∅)
3534neneqd 2828 . . . . . . . 8 ((𝜑 ∧ ran 𝐹 ⊆ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → ¬ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)} = ∅)
361, 2, 3, 4mgmidsssn0 17316 . . . . . . . . . . 11 (𝐺𝑉 → {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)} ⊆ {(0g𝐺)})
375, 36syl 17 . . . . . . . . . 10 (𝜑 → {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)} ⊆ {(0g𝐺)})
38 sssn 4390 . . . . . . . . . 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)}))
3937, 38sylib 208 . . . . . . . . 9 (𝜑 → ({𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)} = ∅ ∨ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)} = {(0g𝐺)}))
4039orcanai 972 . . . . . . . 8 ((𝜑 ∧ ¬ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)} = ∅) → {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)} = {(0g𝐺)})
4135, 40syldan 486 . . . . . . 7 ((𝜑 ∧ ran 𝐹 ⊆ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)} = {(0g𝐺)})
4221, 41syl5eleqr 2737 . . . . . 6 ((𝜑 ∧ ran 𝐹 ⊆ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → (0g𝐺) ∈ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)})
4319, 42sseldi 3634 . . . . 5 ((𝜑 ∧ ran 𝐹 ⊆ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → (0g𝐺) ∈ {𝑥𝐵 ∣ ∀𝑦𝐵 (𝑥 + 𝑦) = 𝑦})
44 oveq1 6697 . . . . . . . . 9 (𝑥 = (0g𝐺) → (𝑥 + 𝑦) = ((0g𝐺) + 𝑦))
4544eqeq1d 2653 . . . . . . . 8 (𝑥 = (0g𝐺) → ((𝑥 + 𝑦) = 𝑦 ↔ ((0g𝐺) + 𝑦) = 𝑦))
4645ralbidv 3015 . . . . . . 7 (𝑥 = (0g𝐺) → (∀𝑦𝐵 (𝑥 + 𝑦) = 𝑦 ↔ ∀𝑦𝐵 ((0g𝐺) + 𝑦) = 𝑦))
4746elrab 3396 . . . . . 6 ((0g𝐺) ∈ {𝑥𝐵 ∣ ∀𝑦𝐵 (𝑥 + 𝑦) = 𝑦} ↔ ((0g𝐺) ∈ 𝐵 ∧ ∀𝑦𝐵 ((0g𝐺) + 𝑦) = 𝑦))
48 oveq2 6698 . . . . . . . 8 (𝑦 = (0g𝐺) → ((0g𝐺) + 𝑦) = ((0g𝐺) + (0g𝐺)))
49 id 22 . . . . . . . 8 (𝑦 = (0g𝐺) → 𝑦 = (0g𝐺))
5048, 49eqeq12d 2666 . . . . . . 7 (𝑦 = (0g𝐺) → (((0g𝐺) + 𝑦) = 𝑦 ↔ ((0g𝐺) + (0g𝐺)) = (0g𝐺)))
5150rspcva 3338 . . . . . 6 (((0g𝐺) ∈ 𝐵 ∧ ∀𝑦𝐵 ((0g𝐺) + 𝑦) = 𝑦) → ((0g𝐺) + (0g𝐺)) = (0g𝐺))
5247, 51sylbi 207 . . . . 5 ((0g𝐺) ∈ {𝑥𝐵 ∣ ∀𝑦𝐵 (𝑥 + 𝑦) = 𝑦} → ((0g𝐺) + (0g𝐺)) = (0g𝐺))
5343, 52syl 17 . . . 4 ((𝜑 ∧ ran 𝐹 ⊆ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → ((0g𝐺) + (0g𝐺)) = (0g𝐺))
5424adantr 480 . . . 4 ((𝜑 ∧ ran 𝐹 ⊆ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → 𝑁 ∈ (ℤ𝑀))
5537ad2antrr 762 . . . . . 6 (((𝜑 ∧ ran 𝐹 ⊆ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) ∧ 𝑧 ∈ (𝑀...𝑁)) → {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)} ⊆ {(0g𝐺)})
5614ffvelrnda 6399 . . . . . 6 (((𝜑 ∧ ran 𝐹 ⊆ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) ∧ 𝑧 ∈ (𝑀...𝑁)) → (𝐹𝑧) ∈ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)})
5755, 56sseldd 3637 . . . . 5 (((𝜑 ∧ ran 𝐹 ⊆ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) ∧ 𝑧 ∈ (𝑀...𝑁)) → (𝐹𝑧) ∈ {(0g𝐺)})
58 elsni 4227 . . . . 5 ((𝐹𝑧) ∈ {(0g𝐺)} → (𝐹𝑧) = (0g𝐺))
5957, 58syl 17 . . . 4 (((𝜑 ∧ ran 𝐹 ⊆ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) ∧ 𝑧 ∈ (𝑀...𝑁)) → (𝐹𝑧) = (0g𝐺))
6053, 54, 59seqid3 12885 . . 3 ((𝜑 ∧ ran 𝐹 ⊆ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → (seq𝑀( + , 𝐹)‘𝑁) = (0g𝐺))
6115, 60eqtr4d 2688 . 2 ((𝜑 ∧ ran 𝐹 ⊆ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → (𝐺 Σg 𝐹) = (seq𝑀( + , 𝐹)‘𝑁))
625adantr 480 . . 3 ((𝜑 ∧ ¬ ran 𝐹 ⊆ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → 𝐺𝑉)
6324adantr 480 . . 3 ((𝜑 ∧ ¬ ran 𝐹 ⊆ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → 𝑁 ∈ (ℤ𝑀))
648adantr 480 . . 3 ((𝜑 ∧ ¬ ran 𝐹 ⊆ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → 𝐹:(𝑀...𝑁)⟶𝐵)
65 simpr 476 . . 3 ((𝜑 ∧ ¬ ran 𝐹 ⊆ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → ¬ ran 𝐹 ⊆ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)})
661, 3, 62, 63, 64, 4, 65gsumval2a 17326 . 2 ((𝜑 ∧ ¬ ran 𝐹 ⊆ {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → (𝐺 Σg 𝐹) = (seq𝑀( + , 𝐹)‘𝑁))
6761, 66pm2.61dan 849 1 (𝜑 → (𝐺 Σg 𝐹) = (seq𝑀( + , 𝐹)‘𝑁))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wo 382  wa 383   = wceq 1523  wcel 2030  wne 2823  wral 2941  {crab 2945  Vcvv 3231  wss 3607  c0 3948  {csn 4210  dom cdm 5143  ran crn 5144   Fn wfn 5921  wf 5922  cfv 5926  (class class class)co 6690  cuz 11725  ...cfz 12364  seqcseq 12841  Basecbs 15904  +gcplusg 15988  0gc0g 16147   Σg cgsu 16148
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-n0 11331  df-z 11416  df-uz 11726  df-fz 12365  df-seq 12842  df-0g 16149  df-gsum 16150
This theorem is referenced by:  gsumprval  17328  gsumwsubmcl  17422  gsumws1  17423  gsumccat  17425  gsumwmhm  17429  gsumval3  18354  gsummptfzcl  18414  gsumncl  30742  gsumnunsn  30743
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