Theorem gsumval2 18615

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	⊢ + = (+g‘𝐺)
gsumval2.g	⊢ (𝜑 → 𝐺 ∈ 𝑉)
gsumval2.n	⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))
gsumval2.f	⊢ (𝜑 → 𝐹:(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.

Step	Hyp	Ref	Expression
1		gsumval2.b	. . . 4 ⊢ 𝐵 = (Base‘𝐺)
2		eqid 2737	. . . 4 ⊢ (0g‘𝐺) = (0g‘𝐺)
3		gsumval2.p	. . . 4 ⊢ + = (+g‘𝐺)
4		eqid 2737	. . . 4 ⊢ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)} = {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}
5		gsumval2.g	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ 𝑉)
6	5	adantr 480	. . . 4 ⊢ ((𝜑 ∧ ran 𝐹 ⊆ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → 𝐺 ∈ 𝑉)
7		ovexd 7395	. . . 4 ⊢ ((𝜑 ∧ ran 𝐹 ⊆ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → (𝑀...𝑁) ∈ V)
8		gsumval2.f	. . . . . . 7 ⊢ (𝜑 → 𝐹:(𝑀...𝑁)⟶𝐵)
9	8	ffnd 6664	. . . . . 6 ⊢ (𝜑 → 𝐹 Fn (𝑀...𝑁))
10	9	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ ran 𝐹 ⊆ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → 𝐹 Fn (𝑀...𝑁))
11		simpr 484	. . . . 5 ⊢ ((𝜑 ∧ ran 𝐹 ⊆ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → ran 𝐹 ⊆ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)})
12		df-f 6497	. . . . 5 ⊢ (𝐹:(𝑀...𝑁)⟶{𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)} ↔ (𝐹 Fn (𝑀...𝑁) ∧ ran 𝐹 ⊆ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}))
13	10, 11, 12	sylanbrc 584	. . . 4 ⊢ ((𝜑 ∧ ran 𝐹 ⊆ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → 𝐹:(𝑀...𝑁)⟶{𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)})
14	1, 2, 3, 4, 6, 7, 13	gsumval1 18612	. . 3 ⊢ ((𝜑 ∧ ran 𝐹 ⊆ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → (𝐺 Σg 𝐹) = (0g‘𝐺))
15		simpl 482	. . . . . . . . 9 ⊢ (((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦) → (𝑥 + 𝑦) = 𝑦)
16	15	ralimi 3074	. . . . . . . 8 ⊢ (∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦) → ∀𝑦 ∈ 𝐵 (𝑥 + 𝑦) = 𝑦)
17	16	a1i 11	. . . . . . 7 ⊢ (𝑥 ∈ 𝐵 → (∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦) → ∀𝑦 ∈ 𝐵 (𝑥 + 𝑦) = 𝑦))
18	17	ss2rabi 4029	. . . . . 6 ⊢ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)} ⊆ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 (𝑥 + 𝑦) = 𝑦}
19		fvex 6848	. . . . . . . 8 ⊢ (0g‘𝐺) ∈ V
20	19	snid 4620	. . . . . . 7 ⊢ (0g‘𝐺) ∈ {(0g‘𝐺)}
21	8	fdmd 6673	. . . . . . . . . . . . 13 ⊢ (𝜑 → dom 𝐹 = (𝑀...𝑁))
22		gsumval2.n	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))
23		eluzfz1 13451	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ (𝑀...𝑁))
24		ne0i 4294	. . . . . . . . . . . . . 14 ⊢ (𝑀 ∈ (𝑀...𝑁) → (𝑀...𝑁) ≠ ∅)
25	22, 23, 24	3syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑀...𝑁) ≠ ∅)
26	21, 25	eqnetrd 3000	. . . . . . . . . . . 12 ⊢ (𝜑 → dom 𝐹 ≠ ∅)
27		dm0rn0 5874	. . . . . . . . . . . . 13 ⊢ (dom 𝐹 = ∅ ↔ ran 𝐹 = ∅)
28	27	necon3bii 2985	. . . . . . . . . . . 12 ⊢ (dom 𝐹 ≠ ∅ ↔ ran 𝐹 ≠ ∅)
29	26, 28	sylib 218	. . . . . . . . . . 11 ⊢ (𝜑 → ran 𝐹 ≠ ∅)
30	29	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ran 𝐹 ⊆ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → ran 𝐹 ≠ ∅)
31		ssn0 4357	. . . . . . . . . 10 ⊢ ((ran 𝐹 ⊆ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)} ∧ ran 𝐹 ≠ ∅) → {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)} ≠ ∅)
32	11, 30, 31	syl2anc 585	. . . . . . . . 9 ⊢ ((𝜑 ∧ ran 𝐹 ⊆ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)} ≠ ∅)
33	32	neneqd 2938	. . . . . . . 8 ⊢ ((𝜑 ∧ ran 𝐹 ⊆ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → ¬ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)} = ∅)
34	1, 2, 3, 4	mgmidsssn0 18601	. . . . . . . . . . 11 ⊢ (𝐺 ∈ 𝑉 → {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)} ⊆ {(0g‘𝐺)})
35	5, 34	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)} ⊆ {(0g‘𝐺)})
36		sssn 4783	. . . . . . . . . 10 ⊢ ({𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)} ⊆ {(0g‘𝐺)} ↔ ({𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)} = ∅ ∨ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)} = {(0g‘𝐺)}))
37	35, 36	sylib 218	. . . . . . . . 9 ⊢ (𝜑 → ({𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)} = ∅ ∨ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)} = {(0g‘𝐺)}))
38	37	orcanai 1005	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)} = ∅) → {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)} = {(0g‘𝐺)})
39	33, 38	syldan 592	. . . . . . 7 ⊢ ((𝜑 ∧ ran 𝐹 ⊆ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)} = {(0g‘𝐺)})
40	20, 39	eleqtrrid 2844	. . . . . 6 ⊢ ((𝜑 ∧ ran 𝐹 ⊆ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → (0g‘𝐺) ∈ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)})
41	18, 40	sselid 3932	. . . . 5 ⊢ ((𝜑 ∧ ran 𝐹 ⊆ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → (0g‘𝐺) ∈ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 (𝑥 + 𝑦) = 𝑦})
42		oveq1 7367	. . . . . . . . 9 ⊢ (𝑥 = (0g‘𝐺) → (𝑥 + 𝑦) = ((0g‘𝐺) + 𝑦))
43	42	eqeq1d 2739	. . . . . . . 8 ⊢ (𝑥 = (0g‘𝐺) → ((𝑥 + 𝑦) = 𝑦 ↔ ((0g‘𝐺) + 𝑦) = 𝑦))
44	43	ralbidv 3160	. . . . . . 7 ⊢ (𝑥 = (0g‘𝐺) → (∀𝑦 ∈ 𝐵 (𝑥 + 𝑦) = 𝑦 ↔ ∀𝑦 ∈ 𝐵 ((0g‘𝐺) + 𝑦) = 𝑦))
45	44	elrab 3647	. . . . . 6 ⊢ ((0g‘𝐺) ∈ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 (𝑥 + 𝑦) = 𝑦} ↔ ((0g‘𝐺) ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ((0g‘𝐺) + 𝑦) = 𝑦))
46		oveq2 7368	. . . . . . . 8 ⊢ (𝑦 = (0g‘𝐺) → ((0g‘𝐺) + 𝑦) = ((0g‘𝐺) + (0g‘𝐺)))
47		id 22	. . . . . . . 8 ⊢ (𝑦 = (0g‘𝐺) → 𝑦 = (0g‘𝐺))
48	46, 47	eqeq12d 2753	. . . . . . 7 ⊢ (𝑦 = (0g‘𝐺) → (((0g‘𝐺) + 𝑦) = 𝑦 ↔ ((0g‘𝐺) + (0g‘𝐺)) = (0g‘𝐺)))
49	48	rspcva 3575	. . . . . 6 ⊢ (((0g‘𝐺) ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ((0g‘𝐺) + 𝑦) = 𝑦) → ((0g‘𝐺) + (0g‘𝐺)) = (0g‘𝐺))
50	45, 49	sylbi 217	. . . . 5 ⊢ ((0g‘𝐺) ∈ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 (𝑥 + 𝑦) = 𝑦} → ((0g‘𝐺) + (0g‘𝐺)) = (0g‘𝐺))
51	41, 50	syl 17	. . . 4 ⊢ ((𝜑 ∧ ran 𝐹 ⊆ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → ((0g‘𝐺) + (0g‘𝐺)) = (0g‘𝐺))
52	22	adantr 480	. . . 4 ⊢ ((𝜑 ∧ ran 𝐹 ⊆ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → 𝑁 ∈ (ℤ≥‘𝑀))
53	35	ad2antrr 727	. . . . . 6 ⊢ (((𝜑 ∧ ran 𝐹 ⊆ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) ∧ 𝑧 ∈ (𝑀...𝑁)) → {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)} ⊆ {(0g‘𝐺)})
54	13	ffvelcdmda 7031	. . . . . 6 ⊢ (((𝜑 ∧ ran 𝐹 ⊆ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) ∧ 𝑧 ∈ (𝑀...𝑁)) → (𝐹‘𝑧) ∈ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)})
55	53, 54	sseldd 3935	. . . . 5 ⊢ (((𝜑 ∧ ran 𝐹 ⊆ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) ∧ 𝑧 ∈ (𝑀...𝑁)) → (𝐹‘𝑧) ∈ {(0g‘𝐺)})
56		elsni 4598	. . . . 5 ⊢ ((𝐹‘𝑧) ∈ {(0g‘𝐺)} → (𝐹‘𝑧) = (0g‘𝐺))
57	55, 56	syl 17	. . . 4 ⊢ (((𝜑 ∧ ran 𝐹 ⊆ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) ∧ 𝑧 ∈ (𝑀...𝑁)) → (𝐹‘𝑧) = (0g‘𝐺))
58	51, 52, 57	seqid3 13973	. . 3 ⊢ ((𝜑 ∧ ran 𝐹 ⊆ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → (seq𝑀( + , 𝐹)‘𝑁) = (0g‘𝐺))
59	14, 58	eqtr4d 2775	. 2 ⊢ ((𝜑 ∧ ran 𝐹 ⊆ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → (𝐺 Σg 𝐹) = (seq𝑀( + , 𝐹)‘𝑁))
60	5	adantr 480	. . 3 ⊢ ((𝜑 ∧ ¬ ran 𝐹 ⊆ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → 𝐺 ∈ 𝑉)
61	22	adantr 480	. . 3 ⊢ ((𝜑 ∧ ¬ ran 𝐹 ⊆ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → 𝑁 ∈ (ℤ≥‘𝑀))
62	8	adantr 480	. . 3 ⊢ ((𝜑 ∧ ¬ ran 𝐹 ⊆ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → 𝐹:(𝑀...𝑁)⟶𝐵)
63		simpr 484	. . 3 ⊢ ((𝜑 ∧ ¬ ran 𝐹 ⊆ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → ¬ ran 𝐹 ⊆ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)})
64	1, 3, 60, 61, 62, 4, 63	gsumval2a 18614	. 2 ⊢ ((𝜑 ∧ ¬ ran 𝐹 ⊆ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}) → (𝐺 Σg 𝐹) = (seq𝑀( + , 𝐹)‘𝑁))
65	59, 64	pm2.61dan 813	1 ⊢ (𝜑 → (𝐺 Σg 𝐹) = (seq𝑀( + , 𝐹)‘𝑁))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∨ wo 848   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ∀wral 3052  {crab 3400  Vcvv 3441   ⊆ wss 3902  ∅c0 4286  {csn 4581  dom cdm 5625  ran crn 5626   Fn wfn 6488  ⟶wf 6489  ‘cfv 6493  (class class class)co 7360  ℤ_≥cuz 12755  ...cfz 13427  seqcseq 13928  Basecbs 17140  +_gcplusg 17181  0_gc0g 17363   Σ_g cgsu 17364

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7682  ax-cnex 11086  ax-resscn 11087  ax-1cn 11088  ax-icn 11089  ax-addcl 11090  ax-addrcl 11091  ax-mulcl 11092  ax-mulrcl 11093  ax-mulcom 11094  ax-addass 11095  ax-mulass 11096  ax-distr 11097  ax-i2m1 11098  ax-1ne0 11099  ax-1rid 11100  ax-rnegex 11101  ax-rrecex 11102  ax-cnre 11103  ax-pre-lttri 11104  ax-pre-lttrn 11105  ax-pre-ltadd 11106  ax-pre-mulgt0 11107

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3062  df-rmo 3351  df-reu 3352  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-iun 4949  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-1st 7935  df-2nd 7936  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-er 8637  df-en 8888  df-dom 8889  df-sdom 8890  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-nn 12150  df-n0 12406  df-z 12493  df-uz 12756  df-fz 13428  df-seq 13929  df-0g 17365  df-gsum 17366

This theorem is referenced by:  gsumsplit1r  18616  gsumprval  18617  gsumwsubmcl  18766  gsumws1  18767  gsumsgrpccat  18769  gsumwmhm  18774  mulgnngsum  19013  gsumval3  19840  gsummptfzcl  19902  gsumncl  34678  gsumnunsn  34679